Existence of Positive Solution for Kirchhoff Problems

Abstract

In this work, we study the following Kirchhoff type problem

$$\begin{aligned} \begin{gathered} -\Big (a+b\int _{\Omega }|\nabla u|^pdx\Big )\Delta _p u =g(x)u^{-\gamma }+\lambda f(x,u),\quad \text {in }\Omega , \\ u=0, \quad \text {on }\partial \Omega , \end{gathered} \end{aligned}$$

where $p\ge 2$, $\Omega$ is a regular bounded domain in $\mathbb {R}^N$, $(N\ge 3)$. Firstly, for $p>2$, we prove under some appropriate conditions on the singularity and the nonlinearity the existence of nontrivial weak solution to this problem. For $p=2$, we show, under supplementary condition, the positivity of this solution. Moreover, in the case $\lambda =0$ we prove an uniqueness result. We use the variational method to prove our main results.

1 Introduction and Main Results

In this article, we consider the Kirchhoff type problem

$$\begin{aligned} (\mathbf P _{\lambda ,p}) \left\{ \begin{aligned} \begin{array}{ll} -\Big (a+b\int _{\Omega }|\nabla u|^pdx\Big )\Delta _p u =g(x)u^{-\gamma }+\lambda f(x,u),&{}\quad \text {in }\Omega , \\ u=0, &{} \quad \text {on }\partial \Omega , \end{array} \end{aligned} \right. \end{aligned}$$

where $p\ge 2$, $\Omega \subset \mathbb {R}^N$, $(N\ge 3)$ is a bounded regular domain, $a,b\ge 0$, $a+b>0$, $0<\gamma <1$ and $\lambda \ge 0$ are parameters.

Note that, the existence and multiplicity of solutions for the following problem

$$\begin{aligned} \left\{ \begin{aligned} \begin{array}{ll} -\Big (a+b\int _{\Omega }|\nabla u|^2dx\Big )\Delta u=f(x,u),&{}\quad \text {in }\Omega , \\ u=0, &{}\quad \text {on }\partial \Omega , \end{array} \end{aligned} \right. \end{aligned}$$

(1.1)

where $\Omega \subset \mathbb {R}^3$ is a smooth bounded domain and $f:\overline{\Omega }\times \mathbb {R}\rightarrow \mathbb {R}$ is a continuous function, has been extensively studied (see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]). This type of problem is related to the stationary analogue of the following problem

$$\begin{aligned} \rho \frac{\partial ^{2} u}{\partial t^{2}}-\left( \frac{\rho _{0}}{h}+\frac{E}{2L}\int _{0}^{L}\left| \frac{\partial u}{\partial x}\right| ^{2}dx\right) \frac{\partial ^{2} u}{\partial x^{2}}=0, \end{aligned}$$

where $\rho ,\; \rho _{0},\; h,\; E,$ and L are constants, which extends the classical d’Alembert wave equation, by considering the effects of the changes in the length of the strings during the vibrations. For more detail we refer the reader to [10] and the references therein. Some important results concerning problem of the form (1.1) are given in [11, 12, 14, 15]. Problems like (1.1) are also introduced as models for other physical phenomena as, biological systems where u describes a process which depends on the average of itself.

An interesting generalization of problem (1.1) is

$$\begin{aligned} \left\{ \begin{aligned} \begin{array}{ll} -\Big (a+b\int _{\Omega }|\nabla u|^pdx\Big )\Delta _p u =f(x,u),&{}\quad \text {in }\Omega , \\ u=0, &{}\quad \text {on }\partial \Omega , \end{array} \end{aligned} \right. \end{aligned}$$

The problems of this type is important and arises in an interesting physical context. Much interest has grown on singular problems (see for example [12, 13, 15]). However, the singular Kirchhoff type problems have few been considered, exept for [9, 10]. Before giving our main results let us recall literature concerning related nonlinear equations. Liu and Sun in [10] have investigated problem (1.1) with $f(x,u)=g(x)u^{-\gamma }+\lambda h(x)\frac{u^p}{|x|^s}$, and $g,h \in C(\overline{\Omega }),0\le s<1,3<p<5-2s$. They proved that the non-degenerate case of problem (1.1) has at least two positive solutions for $\lambda >0$ small enough using the Nehari manifold. By the variational methods, they obtained that problem (1.1) has at least two positive solutions for $\mu >0$ small enough.

