Robin problems with indefinite linear part and competition phenomena

TL;DR

This paper studies a parametric Robin boundary value problem with an indefinite potential and competing nonlinearities, establishing bifurcation results, existence of minimal solutions, and their properties as the parameter varies.

Contribution

It introduces a bifurcation framework for Robin problems with indefinite linear parts and nonlinearities without the Ambrosetti-Rabinowitz condition, including the existence and properties of minimal solutions.

Findings

Bifurcation-type theorem for positive solutions

Existence of a minimal positive solution for each parameter

Monotonicity and continuity of the solution map

Abstract

We consider a parametric semilinear Robin problem driven by the Laplacian plus an indefinite potential. The reaction term involves competing nonlinearities. More precisely, it is the sum of a parametric sublinear (concave) term and a superlinear (convex) term. The superlinearity is not expressed via the Ambrosetti-Rabinowitz condition. Instead, a more general hypothesis is used. We prove a bifurcation-type theorem describing the set of positive solutions as the parameter $\lambda > 0$ varies. We also show the existence of a minimal positive solution $\tilde{u}_\lambda$ and determine the monotonicity and continuity properties of the map $\lambda \mapsto \tilde{u}_\lambda$.

Equations

$$P_{\lambda }$$

$$\lambda x^{q-1}+x^{r-1}\text{mbox}forallx\ge 0\text{mbox}with1<q<2<r<2^{\ast }.$$

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

$$\underset{E_0}{\sup }\phi <\underset{D}{\inf }\phi$$

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

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

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

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

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

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

$${\phi }_0(u)=\frac{1}{2}\vartheta (u)-{\int }_{\Omega }{F}_0(z,u)dz,$$

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

$${\phi }_0(u_0)\le {\phi }_0(u_0+h)\text{mbox}forallh\in C^1(\overline{\Omega })\text{mbox}with||h|{|}_{C^1(\overline{\Omega })}\le {\rho }_0.$$

$${\phi }_0(u_0)\le {\phi }_0(u_0+h)\text{mbox}forallh\in H^1(\Omega )\text{mbox}with||h||\le {\rho }_1.$$

$$-\Delta u(z)+\xi (z)u(z)=\hat{\lambda }u(z)\text{mbox}in\Omega ,\frac{\partial u}{\partial n}+\beta (z)u=0\text{mbox}on\partial \Omega .$$

$$\vartheta (u)+\mu ||u|{|}_2^2\ge c_0||u|{|}^2\text{mbox}forallu\in H^1(\Omega )\text{mbox}andforsome{c}_0>0.$$

$$H^1(\Omega )=\overline{\underset{k\ge 1}{\oplus }E({\hat{\lambda }}_k)}.$$

$$\bullet$$

$${\hat{\lambda }}_m=$$

$$=$$

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

$$‘‘u_n\stackrel{w}{\to }u\text{mbox}inX\text{mbox}and||u_n||\to ||u||\Rightarrow u_n\to u\text{mbox}inX".$$

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

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

$$K_{\phi }=\left\{u\in X:{\phi }^{\prime }(u)=0\right\}\text{mbox}(thecriticalsetof\phi ).$$

$$n_0=\max \left\{k\in \mathbb{N}:{\hat{\lambda }}_k\le 0\right\}.$$

$$g(z,x)\le a_{\rho }(z)\text{mbox}foralmostallz\in \Omega \text{mbox}andallx\in [0,\rho ];$$

$$c_3{x}^{q-1}\le g(z,x)\text{mbox}foralmostallz\in \Omega \text{mbox}andallx\ge 0,$$

$$\underset{x\to 0^{+}}{\mathrm{limsup}}\frac{g(z,x)}{x^{q-1}}\le c_4\text{mbox}uniformlyforalmostallz\in \Omega ;$$

$$x\mapsto g(z,x)+{\hat{\xi }}_{\rho }x$$

$$f(z,x)\ge 0\text{mbox}foralmostallz\in \Omega \text{mbox}andallx\in [0,{\delta }_0];$$

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

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 $\Omega\subseteq\mathbb{R}^{N}$ ($N\geq 2$) be a bounded domain with a $C^{2}$-boundary $\partial\Omega$. In this paper we study the following parametric Robin problem

[TABLE]

In this problem, $\lambda>0$ is a parameter, $\xi\in L^{s}(\Omega)$ ($s>N$) is a potential function which is indefinite (that is, sign changing) and in the reaction, $g(z,x)$ and $f(z,x)$ are Carathéodory functions (that is, for all $x\in\mathbb{R}$, $z\mapsto g(z,x),f(z,x)$ are measurable and for almost all $z\in\Omega$, $x\mapsto g(z,x),f(z,x)$ are continuous). We assume that for almost all $z\in\Omega$, $g(z,\cdot)$ is strictly sublinear near $+\infty$ (concave nonlinearity), while for almost all $z\in\Omega$, $f(z,\cdot)$ is strictly superlinear near $+\infty$ (convex nonlinearity). Therefore the reaction in problem ($P_{\lambda}$) exhibits the combined effects of competing nonlinearities (“concave-convex problem”). The study of such problems was initiated with the well-known work of Ambrosetti, Brezis and Cerami [2], who dealt with a Dirichlet problem with zero potential (that is, $\xi\equiv 0$) and the reaction had the form

[TABLE]

They proved a bifurcation-type result for small values of the parameter $\lambda>0$. The work of Ambrosetti, Brezis and Cerami [2] was extended to more general classes of Dirichlet problems with zero potential by Bartsch and Willem [4], Li, Wu and Zhou [9], and Rădulescu and Repovš [19].

