# Existence and multiplicity of positive solutions for p-Kirchhoff type problem with singularity

## Abstract

In this paper, we consider a class of p-Kirchhoff type problems with a singularity in a bounded domain in $$R^{N}$$. By using the variational method, the existence and multiplicity of positive solutions are obtained.

## 1 Introduction and main results

The purpose of this paper is to investigate the existence of multiple positive solutions to the following problem:

$$\textstyle\begin{cases} -M(\int_{\Omega} \vert \nabla u\vert ^{p} \,dx)\Delta_{p} u =\lambda f(x)u^{-r}+g(x)u^{q-1} \quad\text{in } \Omega,\\ u=0 \quad\text{on } \partial\Omega, \end{cases}$$
(1.1)

where $$\Delta_{p}u= \operatorname{div}(\vert \nabla u\vert ^{p-2}\nabla u)$$, Î© is a smooth bounded domain in $$R^{N}$$, $$0< r<1<p<q<p^{*}$$ ($$p^{*}=\frac {Np}{N-p}$$ if $$N>p$$ and $$p^{*}=\infty$$ if $$N\leq p$$), $$M(s)=as^{p-1}+b$$ and $$a,b,\lambda>0$$. $$f, g \in C(\overline{\Omega})$$ are nontrivial nonnegative functions.

Problem (1.1) is related to the stationary problem introduced by Kirchhoff in [1]. More precisely, it is the model

$$\rho\frac{\partial^{2}u}{\partial t^{2}}- \biggl(\frac{\rho_{0}}{h}+\frac {E}{2L} \int_{0}^{L} \biggl\vert \frac{\partial u}{\partial x} \biggr\vert ^{2} \,d x \biggr)\frac {\partial^{2}u}{\partial x^{2}}=0,$$

where $$\rho,\rho_{0}, E, L$$ are constants, which was proposed as an extension of the classical Dâ€™Alembertâ€™s wave equation for free vibrations of elastic strings to describe transversal oscillations of a stretched string. For more details and backgrounds, we refer to [2, 3].

The existence and multiplicity of solutions for the following problem:

$$\textstyle\begin{cases} -M(\int_{\Omega} \vert \nabla u\vert ^{p} \,dx)\Delta_{p} u = h(x,u) \quad\text{in } \Omega ,\\ u=0 \quad\text{on } \partial\Omega, \end{cases}$$
(1.2)

on a smooth bounded domain $$\Omega\subset R^{N}$$ has been studied in many papers. Liu and Zhao [4] proved (1.2) has at least two nontrivial weak solutions by Morse theory under some restriction on $$M(s)$$ and $$h(x,u)$$. In [5], the authors considered the following problem:

$$\textstyle\begin{cases} -M(\int_{\Omega} \vert \nabla u\vert ^{p} \,dx)\Delta_{p} u = \lambda f(x)\vert u\vert ^{q-2}u+g(x)\vert u\vert ^{r-2}u \quad\text{in } \Omega,\\ u=0 \quad\text{on } \partial\Omega, \end{cases}$$
(1.3)

where $$M(s)=as+b$$, $$1< q< p< r\leq p^{*}$$, they proved the existence of multiplicity nontrivial solutions by using the Nehari manifold when the weight functions $$f(x)$$ and $$g(x)$$ change their signs. For more results, we refer to [6â€“10] and the references therein.

When $$p=2$$ and $$N=3$$, problem (1.1) reduces to the following singular Kirchhoff type problem:

$$\textstyle\begin{cases} -(a+b \int_{\Omega} \vert \nabla u\vert ^{2} \,dx)\Delta u =\lambda f(x) u^{-r}+\mu g(x)u^{q} \quad\text{in } \Omega,\\ u=0\quad\text{on } \partial\Omega, \end{cases}$$
(1.4)

where $$1< q \leq5$$, the existence of solutions for problem (1.4) has been widely studied (see [11â€“13]). When $$3< q < 5$$ and $$\lambda=1$$, Liu and Sun [11] proved that problem (1.4) has at least two positive solutions for $$\mu>0$$ small enough. Liao et al. showed the multiplicity of positive solutions by the Nehari mainfold in the case of $$q=3$$ in [12]. When q is a critical exponents, at least two positive solutions are obtained by variational and perturbation methods in [13].

However, the singular p-Kirchhoff type problems have few been considered, especially $$p\neq2, N\neq3$$. Here we focus on extending the results in [11] and [12]. In fact, the extension is nontrivial and requires a more careful analysis. Our method is based on the Nehari manifold; see [6, 11, 14, 15].

Before starting our main theorems, we make use of the following notations:

â€¢ Let $$W^{1,p}_{0}(\Omega)$$ be the Sobolev space with norm $$\Vert u\Vert =(\int_{\Omega} \vert \nabla u\vert ^{p} \,dx)^{\frac{1}{p}}$$, the norm in $$L^{p}(\Omega)$$ is denoted by $$\parallel\centerdot\parallel_{p}$$;

â€¢ Let $$S_{z}$$ be the best Sobolev constant for the embedding of $$W_{0}^{1,p}(\Omega)$$ in $$L_{z}(\Omega)$$ with $$0< z< p^{*}$$. Then, for all $$u\in W_{0}^{1,p}(\Omega)\backslash\{0\}$$,

$$\Vert u\Vert _{z}\leq S_{z}^{-\frac{1}{p}}\Vert u \Vert .$$

In general, we say that a function $$u\in W_{0}^{1,p}(\Omega)$$ is a weak solution of problem (1.1) if

$$M \bigl(\Vert u\Vert ^{p} \bigr) \int_{\Omega} \vert \nabla u\vert ^{p-2}\nabla u \nabla \varphi \,dx-\lambda \int_{\Omega} f\vert u\vert ^{-r}\varphi \,dx- \int_{\Omega} g\vert u\vert ^{q-1}\varphi \,dx=0$$

for all $$\varphi\in W_{0}^{1,p}(\Omega)$$. Thus, the functional corresponding to problem (1.1) is defined by

$$J(u)=\frac{1}{p} \widehat{M} \bigl(\Vert u\Vert ^{p} \bigr)- \frac{\lambda}{1-r} \int _{\Omega} f\vert u\vert ^{1-r} \,dx- \frac{1}{q} \int_{\Omega}g \vert u\vert ^{q} \,dx, \quad\forall u \in W_{0}^{1,p}(\Omega),$$

where $$\widehat{M}(s)=\int_{0}^{s}M(t) \,dt$$.

To obtain the existence results, we introduce the Nehari manifold:

$$N_{\lambda}= \biggl\{ u \in W_{0}^{1,p}(\Omega)\setminus \{0\}: M \bigl(\Vert u\Vert ^{p} \bigr)\Vert u\Vert ^{p} -\lambda \int_{\Omega} f\vert u\vert ^{1-r} \,dx- \int_{\Omega} g\vert u\vert ^{q} \,dx=0 \biggr\} ,$$

and we define

$$K(u)=(p-1) M \bigl(\Vert u\Vert ^{p} \bigr)\Vert u\Vert ^{p}+ p M' \bigl(\Vert u\Vert ^{p} \bigr) \Vert u\Vert ^{2p}+\lambda r \int_{\Omega} f\vert u\vert ^{1-r} \,dx-(q-1) \int_{\Omega} g\vert u\vert ^{q} \,dx.$$

Now we split $$N_{\lambda}$$ into three disjoint parts as follows:

\begin{aligned} &N_{\lambda}^{+}= \bigl\{ u \in N_{\lambda}:K(u)>0 \bigr\} ; \\ &N_{\lambda}^{0}= \bigl\{ u \in N_{\lambda}:K(u)=0 \bigr\} ; \\ &N_{\lambda}^{-}= \bigl\{ u \in N_{\lambda}:K(u)< 0 \bigr\} . \end{aligned}

Let $$\lambda^{*}=$$ $$\max \{\frac{(1-r)\lambda_{1}(a)}{p^{\frac {p+1}{p}}},\frac{(1-r)\lambda_{2}}{p} \}$$, where $$\lambda_{1}(a)$$ and $$\lambda_{2}$$ are given by

$$\lambda_{1}(a)=\frac{pS_{1-r}^{\frac{1-r}{p}} \sqrt[p]{ab^{p-1}(q-p^{2})(\frac{q-p}{p-1})^{p-1}}}{(q+r-1)\Vert f\Vert _{\infty}} \biggl(\frac{pS_{q}^{\frac{q}{p}}\sqrt[p]{ab^{p-1}(p^{2}+r-1)}}{(q+r-1)\Vert g\Vert _{\infty}} \biggr)^{\frac{2p+r-2}{q-2p+1}}$$

and

$$\lambda_{2}=\frac{bS_{1-r}^{\frac{1-r}{p}}(q-p)}{(q+r-1)\Vert f\Vert _{\infty }} \biggl( \frac{bS_{q}^{\frac{q}{p}}(p+r-1)}{(q+r-1)\Vert g\Vert _{\infty}} \biggr)^{\frac{p +r-1}{q-p}},$$

then we state the main theorems.

### Theorem 1.1

Assume that $$p^{2}< q< p^{*}$$ and $$N<2p$$. Then, for each $$a>0$$ and $$0<\lambda<\lambda^{*}$$, the problem (1.1) has at least two positive solutions $$u_{\lambda}^{+}\in N_{\lambda }^{+}$$ and $$u_{\lambda}^{-}\in N_{\lambda}^{-}$$.

