Open Access

Some properties and applications related to the \((2,p)\)-Laplacian operator

Boundary Value Problems20162016:58

https://doi.org/10.1186/s13661-016-0567-x

Received: 26 November 2015

Accepted: 25 February 2016

Published: 2 March 2016

Abstract

In this paper, we give some properties about the \((2,p)\)-Laplacian operator (\(p>1\), \(p\ne2\)), and consider the existence of solutions to two kinds of partial differential equations related to the \((2,p)\)-Laplacian operator by those properties. Specifically, we establish an existence result of positive solutions using fixed point index theory and an existence result of nodal solutions via the quantitative deformation lemma.

Keywords

properties fixed point index quantitative deformation lemma

MSC

35J05 47H10 47H11

1 Introduction and main results

Recently, much attention has been paid to the existence of solutions to the following quasilinear elliptic problems of \((q,p)\)-Laplacian type:
$$ \left \{ \textstyle\begin{array}{l@{\quad}l} -\Delta_{q}u-\Delta_{p}u=h(x,u),&x\in\Omega, \\ u=0,&x\in\partial\Omega, \end{array}\displaystyle \right . $$
(1.1)
where \(\Omega\subset\mathbb{R}^{N}\), \(N\geqslant1\), is a bounded domain with smooth boundary Ω, \(p,q>1\), \(\Delta_{p} u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)\) is the p-Laplacian of u, and the function \(h:\Omega\times\mathbb{R}\to\mathbb{R}\) is a Carathéodory function. By a solution u of (1.1), we mean that u, belonging to some Sobolev space, solves (1.1) in the weak sense, i.e., u satisfies
$$ \int_{\Omega}|\nabla u|^{q-2}\nabla u\cdot\nabla v+ \int_{\Omega}|\nabla u|^{p-2}\nabla u\cdot\nabla v= \int_{\Omega}h(x,u)v, \quad v\in C^{\infty}_{0}( \Omega). $$
Moreover, by a non-negative nontrivial solution u of (1.1), we mean that u is a solution of (1.1), \(u\ne0\) and \(u(x)\geqslant0\) for \(x\in\Omega\); if u is a solution of (1.1) with \(u^{\pm}\neq0\), where \(u^{+}=\max\{u,0\}\) and \(u^{-}=\max\{ u,0\}\), then we say that u is a sign-changing solution of (1.1). We know that solutions to (1.1) are the steady state solutions of the general reaction-diffusion equation
$$ u_{t}=\operatorname{div} \bigl(H(u)\nabla u \bigr)+h(x,u), $$
(1.2)
where \(H(u)=|\nabla u|^{q-2}+|\nabla u|^{p-2}\). Equation (1.2) has a wide range of applications in physics and related sciences such as biophysics [1], plasma physics [2], and chemical reaction design [3]. The stationary solutions to (1.2) have been studied by many authors; see [49]. When \(q=2\ne p\) and \(h(x,u)=a(x)|u|^{r-2}u + b(x)|u|^{s-2}u\), Sidiropoulos in [7] considered the existence of non-negative solutions to (1.1) with the exponents r, s being subcritical and a, b being essentially bounded functions. Their proofs are variational in character and rely either on the fibering method or on the mountain pass theorem of Ambrosetti-Rabinowitz. In [9], with \(h(x,u)=\theta|u|^{r-2}+|u|^{p^{*}-2}u\), \(p< N\), \(1< q< p< p^{*}=Np/(N-p)\), the authors established the existence of multiple positive solutions of (1.1) by some standard variational methods.

The purpose of this article is to give some properties and applications about the operator \(-\Delta-\Delta_{p}\) (\(p>1\), \(p\neq2\)). Let \(H_{0}^{1}(\Omega)\) and \(W^{1,p}_{0}(\Omega)\) be the usual Sobolev spaces defined as the completion of \(C^{\infty}_{0}(\Omega)\) with respect to the norms \(\|u\|_{H^{1}_{0}(\Omega)}=(\int_{\Omega}|\nabla u|^{2})^{1/2}\) and \(\| u\|_{W^{1,p}_{0}(\Omega)}=(\int_{\Omega}|\nabla u|^{p})^{1/p}\), respectively. When \(1< p<2\), let \(X=H_{0}^{1}(\Omega)\) and \(\|u\|_{X}=\|u\| _{H^{1}_{0}(\Omega)}\), or when \(p>2\), let \(X=W^{1,p}_{0}(\Omega)\) and \(\|u\| _{X}=\|u\|_{W^{1,p}_{0}(\Omega)}\). Denote by \(X^{*}\) the dual space of X. In addition, by \(|\cdot|_{q}\), we denote the usual norm in \(L^{q}(\Omega )\), \(q\geqslant1\).

It is well known that the operator −Δ or \(-\Delta_{p}\) is a homeomorphism in the Sobolev space X. However, we do not know whether the operator \(-\Delta-\Delta_{p}\) is a homeomorphism in the single Sobolev space X and we do not have the related literature in our hands. Similarly, it is obvious that the functional \(\varphi(u)=\frac{1}{2} \int_{\Omega}|\nabla u|^{2}\) and \(\psi(u)=\frac{1}{p} \int_{\Omega}|\nabla u|^{p}\) belong to \(C^{1}(H^{1}_{0}(\Omega), \mathbb{R})\) and \(C^{1}(W^{1,p}_{0}(\Omega), \mathbb{R})\), respectively. Whether the functional \(J=\varphi+\psi\) belongs to \(C^{1}(X, \mathbb{R})\) is an interesting problem. For the reader’s convenience and completeness of the paper, in Section 2, we will answer those questions and obtain the properties (a) and (b):
  1. (a)

    the operator \(-\Delta-\Delta_{p}\) is a homeomorphism from X to \(X^{*}\) (Proposition 2.3);

     
  2. (b)
    the functional
    $$ J(u)= \frac{1}{2} \int_{\Omega}|\nabla u|^{2}+\frac{1}{p} \int_{\Omega}|\nabla u|^{p}, \quad u\in X, $$
    belongs to \(C^{1}(X,\mathbb{R})\) (Proposition 2.4).
     
As an application of the property (a), we will consider in Section 3 the following quasilinear elliptic equation:
$$ \left \{ \textstyle\begin{array}{l@{\quad}l} -\Delta u-\Delta_{p} u=f(x,u)+h(x),&x\in\Omega, \\ u=0,&x\in\partial\Omega, \end{array}\displaystyle \right . $$
(1.3)
where \(h\in L^{\infty}(\Omega)\) and f satisfies the following conditions:
(f1): 

\(f\in C(\overline{\Omega}\times\mathbb{R},\mathbb {R})\), \(f(x,t)\geqslant0\) for all \((x,t)\in\overline{\Omega}\times \mathbb{R}_{+}\) and \(f(x,t)=0\) for all \((x,t)\in\overline{\Omega }\times\mathbb{R}_{-}\), where \(\mathbb{R}_{+}:=[0,\infty)\) and \(\mathbb {R}_{-}:=(-\infty,0]\);

(f2): 
$$ \lim_{t\to\infty}\frac{f(x,t)}{t+t^{p-1}}=f_{\infty}< \infty,\quad \mbox{uniformly for } x\in\overline{\Omega}. $$
The asymptotic behavior of f leads us to define the following two constants:
$$ \lambda_{1}=\inf \biggl\{ \int_{\Omega}|\nabla u|^{2}:u\in H_{0}^{1}( \Omega ), \int_{\Omega}|u|^{2}=1 \biggr\} $$
and
$$ \mu_{1}=\inf \biggl\{ \int_{\Omega}|\nabla u|^{p}:u\in W^{1,p}_{0}( \Omega ), \int_{\Omega}|u|^{p}=1 \biggr\} . $$

Our main result as regards equation (1.3) is the following theorem.

Theorem 1.1

Suppose that f satisfies (f1) and (f2) with \(f_{\infty}<\lambda _{1}\) when \(1< p<2\) or \(f_{\infty}<\mu_{1}\) when \(p>2\). Then equation (1.3) has a non-negative solution. Moreover, equation (1.3) has a non-negative nontrivial solution when \(h\neq0\).

Remark 1.2

When \(h=0\) the conditions in Theorem 1.1 cannot guarantee the existence of nontrivial solution to (1.3). In fact, we can get the following.

Proposition 1.3

Assume that \(f(x,u)=\lambda u^{+}+\mu(u^{+})^{p-1}\) with \(0<\lambda,\mu <\min\{\lambda_{1},\mu_{1}\}\) and \(h=0\). Then (1.3) has only zero solution.

