Nonlinear nonhomogeneous boundary value problems with competition phenomena

TL;DR

This paper investigates a nonlinear boundary value problem with competing nonlinearities, demonstrating the existence of multiple solutions, including constant sign and nodal solutions, using variational and topological methods.

Contribution

It establishes the existence of at least five solutions for small parameters and analyzes their properties, introducing new results for problems with nonhomogeneous operators and competing nonlinearities.

Findings

At least five solutions exist for small parameter values.

Four solutions have constant sign, one is nodal.

Extremal solutions are monotonic and continuous in the parameter.

Abstract

We consider a nonlinear boundary value problem driven by a nonhomogeneous differential operator. The problem exhibits competing nonlinearities with a superlinear (convex) contribution coming from the reaction term and a sublinear (concave) contribution coming from the parametric boundary (source) term. We show that for all small parameter values $\lambda>0$, the problem has at least five nontrivial smooth solutions, four of constant sign and one nodal. We also produce extremal constant sign solutions and determine their monotonicity and continuity properties as the parameter $\lambda>0$ varies. In the semilinear case we produce a sixth nontrivial solution but without any sign information. Our approach uses variational methods together with truncation and perturbation techniques, and Morse theory.

Equations

$$P_{\lambda }$$

$$\frac{\partial u}{\partial n_a}=(a(Du),n{)}_{{\mathbb{R}}^N}\text{mbox}forallu\in W^{1,p}(\Omega ),$$

$$f(z,x)=f(x)=\lambda |x{|}^{q-2}x+|x{|}^{r-2}x\text{mbox}forallx\in \mathbb{R},$$

$$(1+||u_n||){\phi }^{\prime }(u_n)\to 0\text{mbox}in{X}^{\ast }\text{mbox}asn\to \infty ,$$

$$\max \{\phi (u_0),\phi (u_1)\}<\inf [\phi (u):||u-u_0||=\rho ]=m_{\rho }.$$

$$0<\hat{c}⩽\frac{{\vartheta }^{\prime }(t)t}{\vartheta (t)}⩽c_0\text{mbox}and{c}_1{t}^{p-1}⩽\vartheta (t)⩽c_2(1+t^{p-1})\text{mbox}forallt>0,\text{mbox}with{c}_1,c_2>0.$$

$$\underset{t\to 0^{+}}{\lim }\frac{a_0^{\prime }(t)t}{a_0(t)}>-1;$$

$$|\nabla a(y)|⩽c_3\frac{\vartheta (|y|)}{|y|}\text{mbox}forally\in {\mathbb{R}}^N\backslash \{0\};$$

$$t\mapsto G_0(t^{1/\tau })\text{mbox}isconvex,0⩽\underset{t\to 0^{+}}{\mathrm{liminf}}\frac{G_0(t)}{t^{\tau }}⩽\underset{t\to 0^{+}}{\mathrm{limsup}}\frac{G_0(t)}{t^{\tau }}⩽\tilde{c},$$

$$c_4{t}^p⩽a_0(t)t-\tau G_0(t)\text{mbox}forallt>0\text{mbox}andsome{c}_4>0,$$

$$-\bar{c}⩽p{G}_0(t)-a_0(t)t^2\text{mbox}forallt>0\text{mbox}andsome\bar{c}>0.$$

$$G(y)=G_0(|y|)\text{mbox}forally\in {\mathbb{R}}^N.$$

$$\nabla G(0)=0,\nabla G(y)=G_0^{\prime }(|y|)\frac{y}{|y|}=a_0(|y|)y=a(y)\text{mbox}forally\in {\mathbb{R}}^N\backslash \{0\}.$$

$$G(y)⩽(a(y),y{)}_{{\mathbb{R}}^N}\text{mbox}forally\in {\mathbb{R}}^N.$$

$${\Delta }_{p}u=\mathrm{div}(|Du{|}^{p-2}Du)\text{mbox}forallu\in W^{1,p}(\Omega ).$$

$${\Delta }_{p}u+{\Delta }_{\tau }u\text{mbox}forallu\in W^{1,p}(\Omega ).$$

$$\mathrm{div}((1+|Du{|}^2{)}^{\frac{p-2}{2}}Du)\text{mbox}forallu\in W^{1,p}(\Omega ).$$

$${\Delta }_{p}u+\mathrm{div}\left(\frac{|Du{|}^{p-2}Du}{1+|Du{|}^p}\right)\text{mbox}forallu\in W^{1,p}(\Omega ).$$

$${\Delta }_{p}u+\mathrm{div}(\ln (1+|Du{|}^2)Du)\text{mbox}forallu\in W^{1,p}(\Omega ).$$

$$||u||=[||u|{|}_p^p+||Du|{|}_p^p{]}^{1/p}\text{mbox}forallu\in W^{1,p}(\Omega ).$$

$$C_{+}=\{u\in C^1(\overline{\Omega }):u(z)⩾0\text{mbox}forallz\in \overline{\Omega })\}.$$

$$\displaystyle{\rm int}\,C_{+}=\{u\in C_{+}:u(z)>0\ \mbox{for all}\ z\in\Omega,\ \frac{\partial u}{\partial n}\big{|}_{\partial\Omega\cap u^{-1}(0)}<0\ \mbox{if}\ \partial\Omega\cap u^{-1}(0)\not=\emptyset\}.$$

$$D_{+}=\{u\in C_{+}:u(z)>0\text{mbox}forallz\in \overline{\Omega }\}.$$

$$D_{+}\subseteq \mathrm{int}{C}_{+}.$$

$${\gamma }_0(u)=u{|}_{\partial \Omega }\text{mbox}forallu\in W^{1,p}(\Omega )\cap C(\overline{\Omega }).$$

$$\mathrm{im}{\gamma }_0=W^{\frac{1}{p^{\prime }},p}(\partial \Omega ),\text{mbox}where\frac{1}{p}+\frac{1}{p^{\prime }}=1\text{mbox}and\ker {\gamma }_0=W_0^{1,p}(\Omega ).$$

$$u=u^{+}-u^{-},|u|=u^{+}+u^{-},u^{+},u^{-}\in W^{1,p}(\Omega ).$$

$$N_g(u)(\cdot )=g(\cdot ,u(\cdot ))\text{mbox}forallu\in W^{1,p}(\Omega ).$$

$$⟨A(u),h⟩={\int }_{\Omega }(a(Du),Dh{)}_{{\mathbb{R}}^N}dz\text{mbox}forallu,h\in W^{1,p}(\Omega ).$$

$$|f_0(z,x)|⩽a_0(z)(1+|x{|}^{r-1})\text{mbox}foralmostallz\in \Omega \text{mbox}andallx\in \mathbb{R}$$

$$|{\beta }_0(z,x)|⩽c_6(1+|x{|}^{q-1})\text{mbox}forall(z,x)\in \partial \Omega \times \mathbb{R},$$

Full text

aa 2010 AMS Subject Classification: 35J20 (Primary); 35J60, 58E05 (Secondary)

1. Introduction

Let $\displaystyle\Omega\subseteq\mathbb{R}^{N}$ be a bounded domain with a $\displaystyle C^{2}$-boundary $\displaystyle\partial\Omega$. In this paper we study the following nonlinear, nonhomogeneous elliptic problem

[TABLE]

In this problem $\displaystyle a:\mathbb{R}^{N}\rightarrow\mathbb{R}^{N}$ is a strictly monotone, continuous map which satisfies certain other regularity and growth conditions, listed in hypotheses $\displaystyle H(a)$ in Section 2. These hypotheses are general enough to incorporate in our framework several differential operators of interest, such as, e.g., the $\displaystyle p$-Laplacian. The reaction term $\displaystyle f(z,x)$ is a Carathéodory function (that is, for all $\displaystyle x\in\mathbb{R}$, $\displaystyle z\mapsto f(z,x)$ is measurable and for almost all $\displaystyle z\in\Omega$, $\displaystyle x\mapsto f(z,x)$ is continuous) which satisfies the well-known Ambrosetti-Rabinowitz condition (AR-condition for short) in the $\displaystyle x$-variable, hence exhibiting $\displaystyle(p-1)$-superlinear growth near $\displaystyle\pm\infty$. In the boundary condition, $\displaystyle\frac{\partial u}{\partial n_{a}}$ denotes the generalized normal derivative corresponding to the differential operator $\displaystyle u\mapsto{\rm div}\,a(Du)$ and is defined by

[TABLE]

with $\displaystyle n(\cdot)$ being the outward unit normal on $\displaystyle\partial\Omega$. This kind of generalized normal derivative is dictated by the nonlinear Green’s identity (see Gasinski and Papageorgiou [13, p. 210] and it was also used by Lieberman [21]). The boundary function $\displaystyle\beta(z,x)$ is continuous on $\displaystyle\partial\Omega\times\mathbb{R}$ and it satisfies certain other regularity and growth conditions listed in hypotheses $\displaystyle H(\beta)$ in Section 3. In fact, $\displaystyle\beta(z,\cdot)$ exhibits strict $\displaystyle(p-1)$-sublinear growth near $\displaystyle\pm\infty$. So, we see that problem ($\displaystyle P_{\lambda}$) has competing nonlinearities. We refer to a convex (superlinear) input coming from the reaction term $\displaystyle f(z,x)$ and a concave (sublinear) input resulting from the source (boundary) term.

The study of problems with competition phenomena was initiated with the seminal paper of Ambrosetti, Brezis and Cerami [2] for semilinear Dirichlet equations. In their work both competing nonlinearities appear in the reaction term $\displaystyle f(z,x)$, which has the form

[TABLE]

with \displaystyle\lambda>0,\ 1<q<2<r\leqslant 2^{*}=\left\{\begin{array}[]{ll}\frac{2N}{N-2}&\mbox{if}\ N\geqslant 3\\ +\infty&\mbox{if}\ N=1,2\end{array}\right.. Since then there has been a lot of work in this direction, extending the results of [2] to nonlinear equations. In contrast, in the present paper the concave term comes from the boundary condition. The study of such problems is still lagging behind. In this direction, there are only the semilinear works of Furtado and Ruviaro [11], Garcia Azorero, Peral Alonso and Rossi [12] and Hu and Papageorgiou [20].

In this paper, we prove a multiplicity theorem which says that for small values of the parameter $\displaystyle\lambda>0$, the problem has at least five nontrivial smooth solutions, four of constant sign and one nodal. We also show the existence of extremal constant sign solutions, that is, a smallest positive solution $\displaystyle u^{*}_{\lambda}$ and a biggest negative solution $\displaystyle v^{*}_{\lambda}$, and we investigate the monotonicity and continuity properties of the maps $\displaystyle\lambda\mapsto u^{*}_{\lambda}$ and $\displaystyle\lambda\mapsto v^{*}_{\lambda}$. Finally, in the semilinear case, we generate a sixth nontrivial smooth solution (without being able to provide any sign information).

Our approach uses variational methods based on the critical point theory, combined with suitable truncation-perturbation and comparison techniques, and Morse theory.

2. Preliminaries – Hypotheses

In this section we present the main mathematical tools which we will use in the sequel and we prove some auxiliary results which will be needed later. In this section we also fix our notation and we have gathered all the hypotheses on the data of problem ($\displaystyle P_{\lambda}$) which will be used to prove our results. We also state the main results of this work, in order for the reader to have a feeling of what is achieved in this paper.

Let $\displaystyle X$ be a Banach space and let $\displaystyle X^{*}$ be its topological dual. By $\displaystyle\left\langle\cdot,\cdot\right\rangle$ we denote the duality brackets for the pair $\displaystyle(X^{*},X)$. Let $\displaystyle\varphi\in C^{1}(X,\mathbb{R})$. We say that $\displaystyle\varphi$ satisfies the “Cerami condition” (the “C-condition” for short), if the following property holds:

“Every sequence $\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq X$ such that $\displaystyle\{\varphi(u_{n})\}_{n\geqslant 1}\subseteq\mathbb{R}$ is bounded and

[TABLE]

admits a strongly convergent subsequence”.