Define

$$\Lambda=\inf\biggl\{ \Vert u\Vert ^{p^{2}}:u \in W_{0}^{1,p}(\Omega), \int_{\Omega }g\vert u\vert ^{p^{2}} \,dx=1\biggr\} ,$$
(1.5)

then $$\Lambda>0$$ is obtained by some $$\phi_{\Lambda}\in W_{0}^{1,p}(\Omega)$$ with $$\int_{\Omega}g \vert \phi_{\Lambda} \vert ^{p^{2}} \,dx=1$$. In particular,

$$\Lambda \int_{\Omega}g \vert u\vert ^{p^{2}} \,dx \leq \Vert u \Vert ^{p^{2}}$$
(1.6)

and

$$\textstyle\begin{cases} -\Vert u\Vert ^{p}\Delta_{p} u = \mu g\vert u\vert ^{p^{2}-1} \quad\text{in } \Omega,\\ u=0\quad\text{on } \partial\Omega, \end{cases}$$
(1.7)

where Î¼ is an eigenvalue of (1.7), $$u\in W_{0}^{1,p}(\Omega)$$ is nonzero and an eigenvector corresponding to Î¼ such that

$$\Vert u\Vert ^{p} \int_{\Omega} \vert \nabla u\vert ^{p-2}\nabla u \nabla(u\varphi) \,dx=\mu \int_{\Omega}g \vert u\vert ^{p^{2}-1}u\varphi \,dx,\quad \text{for all } \varphi\in W_{0}^{1,p}(\Omega);$$

we write

$$I(u)=\Vert u\Vert ^{p^{2}},\quad \text{for } u \in E= \biggl\{ u\in W_{0}^{1,p}(\Omega): \int _{\Omega}g \vert u\vert ^{p^{2}} \,dx =1 \biggr\} ,$$

and all distinct eigenvalues of (1.7) â€‰denoted by $$0<\mu_{1}<\mu _{2}<\cdots$$â€‰, we have

$$\mu_{1}=\inf_{u\in E}I(u)>0,$$

where $$\mu_{1}$$ is simple, isolated and can be obtained at some $$\psi \in E$$ and $$\psi>0$$ in Î© (see [16]).

### Theorem 1.2

Assume that $$p^{2}=q< p^{*}$$ and $$N<2p$$. Then

1. (i)

for each $$a\geq\frac{1}{\Lambda}$$ and $$\lambda>0$$, the problem (1.1) has at least one positive solution;

2. (ii)

for each $$a<\frac{1}{\Lambda}$$ and $$0<\lambda<\frac{1-r}{p}\hat {\lambda}$$, where

$$\hat{\lambda}=\frac{bS_{1-r}^{\frac{1-r}{p}}(p^{2}-p)}{\Vert f\Vert _{\infty }(p^{2}+r-1)} \biggl(\frac{b\Lambda(p+r-1)}{(1-a\Lambda)(p^{2}+r-1)} \biggr)^{\frac {p+r-1}{p^{2}-p}},$$

the problem (1.1) has at least two positive solutions $$u_{\lambda }^{+}\in N_{\lambda}^{+}$$, $$u_{\lambda}^{-}\in N_{\lambda}^{-}$$ and

$$\lim_{a\to\frac{1}{\Lambda}^{-}}\inf_{u\in N_{\lambda}^{-}} J(u)=\infty.$$

This paper is organized as follows: In SectionÂ 2, we present some lemmas which will be used to prove our main results. In SectionÂ 3 and SectionÂ 4, we will prove Theorems 1.1 and 1.2, respectively.

## 2 Preliminaries

### Lemma 2.1

(i) If $$q\geq p^{2}$$, then the energy functional $$J(u)$$ is coercive and bounded below in $$N_{\lambda}$$;

(ii) if $$q< p^{2}$$, then the energy functional $$J(u)$$ is coercive and bounded below in $$W_{0}^{1,p}(\Omega)$$.

### Proof

(i) For $$u\in N_{\lambda}$$, we have

$$M \bigl(\Vert u\Vert ^{p} \bigr)\Vert u\Vert ^{p} - \lambda \int_{\Omega} f\vert u\vert ^{1-r} \,dx- \int_{\Omega } g\vert u\vert ^{q} \,dx=0.$$

By the Sobolev inequality,

\begin{aligned} J(u)&= \frac{1}{p} \widehat{M}\bigl(\Vert u\Vert ^{p}\bigr)-\frac{\lambda}{1-r} \int _{\Omega} f\vert u\vert ^{1-r} \,dx- \frac{1}{q} \int_{\Omega}g \vert u\vert ^{q} \,dx \\ &= \frac{1}{p} \widehat{M}\bigl(\Vert u\Vert ^{p}\bigr)- \frac{1}{q} M\bigl(\Vert u\Vert ^{p}\bigr)\Vert u\Vert ^{p}-\lambda\frac{q+r-1}{q(1-r)} \int_{\Omega} f\vert u\vert ^{1-r} \,dx \\ &\geq \frac{\Vert u\Vert ^{p}}{pq} \biggl( \frac{a(q-p^{2})}{p}\Vert u\Vert ^{p^{2}-p}+b(q-p) \biggr)-\lambda\frac{q+r-1}{q(1-r)}\Vert f\Vert _{\infty }S_{1-r}^{\frac{r-1}{p}}\Vert u\Vert ^{1-r} \\ &\geq \frac{b(q-p)}{pq}\Vert u\Vert ^{p}-\lambda \frac{q+r-1}{q(1-r)}\Vert f\Vert _{\infty}S_{1-r}^{\frac{r-1}{p}} \Vert u\Vert ^{1-r}. \end{aligned}

Thus, $$J(u)$$ is coercive and bounded below in $$N_{\lambda}$$.

(ii) For $$u\in W_{0}^{1,p}(\Omega)$$, we have

\begin{aligned} J(u)&= \frac{1}{p} \widehat{M}\bigl(\Vert u\Vert ^{p}\bigr)-\frac{\lambda}{1-r} \int _{\Omega} f\vert u\vert ^{1-r} \,dx- \frac{1}{q} \int_{\Omega}g \vert u\vert ^{q} \,dx \\ &\geq \frac{a}{p^{2}} \Vert u\Vert ^{p^{2}}+\frac{b}{p} \Vert u\Vert ^{p}-\frac{\lambda \Vert f\Vert _{\infty}S_{1-r}^{\frac{r-1}{p}} }{1-r}\Vert u\Vert ^{1-r}-\frac{\Vert g\Vert _{\infty}S_{q}^{-\frac{q}{p}} }{q}\Vert u\Vert ^{q} \\ &= \biggl( \frac{a}{p^{2}} \Vert u\Vert ^{p^{2}-q}- \frac{\Vert g\Vert _{\infty}S_{q}^{-\frac {q}{p}} }{q} \biggr)\Vert u\Vert ^{q}+ \biggl( \frac{b}{p}\Vert u\Vert ^{p+r-1}-\frac {\lambda \Vert f\Vert _{\infty}S_{1-r}^{\frac{r-1}{p}} }{1-r} \biggr) \Vert u\Vert ^{1-r}. \end{aligned}

Thus, $$J(u)$$ is coercive and bounded below in $$W_{0}^{1,p}(\Omega)$$.â€ƒâ–¡

### Lemma 2.2

If $$q> p^{2}$$ and $$0<\lambda<\max \{\lambda_{1}(a),\lambda_{2}\}$$, then, for all $$a>0$$,

1. (i)

the submanifold $$N_{\lambda}^{0}=\emptyset$$;

2. (ii)

the submanifold $$N_{\lambda}^{\pm}\neq\emptyset$$.

### Proof

(i) Suppose $$N_{\lambda}^{0}\neq \emptyset$$. Then, for $$u\in N_{\lambda}^{0}$$, we have

\begin{aligned} (q+r-1)\Vert g\Vert _{\infty}S_{q}^{-\frac{q}{p}} \Vert u\Vert ^{q}&\geq (q+r-1) \int _{\Omega}g \vert u\vert ^{q} \,dx \\ &= a\bigl(p^{2}+r-1\bigr)\Vert u\Vert ^{p^{2}}+b(p+r-1) \Vert u\Vert ^{p} \\ &\geq \textstyle\begin{cases} p\sqrt[p]{ab^{p-1}(p^{2}+r-1)}\Vert u\Vert ^{2p-1}, \\ b(p+r-1)\Vert u\Vert ^{p}, \end{cases}\displaystyle \end{aligned}
(2.1)

and

\begin{aligned} \lambda(q+r-1)\Vert f\Vert _{\infty}S_{1-r}^{-\frac{1-r}{p}} \Vert u\Vert ^{1-r}&\geq \lambda(q+r-1) \int_{\Omega}f \vert u\vert ^{1-r} \,dx \\ &= a\bigl(q-p^{2}\bigr)\Vert u\Vert ^{p^{2}}+b(q-p)\Vert u\Vert ^{p} \\ &\geq \textstyle\begin{cases} p\sqrt[p]{ab^{p-1}(q-p^{2})(\frac{q-p}{p-1})^{p-1}}\Vert u\Vert ^{2p-1},\\ b(q-p)\Vert u\Vert ^{p}. \end{cases}\displaystyle \end{aligned}
(2.2)

By (2.1) and (2.2), for all $$u\in N_{\lambda}^{0}$$, we have

$$\biggl( \frac{pS_{q}^{\frac{q}{p}}\sqrt[p]{ab^{p-1}(p^{2}+r-1)}}{(q+r-1)\Vert g\Vert _{\infty}} \biggr) ^{\frac{1}{q-2p+1}}\leq \Vert u\Vert \leq \biggl( \frac{\lambda(q+r-1)\Vert f\Vert _{\infty}}{ pS_{1-r}^{\frac {1-r}{p}}\sqrt[p]{ab^{p-1}(q-p^{2})(\frac{q-p}{p-1})^{p-1}}} \biggr)^{\frac {1}{2p+r-2}}$$

and

$$\biggl( \frac{bS_{q}^{\frac{q}{p}}(p+r-1)}{(q+r-1)\Vert g\Vert _{\infty}} \biggr)^{\frac{1}{q-p}} \leq \Vert u\Vert \leq \biggl( \frac{\lambda(q+r-1)\Vert f\Vert _{\infty}}{ bS_{1-r}^{\frac {1-r}{p}}(q-p)} \biggr)^{\frac{1}{p+r-1}}.$$

Hence, if $$N_{\lambda}^{0}$$ is nonempty, then the inequality $$\lambda\geq\max\{\lambda_{1}(a), \lambda_{2}\}$$ must hold.

