Nodal solutions for nonlinear nonhomogeneous Robin problems

TL;DR

This paper establishes the existence of a sequence of nodal solutions for a nonlinear Robin boundary value problem involving a nonhomogeneous differential operator and indefinite potential, with solutions converging to zero.

Contribution

It introduces new methods to prove the existence of nodal solutions for complex nonlinear Robin problems with indefinite potentials.

Findings

Existence of a sequence of nodal solutions converging to zero.

Solutions are obtained using truncation, comparison, and cut-off techniques.

Solutions converge in the $C^1(ar{ abla})$-norm.

Abstract

We consider the nonlinear Robin problem driven by a nonhomogeneous differential operator plus an indefinite potential. The reaction term is a Carath\'eodory function satisfying certain conditions only near zero. Using suitable truncation, comparison, and cut-off techniques, we show that the problem has a sequence of nodal solutions converging to zero in the $C^1(\overline{\Omega})$-norm.

Equations

$$\left\{\begin{array}[]{l}-{\rm div}\,a(Du(z))+\xi(z)|u(z)|^{p-2}u(z)=f(z,u(z))\ \mbox{in}\ \Omega,\\ \frac{\partial u}{\partial n_{a}}+\beta(z)|u|^{p-2}u=0\ \mbox{on}\ \partial\Omega.\end{array}\right\}$$

$$C^1(\overline{\Omega })\ni u\mapsto (a(Du),n{)}_{{\mathbb{R}}^N},$$

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

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

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

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

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

$$\sup \{\phi (u):u\in V_n\cap \partial B_{{\rho }_n}\}<0,$$

$$\displaystyle\begin{array}[]{lll}&(i)&\ \varphi^{\prime}(u_{n})=0\ \mbox{for all}\ n\in\mathbb{N}\ \mbox{(that is, each $\displaystyle u_{n}$ is a critical point of $\displaystyle\varphi$)};\\ &(ii)&\ \varphi(u_{n})\leqslant 0\ \mbox{for all}\ n\in\mathbb{N};\ and\\ &(iii)&\ u_{n}\rightarrow 0\ \mbox{in}\ X\ \mbox{as}\ n\rightarrow\infty.\end{array}$$

$$K_{\phi }=\{u\in X:{\phi }^{\prime }(u)=0\}.$$

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

$$\begin{array}[]{rr}0<\hat{c}\leqslant\frac{\vartheta^{\prime}(t)t}{\vartheta(t)}\leqslant c_{0}\ \mbox{and}\ c_{1}t^{p-1}\leqslant\vartheta(t)\leqslant c_{2}(t^{\tau-1}+t^{p-1})\\ \mbox{for all}\ t>0,\ \mbox{with}\ c_{1},c_{2}>0,1\leqslant\tau<p.\end{array}$$

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

$$t\mapsto G_0(t^{1/q})\text{mbox}isconvexand\underset{t\to 0^{+}}{\mathrm{limsup}}\frac{q{G}_0(t)}{t^q}<+\infty .$$

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

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

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

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

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

$$(W^{1,p}(\Omega {)}^{\ast },W^{1,p}(\Omega )).$$

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

$$‘‘u_n\stackrel{w}{\to }u\text{mbox}in{W}^{1,p}(\Omega )\text{mbox}and\underset{n\to \infty }{\mathrm{limsup}}⟨A(u_n),u_n-u⟩\Rightarrow u_n\to u".$$

$$\underset{x\to 0}{\lim }\frac{f(z,x)}{|x{|}^{q-2}x}=+\infty \text{mbox}uniformlyforalmostallz\in \Omega ;\text{mbox}and$$

$$x\to f(z,x)+\hat{\xi }|x{|}^{p-2}x$$

$$\hat{f}(z,x)=\gamma (x)f(z,x)+(1-\gamma (x))\xi (z)|x{|}^{p-2}x.$$

$$\hat{f}(z,\cdot ){|}_{[-\epsilon ,\epsilon ]}=f(z,\cdot ){|}_{[-\epsilon ,\epsilon ]}\text{mbox}forallz\in \Omega ;$$

