On semilinear biharmonic equations with concave-convex nonlinearities involving weight functions

Abstract

In this paper, we consider semilinear biharmonic equations with concave-convex nonlinearities involving weight functions, where the concave nonlinear term is $\lambda f(x){|u|}^{q-1}u$ and the convex nonlinear term is $h(x){|u|}^{p-1}u$ with $\lambda \in {\mathbb{R}}^{+}$. By use of the Nehari manifold and the direct variational methods, the existence of multiple positive solutions is established as $\lambda \in (0,{\lambda }_{\ast })$, here the explicit expression of ${\lambda }_{\ast }={\lambda }_{\ast }(f,h,p,q,S)$ is provided.

MSC:35J35, 35J40, 35J65.

1 Introduction

In recent years, there has been extensive attention on semilinear second-order elliptic equations,

$$\{\begin{array}{cc}-\Delta u=g_{\lambda }(x,u),\hfill & \text{in }\mathrm{\Omega },\hfill \\ u=0,\hfill & \text{on }\partial \mathrm{\Omega },\hfill \end{array}$$

(1.1)

here Ω is a bounded smooth domain in ${\mathbb{R}}^N$ ($N⩾3$), $g_{\lambda }:\mathrm{\Omega }\times \mathbb{R}\to \mathbb{R}$ and λ is a positive parameter; see [1–8] and the references therein. As $g_{\lambda }$ is sublinear, say, $g_{\lambda }=\lambda u^q$, $0<q<1$, the monotone iteration scheme or the method of sub-solutions and super-solutions are effective; see [9]. As $g_{\lambda }$ is superlinear, for example, $g_{\lambda }=\lambda u+{|u|}^{p-1}u$, $1<p<\frac{N+2}{N-2}$, variational methods are applicable; see [10]. In contrast with the pure sublinear case and the pure superlinear case, in [2] Ambrosetti et al. considered problem (1.1) when $g_{\lambda }$ is, roughly, the sum of a sublinear and a superlinear term. To be precise, they considered the following problem:

$$\{\begin{array}{cc}-\Delta u=\lambda u^q+u^p,\hfill & \text{in }\mathrm{\Omega },\hfill \\ 0⩽u\in H_0^1(\mathrm{\Omega }),\hfill \end{array}$$

(1.2)

with $0<q<1<p⩽\frac{N+2}{N-2}$. They proved that problem (1.2) admits at least two positive solutions for λ sufficiently small. In [6], Sun and Li considered a similar problem:

$$\{\begin{array}{cc}-\Delta u=u^q+\lambda u^p,\hfill & \text{in }\mathrm{\Omega },\hfill \\ 0⩽u\in H_0^1(\mathrm{\Omega }),\hfill \end{array}$$

with $0<q<1<p=\frac{N+2}{N-2}$, the authors studied the value of Λ, the supremum of the set λ, related to the existence and multiplicity of positive solutions and established uniform lower bounds for Λ. In [8], Wu considered the subcritical case of problem (1.2) with $\lambda u^q$ replaced by $\lambda f(x)u^q$, here $f(x)\in C(\overline{\mathrm{\Omega }})$ is a sign-changing function, and he showed that problem (1.2) has at least two positive solutions as λ is small enough.

Some interesting generalizations of (1.2) have been provided in the framework of quasi-linear elliptic equations or systems, semilinear second-order elliptic systems or fourth-order elliptic equations. More recently, the semilinear fourth-order elliptic equations have been studied by many authors, we refer the reader to [11–13] and the references therein. Motivated by some work in [6, 8, 13], we deal with the following semilinear biharmonic elliptic equation:

$$\{\begin{array}{cc}{\Delta }^{2}u=\lambda f(x){|u|}^{q-1}u+h(x){|u|}^{p-1}u,\hfill & \text{in }\mathrm{\Omega },\hfill \\ u=\Delta u=0,\hfill & \text{on }\partial \mathrm{\Omega },\hfill \end{array}$$

(1.3)

where Ω is a bounded smooth domain in ${\mathbb{R}}^N$ ($N⩾4$), $0<q<1<p<2^{\ast \ast }$ ($2^{\ast \ast }=\frac{N+4}{N-4}$ for $N>4$ and $2^{\ast \ast }=\mathrm{\infty }$ for $N=4$), $\lambda >0$ is a parameter, $f\in C(\overline{\mathrm{\Omega }})$ is a positive or sign-changing weight function and $h\in C(\overline{\mathrm{\Omega }})$ is a positive weight function.

For convenience and simplicity, we introduce some notations. The norm of u in $L^r(\mathrm{\Omega })$ is denoted by ${|u|}_r={({\int }_{\mathrm{\Omega }}{|u(x)|}^r)}^{1/r}$, the norm of u in $C(\overline{\mathrm{\Omega }})$ is denoted by ${|u|}_{\mathrm{\infty }}={max}_{x\in \overline{\mathrm{\Omega }}}|u(x)|$; $H_0^1(\mathrm{\Omega })\cap H^2(\mathrm{\Omega })$ is denoted by $H(\mathrm{\Omega })$, endowed with the norm $\parallel u\parallel ={|\Delta u|}_2$; S denotes the best Sobolev constant for the embedding of $H(\mathrm{\Omega })$ in $L^{p+1}(\mathrm{\Omega })$ (see [14]); to be precise, ${|u|}_{p+1}⩽S\parallel u\parallel$ for all $u\in H(\mathrm{\Omega })$.

Now we define

$$J_{\lambda }(u)=\frac{1}{2}{\parallel u\parallel }^2-\frac{\lambda }{q+1}{\int }_{\mathrm{\Omega }}f(x){|u|}^{q+1}\mathrm{d}x-\frac{1}{p+1}{\int }_{\mathrm{\Omega }}h(x){|u|}^{p+1}\mathrm{d}x,u\in H(\mathrm{\Omega }).$$

It is well known that the weak solutions of problem (1.3) are the critical points of the energy functional $J_{\lambda }$ (see Rabinowitz [15]).

Next, we consider the Nehari minimization problem: for $\lambda >0$,

$${\alpha }_{\lambda }(\mathrm{\Omega })=inf\{J_{\lambda }(u)\mid u\in M_{\lambda }(\mathrm{\Omega })\},$$

where $M_{\lambda }(\mathrm{\Omega })=\{u\in H(\mathrm{\Omega })\mathrm{\setminus }\{0\}\mid 〈J_{\lambda }^{\prime }(u),u〉=0\}$. Define

$${\psi }_{\lambda }(u)=〈J_{\lambda }^{\prime }(u),u〉={\parallel u\parallel }^2-\lambda {\int }_{\mathrm{\Omega }}f(x){|u|}^{q+1}\mathrm{d}x-{\int }_{\mathrm{\Omega }}h(x){|u|}^{p+1}\mathrm{d}x.$$

Then for $u\in M_{\lambda }(\mathrm{\Omega })$,

$$〈{\psi }_{\lambda }^{\prime }(u),u〉=2{\parallel u\parallel }^2-\lambda (q+1){\int }_{\mathrm{\Omega }}f(x){|u|}^{q+1}\mathrm{d}x-(p+1){\int }_{\mathrm{\Omega }}h(x){|u|}^{p+1}\mathrm{d}x.$$

Similarly to the method used in Tarantello [16], we split $M_{\lambda }(\mathrm{\Omega })$ into three parts:

$$\begin{array}{r}{M}_{\lambda }^{+}(\mathrm{\Omega })=\{u\in M_{\lambda }(\mathrm{\Omega })\mid 〈{\psi }_{\lambda }^{\prime }(u),u〉>0\},\\ M_{\lambda }^0(\mathrm{\Omega })=\{u\in M_{\lambda }(\mathrm{\Omega })\mid 〈{\psi }_{\lambda }^{\prime }(u),u〉=0\},\\ M_{\lambda }^{-}(\mathrm{\Omega })=\{u\in M_{\lambda }(\mathrm{\Omega })\mid 〈{\psi }_{\lambda }^{\prime }(u),u〉<0\}.\end{array}$$

Note that all solutions of (1.3) are clearly in the Nehari manifold, $M_{\lambda }(\mathrm{\Omega })$. Hence, our approach to solve problem (1.3) is to analyze the structure of $M_{\lambda }(\mathrm{\Omega })$, and then to deal with the minimization problems for $J_{\lambda }$ on $M_{\lambda }^{+}(\mathrm{\Omega })$ and $M_{\lambda }^{-}(\mathrm{\Omega })$ applying the direct variational method.

