Positive solutions for nonlinear parametric singular Dirichlet problems

TL;DR

This paper studies positive solutions for a nonlinear Dirichlet problem involving the p-Laplace operator with a parametric singular term and a Carathéodory perturbation, providing a bifurcation analysis of solutions depending on the parameter.

Contribution

It introduces a bifurcation theorem characterizing how positive solutions vary with the parameter in a singular p-Laplace problem.

Findings

Established existence of positive solutions for the problem.

Described the exact dependence of solutions on the parameter.

Provided a bifurcation-type theorem for the solution set.

Abstract

We consider a nonlinear parametric Dirichlet problem driven by the $p$-Laplace differential operator and a reaction which has the competing effects of a parametric singular term and of a Carath\'eodory perturbation which is ($p-1$)-linear near $+\infty$. The problem is uniformly nonresonant with respect to the principal eigenvalue of $(-\Delta_p,W^{1,p}_0(\Omega))$. We look for positive solutions and prove a bifurcation-type theorem describing in an exact way the dependence of the set of positive solutions on the parameter $\lambda>0$.

Equations

$$P_{\lambda }$$

$${\Delta }_{p}u=\mathrm{div}(|Du{|}^{p-2}Du)\text{mbox}forallu\in W_0^{1,p}(\Omega ),1<p<\infty .$$

$$u_{\lambda },{\hat{u}}_{\lambda }\in \mathrm{int}{C}_{+},u_{\lambda }\ne {\hat{u}}_{\lambda },u_{\lambda }⩽{\hat{u}}_{\lambda };$$

$$u_{\lambda }^{\ast }\in \mathrm{int}{C}_{+};$$

$$\displaystyle\{\varphi(u_{n})\}_{n\geqslant 1}\subseteq\mathbb{R}\ \mbox{is bounded and}\ \mbox{$\displaystyle(1+||u_{n}||)\varphi^{\prime}(u_{n})\rightarrow 0$ in $\displaystyle X^{*}$ as $\displaystyle n\rightarrow\infty$,}$$

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

$$C_0^1(\overline{\Omega })=\{u\in C^1(\overline{\Omega }):u{|}_{\partial \Omega }=0\}.$$

$$||u||=||Du|{|}_p\text{mbox}forallu\in W_0^{1,p}(\Omega ).$$

$$C_{+}=\{u\in C_0^1(\overline{\Omega }):u(z)⩾0\text{mbox}forallz\in \overline{\Omega }\}.$$

$$\mathrm{int}{C}_{+}=\left\{u\in C_{+}:u(z)>0\text{mbox}forallz\in \Omega ,{\frac{\partial u}{\partial n}|}_{\partial \Omega }<0\right\}.$$

$$-{\Delta }_p{u}_1(z)+\hat{\xi }{u}_1(z{)}^{p-1}-\lambda u_1(z{)}^{-\gamma }=h_1(z),$$

$$-{\Delta }_p{u}_2(z)+\hat{\xi }{u}_2(z{)}^{p-1}-\lambda u_2(z{)}^{-\gamma }=h_2(z)\text{mbox}foralmostallz\in \Omega ,$$

$$⟨A(u),h⟩={\int }_{\Omega }|Du{|}^{p-2}(Du,Dh{)}_{{\mathbb{R}}^N}dz\text{mbox}forallu,h\in W_0^{1,p}(\Omega ).$$

$$-{\Delta }_{p}u(z)=\hat{\lambda }|u(z){|}^{p-2}u(z)\text{mbox}in\Omega ,u{|}_{\partial \Omega }=0.$$

$$\bullet {\hat{\lambda }}_1=\inf \left\{\frac{||Du|{|}_p^p}{||u|{|}_p^p}:u\in W_0^{1,p}(\Omega ),u\ne 0\right\}.$$

$$-{\Delta }_{p}u(z)=\tilde{\lambda }m(z)|u(z){|}^{p-2}u(z)\text{mbox}in\Omega ,u{|}_{\partial \Omega }=0.$$

$${\tilde{\lambda }}_1(m)=\inf \left\{\frac{||Du|{|}_p^p}{{\int }_{\Omega }m(z)|u{|}^{p}dz}:u\in W_0^{1,p}(\Omega ),u\ne 0\right\}.$$

$$u^{\pm }\in W_0^{1,p}(\Omega ),|u|=u^{+}+u^{-},u=u^{+}-u^{-}.$$

$$N_g(u)(\cdot )=g(\cdot ,u(\cdot ))\text{mbox}forallu\in W_0^{1,p}(\Omega ).$$

$$[v,u]=\{y\in W_0^{1,p}(\Omega ):v(z)⩽y(z)⩽u(z)\text{mbox}foralmostallz\in \Omega \}.$$

$$|f(z,x)|⩽a_{\rho }(z)\text{mbox}foralmostallz\in \Omega ,\text{mbox}andall0⩽x⩽\rho ;$$

$$w(z)⩾c_0>0\text{mbox}forallz\in \overline{\Omega },{\Delta }_{p}w\in L^{\infty }(\Omega )\text{mbox}with{\Delta }_{p}w(z)⩽0\text{mbox}foralmostallz\in \Omega ,$$

$$w(z{)}^{-\gamma }+f(z,w(z))⩽-c_K<0\text{mbox}foralmostallz\in K;$$

$$f(z,x)⩾{\hat{c}}_K>0\text{mbox}foralmostallz\in K,\text{mbox}andallx\in \left(0,{\delta }_0\right];$$