$$\hat{f}(z,x)=\xi (z)|x{|}^{p-2}x\text{mbox}forallz\in \Omega ,|x|⩾\eta .$$

$$\hat{f}(z,\eta )-\xi (z){\eta }^{p-1}=0\text{mbox}foralmostallz\in \Omega .$$

$$\hat{f}(z,-\eta )+\xi (z){\eta }^{p-1}=0\text{mbox}foralmostallz\in \Omega .$$

$$f(z,x)x=\hat{f}(z,x)⩾\mu |x{|}^q\text{mbox}foralmostallz\in \Omega ,\text{mbox}andall|x|⩽\delta \text{mbox}(see(\text{ref}eq4)).$$

$$\hat{f}(z,x)x⩾\mu |x{|}^q-c_6|x{|}^r\text{mbox}foralmostallz\in \Omega ,\text{mbox}andallx\in \mathbb{R}.$$

Full text

aa 2010 AMS Subject Classification: 35J20, 35J60

1. Introduction

Let $\displaystyle\Omega\subseteq\mathbb{R}^{N}$ be a bounded domain with a $\displaystyle C^{2}$-boundary $\displaystyle\partial\Omega$. We study the following nonlinear nonhomogeneous Robin problem:

[TABLE]

In this problem, $\displaystyle a:\mathbb{R}^{N}\rightarrow\mathbb{R}^{N}$ is a continuous and strictly monotone map (thus also maximal monotone), which satisfies certain regularity and growth conditions listed in hypotheses $\displaystyle H(a)$ below. These conditions are general and they incorporate in our framework many differential operators of interest, such as the $\displaystyle p$-Laplacian and the $\displaystyle(p,q)$-Laplacian. We stress that $\displaystyle a(\cdot)$ is not homogeneous and this is a source of difficulties in the study of problem (1). The potential function $\displaystyle\xi\in L^{\infty}(\Omega)$ is indefinite (that is, sign changing). The reaction term (the right-hand side of (1)) is a Carathéodory function (that is, for all $\displaystyle x\in\mathbb{R}$, the function $\displaystyle z\mapsto f(z,x)$ is measurable, and for almost all $\displaystyle z\in\Omega$, the function $\displaystyle x\mapsto f(z,x)$) is continuous. We impose conditions on $\displaystyle f(z,\cdot)$ only near zero. In the boundary condition, $\displaystyle\frac{\partial u}{\partial n_{a}}$ denotes the conormal derivative corresponding to the differential operator $\displaystyle u\mapsto{\rm div}\,a(Du)$ and is defined by extension of the map

[TABLE]

with $\displaystyle n(\cdot)$ being the outward unit normal on $\displaystyle\partial\Omega$.

We are looking for nodal (that is, sign-changing) solutions for problem (1). Employing a symmetry condition on $\displaystyle f(z,\cdot)$ near zero and using truncation, perturbation, comparison, and cut-off techniques, and a result of Kajikiya [7], we generate a whole sequence $\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq C^{1}(\overline{\Omega})$ of distinct nodal solutions such that $\displaystyle u_{n}\rightarrow 0$ in $\displaystyle C^{1}(\overline{\Omega})$.

The first result in this direction was produced by Wang [27], who used cut-off techniques to produce an infinity of solutions converging to zero in $\displaystyle H^{1}_{0}({\Omega})$. In Wang [27] the problem is semilinear driven by the Dirichlet Laplacian. There is no potential term (that is, $\displaystyle\xi\equiv 0$). The sequence produced by Wang [27] does not consist of nodal solutions. More recently, Li & Wang [8] produced a sequence of nodal solutions for semilinear Schrödinger equations. For nonlinear equations we mention the recent works of He, Huang, Liang & Lei [5], and Papageorgiou & Rădulescu [19]. In He et al. [5], the problem is Neumann (that is, $\displaystyle\beta\equiv 0$) and the differential operator is the $\displaystyle p$-Laplacian (that is, $\displaystyle a(y)=|y|^{p-2}y$ for all $\displaystyle y\in\mathbb{R}^{N}$, with $\displaystyle 1<p<\infty$). In Papageorgiou & Rădulescu [19], the differential operator is the same as in the present paper, but $\displaystyle\xi\equiv 0$. Also, the hypotheses on $\displaystyle f(z,\cdot)$ near zero are more restrictive. In the present paper we extend the results of all aforementioned works.