The following is our main result.

Theorem 1.1 Let ${\lambda }_{\ast }=\frac{p-1}{p-q}\cdot {[\frac{1-q}{(p-q){|h|}_{\mathrm{\infty }}}]}^{\frac{1-q}{p-1}}{S}^{\frac{2(p-q)}{1-p}}{|f|}_{p^{\ast }}^{-1}$ with $p^{\ast }=\frac{p+1}{p-q}$, then problem (1.3) has at least two positive solutions for any $\lambda \in (0,{\lambda }_{\ast })$.

The paper is organized as follows: in Section 2, we give some lemmas; in Section 3, we prove Theorem 1.1.

2 Preliminaries

In this section, we prove several lemmas.

Lemma 2.1 For $\lambda \in (0,{\lambda }_{\ast })$ (where ${\lambda }_{\ast }$ is given in Theorem  1.1), we have $M_{\lambda }^0(\mathrm{\Omega })=\varphi$.

Proof Suppose that $M_{\lambda }^0(\mathrm{\Omega })\ne \varphi$ for all $\lambda >0$. If $u\in M_{\lambda }^0(\mathrm{\Omega })$, then we have

$${\parallel u\parallel }^2=\lambda {\int }_{\mathrm{\Omega }}f(x){|u|}^{q+1}\mathrm{d}x+{\int }_{\mathrm{\Omega }}h(x){|u|}^{p+1}\mathrm{d}x$$

(2.1)

and

$$2{\parallel u\parallel }^2=\lambda (q+1){\int }_{\mathrm{\Omega }}f(x){|u|}^{q+1}\mathrm{d}x+(p+1){\int }_{\mathrm{\Omega }}h(x){|u|}^{p+1}\mathrm{d}x.$$

(2.2)

By (2.1)-(2.2), the Sobolev inequality, and the Hölder inequality, we get

$${\parallel u\parallel }^2=\frac{p-q}{1-q}{\int }_{\mathrm{\Omega }}h(x){|u|}^{p+1}\mathrm{d}x⩽\frac{p-q}{1-q}{|h|}_{\mathrm{\infty }}{S}^{p+1}{\parallel u\parallel }^{p+1}$$

(2.3)

and

$${\parallel u\parallel }^2=\lambda \cdot \frac{p-q}{p-1}{\int }_{\mathrm{\Omega }}f(x){|u|}^{q+1}\mathrm{d}x⩽\lambda \cdot \frac{p-q}{p-1}{|f|}_{p^{\ast }}{S}^{q+1}{\parallel u\parallel }^{q+1},$$

(2.4)

where $p^{\ast }=\frac{p+1}{p-q}$. Thus, using (2.3) and (2.4), we have

$$\begin{array}{rl}\lambda & ⩾\frac{p-1}{p-q}\cdot {|f|}_{p^{\ast }}^{-1}{S}^{-(q+1)}{\left[\frac{1-q}{p-q}{|h|}_{\mathrm{\infty }}^{-1}{S}^{-(p+1)}\right]}^{\frac{1-q}{p-1}}\\ =\frac{p-1}{p-q}\cdot {\left[\frac{1-q}{(p-q){|h|}_{\mathrm{\infty }}}\right]}^{\frac{1-q}{p-1}}{S}^{\frac{2(p-q)}{1-p}}{|f|}_{p^{\ast }}^{-1}={\lambda }_{\ast }.\end{array}$$

(2.5)

Hence, by (2.5) the desired conclusion yields. □

Lemma 2.2 If $u\in M_{\lambda }^{-}(\mathrm{\Omega })$, then

$$\parallel u\parallel >S^{\frac{1+p}{1-p}}{\left[\frac{1-q}{(p-q){|h|}_{\mathrm{\infty }}}\right]}^{\frac{1}{p-1}}\mathit{\text{and}}{\int }_{\mathrm{\Omega }}h(x){|u|}^{p+1}\mathrm{d}x>{|h|}_{\mathrm{\infty }}^{\frac{2}{1-p}}{\left[\frac{(p-q)S^2}{1-q}\right]}^{\frac{1+p}{1-p}}.$$

Proof From $u\in M_{\lambda }^{-}(\mathrm{\Omega })$, it is easy to see that

$${\parallel u\parallel }^2<\frac{p-q}{1-q}{\int }_{\mathrm{\Omega }}h(x){|u|}^{p+1}\mathrm{d}x.$$

By the Sobolev inequality, we get

$$\parallel u\parallel >S^{\frac{1+p}{1-p}}{\left[\frac{1-q}{(p-q){|h|}_{\mathrm{\infty }}}\right]}^{\frac{1}{p-1}}.$$

In addition,

$${\int }_{\mathrm{\Omega }}h(x){|u|}^{p+1}\mathrm{d}x>{|h|}_{\mathrm{\infty }}^{\frac{2}{1-p}}{\left[\frac{(p-q)S^2}{1-q}\right]}^{\frac{1+p}{1-p}}.$$

The proof is completed. □

By Lemma 2.1, for $\lambda \in (0,{\lambda }_{\ast })$ we write $M_{\lambda }(\mathrm{\Omega })=M_{\lambda }^{+}(\mathrm{\Omega })\cup M_{\lambda }^{-}(\mathrm{\Omega })$ and define

$${\alpha }_{\lambda }^{+}(\mathrm{\Omega })=\underset{u\in M_{\lambda }^{+}(\mathrm{\Omega })}{inf}{J}_{\lambda }(u),{\alpha }_{\lambda }^{-}(\mathrm{\Omega })=\underset{u\in M_{\lambda }^{-}(\mathrm{\Omega })}{inf}{J}_{\lambda }(u).$$

The following lemma shows that the minimizers on $M_{\lambda }(\mathrm{\Omega })$ are ‘usually’ critical points for $J_{\lambda }$.

Lemma 2.3 For $\lambda \in (0,{\lambda }_{\ast })$, if $u_0$ is a local minimizer for $J_{\lambda }$ on $M_{\lambda }(\mathrm{\Omega })$, then $J_{\lambda }^{\prime }(u_0)=0$ in ${[H(\mathrm{\Omega })]}^{\ast }$.

Proof If $u_0$ is a local minimizer for $J_{\lambda }$ on $M_{\lambda }(\mathrm{\Omega })$, then $u_0$ is a solution of the optimization problem

$$\text{minimize}{J}_{\lambda }(u)\text{subject to}{\psi }_{\lambda }(u)=0.$$

Hence, by the theory of Lagrange multipliers, there exists $\theta \in \mathbb{R}$ such that

$$J_{\lambda }^{\prime }(u_0)=\theta {\psi }_{\lambda }^{\prime }(u_0)\text{in }{[H(\mathrm{\Omega })]}^{\ast }.$$

(2.6)

Thus,

$$〈J_{\lambda }^{\prime }(u_0),u_0〉=\theta 〈{\psi }_{\lambda }^{\prime }(u_0),u_0〉.$$

(2.7)

From $u_0\in M_{\lambda }(\mathrm{\Omega })$ and Lemma 2.1, we have $〈J_{\lambda }^{\prime }(u_0),u_0〉=0$ and $〈{\psi }_{\lambda }^{\prime }(u_0),u_0〉\ne 0$. So, by (2.6)-(2.7) we get $J_{\lambda }^{\prime }(u_0)=0$ in ${[H(\mathrm{\Omega })]}^{\ast }$. □

For each $u\in H(\mathrm{\Omega })\mathrm{\setminus }\{0\}$, we write

$$t_{\max }={\left(\frac{(1-q){\parallel u\parallel }^2}{(p-q){\int }_{\mathrm{\Omega }}h(x){|u|}^{p+1}\mathrm{d}x}\right)}^{\frac{1}{p-1}}>0.$$

Then we have the following lemma.