Proof

Suppose that u is a nontrivial solution to (1.3). Then
$$ \int_{\Omega}|\nabla u|^{2}+ \int_{\Omega}|\nabla u|^{p}=\lambda \int _{\Omega}u^{2}+\mu \int_{\Omega}|u|^{p}. $$
Hence,
$$\begin{aligned} 0&< (\lambda_{1}-\lambda) \int_{\Omega}u^{2}\leqslant \int_{\Omega}|\nabla u|^{2}-\lambda \int_{\Omega}u^{2} \\ &=\mu \int_{\Omega}|u|^{p}- \int_{\Omega}|\nabla u|^{2} \leqslant(\mu- \mu_{1}) \int_{\Omega}|u|^{p}< 0, \end{aligned}$$
which is a contradiction. The proof is completed. □
As an application of the property (b), in Section 4, we will investigate the existence of least energy sign-changing solution of the following equation:
$$ \left \{ \textstyle\begin{array}{l@{\quad}l} -\Delta u-\Delta_{p} u=g(u),& x\in\Omega, \\ u=0,& x\in\partial\Omega. \end{array}\displaystyle \right . $$
(1.4)
By the symmetry, we only consider the case \(p>2\). Here, \(g\in C^{1} (\mathbb{R},\mathbb{R})\) satisfies the following hypotheses:
(g1): 

\(\lim_{t\to0}g(t)/t=0\);

(g2): 

for some constant \(q\in(p,p^{*})\), \(\lim_{|t|\to\infty }g(t)/|t|^{q-1}=0\), where \(p^{*}=\infty\) for \(N\leqslant p\), and \(p^{*}=Np/(N-p)\) for \(N>p\);

(g3): 

there exists \(\mu>p\) such that \(\lim_{|t|\to\infty }G(t)/|t|^{\mu}=\infty\), where \(G(t)=\int^{t}_{0} g(s)\, \mathrm{d}s\) for all \(t\in\mathbb{R}\);

(g4): 

\(g(t)/|t|^{\mu-1}\) is increasing on \((-\infty,0)\) and \((0,\infty)\), respectively.

Our main result as regards equation (1.4) is the following theorem.

Theorem 1.4

If the assumptions (g1)-(g4) hold, then the problem (1.4) has one least energy sign-changing solution.

Remark 1.5

In general, in order to obtain sign-changing solutions, it is common to assume that the nonlinearity satisfies \(g(t)t\geqslant0\), \(t\in\mathbb {R}\). However, there are functions satisfying (g1)-(g4) but not having the property \(g(t)t\geqslant0\), \(t\in\mathbb{R}\). For example, we consider the following function:
$$ g(t)=\left \{ \textstyle\begin{array}{l@{\quad}l} t^{3} \log t,& t>0, \\ -t^{4},& t\leqslant0. \end{array}\displaystyle \right . $$
It is obvious that g satisfies (g1)-(g4). But \(g(t)t<0\) when \(0< t<1\). The proof of Theorem 1.4 is based on the ideas from [10, 11] and we put some new ingredients in the proof process.

The paper is organized as follows. First, in Section 2, we prove the two properties related to the operator \(-\Delta-\Delta_{p}\). Next, the applications of the properties (a) and (b) are given in Sections 3 and 4, respectively. In this paper, \(c_{i}\) (\(i=1,2,\ldots\)) and the \(C_{\varepsilon}\) denote various positive constant whose exact values are not essential to the analysis of the relevant problems.

2 Properties of the operator \(-\Delta-\Delta_{p}\)

In this section, we show the properties (a) and (b) for the operator \(-\Delta-\Delta_{p}\). Before completing the proof of the property (a), we introduce some notations and lemmas first. Let \(P=\{u\in X: u(x)\geqslant0, \mbox{a.e. }x\in\Omega\}\) be the positive cone in X and let \(P^{*}=\{h\in X^{*}: \langle h,u \rangle \geqslant0,u\in P\}\) be its dual cone. Define a nonlinear operator \(A:X\to X^{*}\) by
$$ \langle Au, v \rangle= \int_{\Omega}\nabla u\cdot\nabla v+ \int_{\Omega}|\nabla u|^{p-2}\nabla u\cdot\nabla v, \quad u,v\in X. $$

Lemma 2.1

[12]

There exist constants \(c_{i}\) (\(i=1,2,3,4\)) such that, for all \(x_{1},x_{2}\in\mathbb{R}^{N}\), when \(1< p<2\),
$$\begin{aligned}& (x_{2}-x_{1})\cdot\bigl(|x_{2}|^{p-2}x_{2}-|x_{1}|^{p-2}x_{1} \bigr) \geqslant c_{1} \bigl(\vert x_{2}\vert +|x_{1}|\bigr)^{p-2}|x_{2}-x_{1}|^{2}, \end{aligned}$$
(2.1)
$$\begin{aligned}& \bigl\vert |x_{2}|^{p-2}x_{2}-|x_{1}|^{p-2}x_{1} \bigr\vert \leqslant c_{2}|x_{2}-x_{1}|^{p-1}; \end{aligned}$$
(2.2)
when \(p>2\),
$$\begin{aligned}& (x_{2}-x_{1})\cdot\bigl(|x_{2}|^{p-2}x_{2}-|x_{1}|^{p-2}x_{1} \bigr) \geqslant c_{3}|x_{2}-x_{1}|^{p}, \end{aligned}$$
(2.3)
$$\begin{aligned}& \bigl\vert |x_{2}|^{p-2}x_{2}-|x_{1}|^{p-2}x_{1} \bigr\vert \leqslant c_{4} \bigl(\vert x_{2}\vert +|x_{1}|\bigr)^{p-2}|x_{2}-x_{1}|. \end{aligned}$$
(2.4)

Remark 2.2

In (2.1), when \(x_{1}=x_{2}=0\), we define \((|x_{2}|+|x_{1}|)^{p-2}|x_{2}-x_{1}|^{2}=0\).

Proposition 2.3

The operator A is a homeomorphism from X to \(X^{*}\) and \(A^{-1}(P^{*})\subset P\).

Proof

First of all, we show that A is a homeomorphism. When \(1< p<2\), for any \(u, v\in X\), by (2.1), we have
$$\begin{aligned} \begin{aligned} \langle Au-A v, u-v \rangle&= \int_{\Omega}\bigl\vert \nabla(u-v)\bigr\vert ^{2}+ \int _{\Omega}\bigl(\vert \nabla u\vert ^{p-2}\nabla u- \vert \nabla v\vert ^{p-2}\nabla v\bigr)\cdot\nabla (u-v) \\ &\geqslant \int_{\Omega}\bigl\vert \nabla(u-v)\bigr\vert ^{2}+c_{1} \int_{\Omega}\bigl(\vert \nabla u\vert +\vert \nabla v \vert \bigr)^{p-2}\bigl\vert \nabla(u-v)\bigr\vert ^{2} \\ &\geqslant \int_{\Omega}\bigl\vert \nabla(u-v)\bigr\vert ^{2}. \end{aligned} \end{aligned}$$
When \(p>2\), for any \(u, v\in X\), by (2.3), we get
$$\begin{aligned} \langle Au-Av, u-v \rangle =& \int_{\Omega}\bigl\vert \nabla(u-v)\bigr\vert ^{2}+ \int _{\Omega}\bigl(\vert \nabla u\vert ^{p-2}\nabla u- \vert \nabla v\vert ^{p-2}\nabla v\bigr)\cdot\nabla (u-v) \\ \geqslant& \int_{\Omega}\bigl\vert \nabla(u-v)\bigr\vert ^{2}+c_{3} \int_{\Omega}\bigl\vert \nabla (u-v)\bigr\vert ^{p} \\ \geqslant& c_{3} \int_{\Omega}\bigl\vert \nabla(u-v)\bigr\vert ^{p}. \end{aligned}$$
Hence, A is a strongly monotone operator.
Then we claim A is continuous from X to \(X^{*}\). Assume \(u_{n}\to u\) in X. For all \(w\in X\), when \(1< p<2\), by the Hölder inequality, the Sobolev embedding theorem, and (2.2), we obtain
$$\begin{aligned}& \bigl\vert \langle Au_{n}-Au,w \rangle\bigr\vert \\& \quad = \biggl\vert \int_{\Omega}\bigl(|\nabla u_{n}|^{p-2}\nabla u_{n}-|\nabla u|^{p-2}\nabla u\bigr)\cdot\nabla w+ \int_{\Omega}(\nabla u_{n}-\nabla u)\cdot \nabla w\biggr\vert \\& \quad \leqslant \int_{\Omega}\bigl\vert |\nabla u_{n}|^{p-2} \nabla u_{n}-|\nabla u|^{p-2}\nabla u\bigr\vert |\nabla w|+ \int_{\Omega}\bigl\vert \nabla(u_{n}-u)\bigr\vert | \nabla w| \\& \quad \leqslant c_{2} \int_{\Omega}\bigl\vert \nabla(u_{n}-u)\bigr\vert ^{p-1}|\nabla w|+ \int _{\Omega}\bigl\vert \nabla(u_{n}-u)\bigr\vert | \nabla w| \\& \quad \leqslant c_{2} \biggl( \int_{\Omega}\bigl\vert \nabla(u_{n}-u)\bigr\vert ^{2} \biggr)^{(p-1)/2} \biggl( \int_{\Omega}|\nabla w|^{2} \biggr)^{1/2} | \Omega |^{(2-p)/2} \\& \qquad {}+ \biggl( \int_{\Omega}\bigl\vert \nabla(u_{n}-u)\bigr\vert ^{2} \biggr)^{1/2} \biggl( \int_{\Omega}w^{2} \biggr)^{1/2} \\& \quad \leqslant c_{5} \|u_{n}-u\|^{p-1}_{X} \|w\|_{X} +\|u_{n}-u\|_{X}\|w\|_{X}; \end{aligned}$$
(2.5)
similarly, when \(p>2\), by the Hölder inequality, the Sobolev embedding theorem, and (2.4), we can get
$$\begin{aligned}& \bigl|\langle Au_{n}-Au,w \rangle\bigr| \\& \quad \leqslant \int_{\Omega}\bigl\vert |\nabla u_{n}|^{p-2} \nabla u_{n}-|\nabla u|^{p-2}\nabla u\bigr\vert |\nabla w|+ \int_{\Omega}\bigl\vert \nabla(u_{n}-u)\bigr\vert | \nabla w| \\& \quad \leqslant c_{4} \int_{\Omega}\bigl(\vert \nabla u_{n}\vert +|\nabla u|\bigr)^{p-2}\bigl|\nabla (u_{n}-u)\bigr||\nabla w|+ \int_{\Omega}\bigl\vert \nabla(u_{n}-u)\bigr\vert | \nabla w| \\& \quad \leqslant c_{4} \biggl( \int_{\Omega}\bigl(\vert \nabla u_{n}\vert +|\nabla u|\bigr)^{p} \biggr)^{(p-2)/p} \biggl( \int_{\Omega}\bigl\vert \nabla(u_{n}-u)\bigr\vert ^{p} \biggr)^{1/p} \biggl( \int_{\Omega}|\nabla w|^{p} \biggr)^{1/p} \\& \qquad {}+ c_{6}\|u_{n}-u\|_{X} \|w\|_{X} \\& \quad \leqslant c_{7} \bigl(\Vert u_{n}\Vert _{X}+\|u\|_{X} \bigr)^{p-2} \|u_{n}-u \|_{X} \| w\|_{X}+c_{6}\|u_{n}-u \|_{X} \|w\|_{X}. \end{aligned}$$
(2.6)
Thus, \(\|Au_{n}-Au\|\to0\) as \(n\to\infty\). By the strong monotone operator theorem ([13], Theorem 26.A, p.557), A is a homeomorphism.
To show the second part of this proof, we assume that \(w\in P^{*}\). By the first part of the lemma, there exists \(u\in X\) such that
$$ \int_{\Omega}\bigl[|\nabla u|^{p-2}\nabla u\cdot\nabla v+ \nabla u\cdot \nabla v\bigr]=\langle w, v\rangle,\quad v\in X. $$
(2.7)
Taking \(v=u^{-}\) in (2.7), we have \(\int_{\Omega}|\nabla u^{-}|^{p}+\int_{\Omega}|\nabla u^{-}|^{2}\leqslant0\). Hence \(u(x)\geqslant0\) almost everywhere for \(x\in\Omega\), that is, \(u\in P\). Then the proof is completed. □