Liao et al. [13] considered the following problem

$$\begin{aligned} \left\{ \begin{aligned} \begin{array}{ll} -\Big (a+b\int _{\Omega }|\nabla u|^2dx\Big )\Delta u=g(x)u^{-\gamma }-\lambda u^p, &{}\quad \text {in }\Omega , \\ u=0, &{} \quad \text {on }\partial \Omega , \end{array} \end{aligned} \right. \end{aligned}$$

(1.2)

where $\Omega \subset \mathbb {R}^N$$(N\ge 3)$ is a bounded domain, $0<\gamma<1,\lambda \ge 0,0< p\le p^*-1$ and $a,b \ge 0,a+b> 0$ are parameters. The coefficient $\displaystyle g\in L^{\frac{2^*}{2^*+\gamma -1}}(\Omega )$ with $g(x)> 0$ for almost every $x \in \Omega$ and $\displaystyle 2^*=\frac{2N}{N-2}$ denotes the critical Sobolev exponent. Using the minimax method and some analysis techniques, they obtained the uniqueness of positive solutions for problem (1.2).

Inspired by the above articles, in this paper, we would like to generalize problem (1.2). More precisely, we investigate the existence of solutions for problem $(\mathbf P _{\lambda ,p})$ by using variational methods. Under some appropriate conditions we prove the positivity and uniqueness of solution in the case $p=2$.

In the sequel, for Hölder argument reason, we suppose that the function $\displaystyle g\in L^{\frac{p^*}{p^*+\gamma -1}}(\Omega )$ with $g(x)> 0$ for almost every $x \in \Omega$ and $\displaystyle p^*=\frac{Np}{N-p}$ denotes the critical Sobolev exponent for the embedding $W^{1,p}_0({\Omega })$ into $L^{q}(\Omega )$ for $q\in [1,\frac{Np}{N-p}]$. $f\in C\left( \overline{\Omega }\times \mathbb {R},\mathbb {R}\right)$ is positively homogeneous of degree $r-1$ where $1<r<p$. more precisely we assume the following:

(H1):

$f: \overline{\Omega }\times \mathbb {R}\longrightarrow \mathbb {R}$ is a continuous function such that

$$\begin{aligned} f(x,tu) = t^{r-1} f(x,u),\; (t > 0)\;\;\text{ for } \text{ all }\;\; x\in \overline{\Omega },\;u\in \mathbb {R}. \end{aligned}$$

(H2):

$f(x,t)\ge 0$ in $\Omega _1\subset \subset \Omega$ such that $|\Omega _1|>0$.

Note that in the case $1<r<p$ we have the validity of the coercivity properties of the functional energy associated with the problem $(\mathbf P _{\lambda ,p})$.

Remark 1.1

(i):

A more general condition on f is the Ambrosetti–Rabinowitz condition, but in this case, we can’t prove that Lemma 2.2 hold true.

(ii):

Put $F(x,s):=\int _{0}^{s}f(x,t)dt$, then, assumption $\mathbf {(H1)},$f leads to the so-called Euler identity

$$\begin{aligned} u f(x,u)= & {} r F(x,u),\nonumber \\ F(x,u)\le & {} K |u|^r\quad \text{ for } \text{ some } \text{ constant } \text{ K }, \end{aligned}$$

(1.3)

and $f(x,0)=0=(\partial f/\partial t)(x,0)$ for every $t \in \mathbb {R}$.

Our main results are the following:

Theorem 1.2

Assuming the hypotheses $\mathbf {(H1)}$ and $\mathbf {(H2)}$ holds, then for all $\lambda \ge 0$, problem $(\mathbf P _{\lambda ,p})$ has at least one non trivial weak solution with negative energy.

Next, we give tow Theorems concerning the uniqueness and positivity of solution in the special case when $p=2$.

Theorem 1.3

For $\lambda =0$ and under the same assumptions of Theorem 1.2, the solution given in Theorem 1.2 for $(\mathbf P _{0,2})$ is unique.

Theorem 1.4

Under the same assumptions of Theorem 1.2. If $\Omega =\Omega _1$, then, the solution given in Theorem 1.2 for $(\mathbf P _{\lambda ,2})$ is positive.

2 Proof of Theorem 1.2

In this case, we consider the Kirchhoff type problem

$$\begin{aligned} (\mathbf P _{\lambda ,p}) \left\{ \begin{aligned} \begin{array}{ll} -\Big (a+b\int _{\Omega }|\nabla u|^pdx\Big )\Delta _p u =g(x)u^{-\gamma }+\lambda f(x,u),&{}\quad \text {in }\Omega , \\ u>0, &{}\quad \text {in }\Omega ,\\ u=0, &{}\quad \text {on }\partial \Omega , \end{array} \end{aligned} \right. \end{aligned}$$