$$x\mapsto f(z,x)+{\hat{\xi }}_{\rho }{x}^{p-1}$$

$$f(z,x)=0\text{mbox}foralmostallz\in \Omega ,\text{mbox}andallx⩽0.$$

$$\displaystyle f(x)=\left\{\begin{array}[]{ll}x^{\tau-1}-3x^{\vartheta-1}&\mbox{if}\ 0\leqslant x\leqslant 1\\ \eta x^{p-1}-(\eta+2)x^{q-1}&\mbox{if}\ 1<x\end{array}\right\}\ (\mbox{see (\ref{eq4})})$$

$$\displaystyle f(x)=\left\{\begin{array}[]{ll}2\sin(2\pi x)&\mbox{if}\ 0\leqslant x\leqslant 1\\ \eta(x^{p-1}-x^{q-1})&\mbox{if}\ 1<x\end{array}\right.$$

$$A{u}_{\lambda }$$

$${\hat{u}}_1⩽c_1{\tilde{u}}_{\vartheta }^{p^{\prime }},$$

$$g_{\lambda}(z,x)=\left\{\begin{array}[]{ll}\lambda\tilde{u}_{\vartheta}^{-\gamma}&\mbox{if}\ x\leqslant\tilde{u}_{\vartheta}(z)\\ \lambda x^{-\gamma}&\mbox{if}\ \tilde{u}_{\vartheta}(z)<x.\end{array}\right.$$

Full text

aa 2010 AMS Subject Classification: 35J92, 35P30

1. Introduction

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

[TABLE]

In this problem, $\displaystyle\Delta_{p}$ denotes the $\displaystyle p$-Laplacian differential operator defined by

[TABLE]

On the right-hand side of ($\displaystyle P_{\lambda}$) (the reaction of the problem), we have a parametric singular term $\displaystyle u\mapsto\lambda u^{-\gamma}$ with $\displaystyle\lambda>0$ being the parameter and $\displaystyle 0<\gamma<1$. Also, there is a Carathéodory perturbation $\displaystyle f(z,x)$ (that is, for all $\displaystyle x\in\mathbb{R}$ the mapping $\displaystyle z\mapsto f(z,x)$ is measurable and for almost all $\displaystyle z\in\Omega$ the mapping $\displaystyle x\mapsto f(z,x)$ is continuous). We assume that $\displaystyle f(z,\cdot)$ exhibits $\displaystyle(p-1)$-linear growth near $\displaystyle+\infty$.

We are looking for positive solutions of problem ($\displaystyle P_{\lambda}$). Our aim is to describe in a precise way the dependence on the parameter $\displaystyle\lambda>0$ of the set of positive solutions.

We prove a bifurcation-type property, which is the main result of our paper. Concerning the hypotheses $\displaystyle H(f)$ on the perturbation $\displaystyle f(z,x)$ and the other notation used in the statement of the theorem, we refer to Section 2. The main result of the present paper is stated in the following theorem.

Theorem A. If hypotheses $\displaystyle H(f)$ hold, then there exists $\displaystyle\lambda^{*}\in(0,+\infty)$ such that

• (a)

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

[TABLE]

• (b)

for $\displaystyle\lambda=\lambda^{*}$, problem ($\displaystyle P_{\lambda}$) has at least one positive solution

[TABLE]

• (c)

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

In the past, singular problems were studied in the context of semilinear equations (that is, $\displaystyle p=2$). We mention the works of Coclite & Palmieri [2], Ghergu & Rădulescu [5], Hirano, Saccon & Shioji [10], Lair & Shaker [11], Sun, Wu & Long [21]. A detailed bibliography and additional topics on the subject, can be found in the book of Ghergu & Rădulescu [6]. For nonlinear equations driven by the $\displaystyle p$-Laplacian, we mention the works of Giacomoni, Schindler & Takač [7], Papageorgiou, Rădulescu & Repovš [16, 17], Papageorgiou & Smyrlis [18], Perera & Zhang [19]. Of the aforementioned papers, closest to our work here is that of Papageorgiou & Smyrlis [18], where the authors also deal with a parametric singular problem and prove a bifurcation-type result. In their problem, the perturbation $\displaystyle f(z,x)$ is ($\displaystyle p-1$)-superlinear in $\displaystyle x\in\mathbb{R}$ near $\displaystyle+\infty$. So, our present work complements the results of [18], by considering equations in which the reaction has the competing effects of a singular term and of a $\displaystyle(p-1)$-linear term.

Our approach uses variational tools together with suitable truncation and comparison techniques.

2. Preliminaries and hypotheses

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

“Every sequence $\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq X$ such that

[TABLE]

admits a strongly convergent subsequence.”

Using this notion, we can state the “mountain pass theorem”.

Theorem 1**.**

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

[TABLE]

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

The analysis of problem ($\displaystyle P_{\lambda}$) will involve the Sobolev space $\displaystyle W^{1,p}_{0}(\Omega)$ and the Banach space

[TABLE]

We denote by $\displaystyle||\cdot||$ the norm of $\displaystyle W^{1,p}_{0}(\Omega)$. On account of the Poincaré inequality, we have

[TABLE]

The space $\displaystyle C^{1}_{0}(\overline{\Omega})$ is an ordered Banach space with positive (order) cone

[TABLE]

This cone has a nonempty interior given by

[TABLE]

Here, $\displaystyle n(\cdot)$ denotes the outward unit normal on $\displaystyle\partial\Omega$.