Next, we show the property (b) for the operator \(-\Delta-\Delta_{p}\).

Proposition 2.4

The functional
$$ J(u)=\frac{1}{2} \int_{\Omega}|\nabla u|^{2}+\frac{1}{p} \int_{\Omega}|\nabla u|^{p}, \quad u\in X, $$
belongs to \(C^{1}(X,\mathbb{R})\).

Proof

For any \(u\in X\), we define the functional \(J_{1}(u)=\int_{\Omega}|\nabla u|^{2}\) and \(J_{2}(u)=\int_{\Omega}|\nabla u|^{p}\) for convenience. The proof will be completed by considering the following two cases.

(i) When \(1< p<2\), since \(J_{1}(u)=\int_{\Omega}|\nabla u|^{2}=\|u\|_{X}^{2}\), \(J_{1}\) is of \(C^{1}(X,\mathbb{R})\). Hence we only need to show that \(J_{2}\) is of \(C^{1}(X,\mathbb{R})\).

We first show \(J_{2}\) is Gâteaux differentiable. In fact, let \(u,v\in X\). For fixed \(x\in\Omega\) and \(0<|t|<1\), there exists a constant \(\lambda\in[0,1]\) such that
$$ \frac{1}{|t|}{\bigl\vert \bigl\vert \nabla(u+tv)\bigr\vert ^{p}-|\nabla u|^{p} \bigr\vert }\leqslant p\bigl\vert \nabla(u+\lambda tv)\bigr\vert ^{p-1}|\nabla v| \leqslant p\bigl( \vert \nabla u\vert +|\nabla v|\bigr)^{p-1}|\nabla v|. $$
By the Hölder inequality and the Sobolev embedding theorem, we find that
$$\begin{aligned} \int_{\Omega}\bigl\vert \bigl(\vert \nabla u\vert +|\nabla v| \bigr)^{p-1}|\nabla v|\bigr\vert \leqslant& \biggl( \int_{\Omega}\bigl(\vert \nabla u\vert +|\nabla v| \bigr)^{2} \biggr)^{(p-1)/2} \biggl( \int_{\Omega}|\nabla v|^{2} \biggr)^{1/2} | \Omega |^{(2-p)/2} \\ \leqslant&c_{8} \bigl(\Vert u\Vert _{X}+\|v \|_{X}\bigr)^{p-1}\|v\|_{X}< \infty. \end{aligned}$$
Hence, \((|\nabla u|+|\nabla v|)^{p-1}|\nabla v|\in L^{1}\). By the Lebesgue theorem, we obtain
$$\begin{aligned} \bigl\langle J_{2}'(u),v\bigr\rangle =&\lim _{t\to0} \int_{\Omega}\frac{|\nabla(u+tv)|^{p}-|\nabla u|^{p}}{t} \\ =&p\lim_{t\to0} \int_{\Omega}|\nabla u+\lambda t\nabla v|^{p-2}\nabla (u+ \lambda tv)\cdot\nabla v \\ =&p \int_{\Omega}|\nabla u|^{p-2}\nabla u\cdot\nabla v. \end{aligned}$$
Next, we show the continuity of the Gâteaux differentiability. Assume \(u_{n}\to u\) as \(n\to\infty\) in X. Therefore, similar to (2.5), we can deduce
$$\begin{aligned} \bigl\vert \bigl\langle J_{2}'(u_{n})-J_{2}'(u),v \bigr\rangle \bigr\vert \leqslant&p \int_{\Omega}\bigl\vert |\nabla u_{n}|^{p-2} \nabla u_{n}-|\nabla u|^{p-2}\nabla u\bigr\vert |\nabla v| \\ \leqslant&c_{2} \int_{\Omega}\bigl|\nabla(u_{n}-u)\bigr|^{p-1}|\nabla v| \\ \leqslant& c_{5}\|u_{n}-u\|^{p-1}_{X} \|v\|_{X}. \end{aligned}$$
Obviously,
$$ \bigl\Vert J_{2}'(u_{n})-J_{2}'(u) \bigr\Vert \leqslant c_{5}\|u_{n}-u\|^{p-1}_{X} \to0, \quad n\to \infty. $$
Hence \(J_{2}\in C^{1}(X,\mathbb{R})\).
(ii) When \(p>2\), \(J_{2}=\int_{\Omega}|\nabla u|^{p}\) is of \(C^{1}(X,\mathbb {R})\) (see [14], Proposition 1.11, p.9). Hence we only show that \(J_{1}(u)\) is of \(C^{1}(X,\mathbb{R})\). Letting \(u,v\in X\), for fixed \(0<|t|<1\), we have
$$ \frac{1}{|t|}\bigl\vert \bigl\vert \nabla(u+tv)\bigr\vert ^{2}-|\nabla u|^{2}\bigr\vert =\bigl\vert t |\nabla v|^{2}+2 \nabla u \cdot\nabla v\bigr\vert \leqslant|\nabla v|^{2}+2 |\nabla u| |\nabla v|. $$
By the Hölder inequality and the Sobolev embedding theorem, we get
$$ \int_{\Omega}\bigl(|\nabla v|^{2}+2 |\nabla u| |\nabla v| \bigr)\leqslant c_{9} \|v\|_{X}^{2}+2c_{10} \|u\|_{X}\|v\|_{X}< \infty. $$
Hence, \(|\nabla v|^{2}+2 |\nabla u| |\nabla v|\in L^{1}\). Then
$$\begin{aligned} \bigl\langle J_{1}'(u),v\bigr\rangle =&\lim _{t\to0} \int_{\Omega}\frac{|\nabla(u+tv)|^{2}-|\nabla u|^{2}}{t} \\ =&\lim_{t\to0} \int_{\Omega}\bigl(t|\nabla v|^{2}+2\nabla u\cdot\nabla v \bigr) \\ =&2 \int_{\Omega}\nabla u\cdot\nabla v. \end{aligned}$$
Assume \(u_{n}\to u\) as \(n\to\infty\) in X. Therefore, similar to (2.6), we can find that
$$ \bigl\vert \bigl\langle J_{1}'(u_{n})-J_{1}'(u),v \bigr\rangle \bigr\vert \leqslant2 \int_{\Omega}\bigl\vert \nabla (u_{n}-u)\bigr\vert | \nabla v|\leqslant2c_{11}\|u_{n}-u\|_{X}\|v \|_{X}. $$
Then
$$ \bigl\Vert J_{2}'(u_{n})-J_{2}'(u) \bigr\Vert \leqslant2 c_{11}\|u_{n}-u\|_{X}\to0, \quad n\to \infty. $$
Hence, \(J_{1}\in C^{1}(X,\mathbb{R})\). The proof is completed. □

3 An application of the property (a)

In this section, we mainly show Theorem 1.1 by the fixed point index theory. Hence, we first introduce a proposition about the fixed point index.