Our aim in this paper is to extend all the aforementioned results to the more general problem ($P_{\lambda}$). Note that when $\beta\equiv 0$, we recover the Neumann problem with an indefinite potential. Robin and Neumann problems are in principle more difficult to deal with, due to the failure of the Poincaré inequality. Therefore in our problem, the differential operator (left-hand side of the equation) is not coercive (unless $\xi\geq 0$, $\xi\not\equiv 0$). Recently we have examined Robin and Neumann problems with indefinite linear part. We mention the works of Papageorgiou and Rădulescu [13, 14, 16]. In [13] the problem is parametric with competing nonlinearities. The concave term is $-\lambda|x|^{q-2}x$, $1<q<2$, $x\in\mathbb{R}$ (so it enters into the equation with a negative sign) while the perturbation $f(z,x)$ is Carathéodory, asymptotically linear near $\pm\infty$ and resonant with respect to the principal eigenvalue. We proved a multiplicity result for all small values of the parameter $\lambda>0$, producing five nontrivial smooth solutions, four of which have constant sign (two positive and two negative).

In this paper, using variational tools together with truncation, perturbation and comparison techniques, we prove a bifurcation-type theorem, describing the existence and multiplicity of positive solutions as the parameter $\lambda>0$ varies. We also establish the existence of a minimal positive solution $\tilde{u}_{\lambda}$ and determine the monotonicity and continuity properties of the map $\lambda\mapsto\tilde{u}_{\lambda}$.

2. Preliminaries

Let $X$ be a Banach space and $X^{*}$ its topological dual. By $\left\langle\cdot,\cdot\right\rangle$ we denote the duality brackets for the dual 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 property is satisfied:

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

[TABLE]

admits a strongly convergent subsequence”.

This is a compactness-type condition on the functional $\varphi(\cdot)$. It leads to a deformation theorem from which one can derive the minimax theory for the critical values of $\varphi$ (see, for example, Gasinski and Papageorgiou [6]). The following notion is central to this theory.

Definition 2.1**.**

Let $Y$ be a Hausdorff topological space and $E_{0},E,D\subseteq Y$ nonempty, closed sets such that $E_{0}\subseteq E$. We say that the pair $\{E_{0},E\}$ is linking with $D$ in $Y$ if:

• (a)

$E_{0}\cap D=\emptyset$;

• (b)

For any $\gamma\in C(E,Y)$ such that $\gamma\left|{}_{E_{0}}={\rm id}\right|_{E_{0}}$, we have $\gamma(E)\cap D\neq\emptyset$.

Using this topological notion, one can prove the following general minimax principle, known in the literature as the “linking theorem” (see, for example, Gasinski and Papageorgiou [6, p. 644]).

Theorem 2.2**.**

Assume that $X$ is a Banach space, $E_{0},E,D\subseteq X$ are nonempty, closed subsets such that $\{E_{0},E\}$ is linking with $D$ in $X$, $\varphi\in C^{1}(X,\mathbb{R})$ satisfies the $C$-condition

[TABLE]

and $c=\inf\limits_{\gamma\in\Gamma}\sup\limits_{u\in E}\varphi(\gamma(u))$, where $\Gamma=\left\{\gamma\in C(E,X):\gamma\left|{}_{E_{0}}={\rm id}\right|_{E_{0}}\right\}$. Then $c\geq\inf\limits_{D}\varphi$ and $c$ is a critical value of $\varphi$ (that is, there exists $u\in X$ such that $\varphi^{\prime}(u)=0,\ \varphi(u)=c$).

With a suitable choice of the linking sets, we can produce as corollaries of Theorem 2.2, the main minimax theorems of the critical point theory. For future use, we recall the so-called “mountain pass theorem”.

Theorem 2.3**.**

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

[TABLE]

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

Remark 1**.**

Theorem 2.3 can be deduced from Theorem 2.2 if we have $E_{0}=\left\{u_{0},u_{1}\right\}$, $E=\left\{u=(1-t)u_{0}+tu_{1},\ 0\leq t\leq 1\right\}$, $D=\partial B_{\rho}(u_{0})=\{u\in X:||u-u_{0}||$ $=\rho\}.$

In the analysis of problem ($P_{\lambda}$), we will use the following spaces: the Sobolev space $H^{1}(\Omega)$, the Banach space $C^{1}(\overline{\Omega})$ and the boundary Lebesgue spaces $L^{r}(\partial\Omega)$, $1\leq r\leq\infty$.

By $||\cdot||$ we denote the norm of the Sobolev space $H^{1}(\Omega)$. So

[TABLE]

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

[TABLE]

We will use the open set $D_{+}\subseteq C_{+}$ defined 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 $L^{r}(\partial\Omega)$ ($1\leq r\leq\infty$) in the usual way. Recall that 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]

This map is not surjective and it is compact into $L^{q}(\partial\Omega)$ if $1\leq q<\frac{2(N-1)}{N-2}\ if\ N\geq 3$ and into $L^{q}(\partial\Omega)\ \mbox{for all}\ q\geq 1\ \mbox{if}\ N=1,2.$

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

Let $f_{0}:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ be a Carathéodory function such that

[TABLE]

with a_{0}\in L^{\infty}(\Omega),1<r<2^{*}=\left\{\begin{array}[]{ll}\frac{2N}{N-2}&\mbox{if}\ N\geq 3\\ +\infty&\mbox{if}\ N=1,2.\end{array}\right.