Let $\displaystyle h_{1},h_{2}\in L^{\infty}(\Omega)$. We write $\displaystyle h_{1}\prec h_{2}$, if for every compact $\displaystyle K\subseteq\Omega$, we can find $\displaystyle c_{K}>0$ such that $\displaystyle c_{K}\leqslant h_{2}(z)-h_{1}(z)$ for almost all $\displaystyle z\in K$. Note that, if $\displaystyle h_{1},h_{2}\in C(\Omega)$ and $\displaystyle h_{1}(z)<h_{2}(z)$ for all $\displaystyle z\in\Omega$, then $\displaystyle h_{1}\prec h_{2}$.

The next strong comparison principle can be found in Papageorgiou & Smyrlis [18, Proposition 4] (see also Giacomoni, Schindler & Takač [7, Theorem 2.3]).

Proposition 2**.**

If $\displaystyle\hat{\xi}\geqslant 0,h_{1},h_{2}\in L^{\infty}(\Omega)$, $\displaystyle h_{1}\prec h_{2},u_{1}\in C_{+}$ with $\displaystyle u_{1}(z)>0$ for all $\displaystyle z\in\Omega$, $\displaystyle u_{2}\in{\rm int}\,C_{+}$ and

[TABLE]

then $\displaystyle u_{2}-u_{1}\in{\rm int}\,C_{+}.$

We denote by $\displaystyle A:W^{1,p}_{0}(\Omega)\rightarrow W^{-1,p^{\prime}}(\Omega)=W^{1,p}_{0}(\Omega)^{*}\left(\frac{1}{p}+\frac{1}{p^{\prime}}=1\right)$ the nonlinear map defined by

[TABLE]

This map has the following properties (see Motreanu, Motreanu & Papageorgiou [15, p. 40]).

Proposition 3**.**

The map $\displaystyle A:W^{1,p}_{0}(\Omega)\rightarrow W^{-1,p^{\prime}}(\Omega)$ is bounded (that is, $\displaystyle A$ maps bounded sets to bounded sets), continuous, strictly monotone and of type $\displaystyle(S)_{+}$, that is, if $\displaystyle u_{n}\stackrel{{\scriptstyle w}}{{\rightarrow}}u$ in $\displaystyle W^{1,p}_{0}(\Omega)$ and $\displaystyle\limsup\limits_{n\rightarrow\infty}\left\langle A(u_{n}),u_{n}-u\right\rangle\leqslant 0$, then $\displaystyle u_{n}\rightarrow u$ in $\displaystyle W^{1,p}_{0}(\Omega)$.

Consider the following nonlinear eigenvalue problem

[TABLE]

We say that $\displaystyle\hat{\lambda}\in\mathbb{R}$ is an “eigenvalue” of ($\displaystyle-\Delta_{p},W^{1,p}_{0}(\Omega)$) if problem (1) admits a nontrivial solution $\displaystyle\hat{u}\in W^{1,p}_{0}(\Omega)$, known as an “eigenfunction” corresponding to $\displaystyle\hat{\lambda}$. The nonlinear regularity theory (see Gasinski & Papageorgiou [3, pp. 737-738]) implies that $\displaystyle\hat{u}\in C^{1}_{0}(\overline{\Omega})$. There is a smallest eigenvalue $\displaystyle\hat{\lambda}_{1}>0$ with the following properties:

• •

$\displaystyle\hat{\lambda}_{1}>0$ is isolated (that is, if $\displaystyle\hat{\sigma}(p)$ denotes the spectrum of ($\displaystyle-\Delta_{p},W^{1,p}_{0}(\Omega)$) then we can find $\displaystyle\epsilon>0$ such that $\displaystyle(\hat{\lambda}_{1},\hat{\lambda}_{1}+\epsilon)\cap\hat{\sigma}(p)=0$);

• •

$\displaystyle\hat{\lambda}_{1}$ is simple (that is, if $\displaystyle\hat{u},\hat{v}\in C^{1}_{0}(\overline{\Omega})$ are eigenfunctions corresponding to $\displaystyle\hat{\lambda}_{1}$, then $\displaystyle\hat{u}=\xi\hat{v}$ for some $\displaystyle\xi\in\mathbb{R}\backslash\{0\}$);

[TABLE]

It follows from the above properties that the eigenfunctions corresponding to $\displaystyle\hat{\lambda}_{1}$ do not change sign. We denote by $\displaystyle\hat{u}_{1}$ the positive, $\displaystyle L^{p}$-normalized (that is, $\displaystyle||\hat{u}_{1}||_{p}=1$) eigenfunction corresponding to $\displaystyle\hat{\lambda}_{1}>0$. From the nonlinear maximum principle (see, for example, Gasinski & Papageorgiou [3, p. 738]), we have $\displaystyle\hat{u}_{1}\in{\rm int}\,C_{+}$. Any eigenfunction corresponding to an eigenvalue $\displaystyle\hat{\lambda}\neq\hat{\lambda}_{1}$, is nodal (that is, sign-changing). More details about the spectrum of $\displaystyle(-\Delta_{p},W^{1,p}_{0}(\Omega))$ can be found in [3, 15].