Proposition 3.1

(see [15])

Let E be a real Banach space, \(V\subset E\) be a cone, and \(U\subset V\) be a bounded open subset of V. If the completely continuous operator \(B:\overline{U}\to V\) has no fixed point on ∂U, then there exists an integer \(i(B,U,V)\), which is regarded as the fixed point index, and the following statements hold:
  1. (i)

    If \(B:\overline{U}\to U\) is a constant mapping, then \(i(B,U,V)=1\).

     
  2. (ii)

    Assume that \(U_{1}\) and \(U_{2}\) are disjoint open subsets of U and B has no fixed point in \(\overline{U}\setminus(U_{1}\cup U_{2})\), then \(i(B,U,V)=i(B,U_{1},V)+i(B,U_{2},V)\), where \(i(B,U_{i},V)=i(B_{ \overline{U}_{i}},U_{i},V) \), \(i=1,2\).

     
  3. (iii)

    If \(H:[0,1]\times\overline{U} \to V\) is a completely continuous homotopy and \(H(t,u)\neq u\) for any \((t,u)\in[0,1]\times \partial U\), then \(i(H(t,\cdot),U,V)\) is independent of \(t\in[0,1]\).

     
  4. (iv)

    If \(i(B,U,V)\neq0\), then B has a fixed point in U.

     
Before proving Theorem 1.1, we need to give some definitions and a lemma, which will be used to prove Theorem 1.1. We define the operators L and K by
$$\begin{aligned}& \langle Lu,v \rangle= \int_{\Omega}{f(x,u)v},\quad u,v\in X, \\& \langle Ku,v\rangle=\left \{ \textstyle\begin{array}{l@{\quad}l} \int_{\Omega}uv, & u,v\in X,p\in(1,2), \\ \int_{\Omega}|u|^{p-2}uv, & u,v\in X,p\in(2,\infty). \end{array}\displaystyle \right . \end{aligned}$$

Lemma 3.2

Suppose that f satisfies (f1) and (f2). Then when \(1< p<2\), we have
$$ \lim_{\|u\|_{X}\to\infty,u\in P}\frac{Lu-f_{\infty}Ku}{\|u\|_{X}}=0; $$
when \(p>2\), we have
$$ \lim_{\|u\|_{X}\to\infty,u\in P}\frac{Lu-f_{\infty}Ku}{\|u\|^{p-1}_{X}}=0. $$

Proof

When \(1< p<2\), by (f1), (f2), for any \(\varepsilon>0\), there is a constant \(C_{\varepsilon}>0\) such that
$$ \bigl\vert f(x,t)-f_{\infty}t\bigr\vert \leqslant\varepsilon t+C_{\varepsilon},\quad x\in \overline{\Omega},t\geqslant0. $$
For \(u\in P\setminus\{0\}\), letting \(w=u/\|u\|_{X}\), by the Hölder inequality and the Sobolev embedding theorem, we obtain
$$\begin{aligned} \sup_{\|v\|_{X}\leqslant1}\biggl\vert \biggl\langle \frac{Lu-f_{\infty}Ku}{\| u\|_{X}},v \biggr\rangle \biggr\vert \leqslant&\sup_{\|v\|_{X}\leqslant1} \int_{\Omega}\frac {|f(x,u)-f_{\infty}u|}{\|u\|_{X}}|v| \\ \leqslant&\sup_{\|v\|_{X}\leqslant1} \int_{\Omega}\bigl(\varepsilon w|v|+C_{\varepsilon}\|u \|_{X}^{-1}|v|\bigr) \\ \leqslant&\sup_{\|v\|_{X}\leqslant1} \bigl(\varepsilon |w|_{2}|v|_{2}+C_{\varepsilon}\|u\|_{X}^{-1}|v|_{2}|\Omega|^{1/2} \bigr) \\ \leqslant&\sup_{\|v\|_{X}\leqslant1} \bigl(\varepsilon c_{1}\|w \|_{X} \| v\|_{X}+C_{\varepsilon}c_{2}\|u \|_{X}^{-1}\|v\|_{X} \bigr) \\ \leqslant&\varepsilon c_{1}+C_{\varepsilon}c_{2}\|u \|_{X}^{-1}. \end{aligned}$$
Then, when \(\|u\|_{X}\to\infty\), we have
$$ \lim_{\|u\|_{X}\to\infty,u\in P}\frac{Lu-f_{\infty}Ku}{\|u\|_{X}}=0. $$
Similarly, when \(p>2\), we get
$$ \bigl\vert f(x,t)-f_{\infty}t^{p-1}\bigr\vert \leqslant \varepsilon t^{p-1}+C_{\varepsilon}, \quad x\in\overline{\Omega},t \geqslant0. $$
For \(u\in P\setminus\{0\}\), letting \(w=u/\|u\|_{X}\), we can find that
$$\begin{aligned} \sup_{\|v\|_{X}\leqslant1}\biggl\vert \biggl\langle \frac{Lu-f_{\infty}Ku}{\| u\|_{X}^{p-1}},v \biggr\rangle \biggr\vert \leqslant&\sup_{\|v\|_{X}\leqslant1} \int_{\Omega}\frac {|f(x,u)-f_{\infty}u^{p-1}|}{\|u\|^{p-1}_{X}}|v| \\ \leqslant&\sup_{\|v\|_{X}\leqslant1} \int_{\Omega}\bigl(\varepsilon w|v|+C_{\varepsilon}\|u \|_{X}^{1-p}|v| \bigr) \\ \leqslant&C_{\varepsilon}\varepsilon c_{3}+c_{4}\|u \|_{X}^{1-p}. \end{aligned}$$
Then, when \(\|u\|_{X}\to\infty\), we have
$$ \lim_{\|u\|_{X}\to\infty,u\in P}\frac{Lu-f_{\infty}Ku}{\|u\|^{p-1}_{X}}=0. $$
The proof is completed. □

Because of the assumptions (f1) and (f2), f satisfies the subcritical condition. By [16], Proposition B.10, p.90, we know \(L: X\to X^{*}\) is compact. Hence, \(A^{-1}L\) is a completely continuous operator. Since \(A^{-1}h\) is a constant operator, then the operator \(T:X\to X\), where T is defined by \(T=A^{-1}L+A^{-1}h\), is a completely continuous operator.

Now we are ready to prove Theorem 1.1.