This compactness-type condition on the functional $\displaystyle\varphi$ is needed in the critical point theory because the ambient space need not be locally compact (being in general infinite dimensional). Using this compactness-type condition, one can prove a deformation theorem describing the change of the topological structure of the sublevel sets of $\displaystyle\varphi$ along the negative gradient or pseudogradient flow. The deformation theorem leads to the minimax theory of the critical values of $\displaystyle\varphi$. Prominent in that theory is the so-called “mountain pass theorem” due to Ambrosetti and Rabinowitz [3]. Here we state it in a slightly more general form (see, for example, Gasinski and Papageorgiou [13, p. 648]).

Theorem 1**.**

Let $\displaystyle X$ be a Banach space. Suppose that $\displaystyle\varphi\in C^{1}(X,\mathbb{R})$ satisfies the C-condition, and $\displaystyle u_{0},u_{1}\in X$ satisfy $\displaystyle||u_{1}-u_{0}||>\rho>0$

[TABLE]

Let $\displaystyle c=\inf\limits_{\gamma\in\Gamma}\max\limits_{0\leqslant t\leqslant 1}\varphi(\gamma(t))$ with $\displaystyle\Gamma=\{\gamma\in C([0,1],X):\gamma(0)=u_{0},\gamma(1)=u_{1}\}$. Then $\displaystyle c\geqslant m_{\rho}$ and $\displaystyle c$ is a critical value of $\displaystyle\varphi$ (that is, there exists $\displaystyle\hat{u}\in X$ such that $\displaystyle\varphi^{\prime}(\hat{u})=0$ and $\displaystyle\varphi(\hat{u})=c$).

Let $\displaystyle\vartheta\in C^{1}(0,\infty)$ with $\displaystyle\vartheta(t)>0$ for all $\displaystyle t>0$ and assume that

[TABLE]

The hypotheses on the map $\displaystyle y\mapsto a(y)$ involved in the definition of the differential operator in problem ($\displaystyle P_{\lambda}$), are the following:

$\displaystyle H(a):$ $\displaystyle a(y)=a_{0}(|y|)y$ for all $\displaystyle y\in\mathbb{R}^{N}$, with $\displaystyle a_{0}(t)>0$ for all $\displaystyle t>0$ and

• (i)

$\displaystyle a_{0}\in C^{1}(0,+\infty),\ t\mapsto a_{0}(t)t$ is strictly increasing on $\displaystyle(0,+\infty)$, $\displaystyle a_{0}(t)t\rightarrow 0^{+}$ as $\displaystyle t\rightarrow 0^{+}$ and

[TABLE]

• (ii)

there exists $\displaystyle c_{3}>0$ such that

[TABLE]

• (iii)

$\displaystyle(\nabla a(y)\xi,\xi)_{\mathbb{R}^{N}}\geqslant\frac{\vartheta(|y|)}{|y|}|\xi|^{2}$ for all $\displaystyle y\in\mathbb{R}^{N}\backslash\{0\}$ and all $\displaystyle\xi\in\mathbb{R}^{N}$;

• (iv)

if $\displaystyle G_{0}(t)=\int^{t}_{0}a_{0}(s)sds$ for $\displaystyle t>0$, then there exists $\displaystyle\tau\in(1,p)$ such that

[TABLE]

Remark 1**.**

Hypotheses $\displaystyle H(a)(i),(ii),(iii)$ were motivated by the nonlinear regularity theory of Lieberman [21] and the nonlinear maximum principle of Pucci and Serrin [33, pp. 111, 120]. Hypothesis $\displaystyle H(a)(iv)$ serves the particular needs of our problem. However, this is a mild restriction and it is satisfied in all cases of interest, as the examples below show.

Clearly hypotheses $\displaystyle H(a)$ imply that $\displaystyle G_{0}(\cdot)$ is strictly convex and strictly increasing. We set

[TABLE]

Then $\displaystyle G(\cdot)$ is convex, $\displaystyle G(0)=0$, and

[TABLE]

So, $\displaystyle G(\cdot)$ is the primitive of the map $\displaystyle a(\cdot)$. Using the convexity of $\displaystyle G(\cdot)$ and $\displaystyle G(0)=0$, we have

[TABLE]

Hypotheses $\displaystyle H(a)(i),(ii),(iii)$ and (1), lead to the following lemma which summarizes the main properties of the map $\displaystyle y\mapsto a(y)$.

Lemma 2**.**

If hypotheses $\displaystyle H(a)(i),(ii),(iii)$ hold, then

• (a)

$\displaystyle y\mapsto a(y)$* is strictly monotone, continuous, hence also maximal monotone;*

• (b)

$\displaystyle|a(y)|\leqslant c_{4}(1+|y|^{p-1})$* for all $\displaystyle y\in\mathbb{R}^{N}$ and some $\displaystyle c_{4}>0$;*

• (c)

$\displaystyle(a(y),y)_{\mathbb{R}^{N}}\geqslant\frac{c_{1}}{p-1}|y|^{p}$* for all $\displaystyle y\in\mathbb{R}^{N}.$*

This lemma and (2) lead to the following growth estimates for the primitive $\displaystyle G(\cdot)$.

Corollary 3**.**

If hypotheses $\displaystyle H(a)(i),(ii),(iii)$ hold, then $\displaystyle\frac{c_{1}}{p(p-1)}|y|^{p}\leqslant G(y)\leqslant c_{5}(1+|y|^{p})$ for all $\displaystyle y\in\mathbb{R}^{N}$ and some $\displaystyle c_{5}>0$.

The examples which follow illustrate that hypotheses $\displaystyle H(a)$ cover many interesting cases.

Example 1**.**

The following maps satisfy hypotheses $\displaystyle H(a)$:

• (a)

$\displaystyle a(y)=|y|^{p-2}y$* with $\displaystyle 1<p<\infty$.*

This map corresponds to the $\displaystyle p$-Laplacian differential operator defined by

[TABLE]

• (b)

$\displaystyle a(y)=|y|^{p-2}y+|y|^{\tau-2}y$* with $\displaystyle 1<\tau<p$*

This map corresponds to the $\displaystyle(p,\tau)$-differential operator defined by

[TABLE]

Such differential operators arise in problems of mathematical physics. We mention the works of Benci, D’Avenia, Fortunato and Pisani **[4]** (in quantum physics) and Cherfils and Ilyasov **[6]** (in plasma physics). Recently, there have been some existence and multiplicity results for such equations. We mention the works of Cingolani and Degiovanni **[7]**, Gasinski and Papageorgiou **[15]**, Marano, Mosconi and Papageorgiou **[22]**, Mugnai and Papageorgiou **[24]**, Papageorgiou and Rădulescu **[26, 27]**, Papageorgiou, Rădulescu and Repovš **[31]**, Sun **[34]**, and Sun, Zhang and Su **[35]**.

• (c)

$\displaystyle a(y)=(1+|y|^{2})^{\frac{p-2}{2}y}$* with $\displaystyle 1<p<\infty$.*

This map corresponds to the generalized $\displaystyle p$-mean curvature differential operator defined by

[TABLE]

• (d)

$\displaystyle a(y)=|y|^{p-2}y\left[1+\frac{1}{1+|y|^{p}}\right]$* for all $\displaystyle u\in W^{1,p}(\Omega)$.*

This map corresponds to the differential operator

[TABLE]

• (e)

$\displaystyle a(y)=|y|^{p-2}y+\ln(1+|y|^{2})y$* with $\displaystyle 1<p<\infty$.*

This map corresponds to the differential operator

[TABLE]

We will use the Sobolev space $\displaystyle W^{1,p}(\Omega)$, the Banach space $\displaystyle C^{1}(\overline{\Omega})$ and the boundary Lebesgue spaces $\displaystyle L^{q}(\partial\Omega);1\leqslant q\leqslant\infty$. The Sobolev space $\displaystyle W^{1,p}(\Omega)$ is a Banach space for the norm

[TABLE]

The Banach space $\displaystyle C^{1}(\overline{\Omega})$ is an ordered Banach space with positive (order) cone

[TABLE]

This cone has a nonempty interior given by

[TABLE]

This cone contains the open set

[TABLE]

In fact, note that $\displaystyle D_{+}$ is the interior of $\displaystyle C_{+}$ when $\displaystyle C^{1}(\overline{\Omega})$ is furnished with the relative $\displaystyle C(\overline{\Omega})$-topology.

On $\displaystyle C^{1}(\overline{\Omega})$ the $\displaystyle C^{1}(\overline{\Omega})$-norm topology is stronger than the $\displaystyle C(\overline{\Omega})$-norm topology. Therefore we have

[TABLE]

On $\displaystyle\partial\Omega$ we consider the $\displaystyle(N-1)$-dimensional Hausdorff (surface) measure $\displaystyle\sigma(\cdot)$. Using this measure, we can define in the usual way the Lebesgue spaces $\displaystyle L^{q}(\partial\Omega)$, $\displaystyle 1\leqslant q\leqslant\infty$. From the theory of Sobolev spaces we know that there exists a unique continuous linear map $\displaystyle\gamma_{0}:W^{1,p}(\Omega)\rightarrow L^{\tau}(\partial\Omega)$, $\displaystyle\tau=\frac{Np-p}{N-p}$ if $\displaystyle p<N$, and $\displaystyle\tau\geqslant 1$ if $\displaystyle N\leqslant p$, known as the “trace map”, such that

[TABLE]

So, we can understand the trace map as an expression of the “boundary values” of a Sobolev function. We know that

[TABLE]

The trace map $\displaystyle\gamma_{0}$ is compact into $\displaystyle L^{q}(\partial\Omega)$ for all $\displaystyle q\in\left[1,\frac{Np-p}{N-p}\right)$ when $\displaystyle 1<p<N$ and for all $\displaystyle q\geqslant 1$, when $\displaystyle p\geqslant N$. In the sequel, for the sake of notational simplicity we drop the use of the map $\displaystyle\gamma_{0}$. All restrictions of Sobolev functions on $\displaystyle\partial\Omega$ are understood in the sense of traces.

Introducing some more notation, for every $\displaystyle x\in\mathbb{R}$, we set $\displaystyle x^{\pm}=\max\{\pm x,0\}$. Then for $\displaystyle u\in W^{1,p}(\Omega)$ we define $\displaystyle u^{\pm}(\cdot)=u(\cdot)^{\pm}$ and have

[TABLE]

By $\displaystyle|\cdot|_{N}$ we denote the Lebesgue measure on $\displaystyle\mathbb{R}^{N}$ and if $\displaystyle g:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a measurable function (for example, a Carathéodory function), then we define the Nemytskii map corresponding to $\displaystyle g$

[TABLE]

Let $\displaystyle A:W^{1,p}(\Omega)\rightarrow W^{1,p}(\Omega)^{*}$ be the nonlinear map defined by

[TABLE]

The next proposition establishes the main properties of this map. It is a special case of Proposition 3.5 in Gasinski and Papageorgiou [14].

Proposition 4**.**

Assume that hypotheses $\displaystyle H(a)(i),(ii),(iii)$ hold and that $\displaystyle A:W^{1,p}(\Omega)\rightarrow W^{1,p}(\Omega)^{*}$ is the nonlinear map defined by (3). Then $\displaystyle A$ is bounded (that is, maps bounded sets to bounded sets), continuous, monotone (hence also maximal monotone) and of type $\displaystyle(S)_{+}$ (that is, if $\displaystyle u_{n}\stackrel{{\scriptstyle w}}{{\rightarrow}}u$ in $\displaystyle W^{1,p}(\Omega)$ and $\displaystyle\limsup\limits_{n\rightarrow\infty}\left\langle A(u_{n}),u_{n}-u\right\rangle\leqslant 0$, then $\displaystyle u_{n}\rightarrow u$ in $\displaystyle W^{1,p}(\Omega)$).