(ii) Fix $$u\in W_{0}^{1,p}(\Omega)$$. Let

$$h_{a}(t)=at^{p^{2}-(1-r)}\Vert u\Vert ^{p^{2}}+bt^{p-(1-r)} \Vert u\Vert ^{p}-t^{q-(1-r)} \int _{\Omega}g \vert u\vert ^{q} \,dx\quad \text{for } a,t \geq0.$$

We see that $$h_{a}(0)=0$$ and $$h_{a}(t)\rightarrow-\infty$$ as $$t\rightarrow\infty$$. Since $$q> p^{2}$$ and

\begin{aligned} h'_{a}(t)={}&t^{p+r-2} \biggl(a \bigl(p^{2}+r-1 \bigr)t^{p^{2}-p}\Vert u\Vert ^{p^{2}}+b(p+r-1)\Vert u\Vert ^{p}\\ &{} -(q+r-1)t^{q-p} \int_{\Omega}g \vert u\vert ^{q} \,dx \biggr), \end{aligned}

there is a unique $$t_{a,\max}>0$$ such that $$h_{a}(t)$$ reaches its maximum at $$t_{a,\max}$$, increasing for $$t\in[0,t_{a,\max})$$ and decreasing for $$t \in(t_{a,\max},\infty)$$ with $$\lim_{t\rightarrow \infty}h_{a}(t)=-\infty$$. Clearly, if $$tu \in N_{\lambda}$$, then $$tu \in N_{\lambda}^{+}$$ (or $$N_{\lambda}^{-}$$) if and only if $$h'_{a}(t)>0$$ (or <0). Moreover,

$$t_{0,\max}= \biggl(\frac{b(p+r-1)\Vert u\Vert ^{p}}{(q+r-1)\int_{\Omega}g \vert u\vert ^{q} \,dx} \biggr)^{\frac{1}{q-p}}$$

and

\begin{aligned} &h_{0}(t_{0,\max})\\ &\quad= b^{\frac{q+r-1}{q-p}} \biggl[ \biggl( \frac{p+r-1}{q+r-1} \biggr)^{\frac {p+r-1}{q-p}} - \biggl( \frac{p+r-1}{q+r-1} \biggr)^{\frac{q+r-1}{q-p}} \biggr] \frac{\Vert u\Vert ^{\frac{p(q+r-1)}{q-p}}}{ (\int_{\Omega}g \vert u\vert ^{q} \,dx )^{\frac{p+r-1}{q-p}}} \\ &\quad\geq \frac{b(q-p)}{q+r-1} \biggl( \frac{bS_{q}^{\frac {q}{p}}(p+r-1)}{(q+r-1)\Vert g\Vert _{\infty}} \biggr)^{\frac{p+r-1}{q-p}} \Vert u\Vert ^{1-r}. \end{aligned}

On the other hand, since

\begin{aligned} h_{a}(0)=0 &< \lambda \int_{\Omega}f \vert u\vert ^{1-r} \,dx\leq\lambda \Vert f\Vert _{\infty }S_{1-r}^{-\frac{1-r}{p}}\Vert u\Vert ^{1-r} \\ &< \frac{b(q-p)}{q+r-1} \biggl( \frac{bS_{q}^{\frac {q}{p}}(p+r-1)}{(q+r-1)\Vert g\Vert _{\infty}} \biggr)^{\frac{p+r-1}{q-p}}\Vert u \Vert ^{1-r} \\ &\leq h_{0}(t_{0,\max})< h_{a}(t_{a,\max}), \end{aligned}
(2.3)

there exist unique $$t^{+}$$ and $$t^{-}$$ such that $$0< t^{+}< t_{a,\max}< t^{-}$$,

$$h_{a} \bigl(t^{+} \bigr)=\lambda \int_{\Omega}f \vert u\vert ^{1-r} \,dx=h_{a} \bigl(t^{-} \bigr)$$

and

$$h'_{a} \bigl(t^{+} \bigr)>0>h'_{a} \bigl(t^{-} \bigr).$$

That is, $$t^{+}u\in N_{\lambda}^{+}$$ and $$t^{-}u\in N_{\lambda}^{-}$$.â€ƒâ–¡

### Lemma 2.3

(i) If $$q= p^{2}$$ and $$a\geq\frac{1}{\Lambda}$$, then, for all $$\lambda>0$$, $$N_{\lambda}^{+}=N_{\lambda}\neq\emptyset$$;

(ii) if $$q= p^{2}, a<\frac{1}{\Lambda}$$ and $$0<\lambda<\hat{\lambda}$$, then $$N_{\lambda}=N_{\lambda}^{+}\cup N_{\lambda}^{-}$$ and $$N_{\lambda}^{\pm}\neq\emptyset$$.

### Proof

First, we show that $$N_{\lambda}^{+}=N_{\lambda}$$.

Indeed, for all $$u \in N_{\lambda}$$, we have

\begin{aligned} & a\bigl(p^{2}+r-1\bigr)\Vert u\Vert ^{p^{2}}+b(p+r-1)\Vert u\Vert ^{p}-\bigl(p^{2}+r-1 \bigr) \int_{\Omega }g\vert u\vert ^{p^{2}} \,dx \\ &\quad\geq \frac{(a\Lambda-1)(p^{2}+r-1)}{\Lambda} \Vert u\Vert ^{p^{2}}+b(p+r-1)\Vert u \Vert ^{p}>0. \end{aligned}

Therefore, $$u\in N_{\lambda}^{+}$$.

Next, we declare $$N_{\lambda}^{+}\neq\emptyset$$.

Fix $$u\in W_{0}^{1,p}(\Omega)$$. Let

$$\bar{h}(t)=t^{p^{2}+r-1} \biggl(a\Vert u\Vert ^{p^{2}}- \int_{\Omega}g \vert u\vert ^{p^{2}} \,dx \biggr)+bt^{p+r-1}\Vert u\Vert ^{p}\quad \mbox{for } a,t \geq0.$$

Obviously, $$\bar{h}(0)=0$$ and $$\lim_{t \to\infty}\bar{h}(t)=\infty$$. Since

\begin{aligned} \bar{h}'(t)={}& \bigl(p^{2}+r-1 \bigr)t^{p^{2}+r-2} \biggl(a\Vert u\Vert ^{p^{2}}- \int_{\Omega}g \vert u\vert ^{p^{2}} \,dx \biggr)\\ &{}+b(p+r-1)t^{p+r-2}\Vert u\Vert ^{p}, \end{aligned}

we can deduce that $$\bar{h}(t)$$ is increasing for $$t\in[0,\infty)$$. Thus, there is a unique $$t^{+}>0$$ such that $$\bar{h}(t^{+})=\lambda\int_{\Omega}f \vert u\vert ^{1-r} \,dx$$ and $$\bar {h}'(t^{+})>0$$. That is, $$t^{+}u \in N_{\lambda}^{+}$$.

(ii) The proof is similar to Lemma 2.2, we omit it here.â€ƒâ–¡

We write $$N_{\lambda}=N_{\lambda}^{+}\cup N_{\lambda}^{-}$$ and define

$$\alpha^{+}=\inf_{u\in N_{\lambda}^{+}}J(u); \qquad\alpha^{-}= \inf_{u\in N_{\lambda}^{-}}J(u),$$

then we have the following lemma.

### Lemma 2.4

Suppose that $$q>p^{2}$$ and $$0<\lambda<\lambda^{*}$$, then we have

1. (i)

$$\alpha^{+}<0$$;

2. (ii)

$$\alpha^{-}>C_{0}$$, for some $$C_{0}>0$$.

In particular $$\alpha^{+}=\inf_{u\in N_{\lambda}}J(u)$$.

### Proof

(i) Let $$u \in N_{\lambda}^{+}$$, it follows that

$$M \bigl(\Vert u\Vert ^{p} \bigr)\Vert u\Vert ^{p} - \lambda \int_{\Omega} f\vert u\vert ^{1-r} \,dx- \int_{\Omega } g\vert u\vert ^{q} \,dx=0$$

and

$$\lambda(q+r-1) \int_{\Omega}f \vert u\vert ^{1-r} \,dx>a \bigl(q-p^{2}\bigr)\Vert u\Vert ^{p^{2}}+b(q-p)\Vert u \Vert ^{p}.$$

Substituting this into $$J(u)$$, we have

\begin{aligned} J(u)&= \frac{1}{p} \widehat{M}\bigl(\Vert u\Vert ^{p}\bigr)-\frac{\lambda}{1-r} \int _{\Omega} f\vert u\vert ^{1-r} \,dx- \frac{1}{q} \int_{\Omega}g \vert u\vert ^{q} \,dx \\ &= \frac{1}{p} \widehat{M}\bigl(\Vert u\Vert ^{p}\bigr)- \frac{1}{q} M\bigl(\Vert u\Vert ^{p}\bigr)\Vert u\Vert ^{p}-\lambda\frac{q+r-1}{q(1-r)} \int_{\Omega} f\vert u\vert ^{1-r} \,dx \\ &< \frac{a(q-p^{2})(1-r-p^{2})}{p^{2}q(1-r)}\Vert u\Vert ^{p^{2}} +\frac{b(q-p)(1-r-p)}{pq(1-r)}\Vert u\Vert ^{p}< 0, \end{aligned}

and then $$\alpha^{+}<0$$.