Proof of Theorem 1.1

In order to prove Theorem 1.1, we only need to show the fixed point index \(i(T, P_{r}, P)=1\) for large r. To this end, we define a completely continuous homotopy function \(H:[0,1]\times X\to X^{*}\) by
$$ H(t,u)=tLu+th, \quad (t,u)\in[0,1]\times P. $$
We claim that there exists \(R_{0}>0\) such that the operator equation
$$ Au=H(t,u) $$
(3.1)
has no solution on \([0,1]\times\partial P_{r}\) for \(r>R_{0}\). We prove by contradiction. Suppose that there exists a sequence \(\{(t_{n},u_{n})\} \subset[0,1]\times P\) such that
$$ t_{n}\to t_{0}, \qquad \|u_{n}\|\to\infty, \quad n\to\infty, $$
where \((t_{n},u_{n})\) satisfies (3.1), that is,
$$ \int_{\Omega}\nabla u_{n}\cdot\nabla v+ \int_{\Omega}|\nabla u_{n}|^{p-2} \nabla u_{n}\cdot\nabla v=t_{n} \int_{\Omega}f(x,u_{n})v+t_{n} \int_{\Omega}hv, \quad v\in X. $$
Let \(\omega_{n}=u_{n}/\|u_{n}\|_{X}\). Then
$$\begin{aligned}& \|u_{n}\|_{X} \int_{\Omega}\nabla\omega_{n} \cdot\nabla v+ \|u_{n}\| ^{p-1}_{X} \int_{\Omega}|\nabla w_{n}|^{p-2} \nabla w_{n}\cdot\nabla v \\& \quad =t_{n} \int_{\Omega}f(x,u_{n})v+t_{n} \int_{\Omega}hv. \end{aligned}$$
(3.2)
Since \(\{w_{n}\}\) is bounded in P, we may assume for some \(w_{0}\in P\), by passing to a subsequence if necessary, that \(w_{n}\rightharpoonup w_{0}\in P\).
When \(1< p<2\), by (3.2), we can calculate that
$$\begin{aligned}& \int_{\Omega}\nabla\omega_{n} \cdot\nabla v+ \|u_{n}\|^{p-2}_{X} \int _{\Omega}|\nabla w_{n}|^{p-2} \nabla w_{n}\cdot\nabla v \\& \quad = t_{n} \int_{\Omega}\frac{f(x,u_{n})-f_{\infty}u_{n}}{\|u_{n}\|_{X}}v+t_{n} f_{\infty}\int_{\Omega}\omega_{n} v+t_{n} \int_{\Omega}\frac{hv}{\|u_{n}\|_{X}}, \end{aligned}$$
(3.3)
taking \(v=w_{n}\) in (3.3) and letting \(n\to\infty\), by Lemma 3.2, we have
$$ 1=t_{0}f_{\infty}\int_{\Omega}\omega_{0}^{2}. $$
Hence,
$$ \lambda_{1}=t_{0} \lambda_{1} f_{\infty}\int_{\Omega}w_{0}^{2}\leqslant f_{\infty}\lambda_{1} \int_{\Omega}\omega_{0}^{2}\leqslant f_{\infty}, $$
which contradicts \(f_{\infty}<\lambda_{1}\).
When \(p>2\), by (3.2), we can deduce
$$\begin{aligned}& \frac{1}{\|u_{n}\|^{p-2}_{X}} \int_{\Omega}\nabla\omega_{n} \cdot\nabla v+ \int_{\Omega}|\nabla w_{n}|^{p-2} \nabla w_{n}\cdot\nabla v \\& \quad = t_{n} \int_{\Omega}\frac{f(x,u_{n})-f_{\infty}u_{n}^{p-1}}{\|u_{n}\| _{X}^{p-1}}v+t_{n} f_{\infty}\int_{\Omega}\omega_{n}^{p-1} v+t_{n} \int_{\Omega}\frac{ hv}{\|u_{n}\|^{p-1}_{X}}. \end{aligned}$$
Similarly, we get
$$ 1=t_{0}f_{\infty}\int_{\Omega}\omega_{0}^{p}. $$
Hence,
$$ \mu_{1}=t_{0} \mu_{1} f_{\infty}\int_{\Omega}w_{0}^{p}\leqslant f_{\infty}\mu_{1} \int_{\Omega}\omega_{0}^{p}\leqslant f_{\infty}, $$
which contradicts \(f_{\infty}<\mu_{1}\).
Consequently, taking \(r>R_{0}\), we obtain
$$ i(AT,P_{r},P)=i\bigl(A^{-1}H(1, \cdot),P_{r},P\bigr)=i\bigl(A^{-1}H(0,\cdot),P_{r},P \bigr)=1. $$
(3.4)
By (3.4), we know the problem (1.1) has a non-negative solution \(u\in X\). Especially, when \(h\neq0\), it is quite evident that \(u=0\) is not the solution of the problem (1.1). Hence, (1.1) has a non-negative nontrivial solution. The proof is completed. □

4 An application of the property (b)

In this section, we first define the energy functional \(I: X\to\mathbb {R}\) by
$$ I(u)=\frac{1}{2} \int_{\Omega}|\nabla u|^{2}+\frac{1}{p} \int_{\Omega}|\nabla u|^{p}- \int_{\Omega}G(u), \quad v\in X. $$
It is obvious that the functional I is well defined and belongs to \(C^{1}(X,\mathbb{R})\) by Proposition 2.4. Furthermore,
$$ \bigl\langle I'(u),v\bigr\rangle = \int_{\Omega}\nabla u\cdot\nabla v+ \int _{\Omega}|\nabla u|^{p-2}\nabla u\cdot\nabla v- \int_{\Omega}g(u)v,\quad u,v\in X. $$
Clearly, critical points of I are the weak solutions to (1.4). Moreover, if \(u\in X\) is a solution to (1.4) and \(u^{\pm}\neq0\), where \(u^{+}=\max\{u,0\}\) and \(u^{-}=\min\{u,0\}\), then u is a sign-changing solution.
In order to get a sign-changing solution to (1.4), we first need to seek a minimizer of the energy functional I over the constraint:
$$ M=\bigl\{ u\in X: u^{\pm}\neq0, \bigl\langle I'(u), u^{+} \bigr\rangle =\bigl\langle I'(u), u^{-}\bigr\rangle =0\bigr\} , $$
and then we need to prove that the minimizer is a sign-changing solution to (1.4).

Now we first state the following lemmas.

Lemma 4.1

Assume that (g1)-(g4) hold, and \(u\in X\) with \(u^{\pm}\neq0\). Then there is a unique pair \((s_{0},t_{0})\in(0,\infty)\times(0,\infty )\) such that \(s_{0} u^{+}+t_{0} u^{-}\in M\).

Proof

For any \(u\in X\) with \(u^{\pm}\neq0\), we define
$$\begin{aligned} P(s) =&\bigl\langle I'\bigl(su^{+}+tu^{-}\bigr),su^{+} \bigr\rangle \\ =& s^{2} \int_{\Omega}\bigl\vert \nabla u^{+}\bigr\vert ^{2}+s^{p} \int_{\Omega}\bigl\vert \nabla u^{+}\bigr\vert ^{p}- \int _{\Omega}g\bigl(su^{+}\bigr)su^{+} \end{aligned}$$
(4.1)
$$\begin{aligned} =& s^{2} \biggl[ \int_{\Omega}\bigl\vert \nabla u^{+}\bigr\vert ^{2}+s^{p-2} \int_{\Omega}\bigl\vert \nabla u^{+}\bigr\vert ^{p}- \int_{\Omega}\frac{g(su^{+})u^{+}}{s} \biggr]. \end{aligned}$$
(4.2)
By the conditions (g1) and (g2), for any \(\varepsilon>0\), there exists \(C_{\varepsilon}>0\) such that
$$ \bigl\vert g(t)\bigr\vert \leqslant\varepsilon \vert t\vert +C_{\varepsilon} \vert t\vert ^{q-1}, \quad t\in \mathbb{R}. $$
(4.3)
According to the condition (g1), we have, for each \(\eta\in\mathbb{R}\),
$$ \lim_{t\to0}\frac{g(t\eta)}{t}=0. $$
(4.4)
Thus, by (4.3), (4.4), and the Lebesgue theorem, we get
$$ \lim_{s\to0^{+}} \int_{\Omega}\frac{g(s u^{+}) u^{+}}{s}=0. $$
(4.5)
Since \(u^{+}\neq0\), then it follows from (4.2) and (4.5) that \(P(s)>0\) for \(s>0\) small.
By the conditions (g3) and (g4), we find that
$$ \lim_{|t|\to\infty}\frac{g(t)}{|t|^{\mu-2}t}=\infty. $$
(4.6)
By (4.6) and \(g\in C^{1}(\mathbb{R}, \mathbb{R})\), we see, for any \(M>0\), there exists a constant \(c_{1}>0\) such that, for any \(t\in \mathbb{R}\),
$$ g(t)t\geqslant M|t|^{\mu}-c_{1}. $$
(4.7)
According to (4.1) and (4.7), we obtain
$$\begin{aligned} \begin{aligned} P(s)&\leqslant s^{2} \int_{\Omega}\bigl\vert \nabla u^{+}\bigr\vert ^{2}+s^{p} \int_{\Omega}\bigl\vert \nabla u^{+}\bigr\vert ^{p}-M s^{\mu}\int_{\Omega}\bigl\vert u^{+}\bigr\vert ^{\mu}+c_{1} \vert \Omega \vert \\ &= s^{\mu}\biggl[s^{2-\mu} \int_{\Omega}\bigl\vert \nabla u^{+}\bigr\vert ^{2}+s^{p-\mu} \int _{\Omega}\bigl\vert \nabla u^{+}\bigr\vert ^{p}+s^{-\mu}c_{1}\vert \Omega \vert -M \int_{\Omega}\bigl\vert u^{+}\bigr\vert ^{\mu}\biggr]. \end{aligned} \end{aligned}$$
Because of the arbitrariness of M, we see that \(P(s)<0\) for s large. Thus, there exists \(s_{0}>0\) such that \(P(s_{0})=0\).
Similarly, we define
$$ Q(t)=\bigl\langle I'\bigl(su^{+}+tu^{-}\bigr),tu^{-} \bigr\rangle = t^{2} \int_{\Omega}\bigl\vert \nabla u^{-}\bigr\vert ^{2}+t^{p} \int_{\Omega}\bigl\vert \nabla u^{-}\bigr\vert ^{p}- \int_{\Omega}g\bigl(tu^{-}\bigr)tu^{-}. $$
By the similar way, we get there exists \(t_{0}>0\) such that \(Q(t_{0})=0\).
Next, we prove the uniqueness. Suppose that there exist \(s_{1}\), \(s_{2}\) such that \(0< s_{1}< s_{2}\) and \(P(s_{1})=P(s_{2})=0\), that is,
$$ \frac{1}{s_{1}^{\mu-2}} \int_{\Omega}\bigl\vert \nabla u^{+}\bigr\vert ^{2}+ \frac{1}{s_{1}^{\mu -p}} \int_{\Omega}\bigl\vert \nabla u^{+}\bigr\vert ^{p}= \int_{\Omega}\frac{g(s_{1} u^{+})}{\vert s_{1} u^{+}\vert ^{\mu-1}} \bigl\vert u^{+}\bigr\vert ^{\mu}. $$
It also holds if \(s_{1}\) is replaced by \(s_{2}\). Therefore,
$$\begin{aligned}& \biggl(\frac{1}{s_{1}^{\mu-2}}-\frac{1}{s_{2}^{\mu-2}} \biggr) \int_{\Omega}\bigl\vert \nabla u^{+}\bigr\vert ^{2}+ \biggl(\frac{1}{s_{1}^{\mu-p}}-\frac{1}{s_{2}^{\mu-p}} \biggr) \int_{\Omega}\bigl\vert \nabla u^{+}\bigr\vert ^{p} \\& \quad = \int_{\Omega}\biggl[\frac{g(s_{1} u^{+})}{\vert s_{1} u^{+}\vert ^{\mu-1}}-\frac{g(s_{2} u^{+})}{\vert s_{2} u^{+}\vert ^{\mu-1}} \biggr] \bigl\vert u^{+}\bigr\vert ^{\mu}, \end{aligned}$$
which is absurd in view of (g4) and \(0< s_{1}< s_{2}\). Then there exists a unique \(s_{0}\) such that \(P(s_{0})=0\). Similarly, the uniqueness of \(t_{0}\) can be proved. The proof is completed. □

