Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential

TL;DR

This paper investigates the existence and uniqueness of positive solutions for a perturbed Robin eigenvalue problem involving the Laplacian and an indefinite potential, analyzing sublinear and superlinear cases.

Contribution

It provides new results on positive solutions for Robin eigenvalue problems with indefinite potentials, including existence, uniqueness, and properties of minimal solutions.

Findings

Unique positive solution for sublinear perturbations when λ<λ̂₁

Multiple positive solutions for superlinear perturbations when λ<λ̂₁

No positive solutions for λ≥λ̂₁ in both cases

Abstract

We study perturbations of the eigenvalue problem for the negative Laplacian plus an indefinite and unbounded potential and Robin boundary condition. First we consider the case of a sublinear perturbation and then of a superlinear perturbation. For the first case we show that for $\lambda<\widehat{\lambda}_{1}$ ($\widehat{\lambda}_{1}$ being the principal eigenvalue) there is one positive solution which is unique under additional conditions on the perturbation term. For $\lambda\geq\widehat{\lambda}_{1}$ there are no positive solutions. In the superlinear case, for $\lambda<\widehat{\lambda}_{1}$ we have at least two positive solutions and for $\lambda\geq\widehat{\lambda}_{1}$ there are no positive solutions. For both cases we establish the existence of a minimal positive solution $\bar{u}_{\lambda}$ and we investigate the properties of the map $\lambda\mapsto\bar{u}_{\lambda}$.

Equations

$$\left\{\begin{array}[]{ll}-\Delta u(z)+\xi(z)u(z)=\lambda u(z)+f(z,u(z))\ \ \mbox{in}\ \Omega,\\[8.5359pt] \displaystyle\frac{\partial u}{\partial n}+\beta(z)u=0\ \ \mbox{on}\ \partial\Omega,\ u\geq 0.\end{array}\right\}\hskip 56.9055pt(P_{\lambda})$$

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

$$(u,h{)}_{H^1(\Omega )}={\int }_{\Omega }uhdz+{\int }_{\Omega }(Du,Dh{)}_{{\mathbb{R}}^N}dz\text{mbox}forallu,h\in H^1(\Omega ).$$

$$\Vert u\Vert ={\left[\Vert u{\Vert }_2^2+\Vert Du{\Vert }_2^2\right]}^{1/2}\text{mbox}forallu\in H^1(\Omega ).$$

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

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

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

$$\text{im}{\gamma }_0=H^{\frac{1}{2},2}(\partial \Omega )\text{mbox}and\text{ker}{\gamma }_0=H_0^1(\Omega ).$$

$$|f_0(z,x)|\le a_0(z)(1+|x{|}^{r-1})\text{mbox}fora.a.z\in \Omega ,\text{mbox}allx\in \mathbb{R},$$

$${\phi }_0(u)=\frac{1}{2}\gamma (u)-{\int }_{\Omega }{F}_0(z,u(z))dz\text{mbox}forallu\in H^1(\Omega ),$$

$$\gamma (u)=\Vert Du{\Vert }_2^2+{\int }_{\Omega }\xi (z)u^{2}dz+{\int }_{\partial \Omega }\beta (z)u^{2}d\sigma \text{mbox}forallu\in H^1(\Omega ).$$

$$\displaystyle\left\{\begin{array}[]{ll}-\Delta u(z)+\xi(z)u(z)=\widehat{\lambda}u(z)\ \ \mbox{in}\ \Omega,\\[8.5359pt] \displaystyle\frac{\partial u}{\partial n}+\beta(z)u=0\ \ \mbox{on}\ \partial\Omega.\end{array}\right\}$$

$${\hat{\lambda }}_1=\inf \left[\frac{\gamma (u)}{\Vert u{\Vert }_2^2}:u\in H^1(\Omega ),u\ne 0\right].$$

$$J(u)=\gamma (u)-{\int }_{\Omega }\vartheta (z)u^{2}dz\ge \hat{c}\Vert u{\Vert }^2\text{mbox}forallu\in H^1(\Omega ).$$

$$\Vert u_n\Vert =1\text{mbox}foralln\in \mathbb{N}\text{mbox}andJ(u_n)\downarrow 0\text{mbox}asn\to \infty .$$

$$u_n\stackrel{w}{\to }u\text{mbox}in{H}^1(\Omega )\text{mbox}and{u}_n\to u\text{mbox}in{L}^2(\Omega )\text{mbox}andin{L}^2(\partial \Omega ).$$

$$J(u)\le 0,$$

$$\Rightarrow$$

$$\Vert D{u}_n{\Vert }_2\to 0(\text{mbox}see\text{eqref}3),$$

$$\Rightarrow$$

$$\Vert Du{\Vert }_2^2<{\hat{\lambda }}_1\Vert u{\Vert }_2^2,$$

$$\gamma (u)+\mu \Vert u{\Vert }_2^2\ge c_0\Vert u{\Vert }^2\text{mbox}forsome{c}_0>0,\text{mbox}allu\in H^1(\Omega )$$

$$\gamma (u_n)+n\Vert u_n{\Vert }_2^2<\frac{1}{n}\Vert u_n\Vert \text{mbox}foralln\in \mathbb{N}.$$

$$y_n\stackrel{w}{\to }y\text{mbox}in{H}^1(\Omega )\text{mbox}and{y}_n\to y\text{mbox}in{L}^2(\Omega )\text{mbox}andin{L}^2(\partial \Omega ).$$

$$0\le \underset{n\to \infty }{\mathrm{liminf}}\gamma (y_n)\le \underset{n\to \infty }{\mathrm{limsup}}\gamma (y_n)\le \underset{n\to \infty }{\lim }\left[\frac{1}{n}-n\Vert y_n{\Vert }_2^2\right]=0,$$

$$\Rightarrow 0$$

$$\Rightarrow$$

$$u_n\stackrel{w}{\to }u\text{mbox}inX\text{mbox}and\Vert u_n\Vert \to \Vert u\Vert \Rightarrow u_n\to u\text{mbox}inX.$$

$$u^{\pm }(\cdot )=u(\cdot {)}^{\pm }.$$

$$u^{\pm }\in H^1(\Omega ),u=u^{+}-u^{-},|u|=u^{+}+u^{-}\text{mbox}forallu\in H^1(\Omega ).$$

$$N_h(u)(\cdot )=h(\cdot ,u(\cdot ))\text{mbox}forallu\in H^1(\Omega )$$

Full text

Nikolaos S. Papageorgiou

Department of Mathematics

National Technical University, Zografou Campus

Athens 15780, Greece

Vicenţiu D. Rădulescu

Department of Mathematics, Faculty of Sciences

King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Department of Mathematics, University of Craiova

Street A.I. Cuza No 13, 200585 Craiova, Romania

Dušan D. Repovš

Faculty of Education and Faculty of Mathematics and Physics

University of Ljubljana, Kardeljeva ploščad 16

SI-1000 Ljubljana, Slovenia

1. Introduction

Let $\Omega\subseteq\mathbb{R}^{N}$ be a bounded domain with a $C^{2}$-boundary $\partial\Omega$. In this paper we study the following semilinear parametric Robin problem with an indefinite and unbounded potential $\xi(z)$:

[TABLE]

In this problem $\lambda\in\mathbb{R}$ is a parameter and $\xi(\cdot)$ is a potential function which is indefinite (that is, $\xi(\cdot)$ is sign changing) and unbounded from below. We can think of ($P_{\lambda}$) as a perturbation of the standard eigenvalue problem for the differential operator $u\mapsto-\Delta u+\xi(z)u$ with Robin boundary condition. We look for positive solutions and consider two cases: a sublinear perturbation $f(z,\cdot)$ and a superlinear perturbation $f(z,\cdot)$. For both cases we determine the dependence of the set of positive solutions as the parameter $\lambda\in\mathbb{R}$ varies.

We mention that the standard eigenvalue problems for the Robin Laplacian have recently been studied by D’Agui, Marano & Papageorgiou [5] and by Papageorgiou & Rădulescu [12]. Additional existence and multiplicity results for parametric Robin and Neumann problems can be found in the works of Papageorgiou & Rădulescu [13, 14].

Let $\widehat{\lambda}_{1}\in\mathbb{R}$ be the principal eigenvalue of the operator $u\mapsto-\Delta u+\xi(z)u$ with Robin boundary condition. In the sublinear case (that is, when $f(z,\cdot)$ is sublinear near $+\infty$) we show that for $\lambda\geq\widehat{\lambda}_{1}$ problem ($P_{\lambda}$) has no positive solution, whereas for $\lambda<\widehat{\lambda}_{1}$ problem ($P_{\lambda}$) has at least one positive solution. In fact, we show that under an additional monotonicity condition on the quotient $x\mapsto\frac{f(z,x)}{x}$ on $(0,+\infty)$, this positive solution is unique. In the superlinear case (that is, when $f(z,\cdot)$ is superlinear near $+\infty$) the situation changes and uniqueness of the solution fails. In fact, we show that the equation exhibits a kind of bifurcation phenomenon. Namely, for $\lambda\geq\widehat{\lambda}_{1}$ problem ($P_{\lambda}$) has no positive solution, whereas for $\lambda<\widehat{\lambda}_{1}$ it has at least two positive solutions. For both cases, we show that the problem has a minimal (that is, smallest) positive solution $\bar{u}_{\lambda}$ and we determine the monotonicity and continuity properties of the map $\lambda\mapsto\bar{u}_{\lambda}$.