2. Preliminaries and Hypotheses

In the study of problem (1) we will use the following spaces: the Sobolev space $\displaystyle W^{1,p}(\Omega)$, the Banach space $\displaystyle C^{1}(\overline{\Omega})$, and the boundary Lebesgue spaces $\displaystyle L^{r}(\partial\Omega)$, $\displaystyle 1\leqslant r\leqslant\infty$.

We denote by $\displaystyle||\cdot||$ the norm on the Sobolev space $\displaystyle W^{1,p}(\Omega)$ defined by

[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 which contains the open set

[TABLE]

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

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 $\displaystyle L^{r}(\partial\Omega),1\leqslant r\leqslant\infty$. From the theory of Sobolev spaces we know that there exists a unique continuous linear map $\displaystyle\gamma_{0}:W^{1,p}(\Omega)\rightarrow L^{p}(\partial\Omega)$, known as the “trace map”, such that

[TABLE]

So, the trace map assigns “boundary values” to all Sobolev functions. We know that the trace map is compact into $\displaystyle L^{r}(\partial\Omega)$ for all $\displaystyle 1\leqslant r<\frac{(N-1)p}{N-p}$ if $\displaystyle p<N$, and into $\displaystyle L^{r}(\partial\Omega)$ for all $\displaystyle 1\leqslant r<\infty$ if $\displaystyle p\geqslant N$. Furthermore, we have that

[TABLE]

In what follows, for the sake of notational simplicity, we will drop the use of the trace map $\displaystyle\gamma_{0}(\cdot)$. All restrictions of Sobolev functions on $\displaystyle\partial\Omega$, are understood in the sense of traces.

Let $\displaystyle X$ be a Banach space and $\displaystyle\varphi\in C^{1}(X,\mathbb{R})$. We say that $\displaystyle\varphi$ satisfies the “Palais-Smale condition” (the “PS-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 $\displaystyle\varphi^{\prime}(u_{n})\rightarrow 0$ in $\displaystyle X^{*}$ as $\displaystyle n\rightarrow\infty$,

admits a strongly convergent subsequence.”

We shall need the following result of Kajikya [7].

Theorem 1**.**

Assume that $\displaystyle X$ is a Banach space, $\displaystyle\varphi\in C^{1}(X,\mathbb{R})$ satisfies the PS-condition, $\displaystyle\varphi$ is even and bounded below, $\displaystyle\varphi(0)=0$, and for every $\displaystyle n\in\mathbb{N}$, there exists an $\displaystyle n$-dimensional subspace $\displaystyle V_{n}$ of $\displaystyle X$ and $\displaystyle\rho_{n}>0$ such that

[TABLE]

where $\displaystyle\partial B_{\rho_{n}}=\{u\in X:||u||_{X}=\rho_{n}\}$. Then there exists a sequence $\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq X\backslash\{0\}$ such that

[TABLE]

In the sequel, for any $\displaystyle\varphi\in C^{1}(X,\mathbb{R})$, we denote by $\displaystyle K_{\varphi}$ the critical set of $\displaystyle\varphi$, that is,

[TABLE]

For $\displaystyle X\in\mathbb{R}$, we set $\displaystyle x^{\pm}=\max\{\pm x,0\}$. Then for any $\displaystyle u\in W^{1,p}(\Omega)$, we define $\displaystyle u^{\pm}(\cdot)=u(\cdot)^{\pm}$. We know that

[TABLE]

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

[TABLE]

Then the hypotheses on the map $\displaystyle a(\cdot)$ 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)

$\displaystyle|\nabla a(y)|\leqslant c_{3}\frac{\vartheta(|y|)}{|y|}$ for all $\displaystyle y\in\mathbb{R}^{N}\backslash\{0\}$, and some $\displaystyle c_{3}>0$;

• (iii)

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

• (iv)

If $\displaystyle G_{0}(t)=\int_{0}^{t}a_{0}(s)sds$ for all $\displaystyle t>0$, then there exists $\displaystyle q\in(1,p]$ such that

[TABLE]

Remark 1**.**

Hypotheses $\displaystyle H(a)(i),(ii),(iii)$ are dictated by the nonlinear global regularity theory of Lieberman [10] and the nonlinear maximum principle of Pucci & Serrin [24]. Hypothesis $\displaystyle H(a)(iv)$ reflects the particular requirements of our problem. However, $\displaystyle H(a)(iv)$ is not restrictive as the examples below illustrate.