Lemma 4.2

For fixed \(u\in X\) with \(u^{\pm}\neq0\), the vector \((s_{0},t_{0})\) obtained in Lemma  4.1 is the unique maximum point of the function \(\phi: \mathbb{R}_{+}\times\mathbb{R}_{+}\to\mathbb{R}\) defined as \(\phi(s,t)=I(s u^{+}+t u^{-})\).

Proof

From the proof of Lemma 4.1, \((s_{0},t_{0})\) is the unique critical point of ϕ in \((0,\infty)\times(0,\infty)\). By the assumption (g3), we deduce that \(\phi(s,t)\to-\infty\) uniformly as \(|(s,t)|\to\infty\). So it is sufficient to check that a maximum point cannot be achieved on the boundary of \([0,\infty)\times [0,\infty)\). Without loss of generality, we may assume that \((0,\bar{t})\) is a maximum point of ϕ. By (4.5) and
$$\begin{aligned} \phi_{s}'(s,\bar{t}) =&\bigl\langle I' \bigl(s u^{+}+\bar{t} u^{-}\bigr),u^{+}\bigr\rangle \\ =&s \int_{\Omega}\bigl\vert \nabla u^{+}\bigr\vert ^{2}+s^{p-1} \int_{\Omega}\bigl\vert \nabla u^{+}\bigr\vert ^{p}- \int_{\Omega}f\bigl(s u^{+}\bigr)u^{+} \\ =&s \biggl[ \int_{\Omega}\bigl\vert \nabla u^{+}\bigr\vert ^{2}+s^{p-2} \int_{\Omega}\bigl\vert \nabla u^{+}\bigr\vert ^{p}- \int_{\Omega}\frac{f(s u^{+})u^{+}}{s} \biggr], \end{aligned}$$
we have there exists \(\tilde{s}>0\) small enough such that for any \(s\in (0,\tilde{s})\) we have \(\phi_{s}'(s,\bar{t})>0\). It implies that \(\phi (s,t)\) is an increasing function with respect to s when \(s\in (0,\tilde{s})\), that is, the pair \((0,\bar{t})\) is not a maximum point of ϕ in \(\mathbb{R}_{+}\times\mathbb{R}_{+}\). Hence \((s_{0},t_{0})\) is the unique maximum point. The proof is completed. □

Lemma 4.3

Assume that (g1)-(g4) hold. Then \(m:=\inf\{I(u):u\in M\}>0\) can be achieved.