We set $F_{0}(z,x)=\int^{x}_{0}f_{0}(z,s)ds$. Also, let $\xi\in L^{s}(\Omega)$ $(s>N)$ and $\beta\in W^{1,\infty}(\partial\Omega)$ with $\beta(z)\geq 0$ on $\partial\Omega$. We consider the $C^{\prime}$-functional $\varphi_{0}:H^{1}(\Omega)\rightarrow\mathbb{R}$ defined by

[TABLE]

where

[TABLE]

The next result follows from Papageorgiou and Rădulescu [12, Proposition 3] using the regularity theory of Wang [20].

Proposition 1**.**

Let $u_{0}\in H^{1}(\Omega)$ be a local $C^{1}(\overline{\Omega})$-minimizer of $\varphi_{0}$, that is, there exists $\rho_{0}>0$ such that

[TABLE]

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

[TABLE]

We will need some facts concerning the spectrum of $-\Delta$ with Robin boundary condition. Details can be found in Papageorgiou and Rădulescu [12, 16].

So, we consider the following linear eigenvalue problem

[TABLE]

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

[TABLE]

Using (2) and the spectral theorem for compact self-adjoint operators, we generate the spectrum of (1), which consists of a strictly increasing sequence $\{\hat{\lambda}_{k}\}_{k\geq 1}\subseteq\mathbb{R}$ such that $\hat{\lambda}_{k}\rightarrow+\infty$. Also, there is a corresponding sequence $\left\{\hat{u}_{n}\right\}_{n\geq 1}\subseteq H^{1}(\Omega)$ of eigenfunctions which form an orthonormal basis of $H^{1}(\Omega)$ and an orthogonal basis of $L^{2}(\Omega)$. In fact, the regularity theory of Wang [20] implies that $\left\{\hat{u}_{n}\right\}_{n\geq 1}\subseteq C^{1}(\overline{\Omega})$. By $E(\hat{\lambda}_{k})$ (for every $k\in\mathbb{N}$) we denote the eigenspace corresponding to the eigenvalue $\hat{\lambda}_{k},k\in\mathbb{N}$. We have the following orthogonal direct sum decomposition

[TABLE]

Each eigenspace $E(\hat{\lambda}_{k})$ has the so-called “unique continuation property” (UCP for short) which says that if $u\in E(\hat{\lambda}_{k})$ vanishes on a set of positive Lebesgue measure, then $u=0$. The eigenvalues $\{\hat{\lambda}_{k}\}_{k\geq 1}$ have the following properties:

[TABLE]

[TABLE]

In (2) the infimum is realized on $E(\hat{\lambda}_{1})$.

In (2) both the supremum and the infimum are realized on $E(\hat{\lambda}_{m})$.

From these properties, it is clear that the elements of $E(\hat{\lambda}_{1})$ have constant sign while for $m\geq 2$ the elements of $E(\hat{\lambda}_{m})$ are nodal (that is, sign changing). Let $\hat{u}_{1}$ denote the $L^{2}$-normalized (that is, $||\hat{u}_{1}||_{2}=1$) positive eigenfunction corresponding to $\hat{\lambda}_{1}$. As we have already mentioned, $\hat{u}_{1}\in C_{+}\backslash\left\{0\right\}$. Using Harnack’s inequality (see, for example Motreanu, Motreanu and Papageorgiou [11, p. 212]), we have that $\hat{u}_{1}(z)>0$ for all $z\in\Omega$. Moreover, if $\xi^{+}\in L^{\infty}(\Omega)$, then using the strong maximum principle, we have $\hat{u}_{1}\in D_{+}$.

The following useful inequalities are also easy consequences of the above properties.

Proposition 2**.**

• (a)

If $m\in\mathbb{N},\ \eta\in L^{\infty}(\Omega),\ \eta(z)\leq\hat{\lambda}_{m}\ \mbox{for almost all}\ z\in\Omega,\ \eta\not\equiv\hat{\lambda}_{m},$ then $\vartheta(u)-\int_{\Omega}\eta(z)u^{2}dz\geq c_{1}||u||^{2}\ \mbox{for all}\ u\in\overline{\mathop{\oplus}\limits_{k\geq m}E(\hat{\lambda}_{k})}\ \mbox{and for some}\ c_{1}>0$.

• (b)

If $m\in\mathbb{N},\ \eta\in L^{\infty}(\Omega),\ \eta(z)\geq\hat{\lambda}_{m}\ \mbox{for almost all}\ z\in\Omega,\ \eta\not\equiv\hat{\lambda}_{m},$ then $\vartheta(u)-\int_{\Omega}\eta(z)u^{2}dz\leq-c_{2}||u||^{2}\ \mbox{for all}\ u\in\mathrel{\mathop{\oplus}^{m}_{k=1}}E(\hat{\lambda}_{k})\ \mbox{and for some}\ c_{2}>0$.

Finally, let us fix some basic notations and terminology. So, by

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

[TABLE]

A Banach space $X$ is said to have the “Kadec-Klee property” if the following holds

[TABLE]

Locally uniformly convex Banach spaces, in particular Hilbert spaces, have the Kadec-Klee property.

Let $x\in\mathbb{R}$. We set $x^{\pm}=\max\left\{\pm x,0\right\}$ and for $u\in H^{1}(\Omega)$ we define

[TABLE]

We know that

[TABLE]

By $|\cdot|_{N}$ we denote the Lebesgue measure on $\mathbb{R}^{N}$. Also, if $\varphi\in C^{1}(X,\mathbb{R})$ then

[TABLE]

If $p\in\left[1,\infty\right)$, then $p^{\prime}\in(1,+\infty]$ and $\frac{1}{p}+\frac{1}{p^{\prime}}=1$. Finally, we set

[TABLE]

If $\hat{\lambda}_{k}>0$ for all $k\in\mathbb{N}$ (this is the case if $\xi\geq 0$ and $\xi\not\equiv 0$ or $\beta\not\equiv 0$), then we set $n_{0}=0$.