For $u\in W^{1,p}_0({\Omega })$, we define the energy functional associated to the above problem:

$$\begin{aligned} \begin{aligned} I_\lambda (u)&=\frac{a}{p}\int _{\Omega }|\nabla u|^pdx+\frac{b}{2p} \Big (\int _{\Omega }|\nabla u|^pdx\Big )^2 \\&\quad -\frac{1}{1-\gamma }\int _{\Omega }g(x)|u|^{1-\gamma }dx -\lambda \int _{\Omega }F(x,u(x))dx, \end{aligned} \end{aligned}$$

where $W^{1,p}_0({\Omega })$ is a Sobolev space equipped with the norm

$$\begin{aligned} \Vert u\Vert =\left( \int _{\Omega }|\nabla u|^p dx\right) ^{1/p}. \end{aligned}$$

Note that a function u is called a weak solution of $(\mathbf P _{\lambda ,p})$ if $u\in W^{1,p}_0({\Omega })$ satisfies the following:

$$\begin{aligned} \Big (a+b\int _{\Omega }|\nabla u|^pdx\Big ) \int _{\Omega }\mid \nabla u\mid ^{p-1}.\nabla \varphi dx -\int _{\Omega }g(x)|u|^{-\gamma }\varphi dx-\lambda \int _{\Omega }f(x,u)\varphi dx=0, \end{aligned}$$

(2.1)

for all $\varphi \in W^{1,p}_0({\Omega })$.

In order to prove Theorem 1.2, we show firstly that $I_\lambda$ attains his global minimizer in $W^{1,p}_0({\Omega })$. For this purpose, we need the following lemmas:

Lemma 2.1

$I_\lambda$ is coercive and bounded from below on $W^{1,p}_0({\Omega })$.

Proof

Combining Hölder and Sobolev inequalities, it follows that

$$\begin{aligned} \begin{aligned} \displaystyle I_\lambda (u)&=\displaystyle \frac{a}{p}\int _{\Omega }|\nabla u|^pdx+\frac{b}{2p} \Big (\int _{\Omega }|\nabla u|^pdx\Big )^2 \\&\quad -\frac{1}{1-\gamma }\int _{\Omega }g(x)|u|^{1-\gamma }dx -\lambda \int _{\Omega }F(x,u(x))dx,\\ \displaystyle&\ge \displaystyle \frac{a}{p}\Vert u\Vert ^p+ \frac{b}{2p} \Vert u\Vert ^{2p}-\frac{1}{1-\gamma }|g|_{\frac{p^*}{p^*+\gamma -1}}|u|_{p^*}^{1-\gamma }-\lambda K|u|_{p}^{r}|\Omega |^{\frac{p-r}{p}},\\ \displaystyle&\ge \displaystyle \frac{a}{p}\Vert u\Vert ^p+ \frac{b}{2p} \Vert u\Vert ^{2p}-\frac{C}{1-\gamma }|g|_{\frac{p^*}{p^*+\gamma -1}} \Vert u\Vert ^{1-\gamma }-\lambda K\Vert u\Vert ^{r}|\Omega |^{\frac{p-r}{p}},\\ \end{aligned} \end{aligned}$$

where $C> 0$ is a constant. Since $1<r<p$, this ends the proof. $\square$

Thus

$$\begin{aligned} \displaystyle m_\lambda =\inf _{u\in W^{1,p}_0({\Omega })}I_\lambda (u) \end{aligned}$$

is well defined. Let us show that, $m_\lambda < 0$.

Lemma 2.2

There exist $\varphi \in W^{1,p}_0({\Omega })$ such that $\varphi \ge 0, \varphi \not \equiv 0$ and $I_\lambda (t\varphi )< 0$ for $t>0$ and small enough.

Proof

Let $\varphi \in C_0^{\infty }(\Omega )$ such that supp$(\varphi )\subset \Omega _1 \subset \subset \Omega , \varphi =1$ in a subspace $\Omega {'}$$\subset$ supp$(\varphi ),0\le \varphi \le 1$ in $\Omega$, then

$$\begin{aligned} \begin{aligned} \displaystyle I_\lambda (t\varphi )&=\displaystyle a\frac{t^p}{p}\Vert \varphi \Vert ^p+b\frac{t^{2p}}{2p}\Vert \varphi \Vert ^{2p} -\frac{t^{1-\gamma }}{1-\gamma }\int _{\Omega }g(x)|\varphi |^{1-\gamma }dx \\&\quad -\lambda \frac{t^r}{r}\int _{\Omega _1}F(x,\varphi )dx,\\&\le \displaystyle t^p\left[ \frac{a}{p}\Vert \varphi \Vert ^p+\frac{b}{2p}\Vert \varphi \Vert ^{2p}\right] -\frac{t^{1-\gamma }}{1-\gamma }\int _{\Omega }g(x)|\varphi |^{1-\gamma }dx.\\ \end{aligned} \end{aligned}$$