Proof

For every \(u\in M\), we have \(\langle I'(u),u^{\pm}\rangle=0\). Thus by (4.3), we get
$$ \int_{\Omega}|\nabla u|^{2}+ \int_{\Omega}|\nabla u|^{p}= \int_{\Omega}g(u)u\leqslant\varepsilon \int_{\Omega}|u|^{2}+C_{\varepsilon}\int _{\Omega}|u|^{q}. $$
(4.8)
According to the Sobolev embedding theorem, we have
$$ \|u\|_{H_{0}^{1}(\Omega)}^{2}+\|u\|_{X}^{p} \leqslant\varepsilon c_{1}\|u\| _{H_{0}^{1}(\Omega)}^{2}+C_{\varepsilon}c_{2}\|u\|_{X}^{q}. $$
(4.9)
It suggests there exists a constant \(\alpha>0\) such that \(\|u\| _{X}\geqslant\alpha\).
Since the conditions (g3) and (g4) imply that
$$ H(t)=tg(t)-\mu G(t)\geqslant0,\quad t\in\mathbb{R}, $$
(4.10)
and \(H(t)\) is increasing when \(t>0\) and decreasing when \(t<0\), then we can find that
$$\begin{aligned} I(u) =&I(u)-\frac{1}{\mu}\bigl\langle I'(u),u \bigr\rangle \\ =& \biggl(\frac{1}{2}-\frac{1}{\mu}\biggr)\|u \|_{H_{0}^{1}(\Omega)}^{2}+ \biggl(\frac{1}{p}-\frac{1}{\mu}\biggr)\|u\|_{X}^{p}+\frac{1}{\mu}\bigl[g(u)u-\mu G(u) \bigr] \\ \geqslant& \biggl(\frac{1}{p}-\frac{1}{\mu}\biggr)\|u \|_{X}^{p} \\ \geqslant& \biggl(\frac{1}{p}-\frac{1}{\mu}\biggr) \alpha^{p}, \end{aligned}$$
(4.11)
which implies \(m\geqslant(1/p-1/\mu) \alpha^{p}>0\).
Let \(\{u_{n}\}\subset M\) satisfy \(I(u_{n})\to m\). Then \(\{u_{n}\}\) is bounded in X by (4.11). We may assume there exists \(u_{0}\in X\), by passing to a subsequence if necessary, such that \(u_{n}^{\pm}\rightharpoonup u_{0}^{\pm}\) weakly in X. Next, we show \(u_{0}^{\pm}\neq 0\). In fact, since \(u_{n}\in M\), then similar to (4.8), we have
$$ \int_{\Omega}\bigl\vert \nabla u_{n}^{\pm}\bigr\vert ^{2}+ \int_{\Omega}\bigl\vert \nabla u_{n}^{\pm}\bigr\vert ^{p}= \int _{\Omega}g\bigl(u_{n}^{\pm}\bigr) u_{n}^{\pm}\leqslant\varepsilon \int_{\Omega}\bigl\vert u_{n}^{\pm}\bigr\vert ^{2}+C_{\varepsilon}\int_{\Omega}\bigl\vert u_{n}^{\pm}\bigr\vert ^{q}. $$
(4.12)
Similar to the discussion below (4.9) there exists a constant \(\alpha_{1}>0\) such that \(\|u_{n}^{\pm}\|_{X}\geqslant\alpha_{1}\), and then it follows from (4.12) that
$$ \alpha_{1}^{p} \leqslant\varepsilon \int_{\Omega}\bigl\vert u_{n}^{\pm}\bigr\vert ^{2}+C_{\varepsilon}\int_{\Omega}\bigl\vert u_{n}^{\pm}\bigr\vert ^{q}, \quad n=1,2,\ldots. $$
(4.13)
Since \(\{u_{n}\}\) is bounded in X and the embedding \(X\hookrightarrow L^{2}(\Omega)\) holds, there is \(c_{3}>0\) such that
$$ \alpha_{1}^{p} \leqslant\varepsilon c_{3}+C_{\varepsilon}\int_{\Omega}\bigl\vert u_{n}^{\pm}\bigr\vert ^{q}. $$
Choosing \(\varepsilon=\alpha^{p}/(2c_{3})\), we obtain
$$ \int_{\Omega}\bigl\vert u_{n}^{\pm}\bigr\vert ^{q}\geqslant\frac{\alpha_{1}^{p}}{2C_{\varepsilon}}. $$
By \(u_{n}^{\pm}\rightharpoonup u_{0}^{\pm}\) weakly in X and the compactness of the embedding \(X\hookrightarrow L^{q}(\Omega)\), we get
$$ \int_{\Omega}\bigl\vert u_{0}^{\pm}\bigr\vert ^{q}\geqslant\frac{\alpha_{1}^{p}}{2C_{\varepsilon}}, $$
(4.14)
then \(u_{0}^{\pm}\ne0\).
The conditions (g1) and (g2) combined with \(u_{n}^{\pm}\to u_{0}^{\pm}\) in \(L^{q}(\Omega)\) yield
$$ \lim_{n\to\infty} \int_{\Omega}g\bigl(u_{n}^{\pm}\bigr)u_{n}^{\pm}= \int_{\Omega}g\bigl(u_{0}^{\pm}\bigr)u_{0}^{\pm},\qquad \lim_{n\to\infty} \int_{\Omega}G\bigl(u_{n}^{\pm}\bigr)= \int_{\Omega}G\bigl(u_{0}^{\pm}\bigr). $$
(4.15)
By (4.12), (4.15), and the weak lower semicontinuity of the norm, we can deduce
$$\begin{aligned} \int_{\Omega}\bigl\vert \nabla u_{0}^{\pm}\bigr\vert ^{2}+ \int_{\Omega}\bigl\vert \nabla u_{0}^{\pm}\bigr\vert ^{p} \leqslant&\liminf_{n\to\infty} \biggl[ \int_{\Omega}\bigl\vert \nabla u_{n}^{\pm}\bigr\vert ^{2}+ \int_{\Omega}\bigl\vert \nabla u_{n}^{\pm}\bigr\vert ^{p} \biggr] \\ =&\liminf_{n\to \infty} \int_{\Omega}g\bigl(u_{n}^{\pm}\bigr)u_{n}^{\pm}= \int_{\Omega}g\bigl(u_{0}^{\pm}\bigr)u_{0}^{\pm}. \end{aligned}$$
(4.16)
According to Lemma 4.1 and (4.16), there exists \((s_{0},t_{0})\in(0,1]\times(0,1]\) such that \(\tilde{u}=s_{0}u_{0}^{+}+t_{0} u_{0}^{-}\in M\). Since (4.11) and (4.15), we then have
$$\begin{aligned} m \leqslant&I(\tilde{u})-\frac{1}{\mu}\bigl\langle I'( \tilde{u}),\tilde {u}\bigr\rangle \\ =& \biggl(\frac{1}{2}-\frac{1}{\mu}\biggr) \int_{\Omega} \vert \nabla\tilde {u}\vert ^{2}+ \biggl(\frac{1}{p}-\frac{1}{\mu}\biggr) \int_{\Omega} \vert \nabla \tilde{u}\vert ^{p}+ \frac{1}{\mu}\int_{\Omega}\bigl[g(\tilde{u})\tilde{u}-\mu G(\tilde{u})\bigr] \\ =& \biggl(\frac{1}{2}-\frac{1}{\mu}\biggr) \int_{\Omega}\bigl\vert \nabla s_{0}u_{0}^{+} \bigr\vert ^{2}+ \biggl(\frac{1}{2}-\frac{1}{\mu}\biggr) \int_{\Omega}\bigl\vert \nabla t_{0}u_{0}^{-} \bigr\vert ^{2} \\ &{}+ \biggl(\frac{1}{p}-\frac{1}{\mu}\biggr) \int_{\Omega}\bigl\vert \nabla s_{0}u_{0}^{+} \bigr\vert ^{p}+ \biggl(\frac{1}{p}-\frac{1}{\mu}\biggr) \int_{\Omega}\bigl\vert \nabla t_{0}u_{0}^{-} \bigr\vert ^{p} \\ &{}+\frac{1}{\mu}\int_{\Omega}\bigl[g\bigl(s_{0}u_{0}^{+} \bigr)s_{0}u_{0}^{+}-\mu G\bigl(s_{0}u_{0}^{+} \bigr) \bigr]+\frac{1}{\mu}\int_{\Omega}\bigl[g\bigl(t_{0}u_{0}^{-} \bigr)t_{0}u_{0}^{-}-\mu G\bigl(t_{0}u_{0}^{-} \bigr) \bigr] \\ \leqslant& \biggl(\frac{1}{2}-\frac{1}{\mu}\biggr) \int_{\Omega}\bigl\vert \nabla u_{0}^{+}\bigr\vert ^{2}+ \biggl(\frac{1}{2}-\frac{1}{\mu}\biggr) \int_{\Omega}\bigl\vert \nabla u_{0}^{-}\bigr\vert ^{2} \\ &{}+ \biggl(\frac{1}{p}-\frac{1}{\mu}\biggr) \int_{\Omega}\bigl\vert \nabla u_{0}^{+}\bigr\vert ^{p}+ \biggl(\frac{1}{p}-\frac{1}{\mu}\biggr) \int_{\Omega}\bigl\vert \nabla u_{0}^{-}\bigr\vert ^{p} \\ &{}+\frac{1}{\mu}\int_{\Omega}\bigl[g\bigl(u_{0}^{+}\bigr)u_{0}^{+}- \mu G\bigl(u_{0}^{+}\bigr) \bigr]+\frac{1}{\mu}\int_{\Omega}\bigl[g\bigl(u_{0}^{-}\bigr)u_{0}^{-}- \mu G\bigl(u_{0}^{-}\bigr) \bigr] \\ =& \biggl(\frac{1}{2}-\frac{1}{\mu}\biggr) \int_{\Omega} \vert \nabla u_{0}\vert ^{2}+ \biggl(\frac{1}{p}-\frac{1}{\mu}\biggr) \int_{\Omega} \vert \nabla u_{0}\vert ^{p}+ \frac{1}{\mu}\int_{\Omega}\bigl[g(u_{0})u_{0}-\mu G(u_{0}) \bigr] \\ \leqslant&\liminf_{n\to\infty} \biggl[I(u_{n})- \frac{1}{\mu}\bigl\langle I'(u_{n}),u_{n} \bigr\rangle \biggr]=m. \end{aligned}$$
Thus we deduce that \(s_{0}=t_{0}=1\), that is, \(\tilde{u}=u_{0}\) and \(I(u_{0})=m\). Then the proof is completed. □

Proof of Theorem 1.4

In order to prove Theorem 1.4, we need to show \(I'(u_{0})=0\) by the quantitative deformation lemma.

It is clear that \(\langle I'(u_{0}),u_{0}^{+}\rangle=\langle I'(u_{0}),u_{0}^{-}\rangle=0\). It follows from Lemma 4.3 that, for \((s,t)\in\mathbb{R}_{+}\times\mathbb{R}_{+}\) and \((s,t)\ne(1,1)\), we have \(I(su_{0}^{+}+tu_{0}^{-})< I(u_{0}^{+}+u_{0}^{-})=m\). It follows from (4.14) that \(\int_{\Omega}|u_{0}^{\pm}|^{q}\geqslant\alpha _{1}^{p}/(2C_{\varepsilon}):=\beta^{q}\). Then \(|u_{0}^{\pm}|_{q}\geqslant\beta\). We denote
$$ S_{q}=\inf_{u\in X, |u|_{q}=1} \int_{\Omega}|\nabla u|^{q}. $$
(4.17)
We assume that \(I'(u_{0})\ne0\). Then there exist \(r_{0}, \rho>0\) such that \(\|I'(v)\|\geqslant\rho\) for all \(\|v-u_{0}\|_{X}\leqslant r_{0}\). Let \(\delta\in (0, \min\{(\beta S_{q})/2, r_{0}/3\} )\), \(D=(1-\sigma,1+\sigma)\times(1-\sigma,1+\sigma)\), and \(\varphi (s,t)=su_{0}^{+}+tu_{0}^{-}\), where \(0<\sigma<\min\{1/2,\delta/{\|u_{0}\|_{X}}\} \). By Lemma 4.3 again, we get
$$ \bar{m}=\max_{\partial D}I\circ\varphi< m. $$
(4.18)
For \(\varepsilon=\min\{(m-\bar{m})/2,\rho\delta/8\}\) and \(S=B(u_{0},\delta)\), [14], Lemma 2.3, p.38 yields a deformation η such that
  1. (i)

    \(\eta(1,u)=u\), if \(u\notin I^{-1}([m-2\varepsilon, m+2\varepsilon])\cap S_{2\delta}\);

     
  2. (ii)

    \(\eta(1,I^{m+\varepsilon}\cap S)\subset I^{m-\varepsilon}\);

     
  3. (iii)

    \(\|\eta(1,u)-u\|\leqslant\delta\) for all \(u\in X\).

