Open Access

Sign-changing solution and ground state solution for a class of \((p,q)\)-Laplacian equations with nonlocal terms on \(\mathbb{R}^{N}\)

Boundary Value Problems20162016:118

https://doi.org/10.1186/s13661-016-0624-5

Received: 2 March 2016

Accepted: 9 June 2016

Published: 23 June 2016

Abstract

In the paper, we investigate the least energy sign-changing solution and the ground state solution of a class of \((p,q)\)-Laplacian equations with nonlocal terms on \(\mathbb{R}^{N}\). Applying the constraint variational method, the quantitative deformation lemma, and topological degree theory, we see that the equation has one least energy sign-changing solution u. Moreover, we regard c, d as parameters and give a convergence property of such a solution \(u_{c,d}\) as \((c,d)\to 0\). Finally, using the Lagrange multiplier method, we obtain a ground state solution of the equation and show that the energy of u is strictly larger than two times the ground state energy.

Keywords

\((p,q)\)-Laplacian equation sign-changing solution ground state solution nonlocal term

1 Introduction

In this paper, we discuss the existence of a least energy sign-changing solution and a ground state solution of the following equation:
$$\begin{aligned}& - \biggl(a+c \int_{\mathbb{R}^{N}}|\nabla u|^{p} \biggr)\Delta_{p}u- \biggl(b+d \int_{\mathbb{R}^{N}}|\nabla u|^{q} \biggr)\Delta _{q}u+h(x)|u|^{p-2}u+g(x)|u|^{q-2}u \\& \quad =f(u),\quad x\in \mathbb{R}^{N}, \end{aligned}$$
(1.1)
where \(2\leqslant q< p< q^{*}\), \(N<2p\), \(\Delta_{m}=\operatorname{div}(|\nabla u|^{m-2}\nabla u)\) is the m-Laplacian operator, \(m^{*}=\infty\) for \(N\leqslant m\), and \(m^{*}=Nm/(N-m)\) for \(N>m\). a, b are positive constants, \(c,d\geqslant0\). We assume that h, g are continuous, coercive and positive functions.
When \(c=d=0\), equation (1.1) is the following \((p,q)\)-Laplacian equation:
$$ -a\Delta_{p}u-b\Delta_{q}u+h(x)|u|^{p-2}u+g(x)|u|^{q-2}u=f(u), \quad x\in \mathbb{R}^{N}. $$
(1.2)
A special situation for (1.2) is the case where \(p=q>1\), i.e., a single p-Laplacian equation. When \(p=q=2\), (1.2) becomes the nonlinear Laplacian type equation
$$ -\Delta u+au=f(x,u),\quad x\in\mathbb{R}^{N}. $$
(1.3)
Equation (1.2) appears, for example, as the stationary version of a general reaction-diffusion equation
$$ u_{t}=\operatorname{div}\bigl[D(u)\nabla u\bigr]+f(x,u), $$
where u describes a concentration, \(D(u)=|\nabla u|^{p-2}+|\nabla u|^{q-2}\) is the diffusion coefficient, and \(f(x,u)\) is the reaction term connected with source and loss mechanisms. This equation has extensive applications in physics and related sciences such as biophysics, plasma physics, and chemical reaction design. Typically, in chemical and biological applications, the reaction term \(f(x,u)\) is a polynomial of u with variable coefficients (see [14]).

The differential operator \(\Delta_{p}+\Delta_{q}\) is known as the \((p,q)\)-Laplacian operator, if \(p\ne q\). The single p-Laplacian operator has been studied for at least four decades (see [1, 519]), whereas a deeper research involving the \((p,q)\)-Laplacian operator has only arisen in the last decade (see [24, 2022]).