Lemma 2.4 For each $u\in H(\mathrm{\Omega })\mathrm{\setminus }\{0\}$ and $\lambda \in (0,{\lambda }_{\ast })$, we have

• (i) there is a unique $t^{-}=t^{-}(u)>t_{\max }>0$ such that $t^{-}(u)u\in M_{\lambda }^{-}(\mathrm{\Omega })$ and $J_{\lambda }(t^{-}(u)u)={max}_{t⩾0}{J}_{\lambda }(tu)$;

• (ii) $t^{-}(u)$ is a continuous function for nonzero u;

• (iii) $M_{\lambda }^{-}(\mathrm{\Omega })=\{u\in H(\mathrm{\Omega })\mathrm{\setminus }\{0\}\mid \frac{1}{\parallel u\parallel }{t}^{-}(\frac{u}{\parallel u\parallel })=1\}$;

• (iv) if ${\int }_{\mathrm{\Omega }}f(x){|u|}^{q+1}\mathrm{d}x>0$, then there is a unique $0<t^{+}=t^{+}(u)<t_{\max }$ such that $t^{+}(u)u\in M_{\lambda }^{+}(\mathrm{\Omega })$ and $J_{\lambda }(t^{+}(u)u)={min}_{0⩽t⩽t^{-}}{J}_{\lambda }(tu)$.

Proof (i) Fix $u\in H(\mathrm{\Omega })\mathrm{\setminus }\{0\}$. Let

$$s(t)=t^{1-q}{\parallel u\parallel }^2-t^{p-q}{\int }_{\mathrm{\Omega }}h(x){|u|}^{p+1}\mathrm{d}x,t⩾0.$$

Then we have $s(0)=0$, $s(t)\to -\mathrm{\infty }$ as $t\to \mathrm{\infty }$, $s(t)$ is concave and reaches its maximum at $t_{\max }$. Moreover,

$$\begin{array}{rl}s(t_{\max })& =t_{\max }^{1-q}{\parallel u\parallel }^2-t_{\max }^{p-q}{\int }_{\mathrm{\Omega }}h(x){|u|}^{p+1}\mathrm{d}x\\ ={\parallel u\parallel }^{q+1}[{\left(\frac{1-q}{p-q}\right)}^{\frac{1-q}{p-1}}-{\left(\frac{1-q}{p-q}\right)}^{\frac{p-q}{p-1}}]{\left(\frac{{\parallel u\parallel }^{p+1}}{{\int }_{\mathrm{\Omega }}h(x){|u|}^{p+1}\mathrm{d}x}\right)}^{\frac{1-q}{p-1}}\\ ⩾{\parallel u\parallel }^{q+1}\left(\frac{p-1}{p-q}\right){\left(\frac{1-q}{p-q}\right)}^{\frac{1-q}{p-1}}{\left(\frac{1}{{|h|}_{\mathrm{\infty }}{S}^{p+1}}\right)}^{\frac{1-q}{p-1}}.\end{array}$$

(2.8)

Case I. ${\int }_{\mathrm{\Omega }}f(x){|u|}^{q+1}\mathrm{d}x⩽0$.

There is a unique $t^{-}>t_{\max }$ such that $s(t^{-})=\lambda {\int }_{\mathrm{\Omega }}f(x){|u|}^{q+1}\mathrm{d}x$ and $s^{\prime }(t^{-})<0$. Now,

$$\begin{array}{rl}〈J_{\lambda }^{\prime }\left(t^{-}u\right),t^{-}u〉& ={\parallel t^{-}u\parallel }^2-\lambda {\int }_{\mathrm{\Omega }}f(x)|t^{-}u{|}^{q+1}\mathrm{d}x-{\int }_{\mathrm{\Omega }}h(x)|t^{-}u{|}^{p+1}\mathrm{d}x\\ ={\left(t^{-}\right)}^{q+1}[s\left(t^{-}\right)-\lambda {\int }_{\mathrm{\Omega }}f(x){|u|}^{q+1}\mathrm{d}x]\\ =0\end{array}$$

and

$$\begin{array}{rl}〈{\psi }_{\lambda }^{\prime }\left(t^{-}u\right),t^{-}u〉& =(1-q){\parallel t^{-}u\parallel }^2-(p-q){\int }_{\mathrm{\Omega }}h(x)|t^{-}u{|}^{p+1}\mathrm{d}x\\ ={\left(t^{-}\right)}^{2+q}[(1-q){\left(t^{-}\right)}^{-q}{\parallel u\parallel }^2-(p-q){\left(t^{-}\right)}^{p-q-1}{\int }_{\mathrm{\Omega }}h(x){|u|}^{p+1}\mathrm{d}x]\\ ={\left(t^{-}\right)}^{2+q}{s}^{\prime }\left(t^{-}\right)<0.\end{array}$$

Thus, $t^{-}u\in M_{\lambda }^{-}(\mathrm{\Omega })$. In addition,

$$\begin{array}{rl}\frac{\mathrm{d}{J}_{\lambda }(tu)}{\mathrm{d}t}& =t{\parallel u\parallel }^2-\lambda t^q{\int }_{\mathrm{\Omega }}f(x){|u|}^{q+1}\mathrm{d}x-t^p{\int }_{\mathrm{\Omega }}h(x){|u|}^{p+1}\mathrm{d}x\\ =t^{-1}〈J_{\lambda }^{\prime }(tu),tu〉=0\text{if and only if}t=t^{-}\end{array}$$

and

$$\begin{array}{rl}\frac{{\mathrm{d}}^2{J}_{\lambda }(tu)}{\mathrm{d}{t}^2}{|}_{t=t^{-}}& ={\parallel u\parallel }^2-\lambda q{\left(t^{-}\right)}^{q-1}{\int }_{\mathrm{\Omega }}f(x){|u|}^{q+1}\mathrm{d}x-p{\left(t^{-}\right)}^{p-1}{\int }_{\mathrm{\Omega }}h(x){|u|}^{p+1}\mathrm{d}x\\ ={\left(t^{-}\right)}^{-2}〈{\psi }_{\lambda }^{\prime }\left(t^{-}u\right),t^{-}u〉<0.\end{array}$$

Hence, $J_{\lambda }(t^{-}u)={max}_{t⩾0}{J}_{\lambda }(tu)$.

Case II. ${\int }_{\mathrm{\Omega }}f(x){|u|}^{q+1}\mathrm{d}x>0$.