Next, consider a Carathéodory function $\displaystyle f_{0}:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ and a function $\displaystyle\beta_{0}\in C(\partial\Omega\times\mathbb{R})\cap C^{0,\alpha}_{loc}(\partial\Omega\times\mathbb{R})$ with $\displaystyle\alpha\in\left(0,1\right]$ such that

[TABLE]

with \displaystyle a_{0}\in L^{\infty}(\Omega)_{+},p\leqslant r<p^{*}=\left\{\begin{array}[]{ll}\frac{Np}{N-p}&\mbox{if}\ p<N\\ +\infty&\mbox{if}\ p\geqslant N\end{array}\right. and

[TABLE]

with $\displaystyle c_{6}>0,1<q<p$. We set

[TABLE]

and consider the $\displaystyle C^{1}$-functional $\displaystyle\varphi_{0}:W^{1,p}(\Omega)\rightarrow\mathbb{R}$ defined by

[TABLE]

From Papageorgiou and Rădulescu [30] and [28] (the case of the $\displaystyle p$-Laplacian) we obtain the following property.

Proposition 5**.**

Assume that $\displaystyle u_{0}\in W^{1,p}(\Omega)$ is a local $\displaystyle C^{1}(\overline{\Omega})$-minimizer of the functional $\displaystyle\varphi_{0}$, that is, there exists $\displaystyle\rho_{0}>0$ such that

[TABLE]

Then $\displaystyle u_{0}\in C^{1,\mu}(\overline{\Omega})$ with $\displaystyle\mu\in(0,1)$ and $\displaystyle u_{0}$ is also a local $\displaystyle W^{1,p}(\Omega)$-minimizer of $\displaystyle\varphi_{0}$, that is, there exists $\displaystyle\rho_{1}>0$ such that

[TABLE]

Next, let us recall some basic definitions and facts from Morse theory (critical groups) which we will need later.

Given a Banach space $\displaystyle X$, a function $\displaystyle\varphi\in C^{1}(X,\mathbb{R})$ and $\displaystyle c\in\mathbb{R}$, we introduce the following sets:

[TABLE]

Let $\displaystyle(Y_{1},Y_{2})$ be a topological pair such that $\displaystyle Y_{2}\subseteq Y_{1}\subseteq X$. By $\displaystyle H_{k}(Y_{1},Y_{2}),\ k\in\mathbb{N}_{0}$, we denote the $\displaystyle k$th relative singular homology group for the topological pair $\displaystyle(Y_{1},Y_{2})$ with integer coefficients. The critical groups of $\displaystyle\varphi$ at an isolated point $\displaystyle u\in K^{c}_{\varphi}$ are defined by

[TABLE]

Here, $\displaystyle U$ is a neighborhood of $\displaystyle u$ such that $\displaystyle K_{\varphi}\cap\varphi^{c}\cap U=\{u\}$. The excision property of singular homology implies that the above definition of critical groups is independent of the choice of the neighborhood $\displaystyle U$ of $\displaystyle u$.

Suppose that $\displaystyle\varphi$ satisfies the $\displaystyle C$-condition and that $\displaystyle\inf\varphi(K_{\varphi})>-\infty$. Let $\displaystyle c<\inf\varphi(K_{\varphi})$. The critical groups of $\displaystyle\varphi$ at infinity are defined by

[TABLE]

The second deformation theorem (see, for example, Gasinski and Papageorgiou [13, p. 628]) implies that this definition is independent of the level $\displaystyle c<\inf\varphi(K_{\varphi})$.

Suppose that $\displaystyle\varphi\in C^{1}(X,\mathbb{R})$ satisfies the $\displaystyle C$-condition and that $\displaystyle K_{\varphi}$ is finite. We define

[TABLE]

Then the Morse relation says that

[TABLE]

with $\displaystyle Q(t)=\sum\limits_{k\in\mathbb{N}_{0}}\beta_{k}t^{k}$ being a formal series in $\displaystyle t\in\mathbb{R}$ with nonnegative integer coefficients $\displaystyle\beta_{k}$.

Next, we state a strong comparison principle. Our proof uses ideas from Guedda and Véron [17], who were the first to prove a strong comparison principle for the Dirichlet $\displaystyle p$-Laplacian. Recall that $\displaystyle n(\cdot)$ denotes the outward unit normal on $\displaystyle\partial\Omega$.

Proposition 6**.**

Assume that hypotheses $\displaystyle H(a)(i),(ii),(iii)$ hold, $\displaystyle u_{1},u_{2}\in C^{1}(\overline{\Omega}),$ $\displaystyle g_{1},g_{2}\in L^{\infty}(\Omega)$, $\displaystyle u_{1}(z)\leqslant u_{2}(z)$ for all $\displaystyle z\in\overline{\Omega}$, and

[TABLE]

Then $\displaystyle(u_{2}-u_{1})(z)>0$ for all $\displaystyle z\in\Omega$ and $\displaystyle\frac{\partial(u_{2}-u_{1})}{\partial n}(z_{0})<0$ for all $\displaystyle z_{0}\in\Sigma_{0}=\{z\in\partial\Omega:u_{2}(z)=u_{1}(z)\}$.

Proof.

By hypothesis we have

[TABLE]

Let $\displaystyle a=(a_{k})^{N}_{k=1}$ with $\displaystyle a_{k}:\mathbb{R}^{N}\rightarrow\mathbb{R}$ for every $\displaystyle k\in\{1,\ldots,N\}$. Using the mean value theorem, we have

[TABLE]

for all $\displaystyle y=(y_{i})^{N}_{i=1},y^{\prime}=(y^{\prime}_{i})^{N}_{i=1}\in\mathbb{R}^{N}$ and all $\displaystyle k\in\{1,\ldots,N\}$.

We introduce the following coefficient functions

[TABLE]

Using these coefficients, we introduce the following linear differential operator

[TABLE]

Let $\displaystyle v=u_{2}-u_{1}$. Then $\displaystyle v\neq 0$ (recall that $\displaystyle g_{1}\not\equiv g_{2}$) and from (5)–(8) we have

[TABLE]

By hypothesis, we have $\displaystyle\left.\frac{\partial u_{1}}{\partial n}\right|_{\partial\Omega},\left.\frac{\partial u_{2}}{\partial n}\right|_{\partial\Omega}>0$ or $\displaystyle<0$. So, for small $\displaystyle\delta>0$ we have

[TABLE]

with $\displaystyle\Omega_{\delta}=\{z\in\Omega:d(z,\partial\Omega)<\delta\}$. It follows from (8) and (10) that the operator $\displaystyle L$ is strictly elliptic on $\displaystyle\Omega_{\delta}$.

Suppose that $\displaystyle u_{1}|_{\Omega_{\delta}}=u_{2}|_{\Omega_{\delta}}$. Then $\displaystyle g_{1}(z)=g_{2}(z)$ for almost all $\displaystyle z\in\Omega_{\delta}$. We consider a function $\displaystyle\vartheta\in C^{1}(\overline{\Omega})$ such that

[TABLE]

We have

[TABLE]

which is in contradiction with the hypothesis that $\displaystyle g_{1}\not\equiv g_{2}$ (recall $\displaystyle\vartheta>0$, see (11)). So, we have $\displaystyle u_{2}-u_{1}\in C_{+}\backslash\{0\}$.

Then from (9) and the strong maximum principle (see, for example, Gasinski and Papageorgiou [13, p. 738]), we derive

[TABLE]

It follows from (12) that the set $\displaystyle S=\{z\in\Omega:u_{1}(z)=u_{2}(z)\}$ is compact. Hence Corollary 8.23, p. 215, of Motreanu, Motreanu and Papageorgiou [23], implies that

[TABLE]

This completes the proof. ∎

Remark 2**.**

Consider the following order cone in $\displaystyle C^{1}(\overline{\Omega})$:

[TABLE]

where $\displaystyle\Sigma_{0}=\{z\in\partial\Omega:y(z)=0\}$. This cone has a nonempty interior given by

[TABLE]

Then Proposition 6 says that $\displaystyle u_{2}-u_{1}\in int\hat{C}_{+}$.

We will also use the next proposition, which essentially produces an equivalent norm for the Sobolev space $\displaystyle W^{1,p}(\Omega)$. The result is stated in a more general form than the one we will need, because we believe that in this form it is of independent interest and can be used in other circumstances.

Proposition 7**.**

Assume that $\displaystyle\beta\in L^{\infty}(\partial\Omega),\beta(z)\geqslant 0$ for almost all $\displaystyle z\in\partial\Omega,\beta\not\equiv 0$, $\displaystyle 1\leqslant q\leqslant\frac{Np-p}{N-p}$ if $\displaystyle p<N$, and $\displaystyle 1\leqslant p$ if $\displaystyle N\leqslant p$, and $\displaystyle|u|=||Du||_{p}+(\int_{\partial\Omega}\beta(z)|u|^{q}d\sigma)^{1/q}$ for all $\displaystyle u\in W^{1,p}(\Omega)$. Then we can find $\displaystyle 0<c_{7}\leqslant c_{8}$ such that $\displaystyle c_{7}|u|\leqslant||u||\leqslant c_{8}|u|$ for all $\displaystyle u\in W^{1,p}(\Omega)$.

Proof.

Note that

[TABLE]

Next we show that we find $\displaystyle c_{10}>0$ such that

[TABLE]

Suppose that (14) is not true. Then we can find $\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq W^{1,p}(\Omega)$ such that

[TABLE]

Normalizing in $\displaystyle L^{p}(\Omega)$ if necessary, we may assume that $\displaystyle||u_{n}||_{p}=1$ for all $\displaystyle n\in\mathbb{N}$. Then

[TABLE]

Then by passing to a subsequence if necessary, we may assume that

[TABLE]

(here we use the continuity of the trace map). It follows from (2), (16) that

[TABLE]

If $\displaystyle\xi\neq 0$, then by virtue of (2) we have

[TABLE]

a contradiction. Hence $\displaystyle\xi=0$ and so from (16) we have

[TABLE]

which is a contradiction with the fact that $\displaystyle||u_{n}||_{p}=1$ for all $\displaystyle n\in\mathbb{N}$. So, (14) holds and this, combined with (13), implies that the assertion of the proposition is true. ∎

Remark 3**.**

If $\displaystyle\beta\equiv 1$, then Proposition 7 asserts that

[TABLE]

with $\displaystyle q\in\left[1,\frac{Np-p}{N-p}\right]$ if $\displaystyle p<N$, and $\displaystyle q\geqslant 1$ if $\displaystyle N<p$, is an equivalent norm on the Sobolev space $\displaystyle W^{1,p}(\Omega)$ (see also Gasinski and Papageorgiou [13], Proposition 2.5.8, p. 218).

Finally we present all the conditions on the other data of ($\displaystyle P_{\lambda}$) (that is, for $\displaystyle f(z,x)$ and $\displaystyle\beta(z,x)$) which we will use to prove our results and then we have the statements of our main results.

We start with the following hypotheses on the reaction term $\displaystyle f(z,x)$.

$\displaystyle H(f):$ $\displaystyle f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a Carathéordory function such that $\displaystyle f(z,0)=0$ for almost all $\displaystyle z\in\Omega$ and

• (i)

$\displaystyle|f(z,x)|\leqslant a(z)(1+|x|^{r-1})$ for almost all $\displaystyle z\in\Omega$ and all $\displaystyle x\in\mathbb{R}$, with $\displaystyle a\in L^{\infty}(\Omega)_{+}$, $\displaystyle p<r<p^{*}$;

• (ii)

if $\displaystyle F(z,x)=\int^{x}_{0}f(z,s)ds$, then there exist $\displaystyle\eta>p$ and $\displaystyle M>0$ such that

$\displaystyle 0<\eta F(z,x)\leqslant f(z,x)x$ for almost all $\displaystyle z\in\Omega$ and all $\displaystyle|x|\geqslant M$;

$\displaystyle f(z,x)x\leqslant c^{*}_{1}|x|^{r}-c^{*}_{2}|x|^{p}$ for almost all $\displaystyle z\in\Omega$, all $\displaystyle|x|\geqslant M$, and some $\displaystyle c^{*}_{1},c^{*}_{2}>0$;

• (iii)

$\displaystyle\lim\limits_{x\rightarrow 0}\frac{f(z,x)}{|x|^{p-2}x}=0$ uniformly for almost all $\displaystyle z\in\Omega$.