We can also consider a weighted version of the eigenvalue problem (1). So, let $\displaystyle m\in L^{\infty}(\Omega)$, $\displaystyle m(z)\geqslant 0$ for almost all $\displaystyle z\in\Omega,\ m\neq 0$. We consider the following nonlinear eigenvalue problem:

[TABLE]

This problem has the same properties as (1). So, there is a smallest eigenvalue $\displaystyle\tilde{\lambda}_{1}(m)>0$ which is isolated, simple and admits the following variational characterization

[TABLE]

Also the eigenfunctions corresponding to $\displaystyle\tilde{\lambda}_{1}(m)$ have a fixed sign and we denote by $\displaystyle\tilde{u}_{1}(m)$ the positive, $\displaystyle L^{p}$-normalized eigenfunction. We have $\displaystyle\tilde{u}_{1}(m)\in{\rm int}\,C_{+}$. These properties lead to the following monotonicity property of the map $\displaystyle m\mapsto\tilde{\lambda}_{1}(m)$.

Proposition 4**.**

If $\displaystyle m_{1},m_{2}\in L^{\infty}(\Omega),0\leqslant m_{1}(z)\leqslant m_{2}(z)$ for almost all $\displaystyle z\in\Omega$ and both inequalities are strict on the sets of positive measure, then $\displaystyle\tilde{\lambda}_{1}(m_{2})<\tilde{\lambda}_{1}(m_{1})$.

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

[TABLE]

If $\displaystyle g:\Omega\times\mathbb{R}$ is a measurable function (for example, a Carathéodory function) then by $\displaystyle N_{g}(\cdot)$ we denote the Nemytski map corresponding to $\displaystyle g(\cdot,\cdot)$ defined by

[TABLE]

Given $\displaystyle v,u\in W^{1,p}_{0}(\Omega)$ with $\displaystyle v\leqslant u$, we define the order interval $\displaystyle[v,u]$ by

[TABLE]

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

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

• (i)

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

[TABLE]

• (ii)

$\displaystyle\hat{\lambda}_{1}<\eta\leqslant\liminf\limits_{x\rightarrow+\infty}\frac{f(z,x)}{x^{p-1}}\leqslant\limsup\limits_{x\rightarrow+\infty}\frac{f(z,x)}{x^{p-1}}\leqslant\hat{\eta}$ uniformly for almost all $\displaystyle z\in\Omega;$

• (iii)

there exists a function $\displaystyle w\in C^{1}(\overline{\Omega})$ such that

[TABLE]

and for every compact $\displaystyle K\subseteq\Omega$ we can find $\displaystyle c_{K}>0$ such that

[TABLE]

• (iv)

there exists $\displaystyle\delta_{0}\in(0,c_{0})$ such that for every compact $\displaystyle K\subseteq\Omega$

[TABLE]

• (v)

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

[TABLE]

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

Remark 1**.**

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

[TABLE]

Hypothesis $\displaystyle H(f)(iii)$ implies that asymptotically at $\displaystyle+\infty$ we have uniform nonresonance with respect to the principal eigenvalue $\displaystyle\hat{\lambda}_{1}>0$ of $\displaystyle(-\Delta_{p},W^{1,p}_{0}(\Omega))$. The resonant case was recently examined for nonparametric singular Dirichlet problems by Papageorgiou, Rădulescu & Repovš [16].

Example 1**.**

The following functions satisfy hypotheses $\displaystyle H(f)$. For the sake of simplicity we drop the $\displaystyle z$-dependence:

[TABLE]

with $\displaystyle 1<\tau<\vartheta$, $\displaystyle 1<q<p$ and $\displaystyle\eta>\hat{\lambda}_{1}$; and

[TABLE]

*with $\displaystyle\eta>\hat{\lambda}_{1}$, $\displaystyle 1<q<p$. *

3. A purely singular problem

In this section we deal with the following purely singular parametric problem:

[TABLE]

The next proposition establishes the existence and $\displaystyle\lambda$-dependence of the positive solutions for problem ($\displaystyle Au_{\lambda}$).

Proposition 5**.**

For every $\displaystyle\lambda>0$ problem ($\displaystyle Au_{\lambda}$) admits a unique solution $\displaystyle\tilde{u}_{\lambda}\in{\rm int}\,C_{+}$, the map $\displaystyle\lambda\mapsto\tilde{u}_{\lambda}$ is nondecreasing from $\displaystyle(0,\infty)$ into $\displaystyle C^{1}_{0}(\overline{\Omega})$ (that is, if $\displaystyle 0<\vartheta<\lambda$, then $\displaystyle\tilde{u}_{\vartheta}\leqslant\tilde{u}_{\lambda}$) and $\displaystyle||\tilde{u}_{\lambda}||_{C^{1}_{0}(\overline{\Omega})}\rightarrow 0$ as $\displaystyle\lambda\rightarrow 0^{+}$.

Proof.

The existence of a unique solution $\displaystyle\tilde{u}_{\lambda}\in{\rm int}\,C_{+}$ follows from Proposition 5 of Papageorgiou & Smyrlis [18].

Let $\displaystyle 0<\vartheta<\lambda$ and let $\displaystyle\tilde{u}_{\vartheta},\tilde{u}_{\lambda}\in{\rm int}\,C_{+}$ be the corresponding unique solutions of problem ($\displaystyle Au_{\lambda}$). Evidently, $\displaystyle\tilde{u}^{p^{\prime}}_{\vartheta}\in{\rm int}\,C_{+}\left(\frac{1}{p}+\frac{1}{p^{\prime}}=1\right)$ and so by Proposition 2.1 of Marano & Papageorgiou [14], we can find $\displaystyle c_{1}>0$ such that

[TABLE]