Our approach is variational, based on the critical point theory, together with suitable truncation and perturbation techniques. In the next section, for the convenience of the reader, we recall the main mathematical tools which will be used in the sequel.

2. Mathematical background

Let $X$ be a Banach space and $X^{*}$ be its topological dual. By $\langle\cdot,\cdot\rangle$ we denote the duality brackets for the pair $(X^{*},X)$. Given $\varphi\in C^{1}(X,\mathbb{R})$, we say that $\varphi$ satisfies the Cerami condition (the C-condition for short), if the following is true:

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

$(1+\|u_{n}\|)\varphi^{\prime}(u_{n})\rightarrow 0$ in $X^{*}$ as $n\rightarrow\infty$,

                  admits a strongly convergent subsequence.

This compactness-type condition on $\varphi$ replaces the local compactness of the ambient space $X$ (in most applications $X$ is infinite dimensional and so it is not locally compact). It leads to a deformation theorem, from which one can derive the minimax theory of the critical values of $\varphi$. A central result of this theory, is the so-called mountain pass theorem, which we state here in a slightly more general version (see, for example, Gasinski & Papageorgiou [7]).

Theorem 2.1**.**

Assume that $\varphi\in C^{1}(X,\mathbb{R})$ satisfies the C-condition, $u_{0},u_{1}\in X$, $\|u_{1}-u_{0}\|>\rho$,

[TABLE]

and $c=\inf_{\gamma\in\Gamma}\limits\max_{0\leq t\leq 1}\limits\varphi(\gamma(t))$, where $\Gamma=\left\{\gamma\in C([0,1],X):\gamma(0)=u_{0},\gamma(1)=u_{1}\right\}$. Then $c\geq m_{\rho}$ and $c$ is a critical value of $\varphi$.

The analysis of problem ($P_{\lambda}$) will make use of the Sobolev space $H^{1}(\Omega)$. This is a Hilbert space with inner product

[TABLE]

The norm corresponding to this inner product, is given by

[TABLE]

In addition, we will also use the Banach spaces $C^{1}(\overline{\Omega})$ and $L^{\tau}(\partial\Omega)$, $1\leq\tau\leq\infty$.

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

[TABLE]

This cone has a nonempty interior, given by

[TABLE]

On $\partial\Omega$ we consider the $(N-1)$-dimensional Hausdorff (surface) measure $\sigma(\cdot)$. Using this measure we can define the Lebesgue spaces $L^{\tau}(\partial\Omega)$ $(1\leq\tau\leq\infty)$ in the usual way. The theory of Sobolev spaces says that there exists a unique continuous linear map $\gamma_{0}:H^{1}(\Omega)\rightarrow L^{2}(\partial\Omega)$, known as the trace map, such that

[TABLE]

Therefore we can interpret $\gamma_{0}(u)$ as representing the boundary values of $u\in H^{1}(\Omega)$. From the general theory of Sobolev spaces, we know that

[TABLE]

In addition, we know that $\gamma_{0}(\cdot)$ is compact from $H^{1}(\Omega)$ into $L^{\tau}(\partial\Omega)$ with $\tau\in\left[1,\frac{2(N-1)}{N-2}\right)$ if $N\geq 3$ and $\tau\in[1,+\infty)$ if $N=1,2$.

In what follows, for the sake of notational simplicity, we shall drop the use of the map $\gamma_{0}$. All restrictions of functions $u\in H^{1}(\Omega)$ on the boundary $\partial\Omega$, are understood in the sense of traces.

Suppose that $f_{0}:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a Carathéodory function (that is, for all $x\in\mathbb{R}$, $z\mapsto f(z,x)$ is measurable and for a.a. $z\in\Omega$, $x\mapsto f(z,x)$ is continuous) which has subcritical growth in the $x\in\mathbb{R}$ variable. Hence

[TABLE]

with $a_{0}\in L^{\infty}(\Omega)_{+}$, 2\leq r<2^{*}=\left\{\begin{array}[]{ll}\frac{2N}{N-2}\ \ \mbox{if}\ N\geq 3\\ +\infty\ \ \mbox{if}\ N=1,2\end{array}\right. (the critical Sobolev exponent). We set $F_{0}(z,x)=\int_{0}^{x}f_{0}(z,s)\,ds$ and consider the $C^{1}$-functional $\varphi_{0}:H^{1}(\Omega)\rightarrow\mathbb{R}$ defined by

[TABLE]

where $\gamma:H^{1}(\Omega)\rightarrow\mathbb{R}$ is the $C^{1}$-functional defined by

[TABLE]

The next result relates local minimizers of $\varphi_{0}$ in $C^{1}(\overline{\Omega})$ and in $H^{1}(\Omega)$, respectively. It is an outgrowth of the regularity theory for such problems (see Wang [15]) and a more general version of it (with proof) can be found in Papageorgiou & Rădulescu [12]. We mention that the first result of this kind for the space $H^{1}_{0}(\Omega)$ (Dirichlet problems) and $\xi\equiv 0$, was proved by Brezis & Nirenberg [4]. Our conditions on the potential $\xi(\cdot)$ and on the boundary coefficient $\beta(\cdot)$ are:

$H(\xi)$: $\xi\in L^{s}(\Omega)$ with $s>N$ if $N\geq 3$ and $s>1$ if $N=1,2$

$H(\beta)$: $\beta\in W^{1,\infty}(\partial\Omega)$ with $\beta(z)\geq 0$ for all $z\in\partial\Omega$

Remark. If $\beta\equiv 0$, then we recover the Neumann problem. Hence our present paper includes the Neumann problems as a special case.

Proposition 1**.**

Assume that hypotheses $H(\xi)$, $H(\beta)$ hold and $u_{0}\in H^{1}(\Omega)$ is a local $C^{1}(\overline{\Omega})$-minimizer, that is, there exists $\rho_{1}>0$ such that $\varphi_{0}(u_{0})\leq\varphi_{0}(u_{0}+h)$ for all $h\in C^{1}(\overline{\Omega})$ with $\|h\|_{C^{1}(\overline{\Omega})}\leq\rho_{1}$. Then $u_{0}\in C^{1,\alpha}(\overline{\Omega})$ for some $\alpha\in(0,1)$ and $u_{0}$ is a local $H^{1}(\Omega)$-minimizer of $\varphi_{0}$, that is, there exists $\rho_{2}>0$ such that $\varphi_{0}(u_{0})\leq\varphi_{0}(u_{0}+h)$ for all $h\in H^{1}(\Omega)$ with $\|h\|\leq{\rho_{2}}$.

Next we recall some basic facts concerning the spectrum of the differential operator $u\mapsto-\Delta u+\xi(z)u$ with Robin boundary condition. So we consider the following linear eigenvalue problem

[TABLE]

By an eigenvalue we mean a $\widehat{\lambda}\in\mathbb{R}$ for which problem (3) has a nontrivial solution $\hat{u}\in H^{1}(\Omega)$, called an eigenfunction corresponding to $\widehat{\lambda}$. Using the spectral theorem for compact self-adjoint operators on a Hilbert space, we know that the spectrum of (3) consists of a sequence $\{\widehat{\lambda}_{k}\}_{k\geq 1}$ of distinct eigenvalues such that $\widehat{\lambda}_{k}\rightarrow+\infty$ (see [5, 12]). The first eigenvalue $\widehat{\lambda}_{1}\in\mathbb{R}$ has the following properties:

• •

$\widehat{\lambda}_{1}$ is simple with eigenfunctions of constant sign;

• •

we have

[TABLE]

The infimum in (4) is realized on the corresponding one dimensional eigenspace. Let $\hat{u}_{1}$ be the $L^{2}$-normalized (that is, $\|\hat{u}_{1}\|_{2}=1$) positive eigenfunction corresponding to $\widehat{\lambda}_{1}$. If hypotheses $H(\xi)$ and $H(\beta)$ hold, then $\hat{u}_{1}\in C^{1}(\overline{\Omega})$ (see Wang [15]) and by Harnack’s inequality (see Gasinski & Papageorgiou [7, p. 731]), we have $\hat{u}_{1}(z)>0$ for all $z\in\Omega$. Moreover, if in addition we assume that $\xi^{+}\in L^{\infty}(\Omega)$, then $\hat{u}_{1}\in\textrm{int}\,C_{+}$ (via the strong maximum principle, see for example Gasinski & Papageorgiou [7, p. 738]).

As a consequence of these properties, we obtain the following simple lemma.