Consequently, $I_\lambda (t\varphi ) < 0$ for $t< \delta ^{\frac{1}{p-(1-\gamma )}}$ with

$$\begin{aligned} 0<\delta <\min \left\{ 1,\frac{\frac{1}{1-\gamma } \int _{\Omega }g(x)|\varphi |^{1-\gamma }dx}{\displaystyle \frac{a}{p}\Vert \varphi \Vert ^p+\frac{b}{2p}\Vert \varphi \Vert ^{2p}}\right\} \end{aligned}$$

Finally, we point out that $\frac{a}{p}\Vert \varphi \Vert ^p+\frac{b}{2p}\Vert \varphi \Vert ^{2p} >0$. In fact if $\frac{a}{p}\Vert \varphi \Vert ^p+\frac{b}{2p}\Vert \varphi \Vert ^{2p}=0$, then $\varphi =0$ in $\Omega$ which is a contradiction. $\square$

Now, using Lemmas 2.1 and 2.2 one has:

Proposition 2.1

Suppose that $0<\gamma <1$, $\lambda \ge 0$, $a,b\ge 0$ with $a+b> 0$, $\displaystyle g\in L^{\frac{p^*}{p^*+\gamma -1}}(\Omega )$ with $g(x)>0$ for almost every $x\in \Omega$ and assuming the hypotheses $\mathbf {(H1)}$ and $\mathbf {(H2)}$ holds. Then $I_\lambda$ attains his global minimizer in $W^{1,p}_0({\Omega })$, that is, there exists $u_* \in W^{1,p}_0({\Omega })$ such that $I_\lambda (u_*)=m_\lambda <0$.

Proof

Let $\{u_n\}$ be a minimizing sequence, that is to say

$$\begin{aligned} I_\lambda (u_n)\rightarrow m_\lambda . \end{aligned}$$

Suppose $\{u_n\}$ is not bounded, so

$$\begin{aligned} \Vert u_n\Vert \rightarrow +\infty \,\,\,\text{ as }\,\, n \rightarrow +\infty . \end{aligned}$$

Since $I_\lambda$ is coercive then

$$\begin{aligned} I_\lambda (u_n) \rightarrow +\infty \,\,\,\text{ as }\,\, \Vert u_n\Vert \rightarrow +\infty . \end{aligned}$$

This contradicts the fact that $\{u_n\}$ is a minimizing sequence.

So, $\{u_n\}$ is bounded in $W^{1,p}_0({\Omega })$. Therefore up to a subsequence, there exists $u_* \in W^{1,p}_0({\Omega })$ such that

$$\begin{aligned} \left\{ \begin{array}{ll} u_n\rightharpoonup u_* , &{}\quad \text {weakly in } W^{1,p}_0({\Omega }),\\ u_n\rightarrow u_* ,&{}\quad \text {strongly in } L^{s}(\Omega ),\ \ 1\le s< p^{*}\\ u_n(x)\rightarrow u_*(x),&{}\quad \text {a.e. in } \Omega , \end{array} \right. \end{aligned}$$

(2.2)

Let, $M(t)=a+bt,\,\,t\ge 0$ and $\hat{M}(t)=\int _{0}^{t}M(s)ds$. The function M is positive, so $\hat{M}$ is increasing and the weak convergence of $\{u_n\}$ implies

$$\begin{aligned} \Vert u_*\Vert \le \liminf _{n\rightarrow +\infty }\Vert u_n\Vert \end{aligned}$$

from where

$$\begin{aligned} \hat{M}\left( \Vert u_*\Vert ^p\right) \le \hat{M} \left( \left( \liminf _{n\rightarrow +\infty }\Vert u_n\Vert \right) ^p\right) \end{aligned}$$

Since the function $\hat{M}$ is continuous and $\Vert u_n\Vert > 0$ we obtain

$$\begin{aligned} \hat{M}\left( \left( \liminf _{n\rightarrow +\infty } \Vert u_n\Vert \right) ^p\right) =\hat{M}\left( \liminf _{n\rightarrow +\infty }\Vert u_n\Vert ^p\right) =\liminf _{n\rightarrow + \infty }\hat{M} \left( \Vert u_n\Vert ^p\right) . \end{aligned}$$

Hence,

$$\begin{aligned} \displaystyle \hat{M}(\Vert u_*\Vert )^p)\le \liminf _{n\rightarrow +\infty }\hat{M}\left( \Vert u_n\Vert ^p\right) =\liminf _{n\rightarrow +\infty }\left( \frac{a}{p}\Vert u_n\Vert ^p+\frac{b}{2p}\Vert u_n\Vert ^{2p} \right) . \end{aligned}$$

(2.3)