3. Positive solutions

The hypotheses on the data of problem ($P_{\lambda}$) are the following:

$H(\xi):\ \xi\in L^{s}(\Omega)\ \mbox{with}\ s>N\ \mbox{and}\ \xi^{+}\in L^{\infty}(\Omega)$.

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

$H(g):\ g:\Omega\times\mathbb{R}\rightarrow\mathbb{R}\ \mbox{is a Carath\'{e}odory function such that}$

• (i)

for every $\rho>0$, there exists $a_{\rho}\in L^{\infty}(\Omega)_{+}$ such that

[TABLE]

• (ii)

$\lim\limits_{x\rightarrow+\infty}\frac{g(z,x)}{x}=0$ uniformly for almost all $z\in\Omega$;

• (iii)

there exist constants $0<c_{3}<c_{4}$ and $q\in(1,2)$ such that

[TABLE]

[TABLE]

• (iv)

if $G(z,x)=\int^{x}_{0}g(z,s)ds$, then $g(z,x)x-2G(z,x)\leq 0$ for almost all $z\in\Omega$ and all $x\geq 0$;

• (v)

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

[TABLE]

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

$H(f):\ f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a Carathéodory function such that

• (i)

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

• (ii)

$\lim\limits_{x\rightarrow+\infty}\frac{f(z,x)}{x}=+\infty$ uniformly for almost all $z\in\Omega$;

• (iii)

$\lim\limits_{x\rightarrow 0}\frac{f(z,x)}{x}=0$ uniformly for almost all $z\in\Omega$ and there exists $\delta_{0}>0$ such that

[TABLE]

• (iv)

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

[TABLE]

is nondecreasing on $[0,\rho]$

We set $F(z,x)=\int^{x}_{0}f(z,s)ds$ and define

[TABLE]

$H_{0}:$ For every $\lambda>0$, there exists $e_{\lambda}\in L^{1}(\Omega)$ such that

[TABLE]

Remark 2**.**

Since we are looking for positive solutions and all of the above hypotheses concern the positive semi-axis $\mathbb{R}_{+}=[0,+\infty)$, we may assume without any loss of generality that

[TABLE]

(note that hypotheses $H(g)(iii)$ and $H(f)(iii)$ imply that $g(z,0)=f(z,0)=0$ for almost all $z\in\Omega$). Hypothesis $H(g)(ii)$ implies that for almost all $z\in\Omega,\ g(z,\cdot)$ is strictly sublinear near $+\infty$. This, together with hypothesis $H(g)(iii)$, implies that $g(z,\cdot)$ is globally the “concave” contribution to the reaction of problem ($P_{\lambda}$). On the other hand, hypothesis $H(f)(ii)$ implies that for almost all $z\in\Omega,\ f(z,\cdot)$ is strictly superlinear near $+\infty$. Hence $f(z,x)$ is globally the “convex” contribution to the reaction of ($P_{\lambda}$). Therefore on the right-hand side (reaction) of problem ($P_{\lambda}$), we have the competition of concave and convex nonlinearities (“concave-convex problem”). We stress that the superlinearity of $f(z,\cdot)$ is not expressed using the well-known Ambrosetti-Rabinowitz condition (see Ambrosetti and Rabinowitz [3]). Instead, we use hypothesis $H_{0}$, which is a slightly more general version of a condition used by Li and Yang [10]. Hypothesis $H_{0}$ is less restrictive than the Ambrosetti-Rabinowitz superlinearity condition and permits the consideration of superlinear terms with “slower” growth near $+\infty$, which fail to satisfy the AR-condition (see the examples below). Hypothesis $H_{0}$ is a quasimonotonicity condition on $\gamma_{\lambda}(z,\cdot)$ and it is satisfied if there exists $M=M(\lambda)>0$ such that for almost all $z\in\mathbb{R}$,

[TABLE]

is nondecreasing on $[M,+\infty)$ (see [10]).

Examples. The following pair satisfies hypotheses $H(g)$ and $H(f)$:

[TABLE]

with $a,b\in L^{\infty}(\Omega),\ a(z),b(z)>0$ for almost all $z\in\Omega$ and $1<q<2<r<2^{*}$. If $a\equiv b\equiv 1$, this is the reaction pair used by Ambrosetti, Brezis and Cerami [2] in the context of Dirichlet problems with zero potential (that is, $\xi\equiv 0$). The above reaction pair was used by Rădulescu and Repovš [19], again for Dirichlet problems with $\xi\equiv 0$.

Another possibility of a reaction pair which satisfies hypotheses $H(g)$ and $H(f)$ are the following functions (for the sake of simplicity, we drop the $z$-dependence):

[TABLE]

In this pair, the superlinear term $f(x)$ fails to satisfy the Ambrosetti-Rabinowitz condition.

Let $\mu>0$ be as in (2) and $\lambda>0$. Let $k_{\lambda}:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ be the Carathéodory function defined by

[TABLE]

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

[TABLE]

Proposition 3**.**

If hypotheses $H(\xi),\ H(\beta),\ H(g),\ H(f)$ and $H_{0}$ hold, then for every $\lambda>0$ the functional $\hat{\varphi}_{\lambda}$ satisfies the C-condition.

Proof.

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

[TABLE]

By (8) we have

[TABLE]

for all $h\in H^{1}(\Omega)$ with $\epsilon_{n}\rightarrow 0^{+}$.