Remark 4**.**

Hypothesis $\displaystyle H(f)(ii)$ is the well-known Ambrosetti-Rabinowitz condition and it implies that

[TABLE]

From (18) and hypothesis $\displaystyle H(f)(ii)$, we infer that for almost all $\displaystyle z\in\Omega$, $\displaystyle f(z,\cdot)$ is $\displaystyle(p-1)$-superlinear. It would be interesting to know if one can replace the Ambrosetti-Rabinowitz condition by more general superlinearity conditions, like the ones used in Papageorgiou and Rădulescu [29, 30]. Below we give simple examples of functions which satisfy hypotheses $\displaystyle H(f)$ (for the sake of simplicity, we drop the $\displaystyle z$-dependence):

[TABLE]

One of our main results is that for all small $\displaystyle\lambda>0$, problem ($\displaystyle P_{\lambda}$) admits extremal constant sign solutions, that is, there is a smallest positive solution $\displaystyle u_{\lambda}^{*}\in D_{+}$ and a biggest negative solution $\displaystyle v_{\lambda}^{*}\in-D_{+}$. These solutions are crucial in our proof on the existence of nodal (that is, sign changing) solutions (Section 4). To study the maps $\displaystyle\lambda\mapsto u_{\lambda}^{*}$ and $\displaystyle\lambda\mapsto v_{\lambda}^{*}$ and to prove the existence of nodal solutions, we will need to strengthen hypotheses $\displaystyle H(f)$ as follows.

$\displaystyle H(f)^{\prime}:$ $\displaystyle f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a Carathéodory function such that $\displaystyle f(z,0)=0$ for almost all $\displaystyle z\in\Omega$, hypotheses $\displaystyle H(f)^{\prime}(i),(ii),(iii)$ are the same as the corresponding hypotheses $\displaystyle H(f)(i),(ii),(iii)$, and

$\displaystyle(iv)$ for almost all $\displaystyle z\in\Omega,\ f(z,\cdot)$ is strictly increasing.

Remark 5**.**

The reason we impose this extra condition on $\displaystyle f(z,\cdot)$ is to be able to use the strong comparison principle in Proposition 6. The fact that the parameter $\displaystyle\lambda>0$ appears in the boundary and not in the reaction term, leads to stronger conditions on $\displaystyle f(z,\cdot)$.

Finally in Section 5, where we deal with the semilinear problem (that is, $\displaystyle a(y)=y$ for all $\displaystyle y\in\mathbb{R}^{N}$), in order to make use of tools from Morse theory (critical groups), we will need to introduce differentiability conditions on $\displaystyle f(z,\cdot)$. More precisely, the new hypotheses on $\displaystyle f(z,x)$ are:

$\displaystyle H(f)^{\prime\prime}:$ $\displaystyle f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a measurable function such that for almost all $\displaystyle z\in\Omega$, $\displaystyle f(z,0)=0,f(z,\cdot)\in C^{1}(\mathbb{R})$ and

• (i)

$\displaystyle|f^{\prime}_{x}(z,x)|\leqslant a(z)(1+|x|^{r-2})$ for almost all $\displaystyle z\in\Omega$ and all $\displaystyle x\in\mathbb{R}$, with $\displaystyle a\in L^{\infty}(\Omega)$, $\displaystyle 2<r<2^{*}$;

• (ii)

if $\displaystyle F(z,x)=\int^{x}_{0}f(z,s)ds$, then there exist $\displaystyle\eta>2$ and $\displaystyle M>0$ such that

[TABLE]

• (iii)

$\displaystyle f^{\prime}_{x}(z,0)=\lim\limits_{x\rightarrow 0}\frac{f(z,x)}{x}=0$ uniformly for almost all $\displaystyle z\in\Omega$;

• (iv)

for every $\displaystyle\rho>0$, there exists $\displaystyle\hat{\xi}_{\rho}>0$ such that for almost all $\displaystyle z\in\Omega$ the function

[TABLE]

is nondecreasing on $\displaystyle[-\rho,\rho]$.

Remark 6**.**

Here hypothesis $\displaystyle H(f)^{\prime\prime}(iv)$ is much weaker than hypothesis $\displaystyle H(f)^{\prime}(iv)$. The linearity of the differential operator leads to a more general strong comparison principle, which is a trivial consequence of the maximum principle.

It is clear from the above hypotheses that in this paper we deal with subcritical reaction terms.

For the boundary function $\displaystyle\beta(z,x)$, we start with the following conditions.

$\displaystyle H(\beta):$ $\displaystyle\beta\in C(\partial\Omega\times\mathbb{R})\cap C^{0,\alpha}_{loc}(\partial\Omega\times\mathbb{R})$ for some $\displaystyle\alpha\in\left(0,1\right]$, $\displaystyle\beta(z,0)=0$ for all $\displaystyle z\in\partial\Omega$ and

• (i)

$\displaystyle c_{12}|x|^{q}\leqslant\beta(z,x)x$ for all $\displaystyle(z,x)\in\partial\Omega\times\mathbb{R}$ and some $\displaystyle c_{12}>0$, with $\displaystyle q<\tau<p$ (see $\displaystyle H(a)(iv)$);

• (ii)

$\displaystyle\lim\limits_{x\rightarrow\pm\infty}\frac{\beta(z,x)}{|x|^{p-1}x}=0$ uniformly for all $\displaystyle z\in\partial\Omega$;

• (iii)

$\displaystyle\limsup\limits_{x\rightarrow 0}\frac{\beta(z,x)}{|x|^{q-2}x}\leqslant c_{13}$ uniformly for all $\displaystyle z\in\partial\Omega$, with $\displaystyle c_{13}>0$;

• (iv)

if $\displaystyle B(z,x)=\int^{x}_{0}\beta(z,s)ds$ then $\displaystyle c_{14}|x|^{q}\leqslant\tau B(z,x)-\beta(z,x)x$ for all $\displaystyle(z,x)\in\partial\Omega\times\mathbb{R}$ and some $\displaystyle c_{9}>0$ (see $\displaystyle H(a)(iv)$).

Remark 7**.**

The above hypotheses imply that

[TABLE]

So, the boundary term $\displaystyle\beta(z,\cdot)$ is strictly $\displaystyle(p-1)$-superlinear. The typical example of a function satisfying hypotheses $\displaystyle H(\beta)$ above is the following (for the sake of simplicity we again drop the $\displaystyle z$-dependence):

[TABLE]

Other possibilities are the functions

[TABLE]

Later to deal with the semilinear problem we will need a stronger version of these conditions.

$\displaystyle H(\beta)^{\prime}:$ $\displaystyle\beta\in C(\partial\Omega\times\mathbb{R})\cap C^{0,\alpha}_{loc}(\partial\Omega\times\mathbb{R})$ with $\displaystyle\alpha\in(0,1)$, for all $\displaystyle z\in\partial\Omega$, $\displaystyle\beta(z,0)=0$, $\displaystyle\beta(z,\cdot)\in C^{1}(\mathbb{R}\backslash\{0\})$ and

• (i)

$\displaystyle c_{50}|x|^{q}\leqslant\beta(z,x)x$ for all $\displaystyle(z,x)\in\partial\Omega\times\mathbb{R}$, some $\displaystyle c_{50}>0$ and with $\displaystyle q\in(1,2)$;

• (ii)

$\displaystyle\lim\limits_{x\rightarrow\pm\infty}\frac{\beta(z,x)}{x}=0$ uniformly for all $\displaystyle z\in\partial\Omega$;

• (iii)

$\displaystyle\limsup\limits_{x\rightarrow 0}\frac{\beta(z,x)}{|x|^{q-2}x}\leqslant c_{51}$ uniformly for all $\displaystyle z\in\partial\Omega$, with $\displaystyle c_{51}>0$;

• (iv)

if $\displaystyle B(z,x)=\int^{x}_{0}\beta(z,s)ds$, then $\displaystyle c_{52}|x|^{q}\leqslant 2B(z,x)-\beta(z,x)x$ for all $\displaystyle(z,x)\in\partial\Omega\times\mathbb{R}$ and some $\displaystyle c_{52}>0$.

Now we state our main results.

Proposition A. If hypotheses $\displaystyle H(a),H(f),H(\beta)$ hold, then

• (a)

for every $\displaystyle\lambda\in(0,\lambda_{+})$ problem ($\displaystyle P_{\lambda}$) admits two positive solutions

[TABLE]

• (b)

for every $\displaystyle\lambda\in(0,\lambda_{-})$ problem ($\displaystyle P_{\lambda}$) admits two negative solutions

[TABLE]

• (c)

for every $\displaystyle\lambda\in(0,\lambda_{0}=\min\{\lambda_{+},\lambda_{-}\})$ problem ($\displaystyle P_{\lambda}$) admits four nontrivial constant sign solutions

[TABLE]

Proposition B. If hypotheses $\displaystyle H(a),H(f),H(\beta)$ hold, then

• (a)

for every $\displaystyle\lambda\in(0,\lambda_{+})$ problem ($\displaystyle P_{\lambda}$) has a smallest positive solution

[TABLE]

• (b)

for every $\displaystyle\lambda\in(0,\lambda_{-})$ problem ($\displaystyle P_{\lambda}$) has a biggest negative solution

[TABLE]

Theorem C. If hypotheses $\displaystyle H(a),H(f)^{\prime},H(\beta)$ hold, then there exists $\displaystyle\lambda_{0}>0$ such that for every $\displaystyle\lambda\in(0,\lambda_{0})$ problem ($\displaystyle P_{\lambda}$) has at least five nontrivial smooth solutions

[TABLE]

Moreover, for every $\displaystyle\lambda\in(0,\lambda_{0})$, problem ($\displaystyle P_{\lambda}$) has extremal constant sign solutions

[TABLE]

such that $\displaystyle y_{0}\in[v^{*}_{\lambda},u^{*}_{\lambda}]\cap C^{1}(\overline{\Omega})$ and the map $\displaystyle\lambda\mapsto u^{*}_{\lambda}$ is

• •

strictly increasing (that is, $\displaystyle\mu<\lambda\Rightarrow u^{*}_{\lambda}-u^{*}_{\mu}\in{\rm int}\,\hat{C}_{+}$),

• •

left continuous from $\displaystyle(0,\lambda_{0})$ into $\displaystyle C^{1}(\overline{\Omega})$,

while the map $\displaystyle\lambda\mapsto v^{*}_{\lambda}$ is

• •

strictly decreasing (that is, $\displaystyle\mu<\lambda\Rightarrow v^{*}_{\mu}-v^{*}_{\lambda}\in{\rm int}\,\hat{C}_{+}$),

• •

right continuous.

Finally, for the semilinear problem

[TABLE]

we prove the following multiplicity result.

Theorem D. If hypotheses $\displaystyle H(f)^{\prime\prime},\ H(\beta)^{\prime}$ hold, then we can find $\displaystyle\lambda_{0}>0$ such that for every $\displaystyle\lambda\in(0,\lambda_{0})$ problem ($\displaystyle S_{\lambda}$) has at least six nontrivial smooth solutions

[TABLE]

Concluding this section, we point out that we use the word “solution” instead of “weak solution”, since our solution has a pointwise a.e. interpretation (like the Carathéodory or strong solutions from the theory of ordinary differential equations). This pointwise interpretation of the solutions is convenient for the use of strong comparison principles (Proposition 6).

3. Constant Sign Solutions

In this section, we show that for small $\displaystyle\lambda>0$, problem ($\displaystyle P_{\lambda}$) admits at least four nontrivial constant sign smooth solutions (two positive and two negative). We also establish the existence of extremal constant sign solutions $\displaystyle u^{*}_{\lambda}$, $\displaystyle v^{*}_{\lambda}$ and determine the monotonicity and continuity properties of the maps $\displaystyle\lambda\mapsto u^{*}_{\lambda}$ and $\displaystyle\lambda\mapsto v^{*}_{\lambda}$.

The energy (Euler) functional of problem ($\displaystyle P_{\lambda}$) is $\displaystyle\varphi_{\lambda}:W^{1,p}(\Omega)\rightarrow\mathbb{R}$ ($\displaystyle\lambda>0$) and it is defined by

[TABLE]

Evidently, $\displaystyle\varphi_{\lambda}\in C^{1}(W^{1,p}(\Omega))$.