Lemma of Lazer & McKenna [12, p. 726], implies that $\displaystyle\hat{u}_{1}^{-\gamma/p^{\prime}}\in L^{p^{\prime}}(\Omega)$. Therefore $\displaystyle\tilde{u}_{\vartheta}^{-\gamma}\in L^{p^{\prime}}(\Omega)$. We introduce the Carathéodory function $\displaystyle g_{\lambda}(z,x)$ defined by

[TABLE]

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

[TABLE]

Proposition 3 of Papageorgiou & Smyrlis [18] implies that $\displaystyle\hat{\psi}_{\lambda}\in C^{1}(W^{1,p}_{0}(\Omega))$. From (5) and since $\displaystyle\tilde{u}_{\vartheta}^{-\gamma}\in L^{p^{\prime}}(\Omega)$ it follows that $\displaystyle\hat{\psi}_{\lambda}(\cdot)$ is coercive. Also, via the Sobolev embedding theorem, we see that $\displaystyle\hat{\psi}_{\lambda}(\cdot)$ is sequentially weakly lower semicontinuous. So, by the Weierstrass-Tonelli theorem, we can find $\displaystyle\bar{u}_{\lambda}\in W^{1,p}_{0}(\Omega)$ such that

[TABLE]

In (3) we choose $\displaystyle h=(\tilde{u}_{\vartheta}-\bar{u}_{\lambda})^{+}\in W^{1,p}_{0}(\Omega)$. We have

[TABLE]

From (5), (6), (3), we have

[TABLE]

Therefore the map $\displaystyle\lambda\mapsto\tilde{u}_{\lambda}$ is nondecreasing from $\displaystyle(0,+\infty)$ into $\displaystyle C^{1}_{0}(\overline{\Omega})$.

We have

[TABLE]

Choosing $\displaystyle h=\tilde{u}_{\lambda}\in W^{1,p}_{0}(\Omega)$, we obtain

[TABLE]

As in the first part of the proof, using Proposition 2.1 of Marano & Papageorgiou [14], we show that $\displaystyle\tilde{u}_{\lambda}^{-\gamma}\in L^{r}(\Omega)$ for $\displaystyle r>N$. Then Proposition 1.3 of Guedda & Véron [8] implies that

[TABLE]

Let $\displaystyle k_{\lambda}=\lambda\tilde{u}^{-\gamma}_{\lambda}\in L^{r}(\Omega),\lambda\in\left(0,1\right]$ and consider the following linear Dirichlet problem

[TABLE]

Standard existence and regularity theory (see, for example, Struwe [20, p. 218]), implies that problem (10) has a unique solution $\displaystyle v_{\lambda}(\cdot)$ such that

[TABLE]

for some $\displaystyle c_{5}>0$, all $\displaystyle\lambda\in\left(0,1\right]$, and with $\displaystyle\alpha=1-\frac{N}{r}\in(0,1)$ (recall that $\displaystyle r>N$). Let $\displaystyle\beta_{\lambda}(z)=Dv_{\lambda}(z)$. Then $\displaystyle\beta_{\lambda}\in C^{0,\alpha}(\overline{\Omega})$ for every $\displaystyle\lambda\in\left(0,1\right]$. We have

[TABLE]

Then Theorem 1 of Lieberman [13] (see also Corollary 1.1 of Guedda & Véron [8]) and (9), imply that we can find $\displaystyle s\in(0,1)$ and $\displaystyle c_{6}>0$ such that

[TABLE]

Finally, the compact embedding of $\displaystyle C^{1,s}_{0}(\overline{\Omega})$ into $\displaystyle C^{1}_{0}(\overline{\Omega})$ and (8) imply that

[TABLE]

This completes the proof. ∎

4. Bifurcation-type theorem

Let

[TABLE]

$\displaystyle S_{\lambda}=\mbox{the set of positive solutions for problem \eqref{eqp}}$.

Proposition 6**.**

If hypotheses $\displaystyle H(f)$ hold, then $\displaystyle\mathcal{L}\neq\emptyset$.

Proof.

Using Proposition 5, we can find $\displaystyle\lambda_{0}\in\left(0,1\right]$ such that

[TABLE]

Here, $\displaystyle\delta_{0}>0$ is as postulated by hypothesis $\displaystyle H(f)(iv)$.

We fix $\displaystyle\lambda\in\left(0,\lambda_{0}\right]$ and we consider the following truncation of the reaction in problem ($\displaystyle P_{\lambda}$):

[TABLE]

(recall that $\displaystyle\delta_{0}<c_{0}\leqslant w(z)$ for all $\displaystyle z\in\overline{\Omega}$). This is a Carathéodory function. We set $\displaystyle\hat{K}_{\lambda}(z,x)=\int^{x}_{0}\hat{k}_{\lambda}(z,s)ds$ and consider the function $\displaystyle\hat{\varphi}_{\lambda}:W^{1,p}_{0}(\Omega)\rightarrow\mathbb{R}$ defined by

[TABLE]

As before, we have $\displaystyle\hat{\varphi}_{\lambda}\in C^{1}(W^{1,p}_{0}(\Omega))$. Also, it follows from (15) that

[TABLE]

In addition, we have that

[TABLE]

Therefore, we can find $\displaystyle\hat{u}_{\lambda}\in W^{1,p}_{0}(\Omega)$ such that

[TABLE]

In (16) we choose $\displaystyle h=(\tilde{u}_{\lambda}-\hat{u}_{\lambda})^{+}\in W^{1,p}_{0}(\Omega)$. Then

[TABLE]

Next, we choose $\displaystyle h=(\hat{u}_{\lambda}-w)^{+}\in W^{1,p}_{0}(\Omega)$ in (16). Then

[TABLE]

(see hypothesis $\displaystyle H(f)(iii)$ and use the nonlinear Green identity, see [3, p. 211])

[TABLE]

So, we have proved that

[TABLE]

Using (17) and (15), equation (16) becomes

[TABLE]

From (17), (18) and Theorem 1 of Lieberman [13], we infer that

[TABLE]

This completes the proof. ∎

A byproduct of the above proof is the following corollary.