In (9) we choose $h=-u^{-}_{n}\in H^{1}(\Omega)$. Then

[TABLE]

It follows from (7) and (10) that

[TABLE]

If in (9) we choose $h=u^{+}_{n}\in H^{1}(\Omega)$, then

[TABLE]

Adding (11) and (12), we obtain

[TABLE]

Claim. $\left\{u^{+}_{n}\right\}_{n\geq 1}\subseteq H^{1}(\Omega)$* is bounded.*

We argue by contradiction. So, suppose that the claim is not true. By passing to a subsequence if necessary, we may assume that $||u^{+}_{n}||\rightarrow\infty$. Let $y_{n}=\frac{u^{+}_{n}}{||u^{+}_{n}||}$, $n\in\mathbb{N}$. Then

[TABLE]

and so we may assume that

[TABLE]

Suppose that $y\neq 0$ and let $\Omega^{*}=\left\{z\in\Omega:y(z)>0\right\}$. Then $|\Omega^{*}|_{N}>0$ and

[TABLE]

We have

[TABLE]

It follows from (15), (16) and Fatou’s lemma that

[TABLE]

On the other hand, (7) and (10) imply that

[TABLE]

for some $M_{4}$, all $n\in\mathbb{N}$.

Comparing (17) and (18) we obtain a contradiction.

Next, suppose that $y\equiv 0$. For $\eta>0$ we set $\hat{y}_{n}=(2\eta)^{\frac{1}{2}}y_{n},\ n\in\mathbb{N}$. Then $\hat{y}_{n}\rightarrow 0$ in $H^{1}(\Omega)$ and so we have

[TABLE]

Since $||u^{+}_{n}||\rightarrow\infty$, we can find $n_{1}\in\mathbb{N}$ such that

[TABLE]

We choose $t_{n}\in[0,1]$ such that

[TABLE]

From (20), (21) we have

[TABLE]

Since $\eta>0$ is arbitrary, we infer from (3) that

[TABLE]

We know that

[TABLE]

So, (21) implies that

[TABLE]

We have $0\leq t_{n}u^{+}_{n}\leq u^{+}_{n}$. Then hypothesis $H_{0}$ implies that

[TABLE]

We return to (24), add to both sides $-2\int_{\Omega}[\lambda G(z,t_{n}u^{+}_{n})+F(z,t_{n}u^{+}_{n})]dz$ and use (3).

Then

[TABLE]

Comparing (23) and (26) again, we get a contradiction.

This proves the claim.

Then (10) and the claim imply that $\left\{u_{n}\right\}_{n\geq 1}\subseteq H^{1}(\Omega)$ is bounded. So, we may assume that

[TABLE]

In (9) we choose $h=u_{n}-u\in H^{1}(\Omega)$, pass to the limit as $n\rightarrow\infty$ and use (27).

Then

[TABLE]

∎

Let $\mathcal{L}=\{\lambda>0:\mbox{problem \eqref{eqp} admits a positive solution}\}$.

[TABLE]

Proposition 4**.**

If hypotheses $H(\xi),H(\beta),H(g),H(f)$ and $H_{0}$ hold, then $\mathcal{L}\neq\emptyset$ and when $\lambda\in\mathcal{L}$ we have $\left(0,\lambda\right]\subseteq\mathcal{L}$ and for all $\lambda>0$, $S^{\lambda}_{+}\subseteq D_{+}.$

Proof.

Hypotheses $H(g)(i)\rightarrow(iii)$ imply that given $\epsilon>0$, we can find $c_{5}=c_{5}(\epsilon)>0$ such that

[TABLE]

Similarly, hypotheses $H(f)(i),(iii)$ imply that given $\epsilon>0$, we can find $c_{6}=c_{6}(\epsilon)>0$ such that

[TABLE]

We set

[TABLE]

We have

[TABLE]

Recall that $n_{0}=\max\{k\in\mathbb{N}:\hat{\lambda}_{k}\leq 0\}$. We set $n_{0}=0$ if $\hat{\lambda}_{k}>0$ for all $k\in\mathbb{N}$ and this is the case if $\xi\geq 0$ and $\xi\not\equiv 0$ or $\beta\not\equiv 0$. Then $\bar{H}_{n_{0}}=\{0\}$ and $\hat{H}_{n_{0}+1}=H^{1}(\Omega)$.

Let $u\in\hat{H}_{n_{0}+1}$. Then

[TABLE]

see (28) and (29).

Since $u\in\hat{H}_{n_{0}+1}$, from Proposition 2(a) and by choosing $\epsilon\in(0,1)$ small enough, we have

[TABLE]

We use (31) in (3). Then

[TABLE]

Let $\Im_{\lambda}(t)=\lambda t^{q-2}+t^{r-2},\ t>0$. We have $1<q<2<r$. Hence

[TABLE]

So, we can find $t_{0}>0$ such that

[TABLE]

Then we have

[TABLE]

Therefore, we can find $\lambda_{0}>0$ such that

[TABLE]

Returning to (32), we deduce that there exists a positive number $m_{\lambda}$ such that

[TABLE]

On the other hand, hypotheses $H(f)(i),(ii)$ imply that if $\tau>\mu+\hat{\lambda}_{n_{0}+1}$, then we can find $c_{9}=c_{9}(\tau)>0$ such that

[TABLE]

Let $w_{0}\in E(\hat{\lambda}_{n_{0}+1})$ with $||w_{0}||=1$. We consider the space

[TABLE]

This is a finite dimensional subspace of $H^{1}(\Omega)$ and if $u\in Y$, then we can write $u$ in a unique way as