Lemma 2.2**.**

If hypotheses $H(\xi)$, $H(\beta)$ hold, $\vartheta\in L^{\infty}(\Omega)$, $\vartheta(z)\leq\widehat{\lambda}_{1}$ for a.a. $z\in\Omega$ and $\vartheta\not\equiv\widehat{\lambda}_{1}$, then there exists $\hat{c}>0$ such that

[TABLE]

Proof.

From the variational characterization of $\widehat{\lambda}_{1}$ (see (4)), we get that $J\geq 0$. Suppose that the lemma is not true. Exploiting the $2$-homogeneity of $J(\cdot)$, we can then find $\{u_{n}\}_{n\geq 1}\subseteq H^{1}(\Omega)$ such that

[TABLE]

We may assume that

[TABLE]

It follows from (5) and (6) that

[TABLE]

If $\chi=0$, then $u=0$ and so

[TABLE]

which contradicts the fact that $\|u_{n}\|=1$ for all $n\in\mathbb{N}$ (see (5)).

Hence $\chi\neq 0$. Then $u(z)\neq 0$ for all $z\in\Omega$ and by (7) we have

[TABLE]

which contradicts (4). ∎

We observe that there exists $\mu>0$ such that

[TABLE]

(see [5, 14]).

To see (8), we argue by contradiction. So, suppose that the inequality is not true. We can find $\{u_{n}\}_{n\geq 1}\subseteq H^{1}(\Omega)$ such that

[TABLE]

Set $y_{n}=\displaystyle\frac{u_{n}}{\|u_{n}\|}$, $n\in\mathbb{N}$. Then $\|y_{n}\|=1$ for all $n\in\mathbb{N}$ and so we may assume that

[TABLE]

Exploiting the sequential weak lower semicontinuity of $\gamma(\,\cdot\,)$ we see that $y=0$ and $n\|y_{n}\|^{2}_{2}\rightarrow 0$. Therefore $\|y_{n}\|_{2}\rightarrow 0$. Finally we can say that

[TABLE]

We say that a Banach space $X$ has the Kadec-Klee property if the following is true:

[TABLE]

As a consequence of the parallelogram law, we see that every Hilbert space has the Kadec-Klee property.

Given $x\in\mathbb{R}$, we set $x^{\pm}=\max\{\pm x,0\}$. Then for every $u\in H^{1}(\Omega)$ we define

[TABLE]

We know that

[TABLE]

By $|\cdot|_{N}$ we denote the Lebesgue measure on $\mathbb{R}^{N}$ and given a measurable function $h:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ (for example, a Carathéodory function), we set

[TABLE]

(the Nemytskii or superposition map corresponding to the function $h$).

Given $\varphi\in C^{1}(X,\mathbb{R})$ ($X$ a Banach space), we set

[TABLE]

Finally, by $A\in\mathcal{L}(H^{1}(\Omega),H^{1}(\Omega)^{*})$ we denote the linear operator defined by

[TABLE]

3. The sublinear case

In this section we examine problem ($P_{\lambda}$) under the hypothesis that the perturbation term $f(z,\cdot)$ is sublinear near $+\infty$. More precisely, our conditions on the nonlinearity $f(z,x)$ are the following:

$H_{1}$:  $f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a Carathéodory function such that for a.a. $z\in\Omega$, $f(z,0)=0$, $f(z,x)>0$ for all $x>0$ and

1. (i)

for every $\rho>0$, there exists $a_{\rho}\in L^{\infty}(\Omega)_{+}$ such that $f(z,x)\leq a_{\rho}(z)$ for a.a. $z\in\Omega$, all $0\leq x\leq\rho$; 2. (ii)

$\lim_{x\rightarrow+\infty}\limits\frac{f(z,x)}{x}=0$ uniformly for a.a. $z\in\Omega$; 3. (iii)

there exist $\delta>0$ and $q\in(1,2)$ such that $c_{1}x^{q-1}\leq f(z,x)\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ 0\leq x\leq\delta$.

Remarks. Since we are looking for positive solutions and all the above hypotheses concern the positive semiaxis $\mathbb{R}_{+}=[0,+\infty)$, we may assume without any loss of generality that $f(z,x)=0$ for a.a. $z\in\Omega$, all $x\leq 0$. Hypothesis $H_{1}$(ii) implies that $f(z,\cdot)$ is sublinear near $+\infty$. Hypothesis $H_{1}$(iii) says that there is a concave term near the origin.

Examples. The following functions satisfy hypotheses $H_{1}$. For the sake of simplicity we drop the $z$-dependence

[TABLE]

[TABLE]

We can improve the properties of the positive solutions of ($P_{\lambda}$), provided we strengthen hypothesis $H(\xi)$.

$H(\xi)^{\prime}$:  $\xi\in L^{s}(\Omega)$ with $s>N$ if $N\geq 3$, $s>1$ if $N=1,2$ and $\xi^{+}\in L^{\infty}(\Omega)$.

We introduce the following two sets:

[TABLE]

[TABLE]

We start with a simple but useful observation concerning the solution set $S(\lambda)$. For this result the precise behavior of $f(z,\cdot)$ near $+\infty$ and near $0^{+}$ are irrelevant. We only need nonnegativity of $f(z,x)$ and subcritical growth in $x\in\mathbb{R}$. Under these very general conditions, the result is also applicable in the superlinear case (see Section 4). The new hypotheses on the perturbation term $f(z,x)$ are the following:

1. $\widehat{H}$:

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

[TABLE]

with $a\in L^{\infty}(\Omega)_{+}$, $2\leq r<2^{*}$.

Proposition 2**.**

If hypotheses $H(\xi)$ (resp. $H(\xi)^{\prime}$), $H(\beta)$, $\widehat{H}$ hold, then for all $\lambda\in\mathbb{R}$, $S(\lambda)\subseteq C_{+}\setminus\{0\}$ (resp. $S(\lambda)\subseteq\textrm{int}\,C_{+}$) (possibly empty).

Proof.

Let $u\in S(\lambda)$. Then we have

[TABLE]

(see Papageorgiou & Rădulescu [12]).

We introduce the functions

[TABLE]

and

[TABLE]

Evidently, $\vartheta_{\lambda}\in L^{s}(\Omega)$ (see hypotheses $H(\xi)$ and $\widehat{H}$). Also

[TABLE]

Note that if $N\geq 3$ (the cases $N=1,2$ are straightforward), then

[TABLE]

Since $u\in H^{1}(\Omega)$, by the Sobolev embedding theorem, we have

[TABLE]

We rewrite (12) as follows

[TABLE]

By Lemma 5.1 of Wang [15] we have that $u\in L^{\infty}(\Omega)$. Using the Calderon-Zygmund estimates (see Lemma 5.2 of Wang [15]), we obtain $u\in W^{2,s}(\Omega)$. Then the Sobolev embedding theorem implies $u\in C^{1,\alpha}(\overline{\Omega})$ with $\alpha=1-\frac{N}{s}>0$. Therefore we have $u\in C_{+}\setminus\{0\}$. Suppose that $H(\xi)^{\prime}$ holds. From (12) we obtain

[TABLE]

Thus, we have proved that when $H(\xi)^{\prime}$ holds, then $S(\lambda)\subseteq\textrm{int}\,C_{+}$ for all $\lambda\in\mathbb{R}$. ∎

Next, we show that for every $\lambda\geq\widehat{\lambda}_{1}$ problem ($P_{\lambda}$) has no positive solutions (that is, $S(\lambda)=\emptyset$ for all $\lambda\geq\widehat{\lambda}_{1}$).

Proposition 3**.**

If hypotheses $H(\xi)^{\prime}$, $H(\beta)$, $H_{1}$ hold and $\lambda\geq\widehat{\lambda}_{1}$, then $S(\lambda)=\emptyset$.

Proof.

Let $\lambda\geq\widehat{\lambda}_{1}$ and suppose that $S(\lambda)\neq\emptyset$. Let $u\in S(\lambda)$. Due to Proposition 6, we know that $u\in\textrm{int}\,C_{+}$. Let $v\in\textrm{int}\,C_{+}$ and consider the function

[TABLE]

From Picone’s identity (see, for example, Motreanu, Motreanu & Papageorgiou [10, p. 255]), we derive

[TABLE]

Let $v=\hat{u}_{1}\in\textrm{int}\,C_{+}$. Then $0<\gamma(\hat{u}_{1})-\lambda=\widehat{\lambda}_{1}-\lambda\leq 0$ (recall $\|\hat{u}_{1}\|_{2}=1$), a contradiction. Therefore for all $\lambda\geq\widehat{\lambda}_{1}$, we have $S(\lambda)=\emptyset$. ∎

Next, we show that for $\lambda<\widehat{\lambda}_{1}$ there exist positive solutions.

Proposition 4**.**

If hypotheses $H(\xi)^{\prime}$, $H(\beta)$, $H_{1}$ hold and $\lambda<\widehat{\lambda}_{1}$, then $S(\lambda)\neq\emptyset$.