By Vital’s theorem (see [17] pp. 113), we can claim that

$$\begin{aligned} \displaystyle \lim _{n\rightarrow \infty } \int _{\Omega }g(x)\left| u_n\right| ^{1-\gamma }dx=\int _{\Omega }g(x)|u_*|^{1-\gamma }dx. \end{aligned}$$

(2.4)

Indeed, we only need to prove that $\displaystyle \{\smallint \nolimits _{\Omega }g(x)\left| u_n\right| ^{1-\gamma }dx,n\in \mathbb {N}\}$ is equi-absolutely-continuous. Note that $\{u_n\}$ is bounded, by the Sobolev embedding theorem, there exits a constant $C>0$, such that $\mid u_n\mid _{p^*}\le C$. For every $\varepsilon >0$, by the absolutely-continuity of $\displaystyle \smallint \nolimits _{\Omega }|g(x)|^{\frac{p^*}{p^*+\gamma -1}}dx$, there exists $\delta > 0$ such that

$$\begin{aligned} \displaystyle \int _{\Omega }|g(x)|^{\frac{p^*}{p^*+\gamma -1}}dx \le \varepsilon ^{\frac{p^*}{p^*+\gamma -1}} \; \text{ for } \text{ every } \;\;E\subset \Omega \; \text{ with } \;meas E< \delta . \end{aligned}$$

Consequently, by the Hölder inequality, we have:

$$\begin{aligned} \displaystyle \int _{\Omega }|g(x)|\mid u_n\mid ^{1-\gamma }dx \le \left| u_{n}^{1-\gamma }\right| _{p^*}\left( \int _{\Omega }|g(x)|^{\frac{p^*}{p^*+\gamma -1}}dx \right) ^{\frac{p^*+\gamma -1}{p^*}}< C^{1-\gamma }\varepsilon . \end{aligned}$$

Thus, claim (2.4) is valid.

Using inequality (1.3) and the Lebesgue dominated convergence theorem, we have:

$$\begin{aligned} \displaystyle u \mapsto \lambda \int _{\Omega }F(x,u(x))dx \end{aligned}$$

is weakly continuous, then

$$\begin{aligned}&\displaystyle \lim _{n\rightarrow \infty } \frac{1}{1-\gamma }\int _{\Omega }g(x)|u_n|^{1-\gamma }dx-\lambda \int _{\Omega }F(x,u_n(x))dx \nonumber \\&\quad = \frac{1}{1-\gamma }\int _{\Omega }g(x)|u_*|^{1-\gamma }dx-\lambda \int _{\Omega }F(x,u_*(x))dx. \end{aligned}$$

(2.5)

Using (2.3), (2.4) and (2.5) we deduce that $I_\lambda$ is weakly lower semi-continuous and consequently

$$\begin{aligned} m_\lambda \le I_\lambda (u_*)\le \liminf _{n\rightarrow +\infty }I_\lambda (u_n)=m_\lambda , \end{aligned}$$

then

$$\begin{aligned} I_\lambda (u_*)=m_\lambda . \end{aligned}$$

Similar to the arguments in [7, 8, 13], we can prove that $u_*$ is a solution of problem $(\mathbf P _{\lambda ,p})$. This shows that $(\mathbf P _{\lambda ,p})$ has a negative energy solution. This completes the proof of Proposition 2.1 and Theorem 1.2$\square$

3 Proof of Theorems 1.3 and 1.4

In this case, we consider the Kirchhoff type problem for $p=2$

$$\begin{aligned} (\mathbf P _{\lambda ,2}) \left\{ \begin{aligned} \begin{array}{ll} -\Big (a+b\int _{\Omega }|\nabla u|^2dx\Big )\Delta u =g(x)u^{-\gamma }+\lambda f(x,u), &{}\quad \text {in }\;\Omega , \\ u>0, &{}\quad \text {in }\;\Omega ,\\ u=0, &{} \quad \text {on }\;\partial \Omega . \end{array} \end{aligned} \right. \end{aligned}$$

For $u\in H_0^1(\Omega )$, we define

$$\begin{aligned} \begin{aligned} I_\lambda (u)&=\frac{a}{2}\int _{\Omega }|\nabla u|^2dx+\frac{b}{4} \Big (\int _{\Omega }|\nabla u|^2dx\Big )^2 \\&\quad -\frac{1}{1-\gamma }\int _{\Omega }g(x)|u|^{1-\gamma }dx -\lambda \int _{\Omega }F(x,u(x))dx, \end{aligned} \end{aligned}$$

where $H_0^1(\Omega )$ is the Sobolev space equipped with the norm $\Vert u\Vert =\big (\int _{\Omega }|\nabla u|^2dx\big )^{1/2}$. Note that a function u is called a weak solution of $(\mathbf P _{\lambda ,2})$ if $u\in H_0^1(\Omega )$ and satisfies the following:

$$\begin{aligned} \Big (a+b\int _{\Omega }|\nabla u|^2dx\Big ) \int _{\Omega }\nabla u.\nabla \varphi dx -\int _{\Omega }g(x)|u|^{-\gamma }\varphi dx-\lambda \int _{\Omega }f(x,u)\varphi dx=0, \end{aligned}$$

(3.1)

for all $\varphi \in H_0^1(\Omega )$.