[TABLE]

Exploiting the orthogonality of the component spaces and since $G\geq 0$ (see hypothesis $H(g)(iii)$), we have

[TABLE]

Note that

[TABLE]

Returning to (3) and using (37), (38), we obtain

[TABLE]

Since $Y$ is finite dimensional, all norms are equivalent. So, by (3) we have

[TABLE]

But $r>2$. So, it follows from (40) that we can find $\rho\in(0,1)$ small such that

[TABLE]

By (33) and (34) we see that

[TABLE]

Therefore we can find $\lambda_{1}\leq\lambda_{0}$ such that

[TABLE]

We consider the following sets:

[TABLE]

From Gasinski and Papageorgiou [6, p. 643] we know that

[TABLE]

(see Definition 2.1 and recall that $1>\rho>\hat{\rho}_{\lambda}$).

By Proposition 3 we know that for all $\lambda>0$

[TABLE]

On account of (34), (41), (42), (43), we can apply Definition 2.1 (the linking theorem) and find $u\in H^{1}(\Omega)$ such that

[TABLE]

where $m_{\lambda}$ is the same as in relation (34).

It follows from (44) that $u_{0}\neq 0$ and

[TABLE]

In (45) we choose $h=-u^{-}_{0}\in H^{1}(\Omega)$. Then

[TABLE]

So, equation (45) becomes

[TABLE]

(see Papageorgiou and Rădulescu [12]).

We set

[TABLE]

Hypotheses $H(g)(i),(ii)$ and $H(f)(i)$ imply that

[TABLE]

Then we have

[TABLE]

(note that $(r-2)\frac{N}{2}<\left(\frac{2N}{N-2}-2\right)\frac{N}{2}=\frac{2N}{N-2}=2^{*}$ if $N\geq 3$).

We rewrite (46) as

[TABLE]

Using Lemma 5.1 of Wang [20] we have

[TABLE]

Then the Calderon-Zygmund estimates (see Wang [20, Lemma 5.2]) imply that

[TABLE]

Let $\rho=||u_{0}||_{\infty}$. On account of hypotheses $H(g)(v)$ and $H(f)(iv)$, we can find $\bar{\xi}_{\rho}>0$ such that for almost all $z\in\Omega$, $x\mapsto\lambda g(z,x)+f(z,x)+\bar{\xi}_{\rho}x$ is nondecreasing on $[0,\rho]$. Then from (46) we have

[TABLE]

Therefore we have proved that for $\lambda>0$ small enough, we have

[TABLE]

Next, let $\lambda\in\mathcal{L}$ and pick $\tau\in(0,\lambda)$. Since $\lambda\in\mathcal{L}$, we can find $u_{\lambda}\in S^{\lambda}_{+}\subseteq D_{+}$ (see (48)). We have

[TABLE]

We consider the following truncation of the Carathéodory map $k_{\tau}(z,\cdot)$ (see (6))

[TABLE]

We set $\hat{K}_{\tau}(z,x)=\int^{x}_{0}k_{\tau}(z,s)ds$ and consider the $C^{1}$-functional $\hat{\psi}_{\tau}:H^{1}(\Omega)\rightarrow\mathbb{R}$ defined by

[TABLE]

By (50) and (2) it is clear that $\hat{\psi}_{\tau}(\cdot)$ is coercive. In addition, the Sobolev embedding theorem and the compactness of the trace map, imply that $\hat{\psi}_{\tau}(\cdot)$ is sequentially weakly lower semicontinuous. So, by the Weierstrass theorem, we can find $u_{\tau}\in H^{1}(\Omega)$ such that

[TABLE]

With $\delta_{0}>0$ as in hypothesis $H(f)(iii)$, we define

[TABLE]

For $u\in D_{+}$, choose $t\in(0,1)$ so small that

[TABLE]

Then using hypothesis $H(f)(iii)$, we have

[TABLE]

Recall that $q<2$. Then from (3) and by choosing $t\in(0,1)$ even smaller if necessary, we infer that

[TABLE]

By (51) we have

[TABLE]

In (3) we choose $h=-u^{-}_{\tau}\in H^{1}(\Omega)$. Then

[TABLE]

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

[TABLE]

(see (3) and use Green’s identity, see Gasinski and Papageorgiou [6, p. 210]),

[TABLE]

So, we have proved that

[TABLE]

∎

An interesting byproduct of the above proof is the following corollary.

Corollary 1**.**

If hypotheses $H(\xi),H(\beta),H(g),H(f),H_{0}$ hold, $\tau,\lambda\in\mathcal{L}$ with $0<\tau<\lambda$ and $u_{\lambda}\in S^{\lambda}_{+}$, then we can find $u_{\tau}\in S^{\tau}_{+}$ such that $u_{\lambda}-u_{\tau}\in D_{+}$.

Proof.

An inspection of the last part of the proof of Proposition 4 reveals that we can find $u_{\tau}\in S^{\tau}_{+}$ such that

[TABLE]

Let $\rho=||u_{\lambda}||_{\infty}$ and let $\bar{\xi}_{\rho}>0$ be such that for almost all $z\in\Omega$ the function

[TABLE]

is nondecreasing (see hypotheses $H(g)(v),H(f)(iv)$). We have

[TABLE]

∎

Let $\lambda^{*}=\sup\mathcal{L}$.

Proposition 5**.**

If hypotheses $H(\xi),H(\beta),H(g),H(f)$ and $H_{0}$ hold, then $\lambda^{*}<\infty$.

Proof.