Hypotheses $\displaystyle H(a)$ imply that $\displaystyle G_{0}(\cdot)$ is strictly convex and strictly increasing. We set $\displaystyle G(y)=G_{0}(|y|)$ for all $\displaystyle y\in\mathbb{R}^{N}$. Evidently, $\displaystyle G(\cdot)$ is convex and $\displaystyle G(0)=0$. Also, we have

[TABLE]

So, $\displaystyle G(\cdot)$ is the primitive of $\displaystyle a(\cdot)$. Moreover, the convexity of $\displaystyle G(\cdot)$ implies that

[TABLE]

The next lemma summarizes the main properties of the map $\displaystyle a(\cdot)$ and it is an easy consequence of hypotheses $\displaystyle H(a)$ and condition (2) above.

Lemma 2**.**

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

• (a)

$\displaystyle a(\cdot)$* is continuous, strictly monotone, hence maximal monotone, too;*

• (b)

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

• (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 (3) lead to the following growth conditions on $\displaystyle G(\cdot)$.

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 $\displaystyle c_{5}>0$.

Example 1**.**

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

• (a)

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

This map corresponds to the p-Laplace differential operator defined by

[TABLE]

• (b)

$\displaystyle a(y)=|y|^{p-2}y+|y|^{q-2}y,1<q<p<\infty$. This map corresponds to the $\displaystyle(p,q)$-Laplace differential operator defined by

[TABLE]

Such operators arise in problems of mathematical physics. Recently $\displaystyle(p,q)$-equations have been studied by Bobkov & Tanaka **[1]**, Li & Zhang **[9]**, Marano & Mosconi **[11]**, Marano, Mosconi & Papageorgiou **[12, 13]**, Mugnai & Papageorgiou **[16]**, Papageorgiou & Rădulescu **[17]**, Sun, Zhang & Su **[25]**, and Tanaka **[26]**.

• (c)

$\displaystyle a(y)=(1+|y|^{2})^{\frac{p-2}{2}}y,1<p<\infty$. This map corresponds to the generalized p-mean curvature differential operator defined by

[TABLE]

• (d)

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

We denote by $\displaystyle\langle\cdot,\cdot\rangle$ the duality brackets for the pair

[TABLE]

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

[TABLE]

From Gasinski & Papageorgiou [3], we have:

Proposition 4**.**

The map $\displaystyle A:W^{1,p}(\Omega)\rightarrow W^{1,p}(\Omega)^{*}$ is bounded (maps bounded sets to bounded sets), continuous, monotone (hence maximal monotone, too), and of type $\displaystyle(S)_{+}$, that is,

[TABLE]

The hypotheses on the potential function $\displaystyle\xi(\cdot)$ and on the boundary coefficient $\displaystyle\beta(\cdot)$ are the following:

$\displaystyle H(\xi):\xi\in L^{\infty}(\Omega)$.

$\displaystyle H(\beta):\beta\in C^{0,\alpha}(\partial\Omega)\ \mbox{for some}\ \alpha\in(0,1)\ \mbox{and}\ \beta(z)\geqslant 0\ \mbox{for all}\ z\in\partial\Omega.$

Remark 2**.**

If $\displaystyle\beta\equiv 0$, then we recover the Neumann problem.

Finally, we introduce our conditions on the reaction term $\displaystyle f(z,x)$:

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

• (i)

there exists $\displaystyle\eta>0$ such that for almost all $\displaystyle z\in\Omega,f(z,\cdot)|_{[-\eta,\eta]}$ is odd;

• (ii)

$\displaystyle|f(z,x)|\leqslant a_{\eta}(z)$ for almost all $\displaystyle z\in\Omega$, $\displaystyle x\in[-\eta,\eta]$, with $\displaystyle a_{\eta}\in L^{\infty}(\Omega)$;

• (iii)

with $\displaystyle q\in(1,p]$ as in hypothesis $\displaystyle H(a)(iv)$, we have

[TABLE]

• (iv)

there exists $\displaystyle\hat{\xi}>0$ such that for almost all $\displaystyle z\in\Omega$

[TABLE]

is nondecreasing on $\displaystyle[-\eta,\eta]$

Remark 3**.**

We point out that all the above hypotheses concern the behaviour of $\displaystyle f(z,\cdot)$ only near zero.