In what follows, it is very important to mention that the Lemmas 2.1, 2.2 and Proposition 2.1 are also true for $p=2$, so we have automatically the existence result of a solution denoted $u_*$ and we focus the last part of this paper to prove the positivity of this solution when $\Omega =\Omega _1$ and a uniqueness result when $\lambda =0$.

Remark 3.1

On one hand, to the best of our knowledge, the existence and uniqueness of solutions for problem $(\mathbf P _{\lambda ,2})$ has not been studied up to now. The results that we obtain in Theorem 1.2 and Theorem 1.3 holds not only for the degenerate case, but also for the non-degenerate case. On the other hand, in references [7,8,9], problem (1.2) was considered only in dimension $N = 3$. However, we get the existence and uniqueness of solution for problem (1.1) in high dimensions, i.e. $N \ge 3$.

Remark 3.2

When $a=1, b=0$, problem $(\mathbf P _{\lambda ,2})$ reduces to the classic semilinear singular equation. Theorem 1.2 is also true. Moreover, when $\lambda = 0$, our Theorem 1.2 is the corresponding result of [16]. We point out that the condition that $\displaystyle g\in L^{\frac{2^*}{2^*+\gamma -1}}(\Omega )$ is more general than the condition that $\displaystyle g \in L^\infty (\Omega )$ in [16].

3.1 Proof of Theorem 1.3

In this section, we prove that $u_*$ is the unique solution of problem $(\mathbf P _{0,2})$. Assume that $v_*$ is another positive solution of problem $(\mathbf P _{0,2})$. Since $u_*,v_*$ are positive solutions of problem $(\mathbf P _{0,2})$, then it follows from (3.1) that

$$\begin{aligned} \left( a+b\Vert u_*\Vert ^2\right) \int _{\Omega } \left( \nabla u_*,\nabla (u_*-v_*)\right) dx-\int _{\Omega }g(x)u_*^{-\gamma }(u_*-v_*) dx=0, \end{aligned}$$

(3.2)

and

$$\begin{aligned} \left( a+b\Vert v_*\Vert ^2\right) \int _{\Omega } (\nabla v_*, \nabla (u_*-v_*)) dx-\int _{\Omega } g(x)v_ *^{-\gamma }(u_*-v_*) dx=0. \end{aligned}$$

(3.3)

From (3.2) and (3.3), one obtains

$$\begin{aligned} \displaystyle 0&= \left( a+b\Vert u_*\Vert ^2\right) \displaystyle \int _{\Omega }\left( \nabla u_*,\nabla \left( u_*-v_*\right) \right) dx-\displaystyle \left( a+b\Vert v_*\Vert ^2\right) \nonumber \\&\quad \times \int _{\Omega }\left( \nabla v_*,\nabla \left( u_*-v_*\right) \right) dx-\displaystyle \int _{\Omega }g(x)\left( u_*^{-\gamma }-v_*^{-\gamma }\right) \left( u_*-v_*\right) dx,\nonumber \\ \displaystyle&=a\Vert u_*-v_*\Vert ^2+b\Big [ \Vert u_*\Vert ^4-\Vert u_*\Vert ^2\displaystyle \int _{\Omega }\left( \nabla u_*,\nabla v_*\right) dx- \Vert v_*\Vert ^2\displaystyle \nonumber \\&\quad \times \int _{\Omega }\left( \nabla v_*,\nabla v_*\right) dx+\Vert v_*\Vert ^4\Big ]-\displaystyle \int _{\Omega }g(x)\left( u_*^{-\gamma }-v_*^{-\gamma }\right) \left( u_*-v_*\right) dx. \end{aligned}$$

(3.4)

Denote

$$\begin{aligned} J(u_*,v_*)=\Vert u_*\Vert ^4-\Vert u_*\Vert ^2\displaystyle \int _{\Omega }(\nabla u_*,\nabla v_*)dx- \Vert v_*\Vert ^2\displaystyle \int _{\Omega }(\nabla v_*,\nabla v_*)dx+\Vert v_*\Vert ^4. \end{aligned}$$