In [16], the authors investigated the existence of a positive solution of equation (1.3) where \(a>0\) is a constant. In [8], the authors proved the existence of sign-changing solutions of equation (1.3) where \(a\in L_{\mathrm{loc}}^{\infty}(\mathbb{R}^{N})\) and \(\operatorname{ess} \inf a>0\). We also refer the interested reader to more related results as regards equation (1.3) in [9, 10] and the references therein. In [7], the authors proved the existence of least energy positive, negative, and sign-changing solutions for the p-Laplacian equation with potentials vanishing at infinity. In [1], the author obtained multiplicity solutions of the p-Laplacian equation with a critical nonlinearity. Since the \((p,q)\)-Laplacian operator is not homogeneous, some technical difficulties appear when using the common methods of the elliptic equations. The existence of a nontrivial solution to equation (1.2) was obtained in [4, 20, 21]. In [4], the authors dealt with the situation \(2\leqslant q\leqslant p< N\) with \(h\in L_{+}^{N/p}(\mathbb{R}^{N})\) and \(g\in L_{+}^{N/q}(\mathbb{R}^{N})\), whereas in [21] the authors considered the case \(1< q< p< N\), but there h, g are positive constants. In [4, 21], the nonlinearity \(f(x,s)\) was suitably controlled by the variable s as \(s\to0\) and also as \(|s|\to\infty\), uniformly with respect to the variable x. In [20], the authors discussed the case that \(1< q< p< q^{*}\), \(p< N\) with h, g continuous, positive, and coercive functions on \(\mathbb{R}^{N}\) and \(f(x,s)\) a Carathéodory function satisfying some conditions.

To the best of our knowledge, there is little work researching the sign-changing solution and the ground state of the \((p,q)\)-Laplacian equations (1.1). Recently, Shuai in [23] discussed the following Kirchhoff type problem:
$$ \textstyle\begin{cases} - (a+b\int_{\Omega}|\nabla u|^{2} )\Delta u=f(u),&x\in \Omega, \\ u=0,&x\in\partial\Omega. \end{cases} $$
Motivated by [23], we investigate the sign-changing solution and the ground state solution of \((p,q)\)-Laplacian equation with nonlocal terms.

In general, the working space to study \((p,q)\)-Laplacian problems in a bounded domain Ω is \(W_{0}^{1,p}(\Omega)\), by taking advantage of the compact embedding \(W_{0}^{1,p}(\Omega)\hookrightarrow L^{s}(\Omega )\) for all \(s\in[1,p^{*})\). When the domain is the whole \(\mathbb {R}^{N}\), Sobolev’s embedding loses compactness. In order to overcome these difficulties, various methods have been developed. The radically symmetric Sobolev spaces have been applied to (1.3) (see [24, 25]), and the concentration-compactness principle or the constrained minimization method has been used to find solutions in \(W^{1,p}(\mathbb{R}^{N})\cap W^{1,q}(\mathbb{R}^{N})\) (see [1, 3, 4, 21, 26]).