From (2.8) and

$$\begin{array}{rl}s(0)& =0<\lambda {\int }_{\mathrm{\Omega }}f(x){|u|}^{q+1}\mathrm{d}x⩽\lambda {|f|}_{p^{\ast }}{S}^{q+1}{\parallel u\parallel }^{q+1}\\ <{\parallel u\parallel }^{q+1}\left(\frac{p-1}{p-q}\right){\left(\frac{1-q}{p-q}\right)}^{\frac{1-q}{p-1}}{\left(\frac{1}{{|h|}_{\mathrm{\infty }}{S}^{p+1}}\right)}^{\frac{1-q}{p-1}}\\ ⩽s(t_{\max })\text{for }\lambda \in (0,{\lambda }_{\ast }),\end{array}$$

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

$$s\left(t^{+}\right)=\lambda {\int }_{\mathrm{\Omega }}f(x){|u|}^{q+1}\mathrm{d}x=s\left(t^{-}\right)$$

and

$$s^{\prime }\left(t^{+}\right)>0>s^{\prime }\left(t^{-}\right).$$

Similar to the argument in Case I above, we have $t^{+}u\in M_{\lambda }^{+}(\mathrm{\Omega })$, $t^{-}u\in M_{\lambda }^{-}(\mathrm{\Omega })$, and

$$J_{\lambda }\left(t^{-}u\right)=\underset{t⩾0}{max}{J}_{\lambda }(tu),J_{\lambda }\left(t^{+}u\right)=\underset{0⩽t⩽t^{-}}{min}{J}_{\lambda }(tu).$$

• (ii) By the uniqueness of $t^{-}(u)$ and the external property of $t^{-}(u)$, we find that $t^{-}(u)$ is continuous function of $u\ne 0$.

• (iii) For $u\in M_{\lambda }^{-}(\mathrm{\Omega })$, let $v=\frac{u}{\parallel u\parallel }$. By item (i), there is a unique $t^{-}(v)>0$ such that $t^{-}(v)v\in M_{\lambda }^{-}(\mathrm{\Omega })$, that is, $t^{-}(\frac{u}{\parallel u\parallel })\frac{1}{\parallel u\parallel }u\in M_{\lambda }^{-}(\mathrm{\Omega })$. Since $u\in M_{\lambda }^{-}(\mathrm{\Omega })$, we have $t^{-}(\frac{u}{\parallel u\parallel })\frac{1}{\parallel u\parallel }=1$, which implies

  $$M_{\lambda }^{-}(\mathrm{\Omega })\subset \{u\in H(\mathrm{\Omega })\mathrm{\setminus }\{0\}|\frac{1}{\parallel u\parallel }{t}^{-}\left(\frac{u}{\parallel u\parallel }\right)=1\}.$$

Conversely, let $u\in H(\mathrm{\Omega })\mathrm{\setminus }\{0\}$ such that $\frac{1}{\parallel u\parallel }{t}^{-}(\frac{u}{\parallel u\parallel })=1$. Then $t^{-}(\frac{u}{\parallel u\parallel })\frac{u}{\parallel u\parallel }\in M_{\lambda }^{-}(\mathrm{\Omega })$. Therefore,

$$M_{\lambda }^{-}(\mathrm{\Omega })=\{u\in H(\mathrm{\Omega })\mathrm{\setminus }\{0\}|\frac{1}{\parallel u\parallel }{t}^{-}\left(\frac{u}{\parallel u\parallel }\right)=1\}.$$

• (iv) By Case II of item (i). □

By $f\in C(\overline{\mathrm{\Omega }})$ and changes sign in Ω, we have $\Theta =\{x\in \mathrm{\Omega }\mid f(x)>0\}$ is an open set in $R^N$. Without loss of generality, we may assume that Θ is a domain in $R^N$. Consider the following biharmonic equation:

$$\{\begin{array}{cc}{\Delta }^{2}u=h(x){|u|}^{p-1}u,\hfill & \text{in }\Theta ,\hfill \\ u=\Delta u=0,\hfill & \text{on }\partial \Theta .\hfill \end{array}$$

(2.9)

Associated with (2.9), we consider the energy functional

$$K(u)=\frac{1}{2}{\parallel u\parallel }^2-\frac{1}{p+1}{\int }_{\Theta }h(x){|u|}^{p+1}\mathrm{d}x,u\in H(\Theta )$$

and the minimization problem

$$\beta (\Theta )=inf\{K(u)\mid u\in N(\Theta )\},$$

where $N(\Theta )=\{u\in H(\Theta )\mathrm{\setminus }\{0\}\mid 〈K^{\prime }(u),u〉=0\}$. Now we prove that problem (2.9) has a positive solution $w_0$ such that $K(w_0)=\beta (\Theta )>0$.

Lemma 2.5 For any $u\in H(\Theta )\mathrm{\setminus }\{0\}$, there exists a unique $t(u)>0$ such that $t(u)u\in N(\Theta )$. The maximum of $K(tu)$ for $t⩾0$ is reached at $t=t(u)$, the map

$$t:H(\Theta )\mathrm{\setminus }\{0\}\to (0,+\mathrm{\infty });u\mapsto t(u)$$

is continuous and the induced continuous map $u\to t(u)u$ defines a homeomorphism of the unit sphere of $H(\Theta )$ with $N(\Theta )$.

Proof For any given $u\in H(\Theta )\mathrm{\setminus }\{0\}$, consider the function $g(t)=K(tu)$, $t⩾0$. Clearly,

$$g^{\prime }(t)=0\iff tu\in N(\Theta )\iff {\parallel u\parallel }^2=t^{p-1}{\int }_{\mathrm{\Omega }}h(x){|u|}^{p+1}\mathrm{d}x.$$

(2.10)

It is easy to verify that $g(0)=0$, $g(t)>0$ for $t>0$ small and $g(t)<0$ for $t>0$ large. Hence, ${max}_{t⩾0}g(t)$ is reached at a unique $t=t(u)$ such that $g^{\prime }(t(u))=0$ and $t(u)u\in N(\Theta )$. To prove the continuity of $t(u)$, assume that $u_n\to u$ in $H(\Theta )\mathrm{\setminus }\{0\}$. It is easy to verify that $\{t(u_n)\}$ is bounded. If a subsequence of $\{t(u_n)\}$ converges to $t_0$, it follows from (2.10) that $t_0=t(u)$ and then $t(u_n)\to t(u)$. Finally the continuous map from the unit sphere of $H(\Theta )$ to $N(\Theta )$, $u\to t(u)u$, is inverse to the retraction $u\to \frac{u}{\parallel u\parallel }$. □

Define

$$c_{\ast }=\underset{u\in H(\Theta )\mathrm{\setminus }\{0\}}{inf}\underset{t⩾0}{max}K(tu),c=\underset{\gamma \in \Gamma }{inf}\underset{t\in [0,1]}{max}K(\gamma (t)),$$

where $\Gamma =\{\gamma \in C([0,1],H(\Theta ))\mid \gamma (0)=0,K(\gamma (1))<0\}$.

Lemma 2.6 $\beta (\Theta )=c_{\ast }=c>0$ is a critical value of K.

Proof From Lemma 2.5, we know that $\beta (\Theta )=c_{\ast }$. Since $K(tu)<0$ for $u\in H(\Theta )\mathrm{\setminus }\{0\}$ and t large, we obtain $c⩽c_{\ast }$. The manifold $N(\Theta )$ separates $H(\Theta )$ into two components. The component containing the origin also contains a small ball around the origin. Moreover, $K(u)⩾0$ for all u in this component, because $〈K^{\prime }(tu),u〉⩾0$, $\mathrm{\forall }t\in [0,t(u)]$. Then each $\gamma \in \Gamma$ has to cross $N(\Theta )$ and $\beta (\Theta )⩽c$. Since the embedding $H(\Theta )\hookrightarrow L^{p+1}(\Theta )$ is compact (see [14]), it is easy to prove that $c>0$ is a critical value of K and $w_0$ a positive solution corresponding to c. □

With the help of Lemma 2.6, we have the following result.

Lemma 2.7 (i) For $\lambda \in (0,{\lambda }_{\ast })$, there exists $t_{\lambda }>0$ such that

$${\alpha }_{\lambda }(\mathrm{\Omega })⩽{\alpha }_{\lambda }^{+}(\mathrm{\Omega })<-\frac{1-q}{q+1}{t}_{\lambda }^2{\beta }_{\lambda }(\Theta )<0;$$

• (ii) $J_{\lambda }$ is coercive and bounded below on $M_{\lambda }(\mathrm{\Omega })$ for all $\lambda >0$.