By the Hölder inequality, one has

$$\begin{aligned} \begin{aligned} \displaystyle J(u_*,v_*)&\ge \Vert u_*\Vert ^4-\Vert u_*\Vert ^3\Vert v_*\Vert -\Vert v_*\Vert ^3\Vert u_*\Vert +\Vert v_*\Vert ^4\\ \displaystyle&=(\Vert u_*\Vert -\Vert v_*\Vert )^2 \left( \Vert u_*\Vert ^2+\Vert u_*\Vert \Vert v_*\Vert + \Vert v_*\Vert ^2\right) \\ \displaystyle&\ge 0. \end{aligned} \end{aligned}$$

Since $0<\gamma < 1$, we have the following elementary inequality

$$\begin{aligned} (h^{-\gamma }-l^{-\gamma })(h-l))\le 0\;\;\;\forall \; h,l> 0. \end{aligned}$$

Thus

$$\begin{aligned} \displaystyle \int _{\Omega }g(x) \left( u_*^{-\gamma }-v_*^{-\gamma }\right) (u_*-v_*)dx\le 0. \end{aligned}$$

Consequently, it follows from (3.4) that $a\Vert u_*-v_*\Vert ^2\le 0$. If $a=0$, one has $\Vert u_*\Vert =\Vert v_*\Vert$ and $J(u_*,v_*)=0$. As a result,

$$\begin{aligned} J(u_*,v_*)=\Vert u_*\Vert ^2 \left( 2\Vert u_*\Vert ^2-2\displaystyle \int _{\Omega } (\nabla u_*,\nabla v_*)dx\right) =\Vert u_*\Vert ^2\Vert u_*-v_*\Vert ^2=0, \end{aligned}$$

this implies $\Vert u_*-v_*\Vert ^2=0$.

Thus, for every $a\ge 0$, one has $u_*=v_*$. Therefore $u_*$ is the unique solution of problem $(\mathbf P _{\lambda ,2})$. This completes the proof of Theorem 1.3.

3.2 Proof of Theorem 1.4

In what follows and without loss of generality, let us assume that $\Omega =\Omega _1$ and $u_*\ge 0$, we can prove the following:

Proposition 3.3

If the set $\Omega _1$ given by (H2) is such that $\Omega =\Omega _1$, then the solution $u_*$ of problem $(\mathbf P _{\lambda ,2})$ is nonnegative.

Proof

From Proposition 2.1, we have $I_\lambda (u_*)=m_\lambda <0$, so $u_*\ge 0$ and $u_*\not \equiv 0$ in $\Omega$.

We prove that $u_*(x)> 0$ for almost every $x\in \Omega$. Since $u_*(x)\ge 0$ for all $x\in \Omega$, then $\forall \; \phi \in H_0^1(\Omega ),\phi \ge 0$, and $t>0,t\in \mathbb {R}$, such that $u_*+t\phi \in H_0^1(\Omega )$, we have the following

$$\begin{aligned} 0&\le \displaystyle \liminf _{t\rightarrow 0}\frac{I_\lambda (u_*+t\phi )-I_\lambda (u_*)}{t}\nonumber \\ \displaystyle&=a\int _{\Omega }(\nabla u_*,\nabla \phi )dx+b\Vert u_*\Vert ^2\int _{\Omega }(\nabla u_*,\nabla \phi )dx \nonumber \\&\quad -\displaystyle \limsup _{t\rightarrow 0^+} \frac{1}{1-\gamma } \int _{\Omega }g(x) \frac{\left( (u_*+t\phi )^{1-\gamma }-u_*^{1-\gamma }\right) }{t}dx \nonumber \\&\quad -\displaystyle \lim _{t\rightarrow 0^+}\lambda \int _{\Omega }\frac{F(x,u_*+t\phi )-F(x,u_*)}{t}dx. \end{aligned}$$

(3.5)

Obviously, one gets

$$\begin{aligned} \begin{aligned} \displaystyle \int _{\Omega }\frac{F(x,u_*+t\phi )-F(x,u_*)}{t}dx&=\int _{\Omega }f(x,u_*+t\eta \phi )\phi dx,\\ \displaystyle \int _{\Omega }\frac{\left( (u_*+t\phi )^{1-\gamma }-u_*^{1-\gamma }\right) }{t}dx&=(1-\gamma )\int _{\Omega }(u_*+t\phi )^{-\gamma }\phi dx, \end{aligned} \end{aligned}$$

where $0<\eta<1,\theta <1,$ and

$$\begin{aligned} \displaystyle f(x,u_*+\eta t \phi )\phi \rightarrow f(x,u_*)\phi ,\;\;\;\;\;a.e.\; x\in \Omega , \\ \displaystyle (u_*+\theta t\phi )^{-\gamma }\phi \rightarrow u_*^{-\gamma }\phi ,\;\;\;\;\;a.e.\; x\in \Omega , \end{aligned}$$

as $t\rightarrow 0^+$.