Let $\displaystyle\tilde{c}_{2}\in(0,c^{*}_{2})$ and consider the following truncation-perturbation of the reaction term $\displaystyle f(z,\cdot)$:

[TABLE]

Both are Carathéodory functions. We set $\displaystyle\hat{F}_{\pm}(z,x)=\int^{x}_{0}\hat{f}_{\pm}(z,s)ds$. In addition, we introduce the positive and negative truncations of the boundary term $\displaystyle\beta(z,\cdot)$:

[TABLE]

Clearly, $\displaystyle\beta_{\pm}\in C(\partial\Omega\times\mathbb{R})$. We set $\displaystyle B_{\pm}(z,x)=\int^{x}_{0}\beta_{\pm}(z,s)ds$ for all $\displaystyle(z,x)\in\partial\Omega\times\mathbb{R}$. We consider the $\displaystyle C^{1}$-functionals $\displaystyle\hat{\varphi}^{\pm}_{\lambda}:W^{1,p}(\Omega)\rightarrow\mathbb{R}$ $\displaystyle\lambda>0$, defined by

[TABLE]

Proposition 8**.**

If hypotheses $\displaystyle H(a),H(f),H(\beta)$ hold and $\displaystyle\lambda>0$, then the functionals $\displaystyle\hat{\varphi}^{\pm}_{\lambda}$ satisfy the $\displaystyle C$-function.

Proof.

We give the proof for the functional $\displaystyle\hat{\varphi}^{+}_{\lambda}$, the proof for $\displaystyle\hat{\varphi}^{-}_{\lambda}$ being similar.

So, we consider a sequence $\displaystyle\{u_{n}\}\subseteq W^{1,p}(\Omega)$ such that

[TABLE]

From (36) we have

[TABLE]

In (37) we choose $\displaystyle h=-u^{-}_{n}\in W^{1,p}(\Omega)$. Using (24) and (30), we obtain

[TABLE]

Using (20), (30), (24), (38) and hypothesis $\displaystyle H(f)(i)$, we have

[TABLE]

In (37) we choose $\displaystyle h=u^{+}_{n}\in W^{1,p}(\Omega)$. Then

[TABLE]

Adding (39) and (3), we obtain

[TABLE]

From (18) and hypothesis $\displaystyle H(f)(i)$, we see that

[TABLE]

Using (42) in (3) and recalling that $\displaystyle\eta>p$, we obtain

[TABLE]

It follows from (35) and (38) that for all $\displaystyle n\in\mathbb{N}$

[TABLE]

Adding (3) and (44), we have

[TABLE]

Note that

[TABLE]

Also, hypotheses $\displaystyle H(f)(i),(ii)$ imply that

[TABLE]

Moreover, from (20) we have

[TABLE]

Returning to (3) and using (3), (47), (48), we obtain

[TABLE]

It follows from (3) and (49) that

[TABLE]

We can always assume that $\displaystyle\eta\leqslant p^{*}$ (see hypotheses $\displaystyle H(f)(i),(ii)$). We know that

[TABLE]

is an equivalent norm on the Sobolev space $\displaystyle W^{1,p}(\Omega)$ (see Gasinski and Papageorgiou [13, p. 227]). Therefore from (50) we can infer that

[TABLE]

This together with (38) imply that $\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq W^{1,p}(\Omega)$ is bounded and so we may assume that

[TABLE]

In (37) we choose $\displaystyle h=u_{n}-u\in W^{1,p}(\Omega)$, pass to the limit as $\displaystyle n\rightarrow\infty$, and use (51). Then

[TABLE]

Similarly for the functional $\displaystyle\hat{\varphi}^{-}_{\lambda}$. ∎

In a similar fashion, we prove the next property.

Proposition 9**.**

If hypotheses $\displaystyle H(a),H(f),H(\beta)$ hold and $\displaystyle\lambda>0$, then the functional $\displaystyle\varphi_{\lambda}$ satisfies the $\displaystyle C$-condition.

Next, we provide the mountain pass geometry for the functional $\displaystyle\hat{\varphi}^{\pm}_{\lambda}$.

Proposition 10**.**

If hypotheses $\displaystyle H(a),H(f),H(\beta)$ hold, then there exists $\displaystyle\lambda_{\pm}>0$ such that for every $\displaystyle\lambda\in(0,\lambda_{\pm})$ we can find $\displaystyle\rho^{\pm}_{\lambda}$ for which we have

[TABLE]

Proof.

We again present the proof only for $\displaystyle\hat{\varphi}^{+}_{\lambda}$, since the proof for $\displaystyle\hat{\varphi}^{-}_{\lambda}$ is similar.

Hypotheses $\displaystyle H(f)$ imply that given $\displaystyle\epsilon>0$, we can find $\displaystyle\delta>0$ and $\displaystyle c_{26}>0$ such that

[TABLE]

Similarly, hypotheses $\displaystyle H(\beta)(i),(ii),(iii)$, imply that given $\displaystyle\epsilon>0$, we can find $\displaystyle c_{27}>0$ such that

[TABLE]

Then for all $\displaystyle u\in W^{1,p}(\Omega)$, we have

[TABLE]

We have

[TABLE]

Also, we have

[TABLE]

Using these two estimates and choosing small $\displaystyle\epsilon>0$, we obtain

[TABLE]

Note that

[TABLE]

So, given $\displaystyle\vartheta>0$, we can find $\displaystyle\delta_{0}>0$ such that

[TABLE]

Then we have

[TABLE]

Since $\displaystyle c^{*}_{2}>\tilde{c}_{2}$ we choose small $\displaystyle\vartheta>0$ small such that $\displaystyle c^{*}_{2}(1-\vartheta)\geqslant\tilde{c}_{2}$. Then

[TABLE]

Consider the function

[TABLE]

Since $\displaystyle q<p<r$, we see that $\displaystyle\Im_{\lambda}(t)\rightarrow+\infty$ as $\displaystyle t\rightarrow 0^{+}$ and $\displaystyle t\rightarrow+\infty$. So, we can find $\displaystyle t_{0}>0$ such that

[TABLE]

Then $\displaystyle\Im_{\lambda}(t_{0})\rightarrow 0$ as $\displaystyle\lambda\rightarrow 0^{+}$ and so we can find small $\displaystyle\lambda_{+}>0$ such that $\displaystyle\Im_{\lambda}(t_{0})<c_{30}$ for all $\displaystyle\lambda\in(0,\lambda_{+})$. From (56) we see that

[TABLE]

Similarly, we show that there exists $\displaystyle\lambda_{-}>0$ such that for every $\displaystyle\lambda\in(0,\lambda_{-})$ we can find $\displaystyle\rho^{-}_{\lambda}>0$ for which we have

[TABLE]

∎

It is immediate from hypothesis $\displaystyle H(f)(ii)$ (see also (18)) that:

Proposition 11**.**

If hypotheses $\displaystyle H(a),H(f),H(\beta)$ hold, $\displaystyle u\in D_{+}$, and $\displaystyle\lambda>0$, then $\displaystyle\hat{\varphi}^{+}_{\lambda}(tu)\rightarrow-\infty$ as $\displaystyle t\rightarrow+\infty$.

Now, we are ready to produce constant sign solutions.

Proposition 12**.**

If hypotheses $\displaystyle H(a),H(f),H(\beta)$ hold, then

• (a)

for every $\displaystyle\lambda\in(0,\lambda_{+})$ problem ($\displaystyle P_{\lambda}$) admits two positive solutions

[TABLE]

• (b)

for every $\displaystyle\lambda\in(0,\lambda_{-})$ problem ($\displaystyle P_{\lambda}$) admits two negative solutions

[TABLE]

• (c)

for every $\displaystyle\lambda\in(0,\lambda_{0}=\min\{\lambda_{+},\lambda_{-}\})$ problem ($\displaystyle P_{\lambda}$) admits four nontrivial constant sign solutions

[TABLE]

Proof.

(a) Let $\displaystyle\lambda\in(0,\lambda_{+})$ and let $\displaystyle\rho^{+}_{\lambda}$ be as postulated by Proposition 10. We consider the set

[TABLE]

This set is weakly compact in $\displaystyle W^{1,p}(\Omega)$. Moreover, using the Sobolev embedding theorem and the compactness of the trace map, we see that $\displaystyle\hat{\varphi}^{+}_{\lambda}$ is sequentially weakly lower semicontinuous. So, by the Weierstrass theorem, we can find $\displaystyle u_{0}\in W^{1,p}(\Omega)$ such that

[TABLE]

Hypotheses $\displaystyle H(a)(iv),H(f)(iii)$ imply that we can find $\displaystyle\delta\in(0,1)$ such that

[TABLE]

Then for $\displaystyle u\in C_{+}\backslash\{0\}$ with $\displaystyle||u||_{C^{1}(\overline{\Omega})}\leqslant\delta$, we have

[TABLE]

Since $\displaystyle q<\tau<p$, by taking $\displaystyle\delta\in(0,1)$ even smaller if necessary, we infer from (60) that

[TABLE]

It follows from (57) and (61) that

[TABLE]

From (57) and (63) we have

[TABLE]

In (3) we choose $\displaystyle h=-u^{-}_{0}\in W^{1,p}(\Omega)$. Then using Lemma 2 and (24), (30), we obtain

[TABLE]

Hence equation (3) becomes

[TABLE]

From Hu and Papageorgiou [19] and Papageorgiou and Rădulescu [30], we have

[TABLE]

Then invoking the nonlinear regularity theory of Lieberman [21, p. 320], we can infer that

[TABLE]

Hypotheses $\displaystyle H(f)(i),(iii)$ imply that given $\displaystyle\rho>0$, we can find $\displaystyle\hat{\xi}_{\rho}>0$ such that

[TABLE]

If $\displaystyle\rho=||u_{0}||_{\infty}$, from (3) we have

[TABLE]

Let $\displaystyle\mu(t)=a_{0}(t)t,\ t>0$. Then

[TABLE]

Performing integration by parts, we obtain

[TABLE]

We set $\displaystyle H(t)=a_{0}(t)t^{2}-G_{0}(t),H_{0}(t)=c_{4}t^{p}$ for all $\displaystyle t\geqslant 0$. Let $\displaystyle\delta\in(0,1)$ and $\displaystyle s>0$. We introduce the sets

[TABLE]

Then $\displaystyle C_{2}\subseteq C_{1}$ and so $\displaystyle\inf C_{1}\leqslant\inf C_{2}$. Hence

[TABLE]

Because of (66) we can apply the nonlinear strong maximum principle of Pucci and Serrin [33, p. 111], from which we obtain

[TABLE]

Then the boundary point theorem of Pucci and Serrin [33, p. 120], implies that

[TABLE]

Next, note that Propositions 8, 10 and 11 permit the use of Theorem 1 (the mountain pass theorem) on the functional $\displaystyle\hat{\varphi}^{+}_{\lambda}$ $\displaystyle(\lambda\in(0,\lambda_{+}))$. So, we can find $\displaystyle\hat{u}\in W^{1,p}(\Omega)$ such that

[TABLE]

It follows from (67) that

[TABLE]

As before we can easily check that

[TABLE]

(b) Similarly, working this time with the functional $\displaystyle\hat{\varphi}^{-}_{\lambda}\ (\lambda\in(0,\lambda_{-}))$, we produce two negative solutions

[TABLE]

(c) This part follows from (a) and (b) above. ∎

In fact, we can show the existence of extremal constant sign solutions, that is, we will show the following:

• •

for every $\displaystyle\lambda\in(0,\lambda_{+})$, problem ($\displaystyle P_{\lambda}$) has a smallest positive solution $\displaystyle u^{*}_{\lambda}\in D_{+}$;

• •

for every $\displaystyle\lambda\in(0,\lambda_{-})$, problem ($\displaystyle P_{\lambda}$) has a biggest negative solution $\displaystyle v^{*}_{\lambda}\in-D_{+}$.

To this end, note that hypotheses $\displaystyle H(f)$ imply that

[TABLE]

This unilateral growth estimate on the reaction term $\displaystyle f(z,\cdot)$ and hypothesis $\displaystyle H(\beta)(i)$, lead to the following auxiliary nonlinear boundary value problem:

[TABLE]

Proposition 13**.**

If hypotheses $\displaystyle H(a)$ hold and $\displaystyle\lambda>0$ then problem ($\displaystyle Au_{\lambda}$) has a unique positive solution $\displaystyle\bar{u}_{\lambda}\in D_{+}$ and a unique negative solution $\displaystyle\bar{v}_{\lambda}\in-D_{+}$.