Hypotheses $H(g)(iii)$ and $H(f)(ii),(iii)$ imply that we can find $\hat{\lambda}>0$ so big that

[TABLE]

Let $\lambda>\hat{\lambda}$ and assume that $\lambda\in\mathcal{L}$. Then according to Proposition 4 we can find $u\in S^{\lambda}_{+}\subseteq D_{+}$. Then there exists $\eta>0$ such that $\eta\hat{u}_{1}\leq u$. We choose the biggest such $\eta>0$. We have

[TABLE]

But this contradicts the maximality of $\eta>0$. So $\lambda\not\in\mathcal{L}$ and we have

[TABLE]

∎

Proposition 6**.**

If hypotheses $H(\xi),H(\beta),H(g),H(f),H_{0}$ hold and $\lambda\in(0,\lambda^{*})$, then problem ($P_{\lambda}$) admits at least two positive solutions

[TABLE]

Proof.

Let $\nu\in(\lambda,\lambda^{*})$ and let $u_{\nu}\in S^{\nu}_{+}\subseteq D_{+}$ (see Proposition 4). Then

[TABLE]

Let $\hat{k}_{\lambda}(z,x)$ be the Carathéodory function defined in (50), with $\tau$ replaced by $\lambda$ and $u_{\lambda}$ replaced by $u_{\nu}$. We set $\hat{K}_{\lambda}(z,x)=\int^{x}_{0}\hat{k}_{\lambda}(z,x)ds$ and consider the $C^{1}$-functional $\hat{\psi}_{\lambda}:H^{1}(\Omega)\rightarrow\mathbb{R}$ defined by

[TABLE]

As in the proof of Proposition 4, via the Weierstrass theorem, we can find $u_{\lambda}\in H^{1}(\Omega)$ such that

[TABLE]

Using this positive solution, we introduce the following truncation of $k_{\lambda}(z,\cdot)$ (see (6))

[TABLE]

This is a Carathéodory function. We set $K^{*}_{\lambda}(z,x)=\int^{x}_{0}k^{*}_{\lambda}(z,s)ds$ and consider the $C^{1}$-functional $\psi^{*}_{\lambda}:H^{1}(\Omega)\rightarrow\mathbb{R}$ defined by

[TABLE]

As before, using (57) we can verify that

[TABLE]

On account of (58) we see that we may assume that

[TABLE]

Indeed, if (59) is not true, then we have $\hat{u}_{\lambda}\in K_{\psi^{*}_{\lambda}}\cap[0,u_{\nu}]$, $\hat{u}_{\lambda}-u_{\lambda}\in C_{+}\backslash\{0\}$, which is a second positive solution of ($P_{\lambda}$) (see (57), (58)). Moreover, as before, using hypotheses $H(g)(v),H(f)(iv)$ and the strong maximum principle, we have $\hat{u}_{\lambda}-u_{\lambda}\in D_{+}$ and so we are done.

We introduce the following truncation of $k^{*}_{\lambda}(z,\cdot)$:

[TABLE]

This is a Carathéodory function. We set $\hat{K}^{*}_{\lambda}(z,x)=\int^{x}_{0}\hat{k}^{*}_{\lambda}(z,s)ds$ and consider the $C^{1}$-functional $\hat{\psi}^{*}_{\lambda}:H^{1}(\Omega)\rightarrow\mathbb{R}$ defined by

[TABLE]

As in the proof of Proposition 4 we see that

[TABLE]

By (2) and (60) it is clear that $\hat{\psi}^{*}_{\lambda}$ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find $u^{*}_{\lambda}\in H^{1}(\Omega)$ such that

[TABLE]

Note that $\left.(\hat{\psi}^{*}_{\lambda})^{\prime}\right|_{[0,u_{\nu}]}=\left.(\psi^{*}_{\lambda})^{\prime}\right|_{[0,u_{\nu}]}$ (see (57), (60)). Therefore

[TABLE]

Moreover, reasoning as in the proof of Corollary 1, we show that

[TABLE]

We can assume that $K_{\psi^{*}_{\lambda}}$ is finite (otherwise on account of (58) we see that we already have an infinity of positive smooth solutions strictly bigger than $u_{\lambda}$).

Since $K_{\psi^{*}_{\lambda}}$ is finite, we can find $\rho\in(0,1)$ small such that

[TABLE]

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

Due to hypothesis $H(f)(ii)$ and since $G\geq 0$, we have

[TABLE]

Since $k^{*}_{\lambda}(z,\cdot)$ and $k_{\lambda}(z,\cdot)$ coincide on $\left[u_{\lambda}(z)\right)=\{x\in\mathbb{R}_{+}:u_{\lambda}(z)\leq x\}$, we infer that

[TABLE]

(see the proof of Proposition 3).

Then (62), (63), (64) permit the use of Theorem 2.3 (the mountain pass theorem). So, there is $\hat{u}_{\lambda}\in H^{1}(\Omega)$ such that

[TABLE]

Moreover, as in the proof of Corollary 1, using hypotheses $H(g)(v)$ and $H(f)(iv)$ and the strong maximum principle, we obtain

[TABLE]

∎

Proposition 7**.**

If hypotheses $H(\xi),H(\beta),H(g),H(f)$ and $H_{0}$ hold, then $\lambda^{*}\in\mathcal{L}$.

Proof.