(ii) Let $$u\in N_{\lambda}^{-}$$. We divide the proof into two cases.

Case (A): $$\lambda^{*}= \frac{(1-r)\lambda_{2}}{p}$$. Since $$u\in N_{\lambda}^{-}$$, and by the Sobolev inequality,

\begin{aligned} b(p+r-1)\Vert u\Vert ^{p}&\leq a\bigl(p^{2}+r-1 \bigr)\Vert u\Vert ^{p^{2}}+b(p+r-1)\Vert u\Vert ^{p} \\ &< (q+r-1)S^{-\frac{q}{p}}\Vert g\Vert _{\infty} \Vert u\Vert ^{q}, \end{aligned}

which implies

$$\Vert u\Vert > \biggl( \frac{bS_{q}^{\frac{q}{p}}(p+r-1)}{(q+r-1)\Vert g\Vert _{\infty}} \biggr)^{\frac{1}{q-p}}\quad \text{for all } u\in N_{\lambda}^{-}.$$

Hence,

\begin{aligned} J(u)\geq {}&\frac{a(q-p^{2})\Vert u\Vert ^{p^{2}}}{p^{2}q} +\frac{b(q-p)\Vert u\Vert ^{p}}{pq}-\lambda \frac{q+r-1}{q(1-r)}\Vert f\Vert _{\infty}S_{1-r}^{-\frac {1-r}{p}} \Vert u\Vert ^{1-r} \\ \geq{}& \Vert u\Vert ^{1-r} \biggl( \frac{b(q-p)}{pq}\Vert u \Vert ^{p+r-1}-\lambda\frac {q+r-1}{q(1-r)}\Vert f\Vert _{\infty}S_{1-r}^{-\frac{1-r}{p}} \biggr) \\ > {}&\biggl( \frac{bS_{q}^{\frac{q}{p}}(p+r-1)}{(q+r-1)\Vert g\Vert _{\infty}} \biggr)^{\frac{1-r}{q-p}} \biggl[ \frac{b(q-p)}{pq} \biggl( \frac{bS_{q}^{\frac {q}{p}}(p+r-1)}{(q+r-1)\Vert g\Vert _{\infty}} \biggr)^{\frac{p+r-1}{q-p}} \\ &{} -\lambda\frac{q+r-1}{q(1-r)}\Vert f\Vert _{\infty}S_{1-r}^{-\frac{1-r}{p}} \biggr]=C_{0} . \end{aligned}

Thus, if $$0<\lambda<\frac{(1-r)\lambda_{2}}{p}$$, then $$\alpha ^{-}>C_{0}>0$$.

Case (B): $$\lambda^{*}=\frac{(1-r)\lambda_{1}(a)}{p^{\frac {p+1}{p}}}$$. By (2.1), one has

$$p\sqrt[p]{ab^{p-1} \bigl(p^{2}+r-1 \bigr)}\Vert u\Vert ^{2p-1}\leq (q+r-1)\Vert g\Vert _{\infty}S_{q}^{-\frac{q}{p}} \Vert u\Vert ^{q},$$

which implies

$$\Vert u\Vert > \biggl( \frac{p S_{q}^{\frac{q}{p}}\sqrt [p]{ab^{p-1}(p^{2}+r-1)}}{(q+r-1)\Vert g\Vert _{\infty}} \biggr)^{\frac{1}{q-2p+1}} \quad \text{for all } u\in N_{\lambda}^{-}.$$

Repeating the argument of case (A), we conclude if $$\lambda<\frac {(1-r)\lambda_{1}(a)}{p^{\frac{p+1}{p}}}$$, then $$\alpha^{-}>C_{0}$$ for some $$C_{0}>0$$.â€ƒâ–¡

### Lemma 2.5

Suppose that $$q=p^{2},a<\frac {1}{\Lambda}$$ and $$0<\lambda<\frac{1-r}{p}\hat{\lambda}$$, then we have

1. (i)

$$\hat{\alpha}^{+}<0$$;

2. (ii)

$$\hat{\alpha}^{-}>C_{0}$$, for some $$C_{0}>0$$.

In particular $$\hat{\alpha}^{+}=\inf_{u\in N_{\lambda}}J(u)$$.

### Proof

(i) Repeating the same argument of Lemma 2.4(i), we conclude that $$\hat{\alpha}^{+}<0$$.

(ii) Let $$u\in N_{\lambda}^{-}$$. By (1.6), one has

\begin{aligned} b(p+r-1)\Vert u\Vert ^{p}&< \bigl(p^{2}+r-1 \bigr) \biggl( \int_{\Omega}g \vert u\vert ^{p^{2}} \,d x-a\Vert u \Vert ^{p^{2}} \biggr) \\ &\leq \frac{(1-a\Lambda)(p^{2}+r-1)}{\Lambda} \Vert u\Vert ^{p^{2}}, \end{aligned}

which implies that

$$\Vert u\Vert > \biggl(\frac{b\Lambda(p+r-1)}{(1-a\Lambda)(p^{2}+r-1)} \biggr)^{\frac {1}{p^{2}-p}}\quad \text{for all } u\in N_{\lambda}^{-}.$$
(2.4)

Then we have

\begin{aligned} J(u)={}& \frac{1}{p} \widehat{M}\bigl(\Vert u \Vert ^{p}\bigr)-\frac{\lambda}{1-r} \int _{\Omega} f\vert u\vert ^{1-r} \,dx- \frac{1}{q} \int_{\Omega}g \vert u\vert ^{q} \,dx \\ \geq{}& \Vert u\Vert ^{1-r} \biggl(\frac{(p-1)b}{p^{2}}\Vert u \Vert ^{p+r-1}- \lambda\frac{p^{2}+r-1}{p^{2}(1-r)}\Vert f\Vert _{\infty}S_{1-r}^{-\frac {1-r}{p}} \biggr) \\ >{}& \biggl(\frac{b\Lambda(p+r-1)}{(1-a\Lambda)(p^{2}+r-1)} \biggr)^{\frac {1-r}{p^{2}-p}} \biggl[\frac{(p-1)b}{p^{2}} \biggl(\frac{b\Lambda (p+r-1)}{(1-a\Lambda)(p^{2}+r-1)} \biggr)^{\frac{p+r-1}{p^{2}-p}} \\ &{} -\lambda\frac{p^{2}+r-1}{p^{2}(1-r)}\Vert f\Vert _{\infty}S_{1-r}^{-\frac {1-r}{p}} \biggr]. \end{aligned}
(2.5)

Thus, if $$\lambda<\frac{1-r}{p}\hat{\lambda}$$, then $$\hat{\alpha }^{-}>C_{0}$$ for some $$C_{0}>0$$.â€ƒâ–¡

### Lemma 2.6

For each $$u\in N_{\lambda}^{+}$$ (resp. $$u\in N_{\lambda}^{-}$$), there exist $$\varepsilon>0$$ and a continuous function $$f:B(0;\varepsilon)\subset W_{0}^{1,p}(\Omega )\rightarrow R^{+}$$ such that

$$f(0)=1, f(\omega)>0, f(\omega) (u+\omega)\in N_{\lambda}^{+} \bigl(\textit{resp. } u\in N_{\lambda}^{-} \bigr),\quad \textit{for all } \omega\in B(0;\varepsilon),$$

where $$B(0;\varepsilon)=\{\omega\in W_{0}^{1,p}(\Omega):\Vert \omega \Vert <\varepsilon\}$$.

### Proof

For $$u\in N_{\lambda}^{+}$$, define $$F: W_{0}^{1,p}(\Omega)\times R \to R$$ as follows:

\begin{aligned} F(\omega,t)={}& at^{p^{2}+r-1} \biggl( \int_{\Omega}\bigl\vert \nabla(u+\omega)\bigr\vert ^{p} \,d x \biggr)^{p}+ bt^{p+r-1} \int_{\Omega}\bigl\vert \nabla(u+\omega)\bigr\vert ^{p} \,d x \\ &{} -t^{q+r-1} \int_{\Omega}g \vert u\vert ^{q} \,dx -\lambda \int_{\Omega}f \vert u\vert ^{1-r} \,dx. \end{aligned}

Since $$u\in N_{\lambda}^{+}$$, it is easily seen that $$F(0,1)=0$$ and $$F_{t}(0,1)>0$$. Then by the implicit function theorem at the point $$(0,1)$$, we can see that there exist $$\varepsilon>0$$ and a continuous function $$f:B(0;\varepsilon)\subset W_{0}^{1,p}(\Omega)\to R^{+}$$ such that

$$f(0)=1,f(\omega)>0,f(\omega) (u+\omega)\in N_{\lambda}^{+},\quad \text{for all } \omega\in B(0;\varepsilon).$$

In the same way, we can prove the case $$u\in N_{\lambda}^{-}$$.â€ƒâ–¡

### Remark 2.1

The proof of Lemma 2.6 is inspired by [11].