Proof.

First, we establish the existence of a positive solution.

So we consider the $\displaystyle C^{1}$-functional $\displaystyle\psi^{+}_{\lambda}:W^{1,p}(\Omega)\rightarrow\mathbb{R}$ defined by

[TABLE]

Evidently, we can always assume that $\displaystyle c_{33}>1$ (see (68)). Then we have

[TABLE]

Moreover, the Sobolev embedding theorem and the compactness of the trace map, imply that $\displaystyle\psi^{+}_{\lambda}$ is sequentially weakly lower semicontinuous. So, we can find $\displaystyle\bar{u}_{\lambda}\in W^{1,p}(\Omega)$ such that

[TABLE]

As in the proof of Proposition 12, exploiting the fact that $\displaystyle q<\tau<p$, we show that

[TABLE]

From (69) we have

[TABLE]

In (3) we choose $\displaystyle h=-\bar{u}^{-}_{\lambda}\in W^{1,p}(\Omega)$. Then

[TABLE]

Then equation (3) becomes

[TABLE]

As before, the nonlinear regularity theory (see Lieberman [21]) and the nonlinear maximum principle (see [33]), imply that

[TABLE]

Next, we show the uniqueness of this positive solution. To this end, we introduce the integral functional $\displaystyle j:L^{1}(\Omega)\rightarrow\bar{\mathbb{R}}=\mathbb{R}\cup\{+\infty\}$ defined by

[TABLE]

Let $\displaystyle{\rm dom}\,j=\{u\in L^{1}(\Omega):j(u)<+\infty\}$ (the effective domain of $\displaystyle j$) and let $\displaystyle u_{1},u_{2}\in{\rm dom}\,j$. We set

[TABLE]

From Lemma 1 of Diaz and Saa [9], we have

[TABLE]

Then

[TABLE]

Also, recall that $\displaystyle q<\tau$ and so $\displaystyle x\mapsto-x^{q/\tau}$ is convex on $\displaystyle\left[0,+\infty\right)$. Therefore it follows that $\displaystyle j(\cdot)$ is convex. Moreover, Fatou’s lemma implies that $\displaystyle j(\cdot)$ is lower semicontinuous.

Now suppose that $\displaystyle\bar{w}_{\lambda}\in W^{1,p}(\Omega)$ is another positive solution of problem ($\displaystyle Au_{\lambda}$). As above we can show that

[TABLE]

Then for all $\displaystyle h\in C^{1}(\overline{\Omega})$ and for small enogh $\displaystyle|t|\leqslant 1$, we have

[TABLE]

We can easily see that $\displaystyle j(\cdot)$ is Gâteaux differentiable at $\displaystyle\bar{u}^{\tau}_{\lambda},\bar{w}^{\tau}_{\lambda}$ in the direction $\displaystyle h$. Moreover, via the chain rule and the nonlinear Green’s identity (see Gasinski and Papageorgiou [13, p. 210]), we have

[TABLE]

The convexity of $\displaystyle j(\cdot)$ implies that $\displaystyle j^{\prime}(\cdot)$ is monotone. Hence

[TABLE]

Since $\displaystyle x\mapsto x^{p-\tau}$ is strictly increasing on $\displaystyle\left[0,+\infty\right)$ from (74) it follows that

[TABLE]

This proves the uniqueness of the positive solution $\displaystyle\bar{u}_{\lambda}\in D_{+}$ of problem ($\displaystyle Au_{\lambda}$).

The fact that problem ($\displaystyle Au_{\lambda}$) is odd, implies that $\displaystyle\bar{v}_{\lambda}=-\bar{u}_{\lambda}\in-D_{+}$ is the unique negative solution. ∎

In what follows, for every $\displaystyle\lambda>0$, let $\displaystyle S_{+}(\lambda)$ (respectively, $\displaystyle S_{-}(\lambda)$) be the set of positive (respectively, negative) solutions of problem ($\displaystyle P_{\lambda}$). From Proposition 12 and its proof, we know that:

• •

If $\displaystyle\lambda\in(0,\lambda_{+})$, then $\displaystyle S_{+}(\lambda)\neq\emptyset$ and $\displaystyle S_{+}(\lambda)\subseteq D_{+}$.

• •

If $\displaystyle\lambda\in(0,\lambda_{-})$, then $\displaystyle S_{-}(\lambda)\neq\emptyset$ and $\displaystyle S_{-}(\lambda)\subseteq-D_{+}$.

We will use the unique constant sign solutions $\displaystyle\bar{u}_{\lambda}\in D_{+}$ (respectively, $\displaystyle\bar{v}_{\lambda}\in-D_{+}$) of the auxiliary problem ($\displaystyle Au_{\lambda}$) produced in Proposition 13, to provide a lower bound (respectively, upper bound) for the elements of $\displaystyle S_{+}(\lambda)$ (respectively, $\displaystyle S_{-}(\lambda)$).

Proposition 14**.**

If hypotheses $\displaystyle H(a),H(f),H(\beta)$ hold, then

• (a)

for all $\displaystyle\lambda\in(0,\lambda_{+})$ and all $\displaystyle u\in S_{+}(\lambda)$, we have $\displaystyle\bar{u}_{\lambda}\leqslant u$;

• (b)

for all $\displaystyle\lambda\in(0,\lambda_{-})$ and all $\displaystyle v\in S_{-}(\lambda)$, we have $\displaystyle v\leqslant\bar{v}_{\lambda}$.

Proof.

(a) Let $\displaystyle\lambda\in(0,\lambda_{+})$ and $\displaystyle u\in S_{+}(\lambda)$. We introduce the following Carathéodory functions

[TABLE]

We set $\displaystyle\hat{K}_{+}(z,x)=\int^{x}_{0}\hat{k}_{+}(z,s)ds$ and $\displaystyle\hat{B}_{+}(z,x)=\int^{x}_{0}\hat{\beta}_{+}(z,s)ds$ and consider the $\displaystyle C^{1}$-functional $\displaystyle\hat{\psi}^{+}_{\lambda}:W^{1,p}(\Omega)\rightarrow\mathbb{R}$ defined by

[TABLE]

From Corollary 3 and (78), (82), we see that $\displaystyle\hat{\psi}^{+}_{\lambda}$ is coercive. Also, from the Sobolev embedding theorem and the compactness of the trace map, it follows that $\displaystyle\hat{\psi}^{+}_{\lambda}$ is sequentially weakly lower semicontinuous. So, we can find $\displaystyle\tilde{u}_{\lambda}\in W^{1,p}(\Omega)$ such that

[TABLE]

In fact, since $\displaystyle q<\tau<p$, as in the proof of Proposition 12 (see (60) with $\displaystyle\delta\leqslant\min\limits_{\overline{\Omega}}u$ and recall that $\displaystyle u\in D_{+}$), we have

[TABLE]

From (83) we have

[TABLE]

In (3) we first choose $\displaystyle h=-\tilde{u}_{\lambda}^{-}\in W^{1,p}(\Omega)$. Then

[TABLE]

Next, in (3) we choose $\displaystyle h=(\tilde{u}_{\lambda}-u)^{+}\in W^{1,p}(\Omega)$. Then

[TABLE]

So, we have proved that

[TABLE]

Therefore equation (3) becomes

[TABLE]

Since $\displaystyle u\in S_{+}(\lambda)$ is arbitrary, we conclude that

[TABLE]

(b) In a similar fashion, we show that if $\displaystyle\lambda\in(0,\lambda_{-})$, then $\displaystyle v\leqslant\bar{v}_{\lambda}$ for all $\displaystyle v\in S_{-}(\lambda)$. ∎

Using this proposition, we can produce the desired extremal constant sign solutions for problem ($\displaystyle P_{\lambda}$).

As in Filippakis and Papageorgiou [10] (see Lemmata 4.1 and 4.2), we have:

• •

$\displaystyle S_{+}(\lambda)$ is downward directed, that is, if $\displaystyle u_{1},u_{2}\in S_{+}(\lambda)$, then we can find $\displaystyle u\in S_{+}(\lambda)$ such that $\displaystyle u\leqslant u_{1},u\leqslant u_{2}$.

• •

$\displaystyle S_{-}(\lambda)$ is upward directed, that is, if $\displaystyle v_{1},v_{2}\in S_{-}(\lambda)$, then we can find $\displaystyle v\in S_{-}(\lambda)$ such that $\displaystyle v_{1}\leqslant v,v_{2}\leqslant v$.

Proposition 15**.**

If hypotheses $\displaystyle H(a),H(f),H(\beta)$ hold, then

• (a)

for every $\displaystyle\lambda\in(0,\lambda_{+})$ problem ($\displaystyle P_{\lambda}$) has a smallest positive solution

[TABLE]

• (b)

for every $\displaystyle\lambda\in(0,\lambda_{-})$ problem ($\displaystyle P_{\lambda}$) has a biggest negative solution

[TABLE]

Proof.

(a) Using Lemma 3.9, p. 178 of Hu and Papageorgiou [18], we can find a decreasing sequence $\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq S_{+}(\lambda)$ such that

[TABLE]

We have for all $\displaystyle h\in W^{1,p}(\Omega)$ and all $\displaystyle n\in\mathbb{N}$

[TABLE]

Since $\displaystyle 0\leqslant u_{n}\leqslant u_{1}$ for all $\displaystyle n\in\mathbb{N}$, using (85), Corollary 3, hypothesis $\displaystyle H(f)(i)$ and (20), we can infer that $\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq W^{1,p}(\Omega)$ is bounded. So, we may assume that

[TABLE]

In (85) we choose $\displaystyle h=u_{n}-u^{*}_{\lambda}\in W^{1,p}(\Omega)$, pass to the limit as $\displaystyle n\rightarrow\infty$ and use (86). Then we obtain

[TABLE]

So, passing to the limit as $\displaystyle n\rightarrow\infty$ in (85) and using (87), we have

[TABLE]

From Proposition 14 we know that

[TABLE]

(b) Reasoning in a similar fashion, we show that for all $\displaystyle\lambda\in(0,\lambda_{-})$ problem ($\displaystyle P_{\lambda}$) has a biggest negative solution $\displaystyle v^{*}_{\lambda}\in S_{-}(\lambda)$. ∎

In Section 4, using these extremal constant sign solutions, we will produce a nodal (sign changing) solution for problem ($\displaystyle P_{\lambda}$). For the moment, in the remaining part of this section we examine the maps

[TABLE]

The next proposition will be used to prove the monotonicity properties of the maps in (88), (89).

Proposition 16**.**

If hypotheses $\displaystyle H(a),H(f)^{\prime},H(\beta)$ hold, then

• (a)

given $\displaystyle\lambda,\mu\in(0,\lambda_{+})$ with $\displaystyle\mu<\lambda$ and $\displaystyle u_{\lambda}\in S_{+}(\lambda)$, we can find $\displaystyle u_{\mu}\in S_{+}(\mu)$ such that

[TABLE]

• (b)

given $\displaystyle\lambda,\mu\in(0,\lambda_{-})$ with $\displaystyle\mu<\lambda$ and $\displaystyle v_{\lambda}\in S_{-}(\lambda)$, we can find $\displaystyle u_{\mu}\in S_{-}(\mu)$ such that

[TABLE]

Proof.