Corollary 7**.**

If hypotheses $\displaystyle H(f)$ hold, then $\displaystyle S_{\lambda}\subseteq{\rm int}\,C_{+}$ for all $\displaystyle\lambda>0$.

The next proposition shows that $\displaystyle\mathcal{L}$ is an interval.

Proposition 8**.**

If hypotheses $\displaystyle H(f)$ hold, $\displaystyle\lambda\in\mathcal{L}$ and $\displaystyle\vartheta\in(0,\lambda)$, then $\displaystyle\vartheta\in\mathcal{L}.$

Proof.

Since $\displaystyle\lambda\in\mathcal{L}$, we can find $\displaystyle u_{\lambda}\in S_{\lambda}\subseteq{\rm int}\,C_{+}$. Proposition 5 implies that we can find $\displaystyle\tau\in[0,\lambda_{0}]$ (see (11)) such that

[TABLE]

We introduce the Carathéodory function $\displaystyle e(z,x)$ defined by

[TABLE]

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

[TABLE]

We know that $\displaystyle\hat{\psi}_{\vartheta}\in C^{1}(W^{1,p}_{0}(\Omega))$. Moreover, $\displaystyle\hat{\psi}_{\vartheta}$ is coercive (see (19)) and sequentially weakly lower semicontinuous. So, we can find $\displaystyle u_{\vartheta}\in W^{1,p}_{0}(\Omega)$ such that

[TABLE]

In (20) we first choose $\displaystyle h=(\tilde{u}_{\tau}-u_{\vartheta})^{+}\in W^{1,p}_{0}(\Omega)$. Then

[TABLE]

Next, in (20) we choose $\displaystyle h=(u_{\vartheta}-u_{\lambda})^{+}\in W^{1,p}_{0}(\Omega)$. Then

[TABLE]

So, we have proved that

[TABLE]

It follows from (19), (20) and (21) that

[TABLE]

The proof is now complete. ∎

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

Corollary 9**.**

If hypotheses $\displaystyle H(f)$ hold, $\displaystyle\lambda\in\mathcal{L},u_{\lambda}\in S_{\lambda}\subseteq{\rm int}\,C_{+}$, and $\displaystyle\vartheta<\lambda$, then $\displaystyle\vartheta\in\mathcal{L}$ and we can find $\displaystyle u_{\vartheta}\in S_{\vartheta}\subseteq{\rm int}\,C_{+}$ such that $\displaystyle u_{\vartheta}\leqslant u_{\lambda}$.

In fact, we can improve the above result as follows.

Proposition 10**.**

If hypotheses $\displaystyle H(f)$ hold, $\displaystyle\lambda\in\mathcal{L},u_{\lambda}\in S_{\lambda}\subseteq{\rm int}\,C_{+}$, and $\displaystyle\vartheta<\lambda$, then $\displaystyle\vartheta\in\mathcal{L}$ and we can find $\displaystyle u_{\vartheta}\in S_{\vartheta}\subseteq{\rm int}\,C_{+}$ such that $\displaystyle u_{\lambda}-u_{\vartheta}\in{\rm int}\,C_{+}$.

Proof.

From Corollary 9 we know that $\displaystyle\vartheta\in\mathcal{L}$ and we can find $\displaystyle u_{\vartheta}\in S_{\vartheta}\subseteq{\rm int}\,C_{+}$ such that

[TABLE]

Let $\displaystyle\rho=||u_{\lambda}||_{\infty}$ and let $\displaystyle\hat{\xi}_{\rho}>0$ be as postulated by hypothesis $\displaystyle H(f)(v)$. Then

[TABLE]

We set

[TABLE]

We have

[TABLE]

(see (22) and hypotheses $\displaystyle H(f)(v)$).

We can apply Proposition 2 and conclude that

[TABLE]

The proof is now complete. ∎

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

Proposition 11**.**

If hypotheses $\displaystyle h(f)$ hold, then $\displaystyle\lambda^{*}<+\infty$.

Proof.

Let $\displaystyle\epsilon>0$ be such that $\displaystyle\hat{\lambda}_{1}+\epsilon<\eta$ (see hypothesis $\displaystyle H(f)(ii)$). We can find $\displaystyle M>0$ such that

[TABLE]

Also, hypothesis $\displaystyle H(f)(i)$ implies that we can find large enough $\displaystyle\tilde{\lambda}>0$ such that

[TABLE]

It follows from (23) and (24) that

[TABLE]

Let $\displaystyle\lambda>\tilde{\lambda}$ and suppose that $\displaystyle\lambda\in\mathcal{L}$. Then we can find $\displaystyle u_{\lambda}\in S_{\lambda}\subseteq{\rm int}\,C_{+}$. We have

[TABLE]

Since $\displaystyle u_{\lambda}\in{\rm int}\,C_{+}$, we can find $\displaystyle t\in(0,1)$ so small that

[TABLE]

(see Proposition 2.1 of Marano & Papageorgiou [14]). We have

[TABLE]

Using (27), we can define the Carathéodory function $\displaystyle\beta(z,x)$ as follows