## 3 Proof of Theorem 1.1

By Lemma 2.1 and the Ekeland variational principle [17], there exists a minimizing sequence $$\{u_{n}\}\subset N_{\lambda}^{+}$$ such that

\begin{aligned} &\mathrm{(i)}\quad J(u_{n})< \alpha^{+}+ \frac{1}{n}; \\ &\mathrm{(ii)}\quad J(u)>J(u_{n})-\frac{1}{n}\Vert u-u_{n}\Vert ,\quad \forall u\in N_{\lambda}^{+}. \end{aligned}

Note that $$J(\vert u_{n}\vert )=J(u_{n})$$. We may assume that $$u_{n}\geq0$$ in Î©. Using Lemma 2.1 again, we can see that there is a constant $$C_{1}>0$$ such that, for all $$n\in N^{+}$$, $$\Vert u_{n}\Vert \leq C_{1}$$. Thus, there exist a subsequence (still denoted by $$\{u_{n}\}$$) and $$u_{\lambda}^{+}$$ in $$W_{0}^{1,p}(\Omega)$$ such that

\begin{aligned} &u_{n} \rightharpoonup u_{\lambda}^{+} \mbox{ weakly in } W_{0}^{1,p}(\Omega), \\ &u_{n} \rightarrow u_{\lambda}^{+} \mbox{ strongly in } L^{1-r}(\Omega), \\ &u_{n} \rightarrow u_{\lambda}^{+} \mbox{ strongly in } L^{q}(\Omega ), \\ &u_{n} \rightarrow u_{\lambda}^{+} \mbox{ a.e. in } \Omega. \end{aligned}

Now we conclude that $$u_{\lambda}^{+} \in N_{\lambda}^{+}$$ is a positive solution of (1.1). The proof is inspired by Liu and Sun [11]. In order to prove the claim, we divide the arguments into six steps.

Step 1: $$u_{\lambda}^{+}$$ is not identically zero.

Indeed, it is an immediate conclusion of the following inequalities:

$$J \bigl(u_{\lambda}^{+} \bigr)\leq\varliminf_{n \to\infty}J(u_{n})= \alpha^{+}< 0.$$

Step 2: There exists $$C_{2}$$ such that up to a subsequence we have

$$a\bigl(p^{2}+r-1\bigr)\Vert u_{n}\Vert ^{p^{2}}+b(p+r-1)\Vert u_{n}\Vert ^{p}-(q+r-1) \int_{\Omega}g\bigl\vert u_{\lambda}^{+}\bigr\vert ^{q} \,d x >C_{2}.$$
(3.1)

In order to prove (3.1), it suffices to verify

\begin{aligned} a\bigl(p^{2}+r-1\bigr) \varlimsup_{n \to\infty} \Vert u_{n}\Vert ^{p^{2}}+b(p+r-1) \varlimsup_{n \to\infty} \Vert u_{n}\Vert ^{p}>(q+r-1) \int_{\Omega}g\bigl\vert u_{\lambda}^{+}\bigr\vert ^{q} \,d x. \end{aligned}
(3.2)

Since $$u_{n}\in N_{\lambda}^{+}$$,

\begin{aligned} a\bigl(p^{2}+r-1\bigr)\Vert u_{n}\Vert ^{p^{2}}+b(p+r-1)\Vert u_{n}\Vert ^{p} \geq(q+r-1) \int_{\Omega}g\bigl\vert u_{\lambda}^{+}\bigr\vert ^{q} \,d x. \end{aligned}
(3.3)

It follows that

\begin{aligned} a\bigl(p^{2}+r-1\bigr)\varlimsup_{n \to\infty} \Vert u_{n}\Vert ^{p^{2}}+b(p+r-1)\varlimsup_{n \to\infty} \Vert u_{n}\Vert ^{p}\geq(q+r-1) \int_{\Omega}g\bigl\vert u_{\lambda}^{+}\bigr\vert ^{q} \,d x. \end{aligned}

Suppose by contradiction that

\begin{aligned} a\bigl(p^{2}+r-1\bigr)\varlimsup_{n \to\infty} \Vert u_{n}\Vert ^{p^{2}}+b(p+r-1)\varlimsup_{n \to\infty} \Vert u_{n}\Vert ^{p}=(q+r-1) \int_{\Omega}g\bigl\vert u_{\lambda}^{+}\bigr\vert ^{q} \,d x. \end{aligned}
(3.4)

Then, from (3.3) and (3.4), one has

\begin{aligned} a\bigl(p^{2}+r-1\bigr)\lim_{n \to\infty} \Vert u_{n}\Vert ^{p^{2}}+b(p+r-1)\lim_{n \to \infty} \Vert u_{n}\Vert ^{p}=(q+r-1) \int_{\Omega}g\bigl\vert u_{\lambda}^{+}\bigr\vert ^{q} \,d x. \end{aligned}

Thus $$\Vert u_{n}\Vert ^{p}$$ converges to a positive number A that satisfies

\begin{aligned} a\bigl(p^{2}+r-1\bigr)A^{p}+b(p+r-1)A=(q+r-1) \int_{\Omega}g\bigl\vert u_{\lambda}^{+}\bigr\vert ^{q} \,d x \end{aligned}

and

\begin{aligned} a\bigl(q-p^{2}\bigr)A^{p}+b(q-p)A=\lambda(q+r-1) \int_{\Omega}f\bigl\vert u_{\lambda}^{+}\bigr\vert ^{1-r} \,d x. \end{aligned}

On the other hand, by (2.3), we have

\begin{aligned} 0\leq{}& \bigl(\lambda^{*}-\lambda\bigr) \int_{\Omega}f \vert u_{n}\vert ^{1-r} \,d x \\ < {}& b^{\frac{q+r-1}{q-p}} \biggl( \frac{p+r-1}{q+r-1} \biggr)^{\frac {p+r-1}{q-p}} \biggl( \frac{q-p}{q+r-1} \biggr) \frac{\Vert u_{n}\Vert ^{\frac {p(q+r-1)}{q-p}}}{ (\int_{\Omega}g \vert u_{n}\vert ^{q} \,dx )^{\frac {p+r-1}{q-p}}} -\lambda \int_{\Omega}f \vert u_{n}\vert ^{1-r} \,d x \\ \to{}& b^{\frac{q+r-1}{q-p}} \biggl( \frac{p+r-1}{q+r-1} \biggr)^{\frac {p+r-1}{q-p}} \biggl( \frac{q-p}{q+r-1} \biggr) \frac{A^{\frac {q+r-1}{q-p}}}{ (\frac{a(p^{2}+r-1)A^{p}+b(p+r-1)A}{q+r-1} )^{\frac {p+r-1}{q-p}}} \\ &{} -\frac{a(q-p^{2})A^{p}+b(q-p)A}{q+r-1} \\ < {}& b^{\frac{q+r-1}{q-p}} \biggl( \frac{p+r-1}{q+r-1} \biggr)^{\frac {p+r-1}{q-p}} \biggl( \frac{q-p}{q+r-1} \biggr) \frac{A^{\frac {q+r-1}{q-p}}}{ (\frac{b(p+r-1)A}{q+r-1} )^{\frac{p+r-1}{q-p}}} \\ & {}-\frac{a(q-p^{2})A^{p}+b(q-p)A}{q+r-1} \\ ={}& {-}\frac{a(q-p^{2})}{q+r-1}A^{p}< 0, \end{aligned}

which is impossible. Hence, (3.1) and (3.2) must hold.

Step 3: For nonnegative $$\varphi\in W_{0}^{1,p}(\Omega)$$ and $$t>0$$ small, we can find $$f_{n}(t):=f_{n}(t \varphi)$$ such that $$f_{n}(0)=1$$ and $$f_{n}(t)(u_{n}+t\varphi)\in N_{\lambda}^{+}$$ for each $$u_{n}\in N_{\lambda}^{+}$$ by Lemma 2.6. $$f'_{n+}(0)\in[-\infty, \infty ]$$ is denoted by the right derivative of $$f_{n}(t)$$ at zero. We claim that there exists $$C_{3}>0$$ such that $$f'_{n+}(0)>-C_{3}$$ for all $$n\in N^{+}$$. Since $$u_{n}, f_{n}(t)(u_{n}+t\varphi)\in N_{\lambda}$$, we deduce that

$$0=a\Vert u_{n}\Vert ^{p^{2}}+b\Vert u_{n}\Vert ^{p}-\lambda \int_{\Omega}f \vert u_{n}\vert ^{1-r} \,d x- \int_{\Omega}g \vert u_{n}\vert ^{q} \,d x$$

and

\begin{aligned} 0={}& af_{n}^{p^{2}}(t)\Vert u_{n}+t \varphi \Vert ^{p^{2}}+bf_{n}^{p}(t)\Vert u_{n}+t\varphi \Vert ^{p}-\lambda f_{n}^{1-r}(t) \int_{\Omega}f \vert u_{n}+t\varphi \vert ^{1-r} \,d x \\ &{} -f_{n}^{q}(t) \int_{\Omega}g \vert u_{n}+t\varphi \vert ^{q} \,d x. \end{aligned}

Thus

\begin{aligned} 0={}& a\bigl(f_{n}^{p^{2}}(t)-1\bigr)\Vert u_{n}+t\varphi \Vert ^{p^{2}}+a\bigl(\Vert u_{n} +t\varphi \Vert ^{p^{2}}-\Vert u_{n}\Vert ^{p^{2}} \bigr) \\ &{} +b\bigl(f_{n}^{p}(t)-1\bigr)\Vert u_{n}+t \varphi \Vert ^{p}+b\bigl(\Vert u_{n}+t\varphi \Vert ^{p}-\Vert u_{n}\Vert ^{p}\bigr) \\ &{} -\lambda\bigl(f_{n}^{1-r}(t)-1\bigr) \int_{\Omega}f \vert u_{n}+t\varphi \vert ^{1-r} \,d x -\lambda \int_{\Omega}f\bigl(\vert u_{n}+t\varphi \vert ^{1-r}-\vert u_{n}\vert ^{1-r}\bigr) \,d x \\ &{} -\bigl(f_{n}^{q}(t)-1\bigr) \int_{\Omega}g \vert u_{n}+t\varphi \vert ^{q} \,d x - \int_{\Omega}g\bigl(\vert u_{n}+t\varphi \vert ^{q}-\vert u_{n}\vert ^{q}\bigr) \,d x \\ \leq{}& a\bigl(f_{n}^{p^{2}}(t)-1\bigr)\Vert u_{n}+t\varphi \Vert ^{p^{2}} +a\bigl(\Vert u_{n}+t\varphi \Vert ^{p^{2}}-\Vert u_{n}\Vert ^{p^{2}}\bigr) \\ &{} +b\bigl(f_{n}^{p}(t)-1\bigr)\Vert u_{n}+t \varphi \Vert ^{p}+b\bigl(\Vert u_{n}+t\varphi \Vert ^{p}-\Vert u_{n}\Vert ^{p}\bigr) \\ &{} -\lambda\bigl(f_{n}^{1-r}(t)-1\bigr) \int_{\Omega}f \vert u_{n}+t\varphi \vert ^{1-r} \,d x -\bigl(f_{n}^{q}(t)-1\bigr) \int_{\Omega}g \vert u_{n}+t\varphi \vert ^{q} \,d x. \end{aligned}