(a) We introduce the following Carathéodory functions

[TABLE]

We set

[TABLE]

and consider the $\displaystyle C^{1}$-functional $\displaystyle\vartheta^{+}_{\mu}(z,x):W^{1,p}(\Omega)\rightarrow\mathbb{R}$ defined by

[TABLE]

From Corollary 3 and (92), (95), it is clear that the function $\displaystyle\vartheta^{+}_{\mu}$ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find $\displaystyle u_{\mu}\in W^{1,p}(\Omega)$ such that

[TABLE]

Since $\displaystyle q<\tau<p$, we have

[TABLE]

From (96) we have

[TABLE]

In (3) we first choose $\displaystyle h=-u^{-}_{\mu}\in W^{1,p}(\Omega)$. From Lemma 2 and (92), (95) we have

[TABLE]

Next, in (3) we choose $\displaystyle h=(u_{\mu}-u_{\lambda})^{+}\in W^{1,p}(\Omega)$. Then

[TABLE]

So, we have proved that

[TABLE]

Invoking (92), (95), (98), equation (3) becomes

[TABLE]

Evidently, $\displaystyle u_{\mu}\neq u_{\lambda}$ (recall that $\displaystyle\mu<\lambda$ and use hypothesis $\displaystyle H(\beta)(i)$). Then hypothesis $\displaystyle H(f)^{\prime}(iv)$ implies that

[TABLE]

with $\displaystyle g_{\mu},g_{\lambda}\in L^{\infty}(\Omega)$ and $\displaystyle g_{\mu}\not\equiv g_{\lambda}$. Also, we have

[TABLE]

Therefore, we can use Proposition 6 and infer that

[TABLE]

(b) For this part, we consider the following Carathéodory functions

[TABLE]

We set $\displaystyle E_{-}(z,x)=\int^{x}_{0}e_{-}(z,s)ds$ and $\displaystyle D^{-}_{\mu}(z,x)=\int^{x}_{0}d^{-}_{\mu}(z,s)ds$ and consider the $\displaystyle C^{1}$-functional $\displaystyle\vartheta^{-}_{\mu}:W^{1,p}(\Omega)\rightarrow\mathbb{R}$ defined by

[TABLE]

Reasoning as in part (a), we produce some $\displaystyle v_{\mu}\in S_{-}(\mu)$ such that

[TABLE]

∎

Now we can establish the monotonicity and continuity properties of the two maps defined in (88) and (89).

Proposition 17**.**

If hypotheses $\displaystyle H(a),H(f)^{\prime},H(\beta)$ hold, then

• (a)

the map $\displaystyle\lambda\mapsto u^{*}_{\lambda}$ from $\displaystyle(0,\lambda_{+})$ into $\displaystyle C^{1}(\overline{\Omega})$ is strictly increasing in the sense that $\displaystyle\mu<\lambda\Rightarrow u^{*}_{\lambda}-u^{*}_{\mu}\in{\rm int}\,\hat{C}_{+}$ and is left continuous;

• (b)

the map $\displaystyle\lambda\mapsto v^{*}_{\lambda}$ from $\displaystyle(0,\lambda_{-})$ into $\displaystyle C^{1}(\overline{\Omega})$ is strictly decreasing in the sense that $\displaystyle\mu<\lambda\Rightarrow v^{*}_{\mu}-v^{*}_{\lambda}\in{\rm int}\,\hat{C}_{+}$ and is right continuous.

Proof.

(a) Let $\displaystyle\mu,\lambda\in(0,\lambda_{+})$ with $\displaystyle\mu<\lambda$. From Proposition 15, we know that problem ($\displaystyle P_{\lambda}$) has a smallest positive solution $\displaystyle u^{*}_{\lambda}\in S_{+}(\lambda)$. Invoking Proposition 16, we can find $\displaystyle u_{\mu}\in S_{+}(\mu)$ such that

[TABLE]

Next, let $\displaystyle\{\lambda_{n},\lambda\}_{n\geqslant 1}\subseteq(0,\lambda_{+})$ and assume that $\displaystyle\lambda_{n}\rightarrow\lambda^{-}$. We have

[TABLE]

Then from Proposition 15 and the first part of the proof, we have

[TABLE]

Hence the nonlinear regularity theory of Lieberman [21] implies that there exist $\displaystyle\alpha\in(0,1)$ and $\displaystyle c_{37}>0$ such that

[TABLE]

Exploiting the compact embedding of $\displaystyle C^{1,\alpha}(\overline{\Omega})$ into $\displaystyle C^{1}(\overline{\Omega})$ and by passing to a subsequence if necessary, we can say that

[TABLE]

Evidently, we have

[TABLE]

Suppose that $\displaystyle\tilde{u}_{\lambda}\neq u^{*}_{\lambda}$. Then we can find $\displaystyle z_{0}\in\Omega$ such that

[TABLE]

This contradicts the first part (that is, the “monotonicity” part) of the proof. So, $\displaystyle\tilde{u}_{\lambda}=u^{*}_{\lambda}$ and now by Urysohn’s criterion we conclude that for the initial sequence we have

[TABLE]

(b) In a similar fashion we show that the map $\displaystyle\lambda\mapsto v^{*}_{\lambda}$ from $\displaystyle(0,\lambda_{-})$ into $\displaystyle C^{1}(\overline{\Omega})$ is strictly decreasing (in the sense described in the proposition) and right continuous. ∎

4. Nodal Solutions

In this section we turn our attention to the existence of nodal solutions. To do this, we will use a combination of variational methods and Morse theory. So, we start with the computation of the critical groups at the origin of the energy (Euler) functional $\displaystyle\varphi_{\lambda}$.

Proposition 18**.**

If hypotheses $\displaystyle H(a),H(f)^{\prime},H(\beta)$ hold, $\displaystyle\lambda>0$, and $\displaystyle K_{\varphi_{\lambda}}$ is finite, then $\displaystyle C_{k}(\varphi_{\lambda},0)=0$ for all $\displaystyle k\in\mathbb{N}_{0}$.

Proof.

Hypothesis $\displaystyle H(a)(iv)$ and Corollary 3 imply that

[TABLE]

Also, hypotheses $\displaystyle H(f)^{\prime}(i),(ii),(iii)$ (see also (18)) imply that

[TABLE]

Moreover, from (20) we have

[TABLE]

For $\displaystyle u\in W^{1,p}(\Omega)$ and $\displaystyle t>0$, we have

[TABLE]

Since $\displaystyle q<\tau<p<\eta$, from (4) we see that we can find $\displaystyle t^{*}=t^{*}(u)\in(0,1)$ such that

[TABLE]

Now, let $\displaystyle u\in W^{1,p}(\Omega)$ with $\displaystyle 0<||u||\leqslant 1$ and $\displaystyle\varphi_{\lambda}(u)=0$. Then

[TABLE]

Hypothesis $\displaystyle H(a)(iv)$ implies that

[TABLE]

Also, hypotheses $\displaystyle H(f)^{\prime}(i),(iii)$ imply that given $\displaystyle\epsilon>0$, we can find $\displaystyle c_{42}=c_{42}(\epsilon)>0$ such that

[TABLE]

Finally, from hypothesis $\displaystyle H(\beta)(iv)$, we have

[TABLE]

Since $\displaystyle q<p$, for all $\displaystyle||u||_{L^{q}(\partial\Omega)}\leqslant 1$ we have

[TABLE]

From Proposition 7 (see also the remark following that proposition), we know that

[TABLE]

is an equivalent norm on the Sobolev space $\displaystyle W^{1,p}(\Omega)$.

So, returning to (4) and using (115), (116) and (117) and choosing small $\displaystyle\epsilon>0$, we see that for all $\displaystyle u\in W^{1,p}(\Omega)$ with $\displaystyle 0<||u||\leqslant 1$ and $\displaystyle||u||_{L^{q}(\partial\Omega)}\leqslant 1$, $\displaystyle\varphi_{\lambda}(u)=0$, we have

[TABLE]

Recall that $\displaystyle p<r$. Choosing $\displaystyle\rho\in(0,1)$ small, we have

[TABLE]

(recall that via the trace map, $\displaystyle W^{1,p}(\Omega)$ is embedded continuously into $\displaystyle L^{q}(\partial\Omega)$).

Now consider $\displaystyle u\in W^{1,p}(\Omega)$ with $\displaystyle 0<||u||\leqslant\rho,\ \varphi_{\lambda}(u)=0$. We will show that

[TABLE]

If (120) is not true, then we can find $\displaystyle t_{0}\in(0,1)$ such that

[TABLE]

Since $\displaystyle\varphi_{\lambda}(u)=0$ and $\displaystyle\varphi_{\lambda}(\cdot)$ is continuous, we have

[TABLE]

We have

[TABLE]

We set $\displaystyle y=t_{*}u$. Then $\displaystyle 0<||y||\leqslant||u||\leqslant\rho$ and $\displaystyle\varphi_{\lambda}(y)=0$. So, it follows from (119) that

[TABLE]

From (121) we have

[TABLE]

and this implies that

[TABLE]

Comparing (122) and (123), we obtain a contradiction. This proves (120).

We can always choose $\displaystyle\rho\in(0,1)$ small enough so that $\displaystyle K_{\varphi_{\lambda}}\cap\bar{B}_{\rho}=\{0\}$ (here, $\displaystyle\bar{B}_{\rho}=\{v\in W^{1,p}(\Omega):||v||\leqslant\rho\}$). We consider the deformation $\displaystyle h:[0,1]\times(\varphi_{\lambda}^{0}\cap\bar{B}_{\rho})\rightarrow\varphi_{\lambda}^{0}\cap\bar{B}_{\rho}$ defined by

[TABLE]

Using (120), we see easily that this is a well-defined deformation and it implies that the set $\displaystyle\varphi_{\lambda}^{0}\cap\bar{B}_{\rho}$ is contractible in itself.

Let $\displaystyle u\in\bar{B}_{\rho}$ and assume that $\displaystyle\varphi_{\lambda}(u)>0$. We will show that there is a unique $\displaystyle t(u)\in(0,1)$ such that

[TABLE]

From (113) and Bolzano’s theorem, we see that there exists $\displaystyle t(u)\in(0,1)$ such that (124) holds. We need to show that $\displaystyle t(u)\in(0,1)$ is unique. Arguing by contradiction, suppose we can find

[TABLE]

From (120) we have

[TABLE]

which contradicts (119). Therefore $\displaystyle t(u)\in(0,1)$ for which (124) holds is indeed unique. Then

[TABLE]

Consider the function $\displaystyle\vartheta:\bar{B}_{\rho}\backslash\{0\}\rightarrow[0,1]$ defined by

[TABLE]

It is easy to see that $\displaystyle\vartheta(\cdot)$ is continuous. Now let $\displaystyle d:\bar{B}_{\rho}\backslash\{0\}\rightarrow(\varphi_{\lambda}^{\circ}\cap\bar{B}_{\rho})\backslash\{0\}$ be the map defined by

[TABLE]

The continuity of $\displaystyle\vartheta(\cdot)$ implies the continuity of $\displaystyle d(\cdot)$. Note that

[TABLE]

Hence $\displaystyle(\varphi_{\lambda}^{0}\cap\bar{B}_{\rho})\backslash\{0\}$ is a retract of $\displaystyle\bar{B}_{\rho}\backslash\{0\}$ and the latter is contractible. Thus so is the set $\displaystyle(\varphi_{\lambda}^{0}\cap\bar{B}_{\rho})\backslash\{0\}$. Recall that we have established earlier that $\displaystyle\varphi_{\lambda}^{0}\cap\bar{B}_{\rho}$ is contractible. Therefore we have

[TABLE]

∎

Recall that $\displaystyle\lambda_{0}=\min\{\lambda_{+},\lambda_{-}\}$. Next, we show that for every $\displaystyle\lambda\in(0,\lambda_{0})$ problem ($\displaystyle P_{\lambda}$) admits a nodal solution.

Proposition 19**.**

If hypotheses $\displaystyle H(a),H(f)^{\prime},H(\beta)$ hold and $\displaystyle\lambda\in(0,\lambda_{0})$, then problem ($\displaystyle P_{\lambda}$) admits a nodal solution $\displaystyle y_{0}\in[v^{*}_{\lambda},u^{*}_{\lambda}]\cap C^{1}(\overline{\Omega})$.

Proof.