Proof.

Let $\mu>0$ be as in (8) and consider the Carathéodory function $g_{\lambda}:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ defined by

[TABLE]

We set $G_{\lambda}(z,x)=\int_{0}^{x}g_{\lambda}(z,s)\,ds$ and consider the $C^{1}$-functional $\varphi_{\lambda}:H^{1}(\Omega)\rightarrow\mathbb{R}$ defined by

[TABLE]

Hypotheses $H_{1}$(i),(ii) imply that given $\varepsilon>0$, we can find $c_{3}=c_{3}(\varepsilon)>0$ such that

[TABLE]

Using (13), we obtain

[TABLE]

Choosing $\varepsilon\in(0,\widehat{\lambda}_{1}-\lambda)$ (recall $\lambda<\widehat{\lambda}_{1}$) and using Lemma 2.2, from (14) we obtain

[TABLE]

Also, invoking the Sobolev embedding theorem and the compactness of the trace map, we see that $\varphi_{\lambda}$ is sequentially weakly lower semicontinuous. Therefore by the Weierstrass theorem, we can find $u_{\lambda}\in H^{1}(\Omega)$ such that

[TABLE]

Let $t\in(0,1)$ be so small that $t\hat{u}_{1}(z)\in(0,\delta]$ for all $z\in\overline{\Omega}$ (recall that $\hat{u}_{1}\in\textrm{int}\,C_{+}$ and note that $\delta>0$ is as in hypothesis $H_{1}$(iii)). Then

[TABLE]

Since $q<2$, choosing $t\in(0,1)$ even smaller if necessary, from (3) we obtain

[TABLE]

By (15) we have

[TABLE]

[TABLE]

In (17) we choose $h=-u_{\lambda}^{-}\in H^{1}(\Omega)$. Then

[TABLE]

Thus, equation (17) becomes

[TABLE]

∎

In fact, we can show that for every $\lambda<\widehat{\lambda}_{1}$ problem ($P_{\lambda}$) has a smallest positive solution. To this end note that given $\tau\in\left(\frac{2N}{N-1},2^{*}\right)$, because of hypotheses $H_{1}$(i),(iii), we can find $c_{4}(\lambda)>0$ with $\lambda\mapsto c_{4}(\lambda)$ bounded on bounded subsets of $\mathbb{R}$ such that

[TABLE]

This unilateral growth restriction on the reaction term of problem ($P_{\lambda}$) leads to the following auxiliary Robin problem:

[TABLE]

Proposition 5**.**

If hypotheses $H(\xi)^{\prime}$, $H(\beta)$ hold and $\lambda\in\mathbb{R}$, then problem ($Au_{\lambda}$) admits a unique positive solution $u_{*}^{\lambda}\in\textrm{int}\,C_{+}$.

Proof.

First, we establish the existence of a positive solution for problem ($Au_{\lambda}$). So, we introduce the $C^{1}$-functional $\psi_{\lambda}:H^{1}(\Omega)\rightarrow\mathbb{R}$ defined by

[TABLE]

By Lemma 2.2, we have

[TABLE]

Since for $N\geq 3$, $s>N$, by the Sobolev embedding theorem, we have $u^{2}\in L^{s^{\prime}}(\Omega)$ and so due to Hölder’s inequality,

[TABLE]

Note that $s^{\prime}<N^{\prime}=\frac{N}{N-1}$ (recall that for $1\leq\tau<\infty$, $\tau^{\prime}\in(1,+\infty]$ and $\frac{1}{\tau}+\frac{1}{\tau^{\prime}}=1$), hence $2s^{\prime}<\frac{2N}{N-1}<\tau$. Therefore

[TABLE]

Thus we have

[TABLE]

We return to (3), use (3) and recall that $y\mapsto\left[\|y\|_{\tau}^{2}+\|Dy\|_{2}^{2}\right]^{1/2}$ is an equivalent norm on the Sobolev space $H^{1}(\Omega)$ (see, for example, Gasinski & Papageorgiou [7, p. 227]). So, from (3) we infer that $\psi_{\lambda}(\cdot)$ is coercive.

The cases $N=1,2$ are straightforward because

• •

if $N=1$, then $H^{1}(\Omega)\hookrightarrow C(\overline{\Omega})$ (compactly);

• •

if $N=2$, then $H^{1}(\Omega)\hookrightarrow L^{\tau}(\Omega)$ for all $\tau\in[1,+\infty)$ (compactly).

The Sobolev embedding theorem and the compactness of the trace map, imply that $\psi_{\lambda}$ is sequentially weakly lower semicontinuous. So, we can find $u_{*}^{\lambda}\in H^{1}(\Omega)$ such that

[TABLE]

As before (see the proof of Proposition 4), exploiting the fact that $q<2<\tau$, we obtain

[TABLE]

By (21) we have

[TABLE]

In (3) we choose $h=-(u_{*}^{\lambda})^{-}\in H^{1}(\Omega)$. Then

[TABLE]

Next, we infer from (3) that $u_{*}^{\lambda}$ is a positive solution of ($Au_{\lambda}$). Again, using Lemmata 5.1 and 5.2 of Wang [15], we infer that $u_{*}^{\lambda}\in C_{+}\setminus\{0\}$. Moreover, we have

[TABLE]

Next, we prove the uniqueness of this positive solution. So let $v_{*}^{\lambda}\in H^{1}(\Omega)$ be another positive solution for ($Au_{\lambda}$). As above, we show that $v_{*}^{\lambda}\in\textrm{int}\,C_{+}$. Let $t>0$ be the biggest real number such that $tv_{*}^{\lambda}\leq u_{*}^{\lambda}$ (see Marano & Papageorgiou [9, Proposition 2.1]).

Suppose that $0<t<1$. Let $\rho=\|u_{*}^{\lambda}\|_{\infty}$ and let $\hat{\xi_{\rho}}>0$ be such that

[TABLE]

is nondecreasing on $[0,\rho]$. We have

[TABLE]

[TABLE]

However, this contradicts the maximality of $t\in(0,1)$. Hence $t\geq 1$ and so we have $v_{*}^{\lambda}\leq u_{*}^{\lambda}$.

If in the above argument we reverse the roles of $u_{*}$ and $v_{*}$, we obtain

[TABLE]

and this proves the uniqueness of the positive solution of problem ($Au_{\lambda}$). ∎

Remark. We can have an alternative proof of the uniqueness based on the Picone identity. The argument goes as follows. Suppose again that $v_{*}^{\lambda}\in H^{1}(\Omega)$ is another positive solution of ($Au_{\lambda}$). We have $v_{*}^{\lambda}\in\textrm{int}\,C_{+}$. Therefore

[TABLE]

[TABLE]

Interchanging the roles of $u_{*}^{\lambda}$ and $v_{*}^{\lambda}$ in the above argument, we obtain

[TABLE]

We add (3) and (3) and use Picone’s identity. We obtain

[TABLE]

Since the function $x\mapsto\frac{c_{1}}{x^{2-q}}-c_{4}(\lambda)x^{\tau-2}$ is strictly decreasing on $(0,+\infty)$ (recall $q<2<\tau$), it follows from (3) that

[TABLE]

So, we again get the uniqueness of the positive solution of $(Au_{\lambda})$.

Note also that since $\lambda\mapsto c_{4}(\lambda)$ is bounded on bounded sets of $\mathbb{R}$, if $B\subseteq\mathbb{R}$ is bounded and $\hat{c}_{4}\geq c_{4}(\bar{\lambda})$ for all $\bar{\lambda}\in B$, then the unique solution $\bar{u}\in\textrm{int}\,C_{+}$ of

[TABLE]

satisfies $\bar{u}\leq u_{*}^{\bar{\lambda}}$ for all $\bar{\lambda}\in B$.

Using Proposition 5, we can produce a lower bound for the solution set $S(\lambda)$.

Proposition 6**.**

If hypotheses $H(\xi)^{\prime}$, $H(\beta)$, $H_{1}$ hold and $\lambda\in\mathcal{L}=(-\infty,\widehat{\lambda}_{1})$, then $u_{*}^{\lambda}\leq u$ for all $u\in S(\lambda)$.

Proof.