Proof (i) Let $w_0$ be a positive solution of problem (2.9) such that $K(w_0)=\beta (\Theta )$. Then

$${\int }_{\mathrm{\Omega }}f(x)w_0^{q+1}\mathrm{d}x={\int }_{\Theta }f(x)w_0^{q+1}\mathrm{d}x>0.$$

Set $t_{\lambda }=t^{+}(w_0)$ as defined by Lemma 2.4(iv). Hence, $t_{\lambda }{w}_0\in M_{\lambda }^{+}(\mathrm{\Omega })$ and

$$\begin{array}{rl}{J}_{\lambda }(t_{\lambda }{w}_0)& =\frac{1}{2}{\parallel t_{\lambda }{w}_0\parallel }^2-\frac{\lambda }{q+1}{\int }_{\mathrm{\Omega }}f(x){|t_{\lambda }{w}_0|}^{q+1}\mathrm{d}x-\frac{1}{p+1}{\int }_{\mathrm{\Omega }}h(x){|t_{\lambda }{w}_0|}^{p+1}\mathrm{d}x\\ =(\frac{1}{2}-\frac{1}{q+1}){\parallel t_{\lambda }{w}_0\parallel }^2+(\frac{1}{q+1}-\frac{1}{p+1}){\int }_{\mathrm{\Omega }}h(x){|t_{\lambda }{w}_0|}^{p+1}\mathrm{d}x\\ <-\frac{1-q}{q+1}{t}_{\lambda }^2\beta (\Theta )<0.\end{array}$$

This implies

$${\alpha }_{\lambda }(\mathrm{\Omega })⩽{\alpha }_{\lambda }^{+}(\mathrm{\Omega })<-\frac{1-q}{q+1}{t}_{\lambda }^2\beta (\Theta )<0.$$

• (ii) For $u\in M_{\lambda }(\mathrm{\Omega })$, we have ${\parallel u\parallel }^2=\lambda {\int }_{\mathrm{\Omega }}f(x){|u|}^{q+1}\mathrm{d}x+{\int }_{\mathrm{\Omega }}h(x){|u|}^{p+1}\mathrm{d}x$. Then by the Hölder, Sobolev, and Young inequalities,

  $$\begin{array}{rl}{J}_{\lambda }(u)& =\frac{p-1}{2(p+1)}{\parallel u\parallel }^2-\frac{\lambda (p-q)}{(p+1)(q+1)}{\int }_{\mathrm{\Omega }}f(x){|u|}^{q+1}\mathrm{d}x\\ ⩾\frac{p-1}{2(p+1)}{\parallel u\parallel }^2-\frac{\lambda (p-q)}{(p+1)(q+1)}{|f|}_{p^{\ast }}{S}^{q+1}{\parallel u\parallel }^{q+1}\\ ⩾\frac{p-1}{4(p+1)}{\parallel u\parallel }^2-{\lambda }^{\frac{2}{1-q}}C(p,q){\left({|f|}_{p^{\ast }}{S}^{q+1}\right)}^{\frac{2}{1-q}},\end{array}$$

here $C(p,q)={[\frac{p-q}{(p+1)(q+1)}]}^{\frac{2}{1-q}}\cdot {[\frac{4(p+1)}{p-1}]}^{\frac{1+q}{1-q}}$.

Thus, $J_{\lambda }$ is coercive on $M_{\lambda }(\mathrm{\Omega })$ and

$$J_{\lambda }(u)⩾-{\lambda }^{\frac{2}{1-q}}C(p,q){\left({|f|}_{p^{\ast }}{S}^{q+1}\right)}^{\frac{2}{1-q}}$$

for all $\lambda >0$. □

Next, we will use the idea of Tarantello [16] to get the following results.

Lemma 2.8 For $\lambda \in (0,{\lambda }_{\ast })$ and any given $u\in M_{\lambda }(\mathrm{\Omega })$, there exist $ϵ>0$ and a differentiable functional $\xi :B(0;ϵ)\subset H(\mathrm{\Omega })\to {\mathbb{R}}^{+}$ such that $\xi (0)=1$, the function $\xi (v)(u+v)\in M_{\lambda }(\mathrm{\Omega })$ and

$$〈{\xi }^{\prime }(0),v〉=\frac{2{\int }_{\mathrm{\Omega }}\Delta u\Delta v-\lambda (q+1){\int }_{\mathrm{\Omega }}f{|u|}^{q-1}uv-(p+1){\int }_{\mathrm{\Omega }}h{|u|}^{p-1}uv}{(1-q){\parallel u\parallel }^2-(p-q){\int }_{\mathrm{\Omega }}h(x){|u|}^{p+1}\mathrm{d}x}$$

(2.11)

for all $v\in H(\mathrm{\Omega })$.

Proof Define $F:\mathbb{R}\times H(\mathrm{\Omega })\to \mathbb{R}$ as follows:

$$F(\xi ,w)={\xi }^2{\parallel u+w\parallel }^2-\lambda {\xi }^{q+1}{\int }_{\mathrm{\Omega }}f(x){|u+w|}^{q+1}\mathrm{d}x-{\xi }^{p+1}{\int }_{\mathrm{\Omega }}h(x){|u+w|}^{p+1}\mathrm{d}x.$$

Since $F(1,0)=〈J_{\lambda }^{\prime }(u),u〉=0$ and by Lemma 2.1, we obtain

$$\begin{array}{rl}{F}_{\xi }^{\prime }(1,0)& =2{\parallel u\parallel }^2-\lambda (q+1){\int }_{\mathrm{\Omega }}f(x){|u|}^{q+1}\mathrm{d}x-(p+1){\int }_{\mathrm{\Omega }}h(x){|u|}^{p+1}\mathrm{d}x\\ =〈{\psi }_{\lambda }^{\prime }(u),u〉\ne 0,\end{array}$$

we can get the desired results applying the implicit function theorem at the point $(1,0)$. □

Lemma 2.9 For $\lambda \in (0,{\lambda }_{\ast })$ and any given $u\in M_{\lambda }^{-}(\mathrm{\Omega })$, there exist $ϵ>0$ and a differentiable functional ${\xi }^{-}:B(0;ϵ)\subset H(\mathrm{\Omega })\to {\mathbb{R}}^{+}$ such that ${\xi }^{-}(0)=1$, the function ${\xi }^{-}(v)(u+v)\in M_{\lambda }^{-}(\mathrm{\Omega })$ and

$$〈{\left({\xi }^{-}\right)}^{\prime }(0),v〉=\frac{2{\int }_{\mathrm{\Omega }}\Delta u\Delta v-\lambda (q+1){\int }_{\mathrm{\Omega }}f{|u|}^{q-1}uv-(p+1){\int }_{\mathrm{\Omega }}h{|u|}^{p-1}uv}{(1-q){\parallel u\parallel }^2-(p-q){\int }_{\mathrm{\Omega }}h(x){|u|}^{p+1}\mathrm{d}x}$$

(2.12)

for all $v\in H(\mathrm{\Omega })$.

Proof In view of Lemma 2.8, there exist $ϵ>0$ and a differentiable functional ${\xi }^{-}$ such that ${\xi }^{-}(0)=1$, ${\xi }^{-}(v)(u+v)\in M_{\lambda }(\mathrm{\Omega })$ for all $v\in B(0;ϵ)\subset H(\mathrm{\Omega })$ and we have (2.12). By use of $u\in M_{\lambda }^{-}(\mathrm{\Omega })$, we have $〈{\psi }_{\lambda }^{\prime }(u),u〉<0$. In combination with the continuity of the functions ${\psi }_{\lambda }^{\prime }$ and ${\xi }^{-}$, we get $〈{\psi }_{\lambda }^{\prime }({\xi }^{-}(v)(u+v)),{\xi }^{-}(v)(u+v)〉<0$ as ϵ sufficiently small, this implies that ${\xi }^{-}(v)(u+v)\in M_{\lambda }^{-}(\mathrm{\Omega })$. □

3 Proof of Theorem 1.1

Firstly, we provide the existence of minimizing sequences for $J_{\lambda }$ on $M_{\lambda }(\mathrm{\Omega })$ and $M_{\lambda }^{-}(\mathrm{\Omega })$ as λ is sufficiently small.