Finally, we mention that nonlinear problems with an indefinite potential have recently been studied in the context of equations driven by the Neumann $\displaystyle p$-Laplacian by Gasinski & Papageorgiou [4] (resonant problems) and Fragnelli, Mugnai & Papageorgiou [2] (superlinear problems). Also, nodal solutions for nonlinear Robin problems with no potential term, were obtained by Papageorgiou & Rădulescu [21].

3. Nodal solutions

Let $\displaystyle\varepsilon\in(0,\eta)$ and consider an even function $\displaystyle\gamma\in C^{1}(\mathbb{R})$ such that $\displaystyle 0\leqslant\gamma\leqslant 1$, $\displaystyle\gamma|_{[-\varepsilon,\varepsilon]}=1$ and $\displaystyle{\rm supp}\,\gamma\subseteq[-\eta,\eta]$.

We set

[TABLE]

Evidently, $\displaystyle\hat{f}(z,x)$ is a Carathéodory function which is odd in $\displaystyle x\in\mathbb{R}$ and has the following two additional properties:

[TABLE]

It follows from (5) that

[TABLE]

Since $\displaystyle\hat{f}(z,\cdot)$ is odd, we have

[TABLE]

On account of hypothesis $\displaystyle H(f)(iii)$, given any $\displaystyle\mu>0$, we can find $\displaystyle\delta=\delta(\mu)\in(0,\varepsilon)$ such that

[TABLE]

Then (8) combined with hypothesis $\displaystyle H(f)(ii)$ implies that given $\displaystyle r>p$ we can find $\displaystyle c_{6}>0$ such that

[TABLE]

We introduce the following function

[TABLE]

This is a Carathéodory function which is odd in $\displaystyle x\in\mathbb{R}$.

We consider the following auxiliary nonlinear Robin problem:

[TABLE]

Proposition 5**.**

If hypotheses $\displaystyle H(a),\ H(\xi),\ H(\beta)$ hold, then problem (11) admits a unique positive solution

[TABLE]

and since $\displaystyle k(z,\cdot)$ is odd, $\displaystyle v^{*}=-u^{*}\in D_{+}$ is the unique negative solution of (11).

Proof.

We consider the Carathéodory function $\displaystyle\hat{k}(z,x)$ defined by

[TABLE]

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

[TABLE]

From (12) and Corollary 3 it is clear that

[TABLE]

Also, from the Sobolev embedding theorem and the compactness of the trace map, we deduce that

[TABLE]

So, by the Weierstrass-Tonelli theorem, we can find $\displaystyle u^{*}\in W^{1,p}(\Omega)$ such that

[TABLE]

On account of hypothesis $\displaystyle H(a)(iv)$, we can find $\displaystyle c_{7}>0$ such that

[TABLE]

with $\displaystyle\delta>0$ as in (8). Let $\displaystyle u\in D_{+}$. Then we can find $\displaystyle t\in(0,1)$ small such that

[TABLE]

Using (10), (12), (14) and (15), we obtain

[TABLE]

Choosing $\displaystyle\mu>\frac{c_{8}}{c_{9}}$, we infer that

[TABLE]

From (13) we have

[TABLE]

In (3) we choose $\displaystyle h=-(u^{*})^{-}\in W^{1,p}(\Omega)$. Using Lemma 2(c), we obtain

[TABLE]

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

[TABLE]

So, we have proved that

[TABLE]

From (10), (12), (3) and (17), we infer that $\displaystyle u^{*}$ is a positive solution of problem (11). From Papageorgiou & Rădulescu [20], we have

[TABLE]

Now the nonlinear regularity theory of Lieberman [10] implies that

[TABLE]

From (3) and (17), we have

[TABLE]

Next, we show the uniqueness of this solution. To this end, let $\displaystyle\hat{i}:L^{1}(\Omega)\rightarrow\overline{\mathbb{R}}=\mathbb{R}\cup\{+\infty\}$ be the integral functional defined by

[TABLE]

From Papageorgiou & Winkert [23] (see the proof of Proposition 3.3), we know that $\displaystyle\hat{i}(\cdot)$ is convex and if $\displaystyle u^{*},v^{*}\in D_{+}$ are two positive solutions of (11), then

[TABLE]