     
By Lemmas 4.2 and 4.3, for \((s,t)\in\overline {D}\), we know \(I(\varphi(s,t))\leqslant m< m+\varepsilon\), that is, \(\varphi(s,t)\in I^{m+\varepsilon}\). Since
$$\begin{aligned} \bigl\Vert \varphi(s,t)-u_{0}\bigr\Vert _{X}^{p} =& \bigl\Vert su_{0}^{+}+tu_{0}^{-}-u_{0}^{+}-u_{0}^{-} \bigr\Vert _{X}^{p} \\ =& |s-1|^{p}\bigl\Vert u_{0}^{+}\bigr\Vert _{X}^{p}+|t-1|^{p}\bigl\Vert u_{0}^{-} \bigr\Vert _{X}^{p} \\ \leqslant& \sigma^{p} \Vert u_{0}\Vert _{X}^{p} \\ < &\delta^{p}, \end{aligned}$$
we know that \(\varphi(s,t)\in S\). By (ii), we have \(I(\eta(1,\varphi (s,t)))< m-\varepsilon\). It is clear that
$$ \max_{(s,t)\in\overline{D}} I\bigl(\eta\bigl(1,\varphi(s,t)\bigr) \bigr)< m. $$
(4.19)
We claim that \(\eta(1,\varphi(D))\cap M\ne\emptyset\). In fact, define \(\psi(s,t)=\eta(1,\varphi(s,t))\) on ,
$$\begin{aligned}& \Phi(s,t)= \bigl(\bigl\langle I'\bigl(\varphi(s,t) \bigr),su_{0}^{+}\bigr\rangle ,\bigl\langle I'\bigl( \varphi(s,t)\bigr),tu_{0}^{-}\bigr\rangle \bigr) \\& \hphantom{\Phi(s,t)}= \bigl(\bigl\langle I'\bigl(su_{0}^{+}\bigr),su_{0}^{+}\bigr\rangle , \bigl\langle I'\bigl(tu_{0}^{-}\bigr),tu_{0}^{-} \bigr\rangle \bigr), \\& \Psi(s,t)= \bigl(\bigl\langle I'\bigl(\psi(s,t)\bigr),\bigl( \psi(s,t)\bigr)^{+}\bigr\rangle ,\bigl\langle I'\bigl(\psi(s,t)\bigr), \bigl(\psi(s,t)\bigr)^{-}\bigr\rangle \bigr), \end{aligned}$$
and define
$$\begin{aligned}& P(s)=\bigl\langle I'\bigl(su_{0}^{+}\bigr),su_{0}^{+} \bigr\rangle =s^{2} \int_{\Omega}\bigl\vert \nabla u_{0}^{+}\bigr\vert ^{2}+s^{p} \int_{\Omega}\bigl\vert \nabla u_{0}^{+}\bigr\vert ^{p}- \int_{\Omega}g\bigl(su_{0}^{+}\bigr)su_{0}^{+}, \\& Q(t)=\bigl\langle I'\bigl(tu_{0}^{-}\bigr),tu_{0}^{-} \bigr\rangle =t^{2} \int_{\Omega}\bigl\vert \nabla u_{0}^{-}\bigr\vert ^{2}+t^{p} \int_{\Omega}\bigl\vert \nabla u_{0}^{-}\bigr\vert ^{p}- \int_{\Omega}g\bigl(su_{0}^{-}\bigr)su_{0}^{-}, \end{aligned}$$
and the matrix
$$ B=\left [ \textstyle\begin{array}{@{}c@{\quad}c@{}} P'(1)&0\\ 0&Q'(1) \end{array}\displaystyle \right ]. $$
By (g4) and \(g\in C^{1}(\mathbb{R},\mathbb{R})\), we obtain
$$ g'(t)t^{2}\geqslant(\mu-1)g(t)t,\quad t \in{\mathbb{R}}. $$
(4.20)
According to (4.20) and \(\langle I'(u_{0}),u_{0}^{+}\rangle=0\), we get
$$\begin{aligned} P'(1) =&2 \int_{\Omega}\bigl\vert \nabla u_{0}^{+}\bigr\vert ^{2}+p \int_{\Omega}\bigl\vert \nabla u_{0}^{+}\bigr\vert ^{p}- \int_{\Omega}g'\bigl(u_{0}^{+}\bigr) \bigl\vert u_{0}^{+}\bigr\vert ^{2}- \int_{\Omega}g\bigl(u_{0}^{+}\bigr) u_{0}^{+} \\ \leqslant&2 \int_{\Omega}\bigl\vert \nabla u_{0}^{+}\bigr\vert ^{2}+p \int_{\Omega}\bigl\vert \nabla u_{0}^{+}\bigr\vert ^{p}-(\mu-1) \int_{\Omega}g\bigl(u_{0}^{+}\bigr) u_{0}^{+}- \int_{\Omega}g\bigl(u_{0}^{+}\bigr) u_{0}^{+} \\ =&2 \int_{\Omega}\bigl\vert \nabla u_{0}^{+}\bigr\vert ^{2}+p \int_{\Omega}\bigl\vert \nabla u_{0}^{+}\bigr\vert ^{p}-\mu \int_{\Omega}g\bigl(u_{0}^{+}\bigr) u_{0}^{+} \\ =&2 \int_{\Omega}\bigl\vert \nabla u_{0}^{+}\bigr\vert ^{2}+p \int_{\Omega}\bigl\vert \nabla u_{0}^{+}\bigr\vert ^{p}-\mu \int_{\Omega}\bigl\vert \nabla u_{0}^{+}\bigr\vert ^{2}-\mu \int_{\Omega}\bigl\vert \nabla u_{0}^{+}\bigr\vert ^{p} \\ =&(2-\mu) \int_{\Omega}\bigl\vert \nabla u_{0}^{+}\bigr\vert ^{2}+(p-\mu) \int_{\Omega}\bigl\vert \nabla u_{0}^{+}\bigr\vert ^{p}< 0. \end{aligned}$$
Similarly, we have
$$ Q'(1)\leqslant(2-\mu) \int_{\Omega}\bigl\vert \nabla u_{0}^{-}\bigr\vert ^{2}+(p-\mu) \int _{\Omega}\bigl\vert \nabla u_{0}^{-}\bigr\vert ^{p}< 0. $$
Thus, we can deduce
$$ J_{\Phi}(1,1)=\det B>0. $$
Therefore, by the fact that \((1,1)\) is the unique isolated zero point of the \(C^{1}\) function, we have
$$ \deg(\Phi,D,0)=\operatorname{ind}\bigl(\Phi,(1,1)\bigr)=\operatorname{sgn}J_{\Phi}(1,1)=1. $$
It follows from \(\bar{m}< m-2\varepsilon\), (4.17), and (i) above that \(\varphi=\psi\) on ∂D. Thus \(\deg(\Psi,D,0)=\deg (\Phi,D,0)=1\). Hence, there exists a pair \((s_{0},t_{0})\in D\) such that \(\Psi(s_{0},t_{0})=0\). Next we need to prove \(\psi^{\pm}(s_{0},t_{0})\neq0\). We first prove \(\psi^{+}(s_{0},t_{0})\neq0\). Since \(|u_{0}^{\pm}|_{q}\geqslant \beta\), for \((s_{0},t_{0})\in D\), we have \(|\varphi ^{+}(s_{0},t_{0})|_{q}=s_{0}|u^{+}|_{q}\geqslant\beta/2\) and \(|\varphi ^{-}(s_{0},t_{0})|_{q}=t_{0}|u^{-}|_{q}\geqslant\beta/2\). By (iii) and (4.17), we have
$$ \bigl\vert \psi(s_{0},t_{0})-\varphi(s_{0},t_{0}) \bigr\vert _{q}\leqslant S_{q}^{-1}\bigl\Vert \psi (s_{0},t_{0})-\varphi(s_{0},t_{0}) \bigr\Vert _{X}\leqslant S_{q}^{-1}\delta. $$
This implies that
$$ \bigl\vert \psi^{\pm}(s_{0},t_{0})- \varphi^{\pm}(s_{0},t_{0})\bigr\vert _{q}\leqslant\bigl\vert \psi (s_{0},t_{0})- \varphi(s_{0},t_{0})\bigr\vert _{q}\leqslant S_{q}^{-1}\delta. $$
Thus we obtain
$$ \bigl\vert \psi^{\pm}(s_{0},t_{0})\bigr\vert _{q}\geqslant\bigl\vert \varphi^{\pm}(s_{0},t_{0}) \bigr\vert _{q}-S_{q}^{-1}\delta\geqslant \frac{\beta}{2}-S_{q}^{-1}\delta>0, $$
which yields \(\psi^{\pm}(s_{0},t_{0})\ne0\). Thus \(\eta(1,\varphi (s_{0},t_{0}))=\psi(s_{0},t_{0})\in\mathcal{M}\), which is a contradiction to (4.19). Then \(u_{0}\) is a critical point of I, that is, \(u_{0}\) is a least energy sign-changing solution for equation (1.4). The proof is completed. □

Declarations

Acknowledgements

The authors sincerely thank the reviewers for their valuable suggestions and useful comments. This work is partially supported by the National Natural Science Foundation of China (Grant Nos. 11571209 and 11301313) and the Science Council of Shanxi Province (2013021001-4, 2014021009-1, and 2015021007).

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, Shanxi University

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© Liang et al. 2016