Proposition 3.1 Let $\lambda \in (0,{\lambda }_{\ast })$, then

• (i) there exists a minimizing sequence $\{u_n\}\subset M_{\lambda }(\mathrm{\Omega })$ such that

  $$J_{\lambda }(u_n)={\alpha }_{\lambda }(\mathrm{\Omega })+o(1)\mathit{\text{and}}{J}_{\lambda }^{\prime }(u_n)=o(1)\mathit{\text{in }}{[H(\mathrm{\Omega })]}^{\ast };$$

• (ii) there exists a minimizing sequence $\{u_n\}\subset M_{\lambda }^{-}(\mathrm{\Omega })$ such that

  $$J_{\lambda }(u_n)={\alpha }_{\lambda }^{-}(\mathrm{\Omega })+o(1)\mathit{\text{and}}{J}_{\lambda }^{\prime }(u_n)=o(1)\mathit{\text{in }}{[H(\mathrm{\Omega })]}^{\ast }.$$

Proof (i) By Lemma 2.7(ii) and the Ekeland variational principle [17], there exists a minimizing sequence $\{u_n\}\subset M_{\lambda }(\mathrm{\Omega })$ such that

$$J_{\lambda }(u_n)<{\alpha }_{\lambda }(\mathrm{\Omega })+\frac{1}{n}$$

(3.1)

and

$$J_{\lambda }(u_n)<J_{\lambda }(w)+\frac{1}{n}\parallel w-u_n\parallel \text{for each }w\in M_{\lambda }(\mathrm{\Omega }).$$

(3.2)

Taking n large, from Lemma 2.7(i) and (3.1), we have

$$\begin{array}{rl}{J}_{\lambda }(u_n)& =(\frac{1}{2}-\frac{1}{p+1}){\parallel u_n\parallel }^2-(\frac{1}{q+1}-\frac{1}{p+1})\lambda {\int }_{\mathrm{\Omega }}f(x){|u_n|}^{q+1}\mathrm{d}x\\ <{\alpha }_{\lambda }(\mathrm{\Omega })+\frac{1}{n}<-\frac{1-q}{q+1}{t}_{\lambda }^2\beta (\Theta ).\end{array}$$

(3.3)

This implies

$${|f|}_{p^{\ast }}{S}^{q+1}{\parallel u_n\parallel }^{q+1}⩾{\int }_{\mathrm{\Omega }}f(x){|u_n|}^{q+1}\mathrm{d}x>\frac{(p+1)(1-q)}{\lambda (p-q)}{t}_{\lambda }^2\beta (\Theta )>0,$$

(3.4)

that is,

$$\parallel u_n\parallel >{[\frac{(p+1)(1-q)}{\lambda (p-q)}{t}_{\lambda }^2\beta (\Theta )S^{-(q+1)}{|f|}_{p^{\ast }}^{-1}]}^{\frac{1}{q+1}}.$$

(3.5)

Now, we will show that

$$〈J_{\lambda }^{\prime }(u_n),\phi 〉\to 0\text{as }n\to \mathrm{\infty },\mathrm{\forall }\phi \in H(\mathrm{\Omega }).$$

Exactly as in Lemma 2.8 we may apply suitable functionals ${\xi }_n(v)>0$ to $u_n$ and obtain

$${\xi }_n(v)(u_n+v)\in M_{\lambda }(\mathrm{\Omega }),\mathrm{\forall }v\in H(\mathrm{\Omega }),\parallel v\parallel <{ϵ}_n.$$

(3.6)

Hence, if $\phi \in H(\mathrm{\Omega })$ and $s>0$ small, substituting in (3.6) $v=s\phi$ and applying (3.2), we have

$$\begin{array}{c}\frac{1}{n}\left[\right|{\xi }_n(s\phi )-1|\cdot \parallel u_n\parallel +{\xi }_n(s\phi )\parallel s\phi \parallel ]\hfill \\ ⩾J_{\lambda }(u_n)-J_{\lambda }({\xi }_n(s\phi )(u_n+s\phi ))\hfill \\ =\frac{1}{2}{\parallel u_n\parallel }^2-\frac{\lambda }{q+1}{\int }_{\mathrm{\Omega }}f(x){|u_n|}^{q+1}\mathrm{d}x-\frac{1}{p+1}{\int }_{\mathrm{\Omega }}h(x){|u_n|}^{p+1}\mathrm{d}x\hfill \\ -\frac{1}{2}{\xi }_n^2(s\phi ){\parallel u_n+s\phi \parallel }^2+\frac{\lambda }{q+1}{\xi }_n^{q+1}(s\phi ){\int }_{\mathrm{\Omega }}f(x){|u_n+s\phi |}^{q+1}\mathrm{d}x\hfill \\ +\frac{1}{p+1}{\xi }_n^{p+1}(s\phi ){\int }_{\mathrm{\Omega }}h(x){|u_n+s\phi |}^{p+1}\mathrm{d}x\hfill \\ =-\frac{{\xi }_n^2(s\phi )-1}{2}{\parallel u_n+s\phi \parallel }^2-\frac{1}{2}({\parallel u_n+s\phi \parallel }^2-{\parallel u_n\parallel }^2)\hfill \\ +\lambda \frac{{\xi }_n^{q+1}(s\phi )-1}{q+1}{\int }_{\mathrm{\Omega }}f(x){|u_n+s\phi |}^{q+1}\mathrm{d}x\hfill \\ +\frac{\lambda }{q+1}{\int }_{\mathrm{\Omega }}f(x)({|u_n+s\phi |}^{q+1}-{|u_n|}^{q+1})\mathrm{d}x\hfill \\ +\frac{{\xi }_n^{p+1}(s\phi )-1}{p+1}{\int }_{\mathrm{\Omega }}h(x){|u_n+s\phi |}^{p+1}\mathrm{d}x+\frac{1}{p+1}{\int }_{\mathrm{\Omega }}h(x)({|u_n+s\phi |}^{p+1}-{|u_n|}^{p+1})\mathrm{d}x.\hfill \end{array}$$

Dividing by $s>0$ and passing to the limit as $s\to 0$ we derive

$$\begin{array}{c}\frac{1}{n}\left[\right|{\xi }_n^{\prime }(0)\phi |\parallel u_n\parallel +\parallel \phi \parallel ]\hfill \\ ⩾-[{\xi }_n^{\prime }(0)\phi ][{\parallel u_n\parallel }^2-\lambda {\int }_{\mathrm{\Omega }}f(x){|u_n|}^{q+1}\mathrm{d}x-{\int }_{\mathrm{\Omega }}h(x){|u_n|}^{p+1}\mathrm{d}x]\hfill \\ -{\int }_{\mathrm{\Omega }}\Delta u_n\Delta \phi \mathrm{d}x+\lambda {\int }_{\mathrm{\Omega }}f(x){|u_n|}^{q-1}{u}_n\phi \mathrm{d}x+{\int }_{\mathrm{\Omega }}h(x){|u_n|}^{p-1}{u}_n\phi \mathrm{d}x\hfill \\ =-{\int }_{\mathrm{\Omega }}\Delta u_n\Delta \phi \mathrm{d}x+\lambda {\int }_{\mathrm{\Omega }}f(x){|u_n|}^{q-1}{u}_n\phi \mathrm{d}x+{\int }_{\mathrm{\Omega }}h(x){|u_n|}^{p-1}{u}_n\phi \mathrm{d}x.\hfill \end{array}$$

(3.7)

Since

$${\xi }_n^{\prime }(0)\phi =\frac{2{\int }_{\mathrm{\Omega }}\Delta u_n\Delta \phi -\lambda (q+1){\int }_{\mathrm{\Omega }}f{|u_n|}^{q-1}{u}_n\phi -(p+1){\int }_{\mathrm{\Omega }}h{|u_n|}^{p-1}{u}_n\phi }{(1-q){\parallel u_n\parallel }^2-(p-q){\int }_{\mathrm{\Omega }}h(x){|u_n|}^{p+1}\mathrm{d}x},$$

by the boundedness of $u_n$ we get

$$\parallel {\xi }_n^{\prime }(0)\parallel ⩽\frac{C_1}{|(1-q){\parallel u_n\parallel }^2-(p-q){\int }_{\mathrm{\Omega }}h(x){|u_n|}^{p+1}\mathrm{d}x|}$$

(3.8)

for a suitable positive constant $C_1$.