The convexity of $\displaystyle\hat{i}(\cdot)$ implies the monotonicity of $\displaystyle\hat{i}^{\prime}(\cdot)$. Hence

[TABLE]

This proves the uniqueness of the positive solution $\displaystyle u^{*}\in D_{+}$ of (11). Since problem (11) is odd, it follows that $\displaystyle v^{*}=-u^{*}\in-D_{+}$ is the unique negative solution of problem (11). ∎

Consider the following Robin problem:

[TABLE]

We denote by $\displaystyle S^{+}$ (respectively $\displaystyle S^{-}$) the set of positive (respectively negative) solutions of problem (18) which are in the order interval $\displaystyle[0,\eta]=\{u\in W^{1,p}(\Omega):0\leqslant u(z)\leqslant\eta$ for almost all $\displaystyle z\in\Omega\}$ (respectively in $\displaystyle[-\eta,0]=\left\{\right.v\in W^{1,p}(\Omega):-\eta\leqslant v(z)\leqslant 0$ for almost all $\displaystyle z\in\Omega\left.\right\}$). From Papageorgiou, Rădulescu & Repovš [22], we know that

• •

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

• •

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

Moreover, reasoning as in the proof of Proposition 5 (with $\displaystyle k(z,x)$ replaced by $\displaystyle\hat{f}(z,x)$), we show that

[TABLE]

Proposition 6**.**

If hypotheses $\displaystyle H(a),H(\xi),H(\beta),H(f)$ hold, then $\displaystyle u^{*}\leqslant u$ for all $\displaystyle u\in S^{+}$ and $\displaystyle v\leqslant v^{*}$ for all $\displaystyle v\in S^{-}$.

Proof.

Let $\displaystyle u\in S_{+}$ and let $\displaystyle\hat{k}(z,x)$ be given by (12). We introduce the following truncation of $\displaystyle\hat{k}(z,\cdot)$:

[TABLE]

This is a Carathéodory function. We set $\displaystyle E_{+}(z,x)=\int^{x}_{0}e_{+}(z,s)ds$ and consider the $\displaystyle C^{1}$-functional $\displaystyle\Psi_{+}:W^{1,p}(\Omega)\rightarrow\mathbb{R}$ defined by

[TABLE]

Evidently, $\displaystyle\Psi_{+}(\cdot)$ is coercive (see (19)) and sequentially weakly lower semicontinuous. So, we can find $\displaystyle\hat{u}^{*}\in W^{1,p}(\Omega)$ such that

[TABLE]

As in the proof of Proposition 5, using hypotheses $\displaystyle H(a)(iv)$ and $\displaystyle H(f)(iii)$, we show that

[TABLE]

From (20) we have

[TABLE]

In (21), we choose $\displaystyle h=-(u^{*})^{-}\in W^{1,p}(\Omega)$. Then using Lemma 2(c), we have

[TABLE]

Next, in (21) we choose $\displaystyle h=(\hat{u}^{*}-u)^{+}\in W^{1,p}(\Omega)$. We have

[TABLE]

So, we have proved that

[TABLE]

This fact, together with (10), (12), (19), (21), imply that

[TABLE]

Similarly, we show that

[TABLE]

This completes the proof. ∎

Now we can establish the existence of extremal constant sign solutions for problem (18), that is, we show that problem (18) has a smallest positive solution and a biggest negative solution.

Proposition 7**.**

If hypotheses $\displaystyle H(a),H(\beta),H(\xi),H(f)$ hold, then there exists a smallest positive solution $\displaystyle u_{+}\in S^{+}\subseteq D_{+}$ and a biggest negative solution $\displaystyle v_{+}\in S^{-}\subseteq-D_{+}$.

Proof.