For any $x\in \Omega$, put $\displaystyle h(t)=g(x)\displaystyle \frac{[u_*(x)+t\phi (x)]^{1-\gamma }-u_*^{1-\gamma }(x)}{(1-\gamma )t}$. then $h'(t)=g(x)\displaystyle \frac{u_*^{1-\gamma }(x)-[\gamma t\phi (x)+u_*(x)][u_*(x)+t\phi (x)]^{-\gamma }}{(1-\gamma )t^2}\le 0$, which implies that h is non-increasing on $(0,\infty )$. Moreover, one has $\displaystyle \lim _{t\rightarrow 0^+}h(t)=g(x)u_*^{1-\gamma }(x)\phi (x)$ for $x\in \Omega$, which may be $+\infty$ when $u_*(x)=0$ and $\phi (x) > 0$. Consequently, by the Monotone Convergence Theorem (Beppo-Levi), we obtain

$$\begin{aligned} \displaystyle \lim _{t\rightarrow 0^+} \frac{1}{1-\gamma }\int _{\Omega }g(x) \frac{\left( (u_*+t\phi )^{1-\gamma }-u_*^{1-\gamma }\right) }{t}dx =\displaystyle \int _{\Omega }g(x)u_*^{-\gamma }\phi dx, \end{aligned}$$

(3.6)

which possibly equals to $+\infty$.

Moreover, using Lebesgue’s dominated convergence theorem on the function f, one has:

$$\begin{aligned} \displaystyle \lim _{t\rightarrow 0^+} \int _{\Omega }\frac{F(x,u_*+t\phi )-F(x,u_*)}{t}dx&=\displaystyle \int _{\Omega }\lim _{t\rightarrow 0^+}f(x,u_*+t\phi )\phi dx\nonumber \\&=\displaystyle \int _{\Omega }f(x,u_*)\phi dx. \end{aligned}$$

(3.7)

Then, from (3.5),(3.6) and (3.7), we obtain

$$\begin{aligned} \displaystyle \displaystyle \int _{\Omega }g(x)u_*^{-\gamma }\phi dx \le \left( a+b\Vert u_*\Vert ^2\right) \int _{\Omega }(\nabla u_*,\nabla \phi )dx -\displaystyle \int _{\Omega }f(x,u_*)\phi dx , \end{aligned}$$

(3.8)

for all $\phi \in H_0^1(\Omega )$ with $\phi > 0$.

Thus, one has

$$\begin{aligned} \displaystyle \int _{\Omega }(\nabla u_*,\nabla \phi )dx \ge 0. \end{aligned}$$

Since $u_* \ge 0$ and $u_*\not \equiv 0$, by strong maximum principal, it follows that

$$\begin{aligned} u_* > 0,\;\;\;\ \forall x \in \Omega . \end{aligned}$$

This completes the proof of Theorem 1.4. $\square$

4 An Example

In this section, we give an example to illustrate our results. To this aim, we fix $p\ge 2$ and a bounded domain $\Omega \subset \mathbb {R}^3$. Let $\displaystyle g\in L^{\frac{p^*}{p^*+\gamma -1}}(\Omega )$ such that for almost every $x \in \Omega$ we have $g(x)> 0$. We consider the following elliptic problem

$$\begin{aligned} (\mathbf P _{\lambda ,p}) \left\{ \begin{aligned} \begin{array}{ll} -\Big (a+b\int _{\Omega }|\nabla u|^pdx\Big )\Delta _p u =g(x)u^{-\gamma }+\lambda h(x)|u(x)|^{r-2}u(x),&{}\quad \text {in }\;\Omega , \\ u=0, &{} \quad \text {on }\partial \Omega , \end{array} \end{aligned} \right. \end{aligned}$$

where $0<\gamma<1<r<p$, $a,b\ge 0$ such that $a+b>0$ and h be a positive bounded function in $\Omega$. It is easy to see that $f(x,t)=h(x)|t|^{r-2}t$ is positively homogeneous of degree $r-1$, moreover, by a simple calculation we obtain $F(x,t)=h(x)|t|^{r}$ which is positively homogeneous of degree r. That is ($\mathbf H _{1}$) is satisfied. On the other hand, since $h>0$, it is easy to see that for all $x\in \Omega$ , we have $F(x,t)=h(x)|t|^{r}>0$, that is ($\mathbf H _{2}$) is satisfied and $\Omega _1=\Omega$. Hence, all conclusions of Theorems 1.2, 1.3 and 1.4 hold true.