Let $\{\lambda_{n}\}_{n\geq 1}\subseteq(0,\lambda^{*})$ such that $\lambda_{n}\uparrow\lambda^{*}$. As in the second half of the proof of Proposition 4 (see the part of that proof after (48)), we can find $\{u_{n}\}_{n\geq 1}\subseteq D_{+}$ such that

[TABLE]

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

[TABLE]

By (66) we have

[TABLE]

It follows from (67) and (68) that

[TABLE]

Then reasoning as in the proof of Proposition 3 (see the claim) and applying (69), we show that $\{u_{n}\}_{n\geq 1}\subseteq H^{1}(\Omega)$ is bounded. So, 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 (70). Then

[TABLE]

So, if in (65) we pass to the limit as $n\rightarrow\infty$ and use (71), we infer that

[TABLE]

If we show that $u^{*}\neq 0$, then we are finished. To this end, note that we can find $c_{15}>0$ such that

[TABLE]

(see hypothesis $H(g)(iii)$ and hypotheses $H(f)(i),(ii),(iii)$). Let $\lambda=\lambda_{1}\leq\lambda_{n}$ for all $n\in\mathbb{N}$ and consider the following auxiliary Robin problem

[TABLE]

Let $d:H^{1}(\Omega)\rightarrow\mathbb{R}$ be the $C^{1}$-functional defined by

[TABLE]

Using (2) and the fact that $q<2$, we infer that $d(\cdot)$ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find $\bar{u}\in H^{1}(\Omega)$ such that

[TABLE]

Since $q<2$, for $t\in(0,1)$ small enough, we have

[TABLE]

By (74) we have

[TABLE]

In (75) we choose $h=-\bar{u}^{-}\in H^{1}(\Omega)$. Then

[TABLE]

Then (75) becomes

[TABLE]

Moreover, using the regularity results of Wang [20] and the strong maximum principle, we have

[TABLE]

Recall that $u_{n}\in D_{+}$ for all $n\in\mathbb{N}$. So, we can find $\eta_{n}>0$ such that $\eta_{n}\bar{u}\leq u_{n}$. We choose $\eta_{n}$ to be the biggest such positive real and suppose that $\eta_{n}\in(0,1)$. Also, let $\xi^{*}_{n}>c_{15}>0$. Then

[TABLE]

Evidently, $u_{n}\neq\eta_{n}\bar{u}$. So, from (3) and the strong maximum principle, we infer that

[TABLE]

which contradicts the maximality of $\eta_{n}$. Hence $\eta_{n}\geq 1$ and so

[TABLE]

∎

This proposition implies that

[TABLE]

4. Extremal positive solutions - bifurcation theorem

In this section, we first show that for every $\lambda\in(0,\lambda^{*})$ problem ($P_{\lambda}$) has a smallest positive solution $\tilde{u}_{\lambda}\in D_{+}$ and determine the monotonicity and continuity properties of the map $\lambda\mapsto\tilde{u}_{\lambda}.$

Proposition 8**.**

If hypotheses $H(\xi),H(\beta),H(g),H(f)$ and $H_{0}$ hold, then for every $\lambda\in(0,\lambda^{*})$, problem ($P_{\lambda}$) has a smallest positive solution $\tilde{u}_{\lambda}\in D_{+}$ and the map $\lambda\mapsto\tilde{u}_{\lambda}$ is strictly increasing in the sense that

[TABLE]

and it is left continuous from $(0,\lambda^{*})$ into $C^{1}(\overline{\Omega})$.

Proof.

As in Filippakis and Papageorgiou [5, Lemma 4.1], we 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 $u\leq u_{1},u\leq u_{2}$). Invoking Lemma 3.10 of Hu and Papageorgiou [7, p. 178], we can find a decreasing sequence $\{u_{n}\}_{n\geq 1}\subseteq S^{\lambda}_{+}$ such that

[TABLE]

We may assume that

[TABLE]

We have

[TABLE]

Also, by the proof of Proposition 7 and since $\lambda_{1}<\lambda$ (see equation (73)), we have

[TABLE]

If $\tau<\lambda$, then by Corollary 1 we can find $u_{\tau}\in S^{\tau}_{\lambda}$ such that

[TABLE]

Finally, suppose that $\lambda_{n}\rightarrow\lambda^{-}\ (\lambda\in(0,\lambda^{*}))$. From the regularity theory (see Wang [20]), we know that we can find $\alpha\in(0,1)$ and $c_{16}>0$ such that

[TABLE]

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

[TABLE]

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

[TABLE]

which contradicts (79) (recall that $\lambda_{n}\leq\lambda$ for all $n\in\mathbb{N}$). Therefore by the Urysohn criterion, we have for the original sequence

[TABLE]

∎

Summarizing the results of Sections 3 and 4, we can formulate the following bifurcation-type result, describing the behavior of the set of positive solutions of ($P_{\lambda}$) with respect to the parameter $\lambda>0$.

Theorem 4.1**.**

If hypotheses $H(\xi),H(\beta),H(g),H(f)$ and $H_{0}$ hold, then there exists $\lambda^{*}>0$ such that

• (a)

for every $\lambda\in(0,\lambda^{*})$ problem ($P_{\lambda}$) has at least two positive solutions

[TABLE]

it has a smallest positive solution $\tilde{u}_{\lambda}\in D_{+}$ and the map $\lambda\mapsto\tilde{u}_{\lambda}$ from $(0,\lambda^{*})$ into $C^{1}(\overline{\Omega})$ is strictly increasing in the sense that

[TABLE]

and is left continuous;

• (b)

for $\lambda=\lambda^{*}$ problem ($P_{\lambda}$) has at least one positive solution $u^{*}\in D_{+}$;

• (c)

for $\lambda>\lambda^{*}$ problem ($P_{\lambda}$) has no positive solutions.

Acknowledgments

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, project PN-III-P4-ID-PCE-2016-0130. We thank the referee for comments.