Then, dividing by $$t>0$$ and letting $$t\to0$$, we have

\begin{aligned} 0\leq{}& ap^{2}\Vert u_{n}\Vert ^{p^{2}}f'_{n+}(0)+ap^{2}\Vert u_{n}\Vert ^{p^{2}-p} \int_{\Omega }\vert \nabla u_{n}\vert ^{p-2} \nabla u_{n}\nabla\varphi \,dx \\ &{} +bp\Vert u_{n}\Vert ^{p}f'_{n+}(0)+bp \int_{\Omega} \vert \nabla u_{n}\vert ^{p-2} \nabla u_{n}\nabla\varphi \,dx \\ &{} -\lambda(1-r)f'_{n+}(0) \int_{\Omega}f \vert u_{n}\vert ^{1-r} \,d x -qf'_{n+}(0) \int_{\Omega}g \vert u_{n}\vert ^{q} \,d x \\ ={}& f'_{n+}(0) \biggl(ap^{2}\Vert u_{n}\Vert ^{p^{2}}+bp\Vert u_{n}\Vert ^{p}-\lambda (1-r) \int_{\Omega}f \vert u_{n}\vert ^{1-r} \,d x -q \int_{\Omega}g \vert u_{n}\vert ^{q} \,d x \biggr) \\ &{} +ap^{2}\Vert u_{n}\Vert ^{p^{2}-p} \int_{\Omega} \vert \nabla u_{n}\vert ^{p-2} \nabla u_{n}\nabla\varphi \,dx +bp \int_{\Omega} \vert \nabla u_{n}\vert ^{p-2} \nabla u_{n}\nabla\varphi \,dx \\ ={}& f'_{n+}(0) \biggl(a\bigl(p^{2}+r-1\bigr) \Vert u_{n}\Vert ^{p^{2}}+b(p+r-1)\Vert u_{n} \Vert ^{p} -(q+r-1) \int_{\Omega}g \vert u_{n}\vert ^{q} \,d x \biggr) \\ &{} +ap^{2}\Vert u_{n}\Vert ^{p^{2}-p} \int_{\Omega} \vert \nabla u_{n}\vert ^{p-2} \nabla u_{n}\nabla\varphi \,dx +bp \int_{\Omega} \vert \nabla u_{n}\vert ^{p-2} \nabla u_{n}\nabla\varphi \,dx. \end{aligned}

One deduces from (3.1)

$$f'_{n+}(0)\geq-\frac{ap^{2}\Vert u_{n}\Vert ^{p^{2}-p}\int_{\Omega} \vert \nabla u_{n}\vert ^{p-2}\nabla u_{n}\nabla\varphi \,dx +bp\int_{\Omega} \vert \nabla u_{n}\vert ^{p-2}\nabla u_{n}\nabla\varphi \,dx}{a(p^{2}+r-1)\Vert u_{n}\Vert ^{p^{2}}+b(p+r-1)\Vert u_{n}\Vert ^{p} -(q+r-1)\int_{\Omega}g \vert u_{n}\vert ^{q} \,d x }.$$

Therefore, by the boundedness of $$\{u_{n}\}$$, we conclude that $$\{ f'_{n+}(0)\}$$ is bounded from below.

Step 4: Choose $$n^{*}$$ large enough such that $$\frac{(1-r)C_{1}}{n}<\frac {C_{2}}{2}$$ for all $$n>n^{*}$$. Then we claim that there exists $$C_{4}$$ such that $$f'_{n+}(0)< C_{4}$$ for each $$n>n^{*}$$. Without loss of generality, we may suppose $$f'_{n+}(0)\geq0$$. Then from condition (ii), we have

\begin{aligned} & \bigl\vert f_{n}(t)-1\bigr\vert \frac{\Vert u_{n}\Vert }{n}+ \bigl\vert tf_{n}(t)\bigr\vert \frac{\Vert \varphi \Vert }{n}\\ &\quad \geq \frac{1}{n}\bigl\Vert f_{n}(t) (u_{n}+t\varphi)-u_{n}\bigr\Vert \\ &\quad\geq J(u_{n})-J\bigl(f_{n}(t) (u_{n}+t \varphi)\bigr) \\ &\quad= \frac{a(p^{2}+r-1)}{p^{2}(1-r)}\bigl(f_{n}^{p^{2}}(t)-1\bigr) \Vert u_{n} +t\varphi \Vert ^{p^{2}}+\frac{a(p^{2}+r-1)}{p^{2}(1-r)}\bigl( \Vert u_{n} +t\varphi \Vert ^{p^{2}}-\Vert u_{n} \Vert ^{p^{2}}\bigr) \\ &\qquad{} +\frac{b(p+r-1)}{p(1-r)}\bigl(f_{n}^{p}(t)-1\bigr) \Vert u_{n}+t\varphi \Vert ^{p} +\frac{b(p+r-1)}{p(1-r)}\bigl( \Vert u_{n}+t\varphi \Vert ^{p}-\Vert u_{n} \Vert ^{p}\bigr) \\ &\qquad{}-\frac{q+r-1}{q(1-r)} \bigl(f_{n}^{q}(t)-1\bigr) \int_{\Omega}g \vert u_{n}\vert ^{q} \,d x -\frac{q+r-1}{q(1-r)}f_{n}^{q}(t) \int_{\Omega}g\bigl(\vert u_{n}+t\varphi \vert ^{q}-\vert u_{n}\vert ^{q}\bigr) \,d x. \end{aligned}

Then, dividing by $$t>0$$ and letting $$t\to0$$, we deduce

\begin{aligned} & {f'_{n+}(0)} \frac{\Vert u_{n}\Vert }{n}+\frac{\Vert \varphi \Vert }{n} \\ &\quad\geq \frac{a(p^{2}+r-1)}{1-r} f'_{n+}(0)\Vert u_{n}\Vert ^{p^{2}}+\frac {a(p^{2}+r-1)}{1-r}\Vert u_{n}\Vert ^{p^{2}-p} \int_{\Omega} \vert \nabla u_{n}\vert ^{p-2} \nabla u_{n}\nabla\varphi \,dx \\ &\qquad{} +\frac{b(p+r-1)}{1-r}f'_{n+}(0)\Vert u_{n}\Vert ^{p}+\frac{b(p+r-1)}{1-r} \int _{\Omega} \vert \nabla u_{n}\vert ^{p-2} \nabla u_{n}\nabla\varphi \,dx \\ &\qquad{} -\frac{q+r-1}{1-r}f'_{n+}(0) \int_{\Omega}g \vert u_{n}\vert ^{q} \,d x -\frac {q+r-1}{1-r} \int_{\Omega}g \vert u_{n}\vert ^{q-1}\varphi \,d x. \end{aligned}
(3.5)

From (3.5) and the choice of $$n^{*}$$, we have

\begin{aligned} \frac{\Vert \varphi \Vert }{n}\geq{}& \frac{C_{2}}{2(1-r)} f'_{n+}(0)+ \frac{a(p^{2}+r-1)}{1-r}\Vert u_{n}\Vert ^{p^{2}-p} \int_{\Omega} \vert \nabla u_{n}\vert ^{p-2} \nabla u_{n}\nabla\varphi \,dx \\ &{} +\frac{b(p+r-1)}{1-r} \int_{\Omega} \vert \nabla u_{n}\vert ^{p-2} \nabla u_{n}\nabla \varphi \,dx -\frac{q+r-1}{1-r} \int_{\Omega}g \vert u_{n}\vert ^{q-1}\varphi \,d x. \end{aligned}

Namely,

\begin{aligned} \frac{C_{2}}{2(1-r)} f'_{n+}(0)\leq{}& \frac{\Vert \varphi \Vert }{n}- \frac{a(p^{2}+r-1)}{1-r}\Vert u_{n}\Vert ^{p^{2}-p} \int _{\Omega} \vert \nabla u_{n}\vert ^{p-2} \nabla u_{n}\nabla\varphi \,dx \\ &{}- \frac{b(p+r-1)}{1-r} \int_{\Omega} \vert \nabla u_{n}\vert ^{p-2} \nabla u_{n}\nabla \varphi \,dx +\frac{q+r-1}{1-r} \int_{\Omega}g \vert u_{n}\vert ^{q-1}\varphi \,d x. \end{aligned}

Therefore, by the boundedness of $$\{u_{n}\}$$, we conclude $$\{ f'_{n+}(0)\}_{n>n^{*}}$$ is bounded from above.