Let $u\in S(\lambda)$ and consider the following Carathéodory function

[TABLE]

Here $\mu>0$ is as in (8). We set $\widehat{G}_{\lambda}(z,x)=\int_{0}^{x}\hat{g}_{\lambda}(z,s)ds$ and consider the $C^{1}$-functional $\widehat{\psi}_{\lambda}:H^{1}(\Omega)\rightarrow\mathbb{R}$ defined by

[TABLE]

Invoking (8) and (29), we see that $\widehat{\psi}_{\lambda}$ is coercive. It is also sequentially weakly lower semicontinuous. Therefore we can find $\tilde{u}_{*}^{\lambda}\in H^{1}(\Omega)$ such that

[TABLE]

As before (see the proof of Proposition 4), exploiting the fact that $q<2<\tau$, we can show that

[TABLE]

By (30) we have

[TABLE]

In (3) we choose $h=-(\tilde{u}_{*}^{\lambda})^{-}\in H^{1}(\Omega)$ and obtain

[TABLE]

Next, in (3) we choose $(\tilde{u}_{*}^{\lambda}-u)^{+}\in H^{1}(\Omega)$. Then

[TABLE]

[TABLE]

Therefore we have proved that

[TABLE]

By (29) and (32), we see that equation (3) becomes

[TABLE]

Therefore $u_{*}^{\lambda}\leq u$ for all $u\in S(\lambda)$. ∎

This lower bound leads to the existence of a smallest positive solution for problem ($P_{\lambda}$), $\lambda\in\mathcal{L}=(-\infty,\widehat{\lambda}_{1})$.

Proposition 7**.**

If hypotheses $H(\xi)^{\prime}$, $H(\beta)$, $H_{1}$ hold and $\lambda\in\mathcal{L}=(-\infty,\widehat{\lambda}_{1})$, then problem $(P_{\lambda})$ admits a smallest positive solution $\bar{u}_{\lambda}\in S(\lambda)\subseteq\textrm{int}\,C_{+}$.

Proof.

As in Filippakis & Papageorgiou [6], we can show that $S(\lambda)$ is downward directed, that is, if $u_{1},u_{2}\in S(\lambda)$, then we can find $u\in S(\lambda)$ such that $u\leq u_{1}$, $u\leq u_{2}$. For completeness we sketch a proof. So, given $\varepsilon>0$ consider the function

[TABLE]

Evidently, $\vartheta_{\varepsilon}(\cdot)$ is Lipschitz and so $\vartheta_{\varepsilon}((u_{1}-u_{2})^{-})\in H^{1}(\Omega)$. Moreover, the chain rule for Sobolev functions implies that

[TABLE]

Let $\psi\in C^{1}(\overline{\Omega}))$. Then we introduce the test functions

[TABLE]

which belong to $H^{1}(\Omega)\cap L^{\infty}(\Omega)$. We have

[TABLE]

[TABLE]

We add these two equalities and divide by $\varepsilon>0$. Taking into account that

[TABLE]

and

[TABLE]

we obtain

[TABLE]

So, if $\bar{u}=\min\{u_{1},u_{2}\}$, then $\bar{u}$ is an upper solution of ($P_{\lambda}$) and so by standard truncation techniques we can find $u\in S(\lambda)$ such that 0\leq u\leq\bar{u}\leq\left\{\begin{array}[]{l}u_{1}\\ u_{2}\end{array}\right..

Then Lemma 3.10 of Hu & Papageorgiou [8, p. 178], implies that there exist $u_{n}\in S(\lambda)$, $n\in\mathbb{N}$, $\{u_{n}\}_{n\geq 1}$ decreasing such that

[TABLE]

For every $n\in\mathbb{N}$, we have

[TABLE]

Evidently, $\{u_{n}\}_{n\geq 1}\subseteq H^{1}(\Omega)$ is bounded and so we may assume that

[TABLE]

In (33) we pass to the limit as $n\rightarrow\infty$ and use (34). Then

[TABLE]

Due to Proposition 6 we know that

[TABLE]

∎

We examine the monotonicity and continuity properties of the map $\lambda\mapsto\bar{u}_{\lambda}$ from $\mathcal{L}=(-\infty,\widehat{\lambda}_{1})$ into $C^{1}(\overline{\Omega})$.

Proposition 8**.**

If hypotheses $H(\xi)^{\prime}$, $H(\beta)$, $H_{1}$ hold, then the map $\lambda\mapsto\bar{u}_{\lambda}$ is nondecreasing and left continuous from $\mathcal{L}=(-\infty,\widehat{\lambda}_{1})$ into $C^{1}(\overline{\Omega})$.

Proof.

First, we establish the monotonicity of the map $\lambda\rightarrow\bar{u}_{\lambda}$. So let $\lambda<\eta<\widehat{\lambda}_{1}$ and consider $\bar{u}_{\eta}\in S(\eta)\subseteq\textrm{int}\,C_{+}$ the minimal positive solution of problem ($P_{\eta}$). We introduce the following Carathéodory function

[TABLE]

As always, $\mu>0$ is as in (8). We set $E_{\lambda}(z,x)=\int_{0}^{x}e_{\lambda}(z,s)ds$ and consider the $C^{1}$-functional $w_{\lambda}:H^{1}(\Omega)\rightarrow\mathbb{R}$ defined by

[TABLE]

It follows from (8) and (38) that $w_{\lambda}(\cdot)$ is coercive. It is also sequentially weakly lower semicontinuous. So, we can find $\tilde{u}_{\lambda}\in H^{1}(\Omega)$ such that

[TABLE]

By (39) we have $\tilde{u}_{\lambda}\neq 0$ and

[TABLE]

As in the proof of Proposition 6, choosing in (3), first $h=-\tilde{u}_{\lambda}^{-}\in H^{1}(\Omega)$ and then $(\tilde{u}_{\lambda}-\bar{u}_{\eta})^{+}\in H^{1}(\Omega)$, we can show that

[TABLE]

Then we can infer from (38), (3), (41) that

[TABLE]

Next, we establish the left continuity of this map. To this end, let $\lambda_{n}\rightarrow\lambda^{-}$ with $\lambda\in\mathcal{L}=(-\infty,\widehat{\lambda}_{1})$. Evidently, $\{\bar{u}_{\lambda_{n}}\}_{n\geq 1}\subseteq H^{1}(\Omega)$ is bounded and increasing. Therefore we may assume that

[TABLE]

We have

[TABLE]

In (43) we pass to the limit as $n\rightarrow\infty$ and use (42). We obtain

[TABLE]

Set $B=\{\lambda_{n}\}_{n\geq 1}$ and let $\hat{c}_{4}\geq c_{4}(\bar{\lambda})$ for all $\bar{\lambda}\in B$ (recall that $\lambda\mapsto c_{4}(\lambda)$ is bounded on bounded sets of $\mathbb{R}$). Let $\bar{u}\in\textrm{int}\,C_{+}$ be the unique positive solution of the following semilinear Robin problem

[TABLE]

We know that $\bar{u}\leq\bar{u}_{\lambda_{n}}\ \mbox{for all}\ n\in\mathbb{N}$ (see the remark following Proposition 5). Hence

[TABLE]

We claim that $\hat{u}_{\lambda}=\bar{u}_{\lambda}$. If this is not true, then we can find $z_{0}\in\overline{\Omega}$ such that

[TABLE]

From Wang [15], we know that we can find $M_{1}>0$ and $\alpha\in(0,1)$ such that

[TABLE]

Exploiting the compact embedding of $C^{1,\alpha}(\overline{\Omega})$ into $C^{1}(\overline{\Omega})$ and using (42), we obtain

[TABLE]

a contradiction to the monotonicity of $\lambda\mapsto\bar{u}_{\lambda}$ (recall that $\lambda_{n}<\lambda$ for all $n\in\mathbb{N}$). So $\hat{u}_{\lambda}=\bar{u}_{\lambda}$ and this proves the left continuity of $\lambda\mapsto\bar{u}_{\lambda}$ from $\mathcal{L}=(-\infty,\widehat{\lambda}_{1})$ into $C^{1}(\overline{\Omega})$ (see (47)). ∎

We can improve the monotonicity of the map $\lambda\mapsto\bar{u}_{\lambda}$ provided that we strengthen the conditions on $f(z,\cdot)$.

1. $H_{2}$:

$f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a Carathéodory function such that $f(z,0)=0$ for a.a. $z\in\Omega$, hypotheses $H_{2}$(i), (ii), (iii) are the same as hypotheses $H_{1}$(i), (ii), (iii) and

1. (iv)

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

[TABLE]

is nondecreasing on $[0,\rho]$.

Remark. The new condition on $f(z,\cdot)$ is satisfied, if for a.a. $z\in\Omega$, $f(z,\cdot)$ is differentiable and $z\mapsto f_{x}^{\prime}(z,\cdot)$ is locally $L^{\infty}(\Omega)$-bounded (just use the mean value theorem).

Examples. The following functions satisfy hypotheses $H_{2}$. As before, for the sake of simplicity, we drop the $z$-dependence

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Proposition 9**.**

If hypotheses $H(\xi)^{\prime}$, $H(\beta)$, $H_{2}$ hold, then the map $\lambda\mapsto\bar{u}_{\lambda}$ from $\mathcal{L}=(-\infty,\widehat{\lambda}_{1})$ into $C^{1}(\overline{\Omega})$ is strictly increasing (that is, if $\lambda<\eta<\widehat{\lambda}_{1}$, then $\bar{u}_{\eta}-\bar{u}_{\lambda}\in\textrm{int}\,C_{+}$).

Proof.