Let $\displaystyle u^{*}_{\lambda}\in D_{+}$ and $\displaystyle v^{*}_{\lambda}\in-D_{+}$ be the two extremal constant sign solutions of problem ($\displaystyle P_{\lambda}$) produced in Proposition 15. We introduce the following Carathéodory functions

[TABLE]

We set $\displaystyle K(z,x)=\int^{x}_{0}k(z,s)ds$ and $\displaystyle E(z,x)=\int^{x}_{0}e(z,s)ds$ and consider the $\displaystyle C^{1}$-functional $\displaystyle\gamma_{\lambda}:W^{1,p}(\Omega)\rightarrow\mathbb{R}$ defined by

[TABLE]

Also, we consider the positive and negative truncations of $\displaystyle k(z,\cdot),\ e(z,\cdot)$, that is, the Carathéodory functions

[TABLE]

We set $\displaystyle K_{\pm}(z,x)=\int^{x}_{0}k_{\pm}(z,s)ds$ and $\displaystyle E_{\pm}(z,x)=\int^{x}_{0}e_{\pm}(z,s)ds$ and consider the $\displaystyle C^{1}$ - functionals $\displaystyle\gamma^{\pm}_{\lambda}:W^{1,p}(\Omega)\rightarrow\mathbb{R}$ defined by

[TABLE]

Claim 1**.**

$\displaystyle K_{\gamma_{\lambda}}\subseteq[v^{*}_{\lambda},u^{*}_{\lambda}],K_{\gamma^{+}_{\lambda}}=\{0,u^{*}_{\lambda}\},\ K_{\gamma^{-}_{\lambda}}=\{0,v^{*}_{\lambda}\}$.

Suppose that $\displaystyle u\in K_{\gamma_{\lambda}}$. Then

[TABLE]

In (134) first we choose $\displaystyle h=(u-u^{*}_{\lambda})^{+}\in W^{1,p}(\Omega)$. We obtain

[TABLE]

Similarly, if in (134) we choose $\displaystyle h=(v^{*}_{\lambda}-u)^{+}\in W^{1,p}(\Omega)$, then we can show that

[TABLE]

So, we have proved that

[TABLE]

Similarly, we show that

[TABLE]

The extremality of the constant sign solutions $\displaystyle u^{*}_{\lambda}$ and $\displaystyle v^{*}_{\lambda}$, implies that

[TABLE]

This proves Claim 1.

Claim 2**.**

$\displaystyle u^{*}_{\lambda}\in D_{+}$* and $\displaystyle v^{*}_{\lambda}\in-D_{+}$ are local minimizers of $\displaystyle\gamma_{\lambda}$.*

Corollary 3 and (129), (133) imply that $\displaystyle\gamma^{+}_{\lambda}$ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find $\displaystyle\tilde{u}_{\lambda}\in W^{1,p}(\Omega)$ such that

[TABLE]

As before (see the proof of Proposition 12), since $\displaystyle q<\tau<p<\eta$, we have

[TABLE]

From (135) we have $\displaystyle\tilde{u}_{\lambda}\in K_{\gamma^{+}_{\lambda}}$. Then Claim 1 and (136) imply that

[TABLE]

Note that

[TABLE]

Similarly, for $\displaystyle v^{*}_{\lambda}\in-D_{+}$, using this time the functional $\displaystyle\gamma^{-}_{\lambda}$. This proves Claim 2.

Without any loss of generality we may assume that

[TABLE]

The reasoning is similar if the opposite inequality holds.

We assume that $\displaystyle K_{\gamma_{\lambda}}$ is finite. Otherwise, on account of Claim 1 and (129), (133), we already have an infinity of nodal solutions in $\displaystyle C^{1}(\overline{\Omega})$ (by the nonlinear regularity theory of Lieberman [21]). Then since $\displaystyle u^{*}_{\lambda}\in D_{+}$ is a local minimizer of $\displaystyle\gamma_{\lambda}$ (see Claim 2), we can find $\displaystyle\rho\in(0,1)$ so small that

[TABLE]

(see Aizicovici, Papageorgiou and Staicu [1], proof of Proposition 29).

The functional $\displaystyle\gamma_{\lambda}$ is coercive (see (129), (133)). So, we have that

[TABLE]

(see Papageorgiou and Winkert [32]). Then (137), (138) permit the use of Theorem 1 (the mountain pass theorem). So, we can find $\displaystyle y_{0}\in W^{1,p}(\Omega)$ such that

[TABLE]

From (137) and (139) we see that

[TABLE]

Moreover, Corollary 6.81, p. 168 of Motreanu, Motreanu and Papageorgiou [23] implies that

[TABLE]

Claim 3**.**

$\displaystyle C_{k}(\gamma_{\lambda},0)=C_{k}(\varphi_{\lambda},0)$* for all $\displaystyle k\in\mathbb{N}_{0}$.*

We consider the homotopy $\displaystyle h(t,u)$ defined by

[TABLE]

Suppose that we could find $\displaystyle\{t_{n}\}_{n\geqslant 1}\subseteq[0,1]$ and $\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq W^{1,p}(\Omega)$ such that

[TABLE]

From the equality in (142), we have

[TABLE]

It follows (see Papageorgiou and Rădulescu [28]) that

[TABLE]

From (142), (147), we see that we can find $\displaystyle c_{45}>0$ such that

[TABLE]

(see [19] and [30]). This $\displaystyle L^{\infty}$-bound permits the use of the nonlinear regularity theory of Lieberman [21], hence there exist $\displaystyle\alpha\in(0,1)$ and $\displaystyle c_{46}>0$ such that

[TABLE]

From (142), (149) and the compact embedding of $\displaystyle C^{1,\alpha}(\overline{\Omega})$ into $\displaystyle C^{1}(\overline{\Omega})$, we have

[TABLE]

It follows from (129), (133), (147), (150) that

[TABLE]

which contradicts the assumption that $\displaystyle K_{\gamma_{\lambda}}$ is finite. So, (142) cannot occur and we can use Theorem 5.2 of Corvellec and Hantoute [8] (the homotopy invariance of critical groups) and obtain

[TABLE]

This proves Claim 3.

From Claim 3 and Proposition 18, we have

[TABLE]

Comparing (141) and (151), we see that

[TABLE]

∎

Summarizing the situation for problem ($\displaystyle P_{\lambda}$), we can state the following multiplicity theorem.

Theorem 20**.**

If hypotheses $\displaystyle H(a),H(f)^{\prime},H(\beta)$ hold, then there exists $\displaystyle\lambda_{0}>0$ such that for every $\displaystyle\lambda\in(0,\lambda_{0})$ problem ($\displaystyle P_{\lambda}$) has at least five nontrivial smooth solutions

[TABLE]

Moreover, for every $\displaystyle\lambda\in(0,\lambda_{0})$, problem ($\displaystyle P_{\lambda}$) has extremal constant sign solutions

[TABLE]

such that $\displaystyle y_{0}\in[v^{*}_{\lambda},u^{*}_{\lambda}]\cap C^{1}(\overline{\Omega})$ and the map $\displaystyle\lambda\mapsto u^{*}_{\lambda}$ is

• •

strictly increasing (that is, $\displaystyle\mu<\lambda\Rightarrow u^{*}_{\lambda}-u^{*}_{\mu}\in{\rm int}\,\hat{C}_{+}$),

• •

left continuous from $\displaystyle(0,\lambda_{0})$ into $\displaystyle C^{1}(\overline{\Omega})$,

while the map $\displaystyle\lambda\mapsto v^{*}_{\lambda}$ is

• •

strictly decreasing (that is, $\displaystyle\mu<\lambda\Rightarrow v^{*}_{\mu}-v^{*}_{\lambda}\in{\rm int}\,\hat{C}_{+}$),

• •

right continuous.

In the next section, we show that in the semilinear case, we can improve this theorem and produce a sixth nontrivial smooth solution $\displaystyle\hat{y}$, but we cannot provide any sign information for it.

5. Semilinear Problem

In this section we deal with the semilinear problem

[TABLE]

We strengthen the regularity hypotheses on the reaction term $\displaystyle f(z,\cdot)$ and on the boundary (source) term $\displaystyle\beta(z,\cdot)$ and by using Morse theory we are able to generate a sixth nontrivial smooth solution. However, we cannot provide any sign information for this new solution.

In this case the energy (Euler) functional of problem ($\displaystyle S_{\lambda}$) is $\displaystyle\varphi_{\lambda}:H^{1}(\Omega)\rightarrow\mathbb{R}$ defined by

[TABLE]

Hypotheses $\displaystyle H(f)^{\prime\prime}$ and $\displaystyle H(\beta)^{\prime}$ imply that $\displaystyle\varphi_{\lambda}\in C^{2}(H^{1}(\Omega)\backslash\{0\})$. Under these hypotheses we can show that problem ($\displaystyle S_{\lambda}$) has six nontrivial smooth solutions for all small $\displaystyle\lambda>0$.

Theorem 21**.**

If hypotheses $\displaystyle H(f)^{\prime\prime},\ H(\beta)^{\prime}$ hold, then we can find $\displaystyle\lambda_{0}>0$ such that for every $\displaystyle\lambda\in(0,\lambda_{0})$ problem ($\displaystyle S_{\lambda}$) has at least six nontrivial smooth solutions

[TABLE]

Proof.

From Theorem 20, we know that we can find $\displaystyle\lambda_{0}>0$ such that for all $\displaystyle\lambda\in(0,\lambda_{0})$ problem ($\displaystyle S_{\lambda}$) has five nontrivial smooth solutions

[TABLE]

From the proof of Proposition 12, we know that $\displaystyle u_{0}\in D_{+}$ and $\displaystyle v_{0}\in-D_{+}$ are local minimizers of $\displaystyle\varphi_{\lambda}$ and so we have

[TABLE]

Let $\displaystyle\rho=\max\{||u_{0}||_{\infty},||v_{0}||_{\infty}\}$ and let $\displaystyle\hat{\xi}_{\rho}>0$ be as postulated by hypothesis $\displaystyle H(f)(iv)$. We have

[TABLE]

Similarly, we show that

[TABLE]

Therefore we can assert that

[TABLE]

Keeping the notation of the previous section (see the proof of Proposition 19), and assuming without any loss of generality that $\displaystyle u_{0},v_{0}$ are extremal constant sign solutions (see Proposition 15), we have

[TABLE]

Then it follows from (153) that

[TABLE]

(since $\displaystyle C^{1}(\overline{\Omega})$ is dense in $\displaystyle H^{1}(\Omega)$, see Chang [5, p. 14] and Palais [25]),

[TABLE]

(since $\displaystyle y_{0}$ is a critical point of mountain pass type of $\displaystyle\gamma_{\lambda}$).

Since $\displaystyle\varphi_{\lambda}\in C^{2}(H^{1}(\Omega)\backslash\{0\})$, it follows from (154) that

[TABLE]

(see Motreanu, Motreanu and Papageorgiou [23], Corollary 6.102, p. 177).

Similarly, from the proof of Proposition 12 and keeping the notion introduced there,

[TABLE]

Since $\displaystyle\hat{\varphi}^{+}_{\lambda}|_{C_{+}}=\varphi_{\lambda}|_{C_{+}}$ and $\displaystyle\hat{\varphi}^{-}_{\lambda}|_{-C_{+}}=\varphi_{\lambda}|_{-C_{+}}$, as above we have

[TABLE]

From Proposition 18, we know that

[TABLE]

Finally, hypothesis $\displaystyle H(f)^{\prime\prime}(ii)$ implies that

[TABLE]

(see Papageorgiou and Rădulescu [29, Proposition 13]). Suppose that

[TABLE]

Then using (152), (155), (156), (157), (158) and the Morse relation (see (4)) with $\displaystyle t=-1$, we obtain

[TABLE]

So, there exists $\displaystyle\hat{y}\in K_{\varphi_{\lambda}},\hat{y}\notin\{0,u_{0},\hat{u},v_{0},\hat{v},y_{0}\}$. Then $\displaystyle\hat{y}$ is the sixth nontrivial solution of ($\displaystyle S_{\lambda}$) and $\displaystyle\hat{y}\in C^{1}(\overline{\Omega})$ (by regularity theory). ∎

Acknowledgments. The authors wish to thank the two knowledgeable referees for their comments, remarks and constructive criticism. This research was supported by the Slovenian Research Agency grants P1-0292, J1-8131, N1-0083, and N1-0064. V.D. Rădulescu acknowledges the support through a grant of the Romanian Ministry of Research and Innovation, CNCS - UEFISCDI, project number PN-III-P4-ID-PCE-2016-0130, within PNCDI III.