Step 5: $$u_{\lambda}^{+}>0$$ a.e. in Î© and for nonnegative $$\varphi\in W_{0}^{1,p}(\Omega)$$, we have

\begin{aligned} &\bigl(a\bigl\Vert u_{\lambda}^{+}\bigr\Vert ^{p^{2}-p}+b\bigr) \int_{\Omega}\bigl\vert \nabla u_{\lambda }^{+} \bigr\vert ^{p-2}\nabla u_{\lambda}^{+} \nabla\varphi \,dx-\lambda \int _{\Omega} f\bigl\vert u_{\lambda}^{+}\bigr\vert ^{-r}\varphi \,dx- \int_{\Omega} g\bigl\vert u_{\lambda }^{+}\bigr\vert ^{q-1}\varphi \,dx \\ &\quad\geq0. \end{aligned}
(3.6)

Similar to the argument in Step 4, one can obtain

\begin{aligned} & {f'_{n+}(0)}\frac{\Vert u_{n}\Vert }{n}+ \frac{\Vert \varphi \Vert }{n} \\ &\quad\geq -f'_{n+}(0) \biggl(a\Vert u_{n} \Vert ^{p^{2}}+b\Vert u_{n}\Vert ^{p}- \int_{\Omega }g\vert u_{n}\vert ^{q} \,d x -\lambda \int_{\Omega}f \vert u_{n}\vert ^{1-r} \varphi \,d x \biggr) \\ &\qquad{} -a\Vert u_{n}\Vert ^{p^{2}-p} \int_{\Omega} \vert \nabla u_{n}\vert ^{p-2} \nabla u_{n}\nabla\varphi \,dx -b \int_{\Omega} \vert \nabla u_{n}\vert ^{p-2} \nabla u_{n}\nabla\varphi \,dx \\ & \qquad{}+ \int_{\Omega}g \vert u_{n}\vert ^{q-1}\varphi \,d x +\varliminf_{t \to0^{+}}\frac{\lambda}{1-r} \int_{\Omega}\frac {f(\vert u_{n}+t\varphi \vert ^{1-r}-\vert u_{n}\vert ^{1-r})}{t} \,d x \\ &\quad= -a\Vert u_{n}\Vert ^{p^{2}-p} \int_{\Omega} \vert \nabla u_{n}\vert ^{p-2} \nabla u_{n}\nabla\varphi \,dx -b \int_{\Omega} \vert \nabla u_{n}\vert ^{p-2} \nabla u_{n}\nabla\varphi \,dx \\ &\qquad{} + \int_{\Omega}g \vert u_{n}\vert ^{q-1}\varphi \,d x +\varliminf_{t \to0^{+}}\frac{\lambda}{1-r} \int_{\Omega}\frac {f(\vert u_{n}+t\varphi \vert ^{1-r}-\vert u_{n}\vert ^{1-r})}{t} \,d x. \end{aligned}
(3.7)

Since $$f(\vert u_{n}+t\varphi \vert ^{1-r}-\vert u_{n}\vert ^{1-r})\geq0, \forall t>0$$, by Fatouâ€™s lemma, we obtain

\begin{aligned} \int_{\Omega}f \vert u_{n}\vert ^{-r}\varphi \,dx \leq\varliminf_{t \to 0^{+}}\frac{1}{1-r} \int_{\Omega}\frac{f(\vert u_{n}+t\varphi \vert ^{1-r}-\vert u_{n}\vert ^{1-r})}{t} \,d x. \end{aligned}
(3.8)

It follows from (3.7) and (3.8) that

\begin{aligned} &\lambda \int_{\Omega}f \vert u_{n}\vert ^{-r}\varphi \,dx \\ &\quad \leq\frac{1}{n} \bigl({f'_{n+}(0)}\Vert u_{n}\Vert +\Vert \varphi \Vert \bigr)+a\Vert u_{n} \Vert ^{p^{2}-p} \int_{\Omega} \vert \nabla u_{n}\vert ^{p-2} \nabla u_{n}\nabla\varphi \,dx \\ &\qquad{} +b \int_{\Omega} \vert \nabla u_{n}\vert ^{p-2} \nabla u_{n}\nabla\varphi \,dx - \int _{\Omega}g \vert u_{n}\vert ^{q-1} \varphi \,d x \\ &\quad \leq\frac{C_{1}\cdot\max\{C_{3},C_{4}\}+\Vert \varphi \Vert }{n}+a\Vert u_{n}\Vert ^{p^{2}-p} \int_{\Omega} \vert \nabla u_{n}\vert ^{p-2} \nabla u_{n}\nabla\varphi \,dx \\ & \qquad{}+b \int_{\Omega} \vert \nabla u_{n}\vert ^{p-2} \nabla u_{n}\nabla\varphi \,dx - \int _{\Omega}g \vert u_{n}\vert ^{q-1} \varphi \,d x, \end{aligned}

for all $$n>n^{*}$$.

Passing to the limit as $$n\to\infty$$, one has

\begin{aligned} \varliminf_{n \to\infty}\lambda \int_{\Omega}f \vert u_{n}\vert ^{-r}\varphi \,dx \leq{}& a\varliminf_{n \to\infty} \Vert u_{n}\Vert ^{p^{2}-p} \int_{\Omega}\bigl\vert \nabla u_{\lambda}^{+} \bigr\vert ^{p-2}\nabla u_{\lambda}^{+}\nabla\varphi \,dx \\ &{} +b \int_{\Omega}\bigl\vert \nabla u_{\lambda}^{+} \bigr\vert ^{p-2}\nabla u_{\lambda }^{+}\nabla\varphi \,dx - \int_{\Omega}g\bigl\vert u_{\lambda}^{+}\bigr\vert ^{q-1}\varphi \,d x. \end{aligned}

Then using Fatouâ€™s lemma again, we infer that

\begin{aligned} &\lambda \int_{\Omega}f\bigl\vert u_{\lambda}^{+}\bigr\vert ^{-r}\varphi \,dx \\ & \quad\leq a\varliminf_{n \to\infty} \Vert u_{n}\Vert ^{p^{2}-p} \int_{\Omega}\bigl\vert \nabla u_{\lambda}^{+} \bigr\vert ^{p-2}\nabla u_{\lambda}^{+}\nabla\varphi \,dx \\ &\qquad{} +b \int_{\Omega}\bigl\vert \nabla u_{\lambda}^{+} \bigr\vert ^{p-2}\nabla u_{\lambda }^{+}\nabla\varphi \,dx - \int_{\Omega}g\bigl\vert u_{\lambda}^{+}\bigr\vert ^{q-1}\varphi \,d x. \end{aligned}
(3.9)

Since $$u_{n} \to u_{\lambda}^{+}$$ a.e. in Î©, we get $$u_{\lambda }^{+}\geq0$$ a.e. in Î©. Thus, one infers from (3.9) that

\begin{aligned} \lambda \int_{\Omega}f\bigl\vert u_{\lambda}^{+}\bigr\vert ^{1-r} \,dx&\leq a\varliminf_{n \to\infty} \Vert u_{n}\Vert ^{p^{2}-p}\bigl\Vert u_{\lambda}^{+} \bigr\Vert ^{p}+b\bigl\Vert u_{\lambda}^{+}\bigr\Vert ^{p} - \int_{\Omega}g\bigl\vert u_{\lambda}^{+}\bigr\vert ^{q} \,d x. \end{aligned}
(3.10)

On the other hand

\begin{aligned} &a\varliminf_{n \to\infty} \Vert u_{n} \Vert ^{p^{2}-p}\bigl\Vert u_{\lambda}^{+}\bigr\Vert ^{p}+b\bigl\Vert u_{\lambda}^{+}\bigr\Vert ^{p} \leq a\varlimsup_{n \to\infty} \Vert u_{n}\Vert ^{p^{2}}+b\varlimsup_{n \to\infty} \Vert u_{n}\Vert ^{p} \\ &\quad=\lambda \int_{\Omega}f\bigl\vert u_{\lambda}^{+}\bigr\vert ^{1-r} \,dx+ \int_{\Omega }g\bigl\vert u_{\lambda}^{+}\bigr\vert ^{q} \,d x. \end{aligned}
(3.11)

Combining (3.10) and (3.11), we have

\begin{aligned} \varliminf_{n \to\infty} \Vert u_{n}\Vert ^{p}= \varlimsup_{n \to\infty} \Vert u_{n}\Vert ^{p}=\bigl\Vert u_{\lambda}^{+}\bigr\Vert ^{p}. \end{aligned}
(3.12)

Thus, (3.6) can be obtained by inserting (3.12) into (3.9). Moreover, from (3.6), one has

$$\int_{\Omega} \bigl\vert \nabla u_{\lambda}^{+} \bigr\vert ^{p-2}\nabla u_{\lambda}^{+} \nabla\varphi \,dx \geq0,\quad \forall \varphi\in W_{0}^{1,p}(\Omega ), \varphi\geq0.$$

Therefore, using the strong maximum principle for weak solutions (see [18]), we obtain $$u_{\lambda}^{+}> 0$$ a.e. in Î©.