Let $\lambda<\eta<\widehat{\lambda}_{1}$ and let $\bar{u}_{\lambda}\in S(\lambda)\subseteq\textrm{int}\,C_{+}$, $\bar{u}_{\eta}\in S(\eta)\subseteq\textrm{int}\,C_{+}$ be the corresponding minimal positive solutions of problems ($P_{\lambda}$) and ($P_{\eta}$), respectively. By Proposition 8 we know that

[TABLE]

Let $\rho=\|\bar{u}_{\eta}\|_{\infty}$ and let $\hat{\xi}_{\rho}>0$ be as postulated by hypothesis $H_{2}$(iv). We have

[TABLE]

This proves the strict monotonicity of $\lambda\mapsto\bar{u}_{\lambda}$. ∎

In fact, under a monotonicity restriction on the quotient $\frac{f(z,x)}{x}$, we can conclude that for all $\lambda\in\mathcal{L}=(-\infty,\widehat{\lambda}_{1})$ problem ($P_{\lambda}$) admits a unique positive solution.

Hence the new conditions on the perturbation $f(z,x)$ are the following:

1. $H_{3}$:

$f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a Carathéodory function such that $f(z,0)=0$ for a.a. $z\in\Omega$, hypotheses $H_{3}$(i), (ii), (iii) are the same as hypotheses $H_{1}$(i), (ii), (iii) and

1. (iv)

for a.a. $z\in\Omega$, $x\mapsto\frac{f(z,x)}{x}$ is strictly decreasing on $(0,+\infty)$.

Examples. The following functions satisfy the new conditions. Again, for the sake of simplicity we drop the $z$-dependence:

[TABLE]

[TABLE]

Proposition 10**.**

If hypotheses $H(\xi)^{\prime}$, $H(\beta)$, $H_{3}$ hold and $\lambda\in\mathcal{L}=(-\infty,\widehat{\lambda}_{1})$, then $S(\lambda)$ is a singleton, that is, $S(\lambda)=\{\bar{u}_{\lambda}\}$ and $\lambda\mapsto\bar{u}_{\lambda}$ is nondecreasing and continuous from $(-\infty,\widehat{\lambda}_{1})$ into $C^{1}(\overline{\Omega})$.

Proof.

The nonemptiness of $S(\lambda)$ follows from Proposition 4. Suppose that $\bar{u}_{\lambda},\bar{v}_{\lambda}\in S(\lambda)\subseteq\textrm{int}\,C_{+}$. We have

[TABLE]

[TABLE]

We multiply (49) with $\bar{v}_{\lambda}(z)$ and (50) with $\bar{u}_{\lambda}(z)$, then integrate both equations over $\Omega$ and use Green’s identity. We obtain

[TABLE]

[TABLE]

We subtract (3) from (3) and obtain

[TABLE]

This proves the uniqueness of the positive solution of problem ($P_{\lambda}$). The uniqueness of this positive solution, together with Proposition 8, imply that the map $\lambda\mapsto\bar{u}_{\lambda}$ from $\mathcal{L}=(-\infty,\widehat{\lambda}_{1})$ into $C^{1}(\overline{\Omega})$ is nondecreasing and continuous. ∎

As before, by strengthening the conditions on the perturbation $f(z,\cdot)$ we can improve the monotonicity of the map $\lambda\rightarrow\bar{u}_{\lambda}$.

The new condition on $f(z,x)$ are the following:

1. $H_{4}$:

$f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a Carathéodory function such that $f(z,0)=0$ for a.a. $z\in\Omega$, hypotheses $H_{4}$(i), (ii), (iii), (iv) are the same as the corresponding hypotheses $H_{3}$(i), (ii), (iii), (iv) and

1. (v)

for every $\rho>0$, there exists $\hat{\xi}_{\rho}>0$ such that for a.a. $z\in\Omega$ the function $x\mapsto f(z,x)+\hat{\xi}_{\rho}x$ is nondecreasing on $[0,\rho]$.

Remark. The examples after hypotheses $H_{3}$, also satisfy hypotheses $H_{4}$. Then Propositions 9 and 10 imply the following result.

Proposition 11**.**

If hypotheses $H(\xi)^{\prime}$, $H(\beta)$, $H_{4}$ hold and $\lambda\in\mathcal{L}=(-\infty,\widehat{\lambda}_{1})$, then the set $S(\lambda)$ is a singleton $\{\bar{u}_{\lambda}\}$ and the map $\lambda\mapsto\bar{u}_{\lambda}$ from $\mathcal{L}$ into $C^{1}(\overline{\Omega})$ is strictly increasing (that is, if $\lambda<\eta<\widehat{\lambda}_{1}$, then $\bar{u}_{\eta}-\bar{u}_{\lambda}\in\textrm{int}\,C_{+}$).

Summarizing the situation for problem ($P_{\lambda}$) when the perturbation term $f(z,\cdot)$ is sublinear, we can state the following theorem.

Theorem 3.1**.**

1. (a)

If hypotheses $H(\xi)^{\prime}$, $H(\beta)$, $H_{1}$ hold, then for every $\lambda\geq\widehat{\lambda}_{1}$ we have $S(\lambda)=\emptyset$, while for every $\lambda<\widehat{\lambda}_{1}$, $S(\lambda)\neq\emptyset$, $S(\lambda)\subseteq\textrm{int}\,C_{+}$, problem ($P_{\lambda}$) admits a smallest positive solution $\bar{u}_{\lambda}\in\textrm{int}\,C_{+}$ and the map $\lambda\mapsto\bar{u}_{\lambda}$ from $\mathcal{L}=(-\infty,\widehat{\lambda}_{1})$ into $C^{1}(\overline{\Omega})$ is nondecreasing (that is, $\lambda<\eta<\lambda_{1}$ implies $\bar{u}_{\lambda}\leq\bar{u}_{\eta}$) and left continuous. 2. (b)

If hypotheses $H(\xi)^{\prime}$, $H(\beta)$, $H_{2}$ hold, then the map $\lambda\mapsto\bar{u}_{\lambda}$ from $\mathcal{L}=(-\infty,\widehat{\lambda}_{1})$ into $C^{1}(\overline{\Omega})$ is strictly increasing (that is, $\lambda<\eta<\widehat{\lambda}_{1}$ implies $\bar{u}_{\eta}-\bar{u}_{\lambda}\in\textrm{int}\,C_{+}$). 3. (c)

If hypotheses $H(\xi)^{\prime}$, $H(\beta)$, $H_{3}$ hold and $\lambda\in\mathcal{L}=(-\infty,\widehat{\lambda}_{1})$, then $S(\lambda)$ is a singleton $\{\bar{u}_{\lambda}\}$ ($\bar{u}_{\lambda}\in\textrm{int}\,C_{+}$) and the map $\lambda\mapsto\bar{u}_{\lambda}$ from $\mathcal{L}=(-\infty,\widehat{\lambda}_{1})$ into $C^{1}(\overline{\Omega})$ is nondecreasing and continuous. 4. (d)

If hypotheses $H(\xi)^{\prime}$, $H(\beta)$, $H_{4}$ hold, then the map $\lambda\mapsto\bar{u}_{\lambda}$ is strictly increasing from $\mathcal{L}=(-\infty,\widehat{\lambda}_{1})$ into $C^{1}(\overline{\Omega})$.

4. The superlinear case

In this section, we investigate the case of a superlinear perturbation $f(z,\cdot)$. Now we cannot have uniqueness of the positive solution and in fact we show that the problem exhibits a kind of bifurcation phenomenon, that is, for $\lambda<\widehat{\lambda}_{1}$, problem ($P_{\lambda}$) has at least two positive solutions, while for $\lambda\geq\widehat{\lambda}_{1}$, $S(\lambda)=\emptyset$. We stress that for the superlinearity of $f(z,\cdot)$ we do not use the Ambrosetti-Rabinowitz condition (AR-condition for short).

The hypotheses on the perturbation $f(z,x)$ are the following:

1. $H_{5}$:

$f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a Carathéodory function such that for a.a. $z\in\Omega$, $f(z,0)=0$, $f(z,x)\geq 0$ for all $x\geq 0$, $f(z,x)>0$ for a.a. $z\in\Omega_{0}\subseteq\Omega$ with $|\Omega_{0}|_{N}>0$, all $x>0$ and

1. (i)

$f(z,x)\leq a(z)(1+x^{r-1})$ for a.a. $z\in\Omega$, all $x\geq 0$, with $a\in L^{\infty}(\Omega)_{+}$ and $2<r<2^{*}$; 2. (ii)

if $F(z,x)=\int_{0}^{x}f(z,s)ds$, then

[TABLE]

and there exists $q\in\left(\max\left\{1,(r-2)\frac{N}{2}\right\},2^{*}\right)$ such that

[TABLE] 3. (iii)

$\displaystyle\lim_{x\rightarrow 0^{+}}\limits\frac{f(z,x)}{x}=0$ uniformly for a.a. $z\in\Omega$.