Invoking Lemma 3.10 of Hu & Papageorgiou [6, p. 178], we can find a decreasing sequence $\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq S^{+}$ such that

[TABLE]

Evidently, $\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq W^{1,p}(\Omega)$ is bounded. So, we may assume that

[TABLE]

We have

[TABLE]

In (23) we choose $\displaystyle h=u_{n}-u_{+}\in W^{1,p}(\Omega)$, pass to the limit as $\displaystyle n\rightarrow\infty$ and use (22). Then

[TABLE]

In (23) we pass to the limit as $\displaystyle n\rightarrow\infty$ and use (24). Then

[TABLE]

From Proposition 6, we have

[TABLE]

It follows from (25) and (26) that

[TABLE]

Similarly, we produce

[TABLE]

∎

Let $\displaystyle\tau>||\xi||_{\infty}$ and consider the following truncation-perturbation of $\displaystyle\hat{f}(z,\cdot)$:

[TABLE]

We set $\displaystyle F_{0}(z,x)=\int_{0}^{x}f_{0}(z,s)ds$ and consider the $\displaystyle C^{1}$-functional $\displaystyle\varphi_{0}:W^{1,p}(\Omega)\rightarrow\mathbb{R}$ defined by

[TABLE]

Evidently, $\displaystyle\varphi_{0}(\cdot)$ is coercive (see (27) and recall that $\displaystyle\tau>||\xi||_{\infty}$). So, $\displaystyle\varphi_{0}(\cdot)$ is bounded below and satisfies the PS-condition (see Marano & Papageorgiou [14, 15]).

Proposition 8**.**

If hypotheses $\displaystyle H(a),H(\xi),H(\beta),H(f)$ hold and $\displaystyle V\subseteq W^{1,p}(\Omega)$ is a finite dimensional linear subspace, then there exists $\displaystyle\rho_{V}>0$ such that

[TABLE]

Proof.

Recall that $\displaystyle u_{+}\in D_{+}$ and $\displaystyle v_{-}\in-D_{+}$. So, $\displaystyle m_{0}=\min\{\min_{\overline{\Omega}}u_{+},-\max_{\overline{\Omega}}v_{-}\}>0$. We set $\displaystyle\epsilon_{0}=\min\{\epsilon,m_{0}\}$ (where $\displaystyle\epsilon>0$ is from (4)). On account of hypothesis $\displaystyle H(f)(iii)$, given any $\displaystyle\mu>0$, we can find $\displaystyle\delta=\delta(\mu)>0\in(0,\epsilon_{0})$ such that

[TABLE]

Moreover, on account of hypothesis $\displaystyle H(a)(iv)$ and Corollary 3, we have

[TABLE]

Since the subspace $\displaystyle V\subseteq W^{1,p}(\Omega)$ is finite dimensional, all norms are equivalent. So, we can find $\displaystyle\rho_{V}\in(0,1]$ such that

[TABLE]

Then for every $\displaystyle u\in V$ with $\displaystyle||u||\leqslant\rho_{V}$, we have

[TABLE]

Since $\displaystyle\mu>0$ is arbitrary, we choose $\displaystyle\mu>\frac{c_{11}}{c_{12}}$ and conclude that

[TABLE]

The proof is now complete. ∎

We now obtain the following multiplicity theorem for the nodal solutions of problem (1).

Theorem 9**.**

Assume that hypotheses $\displaystyle H(a),H(\xi),H(\beta),H(f)$ hold. Then there exists a sequence $\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq C^{1}(\overline{\Omega})$ of nodal solutions of problem (1) such that

[TABLE]

Proof.

We know that $\displaystyle\varphi_{0}(\cdot)$ is even, bounded below, satisfies the PS-condition, and $\displaystyle\varphi_{0}(0)=0$. Moreover, using (27) as before, we can check that

[TABLE]

The aforementioned properties of $\displaystyle\varphi_{0}(\cdot)$ and Proposition 8 permit us to apply Theorem 1. So, we can find a sequence $\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq W^{1,p}(\Omega)$ such that

[TABLE]

The nonlinear regularity theory of Lieberman [10] implies that we can find $\displaystyle\gamma\in(0,1)$ and $\displaystyle c_{13}>0$ such that

[TABLE]

We know that $\displaystyle C^{1,\gamma}(\overline{\Omega})$ is compactly embedded in $\displaystyle C^{1}(\overline{\Omega})$. So, it follows from (32) and (33) that

[TABLE]

From (4), (32) and the extremality of $\displaystyle u_{+},v_{-}$, we get that $\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq C^{1}(\overline{\Omega})$ are nodal solutions of (1) and we have $\displaystyle u_{n}\rightarrow 0$ in $\displaystyle C^{1}(\overline{\Omega})$. ∎

Acknowledgements. This research was supported by the Slovenian Research Agency grants P1-0292, J1-8131, J1-7025, N1-0064, and N1-0083. 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.