Next, we show that $|(1-q){\parallel u_n\parallel }^2-(p-q){\int }_{\mathrm{\Omega }}h(x){|u_n|}^{p+1}\mathrm{d}x|$ is bounded away from zero. Arguing by contradiction, assume that

$$(1-q){\parallel u_n\parallel }^2-(p-q){\int }_{\mathrm{\Omega }}h(x){|u_n|}^{p+1}\mathrm{d}x=o(1),n\to \mathrm{\infty }.$$

(3.9)

Since $u_n\in M_{\lambda }(\mathrm{\Omega })$, we have

$${\parallel u_n\parallel }^2=\lambda {\int }_{\mathrm{\Omega }}f(x){|u_n|}^{q+1}\mathrm{d}x+{\int }_{\mathrm{\Omega }}h(x){|u_n|}^{p+1}\mathrm{d}x,$$

and consequently by (3.9),

$$\frac{p-1}{1-q}{\int }_{\mathrm{\Omega }}h(x){|u_n|}^{p+1}\mathrm{d}x=\lambda {\int }_{\mathrm{\Omega }}f(x){|u_n|}^{q+1}\mathrm{d}x+o(1),n\to \mathrm{\infty }.$$

(3.10)

Then by (3.4), the Hölder inequality, Sobolev inequality and (3.9)-(3.10), we obtain

$$\begin{array}{rl}0& <({\lambda }_{\ast }-\lambda ){\int }_{\mathrm{\Omega }}f(x){|u_n|}^{q+1}\mathrm{d}x\\ ⩽\frac{p-1}{1-q}{\int }_{\mathrm{\Omega }}h(x){|u_n|}^{p+1}\mathrm{d}x{\left[\frac{(p-q){\int }_{\mathrm{\Omega }}h(x){|u_n|}^{p+1}\mathrm{d}x}{(1-q){\parallel u_n\parallel }^2}\right]}^{\frac{q-p}{p-1}}-\lambda {\int }_{\mathrm{\Omega }}f(x){|u_n|}^{q+1}\mathrm{d}x\\ =o(1),\end{array}$$

moreover, $\parallel u_n\parallel =o(1)$, which contradicts (3.5).

Thus, we get from (3.8) that

$$\parallel {\xi }_n^{\prime }(0)\parallel ⩽C_2,\text{independent of }n.$$

Hence, by (3.7) it follows that

$${\int }_{\mathrm{\Omega }}\Delta u_n\Delta \phi \mathrm{d}x-\lambda {\int }_{\mathrm{\Omega }}f(x){|u_n|}^{q-1}{u}_n\phi \mathrm{d}x-{\int }_{\mathrm{\Omega }}h(x){|u_n|}^{p-1}{u}_n\phi \mathrm{d}x⩾-\frac{C_3}{n},$$

which implies that $〈J_{\lambda }^{\prime }(u_n),\phi 〉\to 0$, as $n\to \mathrm{\infty }$.

• (ii) Similar to the arguments in (i), by Lemma 2.9 and Lemma 2.2, we can prove (ii). □

Now, we establish the existence of a local minimum for $J_{\lambda }$ on $M_{\lambda }^{+}(\mathrm{\Omega })$.

Theorem 3.1 Let $\lambda \in (0,{\lambda }_{\ast })$, then the functional $J_{\lambda }$ has a minimizer $u_0^{+}$ in $M_{\lambda }^{+}(\mathrm{\Omega })$ and it satisfies

• (i) $J_{\lambda }(u_0^{+})={\alpha }_{\lambda }(\mathrm{\Omega })={\alpha }_{\lambda }^{+}(\mathrm{\Omega })$;

• (ii) $u_0^{+}$ is a positive solution of problem (1.3);

• (iii) $J_{\lambda }(u_0^{+})\to 0$ as $\lambda \to 0$.

Proof By Proposition 3.1(i), there is a minimizing sequence $\{u_n\}$ for $J_{\lambda }$ on $M_{\lambda }(\mathrm{\Omega })$ such that

$$J_{\lambda }(u_n)={\alpha }_{\lambda }(\mathrm{\Omega })+o(1)\text{and}{J}_{\lambda }^{\prime }(u_n)=o(1)\text{in }{[H(\mathrm{\Omega })]}^{\ast }.$$

(3.11)

Then by Lemma 2.7 and the compact imbedding theorem, there exist a subsequence $\{u_n\}$ and $u_0^{+}\in H(\mathrm{\Omega })$ such that

$$u_n\rightharpoonup u_0^{+}\text{weakly in }H(\mathrm{\Omega }),$$

(3.12)

$$u_n\to u_0^{+}\text{strongly in }{L}^{p+1}(\mathrm{\Omega })$$

(3.13)

and

$$u_n\to u_0^{+}\text{strongly in }{L}^{q+1}(\mathrm{\Omega }).$$

(3.14)

First, we claim that

$${\int }_{\mathrm{\Omega }}f(x)|u_0^{+}{|}^{q+1}\mathrm{d}x>0.$$

If not, by (3.14) we conclude that

$${\int }_{\mathrm{\Omega }}f(x){|u_n|}^{q+1}\mathrm{d}x\to {\int }_{\mathrm{\Omega }}f(x)|u_0^{+}{|}^{q+1}\mathrm{d}x⩽0\text{as }n\to \mathrm{\infty }.$$

Therefore, as $n\to \mathrm{\infty }$,

$$\begin{array}{rl}{J}_{\lambda }(u_n)& =\frac{1}{2}{\parallel u_n\parallel }^2-\frac{\lambda }{q+1}{\int }_{\mathrm{\Omega }}f(x){|u_n|}^{q+1}\mathrm{d}x-\frac{1}{p+1}{\int }_{\mathrm{\Omega }}h(x){|u_n|}^{p+1}\mathrm{d}x\\ =(\frac{1}{2}-\frac{1}{q+1})\lambda {\int }_{\mathrm{\Omega }}f(x){|u_n|}^{q+1}\mathrm{d}x+(\frac{1}{2}-\frac{1}{p+1}){\int }_{\mathrm{\Omega }}h(x){|u_n|}^{p+1}\mathrm{d}x\\ =(\frac{1}{2}-\frac{1}{q+1})\lambda {\int }_{\mathrm{\Omega }}f(x){\left|u_0^{+}\right|}^{q+1}\mathrm{d}x+(\frac{1}{2}-\frac{1}{p+1}){\int }_{\mathrm{\Omega }}h(x){\left|u_0^{+}\right|}^{p+1}\mathrm{d}x+o(1),\end{array}$$

this contradicts $J_{\lambda }(u_n)\to {\alpha }_{\lambda }(\mathrm{\Omega })<0$ as $n\to \mathrm{\infty }$.