Remarks. As in the sublinear case, since we are looking for positive solutions and the above hypotheses concern the positive semiaxis $\mathbb{R}_{+}=[0,+\infty)$, we may assume without any loss of generality that $f(z,x)=0$ for a.a. $z\in\Omega$, all $x\leq 0$. Hypothesis $H_{5}$(ii) implies that for a.a. $z\in\Omega$, $f(z,\cdot)$ is superlinear near $+\infty$. However, we do not assume the usual for superlinear problems AR-condition. Recall that the AR-condition (unilateral version since it is imposed only on the positive semiaxis), says that there exist $\tau>2$ and $M_{2}>0$ such that

[TABLE]

[TABLE]

(see Ambrosetti & Rabinowitz [3] and Mugnai [11]). Integrating (53) and using (54), we obtain

[TABLE]

From (53) and (55) it follows that for a.a. $z\in\Omega$, $f(z,\cdot)$ has at least $(\tau-1)$-polynomial growth near $+\infty$. Our hypothesis is implied by the AR-condition. We may assume that $\tau>\max\left\{1,(r-2)\frac{N}{2}\right\}$. We have

[TABLE]

Thus, hypothesis $H_{5}$(ii) is satisfied (recall that $\tau>2$). This more general superlinearity condition, incorporates in our framework superlinear nonlinearities with slower growth near $+\infty$, which fail to satisfy the AR-condition (unilateral version).

Examples. The following functions satisfy hypotheses $H_{5}$. As in the previous examples, for the sake of simplicity we drop the $z$-dependence.

[TABLE]

[TABLE]

Note that $f_{1}$ satisfies the unilateral AR-condition, but $f_{2}$ does not.

Using Proposition 2 and reasoning exactly as in the proof of Proposition 3, we obtain the following result.

Proposition 12**.**

If hypotheses $H(\xi)^{\prime}$, $H(\beta)$, $H_{5}$ hold and $\lambda\geq\lambda_{1}$, then $S(\lambda)=\emptyset$.

So according to this proposition, we have $\mathcal{L}\subseteq(-\infty,\widehat{\lambda}_{1})$. In fact we will show that equality holds.

Proposition 13**.**

If hypotheses $H(\xi)^{\prime}$, $H(\beta)$, $H_{5}$ hold, then $\mathcal{L}=(-\infty,\widehat{\lambda}_{1})$.

Proof.

We fix $\lambda\in(-\infty,\widehat{\lambda}_{1})$ and consider the Carathéodory function

[TABLE]

We set $K_{\lambda}(z,x)=\int_{0}^{x}k_{\lambda}(z,s)ds$ and consider the $C^{1}$-functional $\varphi_{\lambda}:H^{1}(\Omega)\rightarrow\mathbb{R}$ defined by

[TABLE]

Here, $\mu>0$ is as in (8). Hypotheses $H_{5}$(i), (iii) imply that given $\varepsilon>0$, we can find $c_{9}=c_{9}(\varepsilon)>0$ such that

[TABLE]

Choosing $\varepsilon\in(0,\widehat{\lambda}_{1}-\lambda)$ (recall $\lambda<\widehat{\lambda}_{1}$), for all $u\in H^{1}(\Omega)$ we obtain

[TABLE]

Since $r>2$, from (4) we infer that $u=0$ is a strict local minimizer of $\varphi_{\lambda}$. It is easy to see that $K_{\varphi_{\lambda}}\subseteq C_{+}$ (see (58)). Thus, we may assume that $u=0$ is an isolated critical point of $\varphi_{\lambda}$ or otherwise we already have a whole sequence of distinct positive solutions of ($P_{\lambda}$) which converge to zero in $H^{1}(\Omega)$. Therefore we can find $\rho\in(0,1)$ small such that

[TABLE]

Hypothesis $H_{5}$(ii) implies that given any $u\in\textrm{int}\,C_{+}$, we have

[TABLE]

Claim. $\varphi_{\lambda}$ satisfies the C-condition.

Let $\{u_{n}\}_{n\geq 1}\subseteq H^{1}(\Omega)$ such that

[TABLE]

[TABLE]

By (64) we have

[TABLE]

In (65) we choose $h=-u_{n}^{-}\in H^{1}(\Omega)$. Using (58), we obtain

[TABLE]

It follows from (63) and (4) that

[TABLE]

On the other hand, if in (65) we choose $h=u_{n}^{+}\in H^{1}(\Omega)$, then

[TABLE]

Adding (67) and (68), we obtain

[TABLE]

Hypotheses $H_{5}$(i), (ii) imply that we can find $\tilde{\xi}_{0}\in(0,\xi_{0})$ and $c_{12}>0$ such that

[TABLE]

We use (70) in (69) and infer that

[TABLE]

First, suppose that $N\neq 2$. By hypothesis $H_{5}$(ii), we see that we may assume without loss of generality that $q<r<2^{*}$. So, we can find $t\in(0,1)$ such that

[TABLE]

Invoking the interpolation inequality (see, for example, Gasinski & Papageorgiou [7, p. 905]), we have

[TABLE]

In (65) we choose $h=u_{n}^{+}\in H^{1}(\Omega)$. Then

[TABLE]

By hypothesis $H_{5}$(i), we see that we can always assume that $r$ is close to $2^{*}$, hence $q\geq 2$ (see hypothesis $H_{5}$(ii)). Then (71) implies that $\{u_{n}^{+}\}_{n\geq 1}\subseteq L^{2}(\Omega)$ is bounded and so by (4) we have

[TABLE]

Due to (72) and hypothesis $H_{5}$(ii), we see that

[TABLE]

If $N=2$, then $2^{*}=+\infty$ and the Sobolev embedding theorem says that $H^{1}(\Omega)\hookrightarrow L^{\eta}(\Omega)$ for all $\eta\in[1,+\infty)$. Let $\eta>r>q$ and $t\in(0,1)$ such that

[TABLE]

Note that

[TABLE]

Therefore the previous argument works if instead of $2^{*}$ we use $\eta>1$ big such that $t\,r<2$. We again obtain (4). It follows from (4) and (4) that

[TABLE]

Thus, we may assume that

[TABLE]

In (65) we choose $h=u_{n}-u\in H^{1}(\Omega)$, pass to the limit as $n\rightarrow\infty$ and use (77). Then

[TABLE]

[TABLE]

Therefore $\varphi_{\lambda}$ satisfies the C-condition and this proves the Claim.

Then (61), (62) and the Claim, permit the use of Theorem 2.1 (the mountain pass theorem) and so we can find $u_{\lambda}\in H^{1}(\Omega)$ such that

[TABLE]

∎

Next, we show that for every $\lambda\in(-\infty,\widehat{\lambda}_{1})$, problem ($P_{\lambda}$) has a smallest positive solution.

First, let us recall the following lemma from Hu and Papageorgiou [8, p. 178].

Lemma A. *If $(\Omega,\sum,\mu)$ is a finite measure space and ${\mathcal{D}}$ is a family of $\mathbb{R}_{+}$-valued measurable functions which is downward directed, then there exists a unique (modulo equality $\mu$-a.e.) function $h:\Omega\rightarrow\mathbb{R}_{+}$ such that

(a) $h(\omega)\leq u(\omega)$ $\mu$-a.e. in $\Omega$ for all $u\in{\mathcal{D}}$;

(b) if $g:\Omega\rightarrow\mathbb{R}$ is a measurable function such that*

[TABLE]

then $g(\omega)\leq h(\omega)$ $\mu$-a.e., that is, $h=\inf{\mathcal{D}}$. Moreover, there is a decreasing sequence $\{u_{n}\}_{n\geq 1}\subseteq{\mathcal{D}}$ such that $\inf_{n\geq 1}u_{n}=h=\inf{\mathcal{D}}$.

Proposition 14**.**

If hypotheses $H(\xi)^{\prime}$, $H(\beta)$, $H_{5}$ hold and $\lambda\in\mathcal{L}=(-\infty,\widehat{\lambda}_{1})$, then problem ($P_{\lambda}$) admits a smallest positive solution $\bar{u}_{\lambda}\in\textrm{int}\,C_{+}$.

Proof.