[TABLE]

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

[TABLE]

From (32) it is clear that $\displaystyle\sigma(\cdot)$ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find $\displaystyle\bar{u}\in W^{1,p}_{0}(\Omega)$ such that

[TABLE]

In (33) we first choose $\displaystyle h=(\hat{y}_{1}-\bar{u})^{+}\in W^{1,p}_{0}(\Omega)$. Then

[TABLE]

Also, in (33) we choose $\displaystyle h=(\bar{u}-u_{\lambda})^{+}\in W^{1,p}_{0}(\Omega)$. Then

[TABLE]

So, we have proved that

[TABLE]

It follows from (32), (33) and (34) that

[TABLE]

Therefore we have $\displaystyle\lambda^{*}\leqslant\tilde{\lambda}<+\infty$. ∎

Next, we show that the critical parameter $\displaystyle\lambda^{*}>0$ is admissible.

Proposition 12**.**

If hypotheses $\displaystyle H(f)$ hold, then $\displaystyle\lambda^{*}\in\mathcal{L}$.

Proof.

Let $\displaystyle\{\lambda_{n}\}_{n\geqslant 1}\subseteq(0,\lambda^{*})$ and assume that $\displaystyle\lambda_{n}\rightarrow(\lambda^{*})^{-}$ as $\displaystyle n\rightarrow\infty$. We can find $\displaystyle u_{n}=u_{\lambda_{n}}\in S_{\lambda_{n}}\subseteq{\rm int}\,C_{+}$ for all $\displaystyle n\in\mathbb{N}$. Then

[TABLE]

Suppose that $\displaystyle||u_{n}||\rightarrow\infty$. We set $\displaystyle y_{n}=\frac{u_{n}}{||u_{n}||}\ n\in\mathbb{N}$. Then $\displaystyle||y_{n}||=1,y_{n}\geqslant 0$ for all $\displaystyle n\in\mathbb{N}$. So, we may assume that

[TABLE]

From (35) we have

[TABLE]

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

[TABLE]

This growth condition implies that

[TABLE]

Then (38) and hypothesis $\displaystyle H(f)(ii)$ imply that at least for a subsequence, we have

[TABLE]

In (37) we choose $\displaystyle h=y_{n}-y\in W^{1,p}_{0}(\Omega)$, pass to the limit as $\displaystyle n\rightarrow\infty$, and use (36) and (38). Then

[TABLE]

Therefore, if in (37) we pass to the limit as $\displaystyle n\rightarrow\infty$ and use (40) and (39), then

[TABLE]

Since $\displaystyle\eta\leqslant\eta_{0}(z)\leqslant\hat{\eta}$ for almost all $\displaystyle z\in\Omega$ (see (39)), using Proposition 4, we have

[TABLE]

So, from (41) and since $\displaystyle||y||=1$ (see (40)), it follows that $\displaystyle y$ must be nodal, a contradiction (see (40)). Therefore

[TABLE]

Hence, we may assume that

[TABLE]

On account of Corollary 9, we may assume that $\displaystyle\{u_{n}\}_{n\geqslant 1}$ is nondecreasing. Therefore $\displaystyle u^{*}\neq 0$. Also, we have

[TABLE]

From (42) and by passing to a subsequence if necessary, we can say that

[TABLE]

From (43), (44) and Problem 1.19 of Gasinski & Papageorgiou [4], we have that

[TABLE]

If in (35) we choose $\displaystyle h=u_{n}-u^{*}\in W^{1,p}_{0}(\Omega)$, pass to the limit as $\displaystyle n\rightarrow\infty$ and use (45) and the fact that $\displaystyle\{N_{f}(u_{n})\}_{n\geqslant 1}\subseteq L^{p^{\prime}}(\Omega)$ is bounded, then

[TABLE]

Finally, in (35) we pass to the limit as $\displaystyle n\rightarrow\infty$ and use (45) and (46). We obtain

[TABLE]

This completes the proof. ∎

We have proved that

[TABLE]

Proposition 13**.**

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

[TABLE]

Proof.

Let $\displaystyle u^{*}\in S_{\lambda^{*}}\subseteq{\rm int}\,C_{+}$ (see Proposition 12). Invoking Proposition 10, we can find $\displaystyle u_{\lambda}\in S_{\lambda}\subseteq{\rm int}\,C_{+}$ such that

[TABLE]

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

[TABLE]

Recall that $\displaystyle u_{\lambda}^{-\gamma}\in L^{p^{\prime}}(\Omega)$ (see the proof of Proposition 5). We set $\displaystyle T_{\lambda}(z,x)=\int^{x}_{0}\tau_{\lambda}(z,s)ds$ and consider the functional $\displaystyle\tilde{\varphi}_{\lambda}:W^{1,p}_{0}(\Omega)\rightarrow\mathbb{R}$ defined by

[TABLE]

We know that $\displaystyle\tilde{\varphi}_{\lambda}\in C^{1}(W^{1,p}_{0}(\Omega))$. Let $\displaystyle K_{\tilde{\varphi}_{\lambda}}=\{u\in W^{1,p}_{0}(\Omega):\tilde{\varphi}_{\lambda}^{\prime}(u)=0\}$ (the critical set of $\displaystyle\tilde{\varphi}_{\lambda}$). Also, for $\displaystyle u\in W^{1,p}_{0}(\Omega)$, we set

[TABLE]

Claim 1**.**

$\displaystyle K_{\tilde{\varphi_{\lambda}}}\subseteq[u_{\lambda})\cap{\rm int}\,C_{+}$.