Step 6: $$u_{\lambda}^{+}$$ is a weak solution of (1.1), and $$u_{\lambda}^{+}\in N_{\lambda}^{+}$$. By (3.12), we have $$u_{n}\to u_{\lambda}^{+}$$ strongly in $$W_{0}^{1,p}(\Omega)$$, and so $$u_{\lambda}^{+}\in N_{\lambda}^{+}$$. Assume $$\phi\in W_{0}^{1,p}(\Omega)$$ and $$\varepsilon>0$$, define $$\Psi\in W_{0}^{1,p}(\Omega)$$ by $$\Psi :=(u_{\lambda}^{+}+\varepsilon\phi)^{+}$$. Then from Step 5 it follows

\begin{aligned} 0\leq{}& \int_{\Omega} \bigl[\bigl(a\bigl\Vert u_{\lambda}^{+} \bigr\Vert ^{p^{2}-p}+b\bigr)\bigl\vert \nabla u_{\lambda}^{+} \bigr\vert ^{p-2}\nabla u_{\lambda}^{+}\nabla\Psi- \lambda f\bigl\vert u_{\lambda}^{+}\bigr\vert ^{-r} \Psi-g\bigl\vert u_{\lambda}^{+}\bigr\vert ^{q-1}\Psi \bigr] \,dx \\ ={}& \int_{[u_{\lambda}^{+}+\varepsilon\phi>0]} \bigl[\bigl(a\bigl\Vert u_{\lambda}^{+} \bigr\Vert ^{p^{2}-p}+b\bigr)\bigl\vert \nabla u_{\lambda}^{+} \bigr\vert ^{p-2}\nabla u_{\lambda}^{+}\nabla \bigl(u_{\lambda}^{+}+\varepsilon\phi\bigr) \\ &{} -\lambda f\bigl\vert u_{\lambda}^{+}\bigr\vert ^{-r}\bigl(u_{\lambda}^{+}+\varepsilon\phi\bigr) -g\bigl\vert u_{\lambda}^{+}\bigr\vert ^{q-1} \bigl(u_{\lambda}^{+}+\varepsilon\phi\bigr) \bigr] \,dx \\ ={}& \biggl( \int_{\Omega}- \int_{[u_{\lambda}^{+}+\varepsilon\phi\leq 0]} \biggr) \bigl[\bigl(a\bigl\Vert u_{\lambda}^{+}\bigr\Vert ^{p^{2}-p}+b\bigr)\bigl\vert \nabla u_{\lambda }^{+}\bigr\vert ^{p-2}\nabla u_{\lambda}^{+}\nabla\bigl(u_{\lambda}^{+}+ \varepsilon \phi\bigr) \\ &{} -\lambda f\bigl\vert u_{\lambda}^{+}\bigr\vert ^{-r}\bigl(u_{\lambda}^{+}+\varepsilon\phi\bigr) -g\bigl\vert u_{\lambda}^{+}\bigr\vert ^{q-1} \bigl(u_{\lambda}^{+}+\varepsilon\phi\bigr) \bigr] \,dx \\ ={}& a\bigl\Vert u_{\lambda}^{+}\bigr\Vert ^{p^{2}}+b \bigl\Vert u_{\lambda}^{+}\bigr\Vert ^{p}-\lambda \int _{\Omega} f\bigl\vert u_{\lambda}^{+}\bigr\vert ^{1-r} \,dx- \int_{\Omega} g\bigl\vert u_{\lambda }^{+}\bigr\vert ^{q} \,dx \\ &{}+\varepsilon \int_{\Omega} \bigl[\bigl(a\bigl\Vert u_{\lambda}^{+} \bigr\Vert ^{p^{2}-p}+b\bigr)\bigl\vert \nabla u_{\lambda}^{+} \bigr\vert ^{p-2}\nabla u_{\lambda}^{+}\nabla\phi- \lambda f\bigl\vert u_{\lambda}^{+}\bigr\vert ^{-r} \phi-g\bigl\vert u_{\lambda}^{+}\bigr\vert ^{q-1}\phi \bigr] \,dx \\ &{} - \int_{[u_{\lambda}^{+}+\varepsilon\phi\leq0]} \bigl[\bigl(a\bigl\Vert u_{\lambda }^{+} \bigr\Vert ^{p^{2}-p}+b\bigr)\bigl\vert \nabla u_{\lambda}^{+} \bigr\vert ^{p-2}\nabla u_{\lambda }^{+}\nabla \bigl(u_{\lambda}^{+}+\varepsilon\phi\bigr) \\ &{} -\lambda f\bigl\vert u_{\lambda}^{+}\bigr\vert ^{-r}\bigl(u_{\lambda}^{+}+\varepsilon\phi\bigr) -g\bigl\vert u_{\lambda}^{+}\bigr\vert ^{q-1} \bigl(u_{\lambda}^{+}+\varepsilon\phi\bigr)\bigr] \,dx \\ ={}& \varepsilon \int_{\Omega} \bigl[\bigl(a\bigl\Vert u_{\lambda}^{+} \bigr\Vert ^{p^{2}-p}+b\bigr)\bigl\vert \nabla u_{\lambda}^{+} \bigr\vert ^{p-2}\nabla u_{\lambda}^{+}\nabla \phi- \lambda f\bigl\vert u_{\lambda}^{+}\bigr\vert ^{-r} \phi-g\bigl\vert u_{\lambda}^{+}\bigr\vert ^{q-1}\phi \bigr] \,dx \\ &{} - \int_{[u_{\lambda}^{+}+\varepsilon\phi\leq0]} \bigl[\bigl(a\bigl\Vert u_{\lambda }^{+} \bigr\Vert ^{p^{2}-p}+b\bigr)\bigl\vert \nabla u_{\lambda}^{+} \bigr\vert ^{p-2}\nabla u_{\lambda }^{+}\nabla \bigl(u_{\lambda}^{+}+\varepsilon\phi\bigr) \\ &{} -\lambda f\bigl\vert u_{\lambda}^{+}\bigr\vert ^{-r}\bigl(u_{\lambda}^{+}+\varepsilon\phi\bigr) -g\bigl\vert u_{\lambda}^{+}\bigr\vert ^{q-1} \bigl(u_{\lambda}^{+}+\varepsilon\phi\bigr) \bigr] \,dx \\ \leq{}& \varepsilon \int_{\Omega} \bigl[\bigl(a\bigl\Vert u_{\lambda}^{+} \bigr\Vert ^{p^{2}-p}+b\bigr)\bigl\vert \nabla u_{\lambda}^{+} \bigr\vert ^{p-2}\nabla u_{\lambda}^{+}\nabla \phi- \lambda f\bigl\vert u_{\lambda}^{+}\bigr\vert ^{-r} \phi-g\bigl\vert u_{\lambda}^{+}\bigr\vert ^{q-1}\phi \bigr] \,dx \\ &{} -\varepsilon\bigl(a\bigl\Vert u_{\lambda}^{+}\bigr\Vert ^{p^{2}-p}+b\bigr) \int_{[u_{\lambda }^{+}+\varepsilon\phi\leq0]} \bigl\vert \nabla u_{\lambda}^{+} \bigr\vert ^{p-2}\nabla u_{\lambda}^{+}\nabla\phi \,dx. \end{aligned}

Since the measure of the domain of integration $$[u_{\lambda }^{+}+\varepsilon\phi\leq0]$$ tends to zero as $$\varepsilon\to0$$, it follows $$\int_{[u_{\lambda}^{+}+\varepsilon\phi\leq0]} \vert \nabla u_{\lambda}^{+}\vert ^{p-2}\nabla u_{\lambda}^{+}\nabla\phi \,dx \to0$$. Dividing by Îµ and letting $$\varepsilon\to0$$, we have

\begin{aligned} \bigl(a\bigl\Vert u_{\lambda}^{+}\bigr\Vert ^{p^{2}-p}+b\bigr) \int_{\Omega}\bigl\vert \nabla u_{\lambda }^{+} \bigr\vert ^{p-2}\nabla u_{\lambda}^{+} \nabla\phi \,dx- \lambda \int_{\Omega } f\bigl\vert u_{\lambda}^{+}\bigr\vert ^{-r}\phi \,dx- \int_{\Omega} g\bigl\vert u_{\lambda }^{+}\bigr\vert ^{q-1}\phi \,dx\geq0. \end{aligned}

Notice that Ï• is arbitrary, the inequality also holds for âˆ’Ï•, so it follows that $$u_{\lambda}^{+}$$ is a weak solution of (1.1). Moreover, from (3.2) and (3.12), we deduce that $$u_{\lambda}^{+}\in N_{\lambda}^{+}$$.

A similar argument shows that there exists another solution $$u_{\lambda }^{-}\in N_{\lambda}^{-}$$.

## 4 Proof of Theorem 1.2

(i) By Lemma 2.3(i), we write $$N_{\lambda}=N^{+}_{\lambda}$$ and define

$$\theta^{+}=\inf_{u\in N^{+}_{\lambda}}J(u).$$

Similar to Lemma 2.5(i), we have $$\theta^{+}<0$$. Applying Lemma 2.2(i) and the Ekeland variational principle, we see that there exists a minimizing sequence $$\{u_{n}\}$$ for $$J(u)$$ in $$N^{+}_{\lambda}$$ such that

\begin{aligned} &\mathrm{(i)}\quad J(u_{n})< \theta^{+}+ \frac{1}{n}; \\ &\mathrm{(ii)}\quad J(u)>J(u_{n})-\frac{1}{n}\Vert u-u_{n}\Vert ,\quad \forall u\in N^{+}_{\lambda}. \end{aligned}

Repeating the same argument as Theorem 1.1, we can see that $$u_{\lambda }\in N^{+}_{\lambda}$$ is a positive solution of the problem (1.1).

(ii) Similar to the proof of Theorem 1.1, we know that the problem (1.1) has at least two positive solutions $$u_{\lambda}^{+}\in N_{\lambda }^{+}$$ and $$u_{\lambda}^{-}\in N_{\lambda}^{-}$$. Moreover, combining (2.4) with (2.5), we have

$$\lim_{a\to\frac{1}{\Lambda}^{-}} \bigl\Vert u_{\lambda}^{-} \bigr\Vert =\infty$$

and

$$\lim_{a\to\frac{1}{\Lambda}^{-}}\inf_{u\in N_{\lambda}^{-}}J(u)=\infty.$$

This completes the proof of Theorem 1.2.

### Remark 4.1

The results of Theorems 1.1 and 1.2 extend the results of [11, 12]. The results from the cited work correspond to our results for the case $$p=2$$ and $$N=3$$. From these two references, we obtained the motivation for this paper.

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## Acknowledgements

This work was supported by Young Award of Shandong Province (ZR2013AQ008), NNSF (61603226), the Fund of Science and Technology Plan of Shandong Province (2014GGH201010) and NSFC (11671237).

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Wang, D., Yan, B. Existence and multiplicity of positive solutions for p-Kirchhoff type problem with singularity. Bound Value Probl 2017, 38 (2017). https://doi.org/10.1186/s13661-017-0771-3