We again have that $S(\lambda)$ is downward directed (that is, if $u_{1},u_{2}\in S(\lambda)$, then we can find $u\in S(\lambda)$ such that $u\leq u_{1}$, $u\leq u_{2}$; see Filippakis & Papageorgiou [6]). Therefore Lemma A above implies that we can find a decreasing sequence $\{u_{n}\}_{n\geq 1}\subseteq S(\lambda)$ such that

[TABLE]

We have

[TABLE]

Choosing $h=u_{n}\in H^{1}(\Omega)$ in (78), we obtain

[TABLE]

Since $u_{n}\leq u_{1}\in\textrm{int}\,C_{+}$ for all $n\in\mathbb{N}$ (recall that $\{u_{n}\}_{n\geq 1}\subseteq H^{1}(\Omega)$ is decreasing) and by (79), (8) and hypotheses $H(\beta)^{\prime}$, $H(\beta)$, $H_{5}$(i), we can conclude that

[TABLE]

Hence, we may assume that

[TABLE]

Returning to (78), passing to the limit as $n\rightarrow\infty$ and using (80), we obtain

[TABLE]

We will show that $\bar{u}_{\lambda}\neq 0$. Arguing by contradiction, suppose that $\bar{u}_{\lambda}=0$. Let $y_{n}=\frac{u_{n}}{\|u_{n}\|}$, $n\in\mathbb{N}$. Then $\|y_{n}\|=1$, $y_{n}\in\textrm{int}\,C_{+}$ for all $n\in\mathbb{N}$. So we may assume that

[TABLE]

By (78) we have

[TABLE]

Let $\rho=\|u_{1}\|_{\infty}$. Hypotheses $H_{5}$(i), (iii) imply that

[TABLE]

By passing to a subsequence if necessary and using hypothesis $H_{5}$(iii) we infer that

[TABLE]

(see Aizicovici, Papageorgiou & Staicu [2], proof of Proposition 14). In (82), we first choose $h=y_{n}-y\in H^{1}(\Omega)$, pass to the limit as $n\rightarrow\infty$ and use (81) and (4). Then

[TABLE]

Next in (82) we choose $h=y_{n}\in H^{1}(\Omega)$ and pass to the limit as $n\rightarrow\infty$. Using (84) and (85) we obtain

[TABLE]

a contradiction to (4). Hence, $\bar{u}_{\lambda}\neq 0$ and therefore

[TABLE]

∎

Reasoning as in the proof of Proposition 8, we can establish the monotonicity and continuity properties of the map $\lambda\mapsto\bar{u}_{\lambda}$.

Proposition 15**.**

If hypotheses $H(\xi)^{\prime}$, $H(\beta)$, $H_{5}$ hold, then the map $\lambda\mapsto\bar{u}_{\lambda}$ from $\mathcal{L}=(-\infty,\widehat{\lambda}_{1})$ into $C^{1}(\overline{\Omega})$ is nondecreasing and continuous.

As before (see Proposition 9 and hypotheses $H_{2}$), by strengthening the conditions on $f(z,\cdot)$, we can improve the monotonicity of $\lambda\mapsto\bar{u}_{\lambda}$.

The new conditions on $f(z,x)$ are the following:

1. $H_{6}$:

$f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a Carathéodory function such that $f(z,0)=0$ for a.a. $z\in\Omega$, hypotheses $H_{6}$(i), (ii), (iii) are the same as the corresponding hypotheses $H_{5}$(i), (ii), (iii) and

1. (iv)

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

[TABLE]

is nondecreasing on $[0,\rho]$.

Reasoning as in the proof of Proposition 7, we obtain.

Proposition 16**.**

If hypotheses $H(\xi)^{\prime}$, $H(\beta)$, $H_{6}$ hold, then the map $\lambda\mapsto\bar{u}_{\lambda}$ from $\mathcal{L}=(-\infty,\widehat{\lambda}_{1})$ into $C^{1}(\overline{\Omega})$ is strictly increasing (that is, if $\lambda<\eta<\widehat{\lambda}_{1}$, then $\bar{u}_{\eta}-\bar{u}_{\lambda}\in\textrm{int}\,C_{+}$).

Next, we show that for all admissible $\lambda\in\mathcal{L}=(-\infty,\widehat{\lambda}_{1})$, problem ($P_{\lambda}$) has at least two positive solutions, which are ordered.

Proposition 17**.**

If hypotheses $H(\xi)^{\prime}$, $H(\beta)$, $H_{6}$ hold and $\lambda\in\mathcal{L}=(-\infty,\widehat{\lambda}_{1})$, then problem ($P_{\lambda}$) admits at least two positive solutions

[TABLE]

Proof.

By Proposition 13 we already have one positive solution $u_{\lambda}\in\textrm{int}\,C_{+}$. We may assume that $u_{\lambda}$ is the minimal positive solution (that is, $u_{\lambda}=\bar{u}_{\lambda}$, see Proposition 14). We consider the following Carathéodory function

[TABLE]

We set $\widetilde{G}_{\lambda}(z,x)=\int_{0}^{x}\tilde{g}_{\lambda}(z,s)ds$ and consider the $C^{1}$-functional $\widetilde{\psi}_{\lambda}:H^{1}(\Omega)\rightarrow\mathbb{R}$ defined by

[TABLE]

From (89), we see that on $[u_{\lambda})=\{u\in H^{1}(\Omega):\ u_{\lambda}(z)\leq u(z)\ \mbox{for a.a.}\ z\in\Omega\}$ we have

[TABLE]

Claim. We may assume that $u_{\lambda}\in\textrm{int}\,C_{+}$ is a local minimizer of the functional $\widetilde{\psi}_{\lambda}$.

Let $\lambda<\eta<\widehat{\lambda}_{1}$ and let $u_{\eta}\in S(\eta)$. From Proposition 16, we obtain $u_{\eta}-u_{\lambda}\in\textrm{int}\,C_{+}$. We introduce the following truncation of $\tilde{g}_{\lambda}(z,\cdot)$:

[TABLE]

Evidently, this is a Carathéodory function. We set $G_{\lambda}^{*}(z,x)=\int_{0}^{x}g_{\lambda}^{*}(z,s)ds$ and consider the $C^{1}$-functional $\psi_{\lambda}^{*}:H^{1}(\Omega)\rightarrow\mathbb{R}$ defined by

[TABLE]

As before (see the proof of Proposition 8), we can show that

[TABLE]

Moreover, by (8) and (93), we see that $\psi_{\lambda}^{*}$ is coercive. It is also sequentially weakly lower semicontinuous. So we can find $u_{\lambda}^{*}\in H^{1}(\Omega)$ such that

[TABLE]

If $u_{\lambda}^{*}\neq u_{\lambda}$, then this is the second positive solution of problem ($P_{\lambda}$) and, as we will see in the last part of the proof, we have $u_{\lambda}^{*}-u_{\lambda}\in\textrm{int}\,C_{+}$, so we are done. So, suppose $u_{\lambda}^{*}=u_{\lambda}$. We have

[TABLE]

Since $u_{\lambda}=u_{\lambda}^{*}$ and $u_{\eta}-u_{\lambda}\in\textrm{int}\,C_{+}$ (see Proposition 16), it follows that

[TABLE]

This proves the Claim.

We assume that $K_{\psi_{\lambda}}$ is finite or otherwise we already have a sequence of distinct positive solutions, all strictly bigger than $u_{\lambda}$ (note that $K_{\widetilde{\psi}_{\lambda}}\subseteq[u_{\lambda})=\left\{u\in H^{1}(\Omega):\,u_{\lambda}(z)\leq u(z)\right.$ for a.a. $\left.z\in\Omega\right\}$) and so we are done. Using the Claim, we can find $\rho\in(0,1)$ small such that

[TABLE]

Hypothesis $H_{6}$(ii) implies that

[TABLE]

Then (4), (95), (96) permit the use of Theorem 2.1 (the mountain pass theorem). Therefore we can find $\hat{u}_{\lambda}\in H^{1}(\Omega)$ such that

[TABLE]

Moreover, if $\rho=\|\hat{u}_{\lambda}\|_{\infty}$ and $\hat{\xi}_{\rho}>0$ is as postulated by hypothesis $H_{6}$(iv), then

[TABLE]

[TABLE]

∎

Summarizing the situation for problem ($P_{\lambda}$) when the perturbation $f(z,\cdot)$ is superlinear, we can state the following theorem.

Theorem 4.1**.**

If hypotheses $H(\xi)^{\prime}$, $H(\beta)$, $H_{6}$ hold, then

1. (a)

for all $\lambda\geq\widehat{\lambda}_{1}$ problem ($P_{\lambda}$) has no positive solution (that is, $S(\lambda)=\emptyset$); 2. (b)

for every $\lambda\in\mathcal{L}=(-\infty,\widehat{\lambda}_{1})$ problem ($P_{\lambda}$) has at least two positive solutions $u_{\lambda}$, $\hat{u}_{\lambda}\in\textrm{int}\,C_{+}$, $\hat{u}_{\lambda}-u_{\lambda}\in\textrm{int}\,C_{+}$; 3. (c)

for every $\lambda\in\mathcal{L}=(-\infty,\widehat{\lambda}_{1})$ problem ($P_{\lambda}$) has a smallest positive solution $\bar{u}_{\lambda}$ and the map $\lambda\mapsto\bar{u}_{\lambda}$ from $\mathcal{L}=(-\infty,\widehat{\lambda}_{1})$ is strictly increasing and left continuous.

Acknowledgments

The authors wish to thank a knowledgeable referee for remarks which helped them to improve the presentation. This research was supported by the Slovenian Research Agency grants P1-0292, J1-7025, and J1-6721, and the Romanian National Authority for Scientific Research and Innovation, CNCS-UEFISCDI grant PN-II-PT-PCCA-2013-4-0614.