Let $\displaystyle u\in K_{\tilde{\varphi}_{\lambda}}$. We have

[TABLE]

We choose $\displaystyle h=(u_{\lambda}-u)^{+}\in W^{1,p}_{0}(\Omega)$. Then

[TABLE]

From (48), (49) and (50), we obtain

[TABLE]

This proves Claim 1.

Note that $\displaystyle u_{\lambda}\in K_{\tilde{\varphi}_{\lambda}}$. We may assume that

[TABLE]

or otherwise we already have a second positive smooth solution for problem ($\displaystyle P_{\lambda}$) (see (48)) and so we are done.

We introduce the following Carathéodory function

[TABLE]

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

[TABLE]

This functional is coercive (see (52)) and sequentially weakly lower semicontinuous. Hence we can find $\displaystyle\tilde{u}_{\lambda}\in W^{1,p}_{0}(\Omega)$ such that

[TABLE]

In (53) we choose $\displaystyle h=(u_{\lambda}-\tilde{u}_{\lambda})^{+}\in W^{1,p}_{0}(\Omega)$ and $\displaystyle h=(\tilde{u}_{\lambda}-u^{*})^{+}\in W^{1,p}_{0}(\Omega)$ and obtain that

[TABLE]

From (52), (53), (54) we infer that

[TABLE]

From (48) and (52) it is clear that

[TABLE]

Also, $\displaystyle u_{\lambda}$ is a minimizer of $\displaystyle\hat{\varphi}_{\lambda}$. Since $\displaystyle u^{*}-u_{\lambda}\in{\rm int}\,C_{+}$ (see (47)), it follows that

[TABLE]

We assume that $\displaystyle K_{\tilde{\varphi}_{\lambda}}$ is finite or otherwise on account of Claim 1, we already have an infinity of positive smooth solutions for problem ($\displaystyle P_{\lambda}$) bigger than $\displaystyle u_{\lambda}$ and so we are done. Because of (4), we can find $\displaystyle\rho\in(0,1)$ small such that

[TABLE]

Hypothesis $\displaystyle H(f)(ii)$ implies that

[TABLE]

Claim 2**.**

$\displaystyle\tilde{\varphi}_{\lambda}$* satisfies the $\displaystyle C$-condition.*

Let $\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq W^{1,p}_{0}(\Omega)$ such that $\displaystyle\{\tilde{\varphi}_{\lambda}(u_{n})\}_{n\geqslant 1}\subseteq\mathbb{R}$ is bounded and

[TABLE]

We have

[TABLE]

We choose $\displaystyle h=-u^{-}_{n}\in W^{1,p}_{0}(\Omega)$ in (58) and also use (48). Then

[TABLE]

Suppose that $\displaystyle||u^{+}_{n}||\rightarrow\infty$ and let $\displaystyle y_{n}=\frac{u^{+}_{n}}{||u^{+}_{n}||}\ n\in\mathbb{N}$. Then $\displaystyle||y_{n}||=1,y_{n}\geqslant 0$ for all $\displaystyle n\in\mathbb{N}$. So, we may assume that

[TABLE]

From (58) and (59), we have

[TABLE]

From (48) and hypothesis $\displaystyle H(f)(ii)$, we have

[TABLE]

In (61) we choose $\displaystyle h=y_{n}-y\in W^{1,p}_{0}(\Omega)$ and pass to the limit as $\displaystyle n\rightarrow\infty$. Then

[TABLE]

Then passing to the limit as $\displaystyle n\rightarrow\infty$ in (61) and using (62) and (63), we obtain

[TABLE]

As before, using Proposition 4, we have

[TABLE]

This proves that $\displaystyle\{u^{+}_{n}\}_{n\geqslant 1}\subseteq W^{1,p}_{0}(\Omega)$ is bounded. Hence

[TABLE]

So, we may assume that

[TABLE]

In (58) we choose $\displaystyle h=u_{n}-u\in W^{1,p}_{0}(\Omega)$, pass to the limit as $\displaystyle n\rightarrow\infty$ and use (65). Then

[TABLE]

This proves Claim 2.

On account of (56), (57) and Claim 2 we can apply Theorem 1 (the mountain pass theorem) and find $\displaystyle\hat{u}_{\lambda}\in W^{1,p}_{0}(\Omega)$ such that

[TABLE]

Therefore $\displaystyle\hat{u}_{\lambda}\in{\rm int}\,C_{+}$ is the second positive solution of ($\displaystyle P_{\lambda}$) and

[TABLE]

The proof is now complete. ∎

Therefore we have also proved Theorem A, which is the main result of this paper.

Remark 2**.**

An interesting open problem is whether there is such a bifurcation-type theorem for resonant problems, that is,

[TABLE]

or even for the nonuniformly nonresonant problems, that is,

[TABLE]

with $\displaystyle\eta\in L^{\infty}(\Omega)$ such that

[TABLE]

In both cases it seems to be difficult to show that $\displaystyle\lambda^{*}<\infty$. Additional conditions on $\displaystyle f(z,\cdot)$ might be needed.

Acknowledgements. The authors wish to thank the referee for his/her remarks and suggestions. This research was supported by the Slovenian Research Agency grants P1-0292, J1-8131, J1-7025, N1-0064, and N1-0083. V.D. Rădulescu acknowledges the support through a grant of the Romanian Ministry of Research and Innovation, CNCS-UEFISCDI, project number PN-III-P4-ID-PCE-2016-0130, within PNCDI III.