Open Access

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

Boundary Value Problems20172017:38

https://doi.org/10.1186/s13661-017-0771-3

Received: 11 December 2016

Accepted: 9 March 2017

Published: 20 March 2017

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.

Keywords

p-Kirchhoff type problemsingularityNehari manifold

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 \(\Delta_{p}u= \operatorname{div}(\vert \nabla u\vert ^{p-2}\nabla u)\), Ω 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 D’Alembert’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 [610] 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 [1113]). 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:

• 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 \(\parallel\centerdot\parallel_{p}\);

• 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 μ is an eigenvalue of (1.7), \(u\in W_{0}^{1,p}(\Omega)\) is nonzero and an eigenvector corresponding to μ 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)  denoted by \(0<\mu_{1}<\mu _{2}<\cdots\) , 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 Ω (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 Section 2, we present some lemmas which will be used to prove our main results. In Section 3 and Section 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 \(W_{0}^{1,p}(\Omega)\). □

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 \(t^{-}u\in N_{\lambda}^{-}\). □

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. □

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 \(C_{0}>0\). □

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 \(C_{0}>0\). □

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 \(u\in N_{\lambda}^{-}\). □

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 Ω. 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 Ω 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 Fatou’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 Fatou’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 Ω, we get \(u_{\lambda }^{+}\geq0\) a.e. in Ω. 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 Ω.
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 ε 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 ϕ is arbitrary, the inequality also holds for −ϕ, 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.

Declarations

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).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Mathematical Sciences, Shandong Normal University

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