In combination with (3.11)-(3.14), it is easy to verify that $u_0^{+}\in M_{\lambda }(\mathrm{\Omega })$ is a nontrivial weak solution of problem (1.3).

Now we prove that $u_n\to u_0^{+}$ strongly in $H(\mathrm{\Omega })$. Supposing the contrary, then $\parallel u_0^{+}\parallel <{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}\parallel u_n\parallel$ and so

$$\begin{array}{c}{\parallel u_0^{+}\parallel }^2-\lambda {\int }_{\mathrm{\Omega }}f(x)|u_0^{+}{|}^{q+1}\mathrm{d}x-{\int }_{\mathrm{\Omega }}h(x)|u_0^{+}{|}^{p+1}\mathrm{d}x\hfill \\ <\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}({\parallel u_n\parallel }^2-\lambda {\int }_{\mathrm{\Omega }}f(x){|u_n|}^{q+1}\mathrm{d}x-{\int }_{\mathrm{\Omega }}h(x){|u_n|}^{p+1}\mathrm{d}x)=0,\hfill \end{array}$$

this contradicts $u_0^{+}\in M_{\lambda }(\mathrm{\Omega })$. Hence, $u_n\to u_0^{+}$ strongly in $H(\mathrm{\Omega })$. This implies

$$J_{\lambda }(u_n)\to J_{\lambda }\left(u_0^{+}\right)={\alpha }_{\lambda }(\mathrm{\Omega })\text{as }n\to \mathrm{\infty }.$$

Moreover, we have $u_0^{+}\in M_{\lambda }^{+}(\mathrm{\Omega })$. In fact, if $u_0^{+}\in M_{\lambda }^{-}(\mathrm{\Omega })$, by Lemma 2.4, there exist unique $t_0^{+}$ and $t_0^{-}$ such that $t_0^{+}{u}_0^{+}\in M_{\lambda }^{+}(\mathrm{\Omega })$ and $t_0^{-}{u}_0^{+}\in M_{\lambda }^{-}(\mathrm{\Omega })$, we get $t_0^{+}<t_0^{-}=1$. Since

$$\frac{\mathrm{d}{J}_{\lambda }(tu)}{\mathrm{d}t}=0\text{if and only if}t=t_0^{+}\text{ and }{t}_0^{-}$$

and

$$\frac{{\mathrm{d}}^2{J}_{\lambda }(tu)}{\mathrm{d}{t}^2}{|}_{t=t_0^{+}}>0,\frac{{\mathrm{d}}^2{J}_{\lambda }(tu)}{\mathrm{d}{t}^2}{|}_{t=t_0^{-}}<0,$$

there exists $\tilde{t}\in (t_0^{+},t_0^{-}]$ such that $J_{\lambda }(t_0^{+}{u}_0^{+})<J_{\lambda }(\tilde{t}{u}_0^{+})$. By Lemma 2.4,

$$J_{\lambda }\left(t_0^{+}{u}_0^{+}\right)<J_{\lambda }\left(\tilde{t}{u}_0^{+}\right)⩽J_{\lambda }\left(t_0^{-}{u}_0^{+}\right)=J_{\lambda }\left(u_0^{+}\right),$$

which is a contradiction. Since $J_{\lambda }(u_0^{+})=J_{\lambda }(|u_0^{+}|)$ and $|u_0^{+}|\in M_{\lambda }^{+}(\mathrm{\Omega })$, by Lemma 2.3 we may assume that $u_0^{+}$ is a nonnegative weak solution to problem (1.3). Applying the regularity theory and strong maximum principle of elliptic equations, we find that $u_0^{+}$ is one positive solution of problem (1.3). In addition, by Lemma 2.7,

$$0>J_{\lambda }\left(u_0^{+}\right)⩾-{\lambda }^{\frac{2}{1-q}}C(p,q){\left({|f|}_{p^{\ast }}{S}^{q+1}\right)}^{\frac{2}{1-q}},$$

which implies that $J_{\lambda }(u_0^{+})\to 0$ as $\lambda \to 0$. □

Next, we establish the existence of a local minimum for $J_{\lambda }$ on $M_{\lambda }^{-}(\mathrm{\Omega })$.

Theorem 3.2 Let $\lambda \in (0,{\lambda }_{\ast })$, then the functional $J_{\lambda }$ has a minimizer $u_0^{-}$ in $M_{\lambda }^{-}(\mathrm{\Omega })$ and it satisfies

• (i) $J_{\lambda }(u_0^{-})={\alpha }_{\lambda }^{-}(\mathrm{\Omega })$;

• (ii) $u_0^{-}$ is a positive solution of problem (1.3).

Proof By Proposition 3.1(ii), there is a minimizing sequence $\{u_n\}$ for $J_{\lambda }$ on $M_{\lambda }^{-}(\mathrm{\Omega })$ such that

$$J_{\lambda }(u_n)={\alpha }_{\lambda }^{-}(\mathrm{\Omega })+o(1)\text{and}{J}_{\lambda }^{\prime }(u_n)=o(1)\text{in }{[H(\mathrm{\Omega })]}^{\ast }.$$

Then by Lemma 2.7 and the compact imbedding theorem, there exist a subsequence $\{u_n\}$ and $u_0^{-}\in H(\mathrm{\Omega })$ such that

$$\begin{array}{c}{u}_n\rightharpoonup u_0^{-}\text{weakly in }H(\mathrm{\Omega }),\hfill \\ u_n\to u_0^{-}\text{strongly in }{L}^{p+1}(\mathrm{\Omega })\hfill \end{array}$$

and

$$u_n\to u_0^{-}\text{strongly in }{L}^{q+1}(\mathrm{\Omega }).$$

Connecting with Lemma 2.2, it is easy to see that $u_0^{-}\in M_{\lambda }(\mathrm{\Omega })$ is a nontrivial weak solution of problem (1.3).

Next we prove that $u_n\to u_0^{-}$ strongly in $H(\mathrm{\Omega })$. Supposing the contrary, then $\parallel u_0^{-}\parallel <{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}\parallel u_n\parallel$ and so

$$\begin{array}{c}{\parallel u_0^{-}\parallel }^2-\lambda {\int }_{\mathrm{\Omega }}f(x)|u_0^{-}{|}^{q+1}\mathrm{d}x-{\int }_{\mathrm{\Omega }}h(x)|u_0^{-}{|}^{p+1}\mathrm{d}x\hfill \\ <\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}({\parallel u_n\parallel }^2-\lambda {\int }_{\mathrm{\Omega }}f(x){|u_n|}^{q+1}\mathrm{d}x-{\int }_{\mathrm{\Omega }}h(x){|u_n|}^{p+1}\mathrm{d}x)=0,\hfill \end{array}$$

this contradicts $u_0^{-}\in M_{\lambda }(\mathrm{\Omega })$. Hence, $u_n\to u_0^{-}$ strongly in $H(\mathrm{\Omega })$. This implies

$$J_{\lambda }(u_n)\to J_{\lambda }\left(u_0^{-}\right)={\alpha }_{\lambda }^{-}(\mathrm{\Omega })\text{as }n\to \mathrm{\infty }.$$

In addition, from Lemma 2.4(ii)-(iii), we have $u_0^{-}\in M_{\lambda }^{-}(\mathrm{\Omega })$. Since $J_{\lambda }(u_0^{-})=J_{\lambda }(|u_0^{-}|)$ and $|u_0^{-}|\in M_{\lambda }^{-}(\mathrm{\Omega })$, by Lemma 2.3 we may assume that $u_0^{-}$ is a nonnegative weak solution to problem (1.3). Applying the regularity theory and strong maximum principle of elliptic equations, we see that $u_0^{-}$ is one positive solution of problem (1.3). □

Proof of Theorem 1.1 It is an immediate consequence of Theorems 3.1 and 3.2. □