In this paper, we intend to choose an appropriate approach by taking into account the Banach space,
$$ W= \biggl\{ u\in{\mathcal{D}}^{1,p}\bigl(\mathbb{R}^{N}\bigr) \cap{\mathcal {D}}^{1,q}\bigl(\mathbb{R}^{N}\bigr): \int_{\mathbb{R}^{N}}h|u|^{p}, \int_{\mathbb {R}^{N}}g|u|^{q}< \infty \biggr\} . $$
We recall that the space \({\mathcal{D}}^{1,m}(\mathbb{R}^{N})\) is a reflexive Banach space which is characterized by (see [11])
$$ {\mathcal{D}}^{1,m}\bigl(\mathbb{R}^{N}\bigr)= \biggl\{ u\in L^{m^{*}}\bigl(\mathbb {R}^{N}\bigr):\frac{\partial u}{\partial x_{i}}\in L^{m}\bigl(\mathbb{R}^{N}\bigr) \biggr\} $$
and its norm is equivalent to the norm \(\|\nabla u\|_{L^{m}(\mathbb {R}^{N})}\). We denote the norm of \(L^{m}(\mathbb{R}^{N})\) as \(|\cdot|_{m}\) hereafter. Moreover, \(W^{1,m}(\mathbb{R}^{N})\subset{\mathcal {D}}^{1,m}(\mathbb{R}^{N})\hookrightarrow L^{m^{*}}(\mathbb{R}^{N})\).
We take \(h,g\) as continuous, coercive, and positive functions on \(\mathbb{R}^{N}\) and define normed spaces \((W_{p,a,h},\|\cdot\| _{1} )\) and \((W_{q,b,g},\|\cdot\|_{2} )\), respectively, by
$$\begin{aligned}& W_{p,a,h}= \biggl\{ u\in{\mathcal{D}}^{1,p}\bigl( \mathbb{R}^{N}\bigr): \int _{\mathbb{R}^{N}}h|u|^{p}< \infty \biggr\} , \\& W_{q,b,g}= \biggl\{ u\in {\mathcal{D}}^{1,q}\bigl( \mathbb{R}^{N}\bigr): \int_{\mathbb{R}^{N}}g|u|^{q}< \infty \biggr\} , \end{aligned}$$
with norms
$$\begin{aligned}& \|u\|_{1}= \biggl( \int_{\mathbb{R}^{N}}\bigl[a|\nabla u|^{p}+h|u|^{p} \bigr] \biggr)^{1/p}, \\& \|u\|_{2}= \biggl( \int_{\mathbb{R}^{N}}\bigl[b|\nabla u|^{q}+g|u|^{q} \bigr] \biggr)^{1/q}. \end{aligned}$$
Then \(W_{p,a,h}\) and \(W_{q,b,g}\) are reflexive Banach spaces. The embedding \(W_{p,a,h}\hookrightarrow L^{s}(\mathbb{R}^{N})\) is continuous for all \(s\in[p,p^{*}]\) and compact for all \(s\in[p,p^{*})\). Similarly, the embedding \(W_{q,b,g}\hookrightarrow L^{s}(\mathbb{R}^{N})\) is continuous if \(s\in[q,q^{*}]\) and compact if \(s\in[q,q^{*})\) (see [20]).
Now we can define our working space W:
$$ W=W_{p,a,h}\cap W_{q,b,g} $$
endowed with the norm
$$ \|u\|=\|u\|_{1}+\|u\|_{2}. $$
Then it is easy to see that W is a reflexive Banach space and the embedding \(W\hookrightarrow L^{s}(\mathbb{R}^{N})\) is continuous if \(s\in [q,p^{*}]\) and compact if \(s\in[q,p^{*})\).

For brevity, we omit the integral domain \(\mathbb{R}^{N}\) when no confusion arises hereafter.

We assume that \(f\in C^{1}(\mathbb{R},\mathbb{R})\) satisfies the following hypotheses:
(f1): 

\(\lim_{s\to0}f(s)/|s|^{q-1}=0\);

(f2): 

for some constant \(r\in(2p,p^{*})\), \(\lim_{|s|\to\infty }f(s)/|s|^{r-1}=0\);

(f3): 

\(\lim_{|s|\to\infty}F(s)/|s|^{2p}=\infty\), where \(F(s)=\int_{0}^{t}f(t)\, \mathrm{d}t\) for all \(s\in\mathbb{R}\);

(f4): 

\(f(s)/|s|^{2p-1}\) is increasing on \((-\infty,0)\) and \((0,\infty)\), respectively.

Define the energy functional \(I:W\to\mathbb{R}\) of (1.1) by
$$ I(u)=\frac{1}{p}\|u\|_{1}^{p}+ \frac{1}{q}\|u\|_{2}^{q}+\frac{c}{2p} \biggl( \int|\nabla u|^{p} \biggr)^{2}+\frac{d}{2q} \biggl( \int|\nabla u|^{q} \biggr)^{2}- \int F(u), \quad u\in W. $$
(1.4)
Then the functional I is well defined on W and belongs to \(C^{1}(W,\mathbb{R})\). Moreover, for any \(u,\varphi\in W\), we have
$$\begin{aligned} \bigl\langle I'(u),\varphi\bigr\rangle =& \int\bigl[a|\nabla u|^{p-2}\nabla u\cdot \nabla \varphi+h|u|^{p-2}u\varphi\bigr]+ \int\bigl[b|\nabla u|^{q-2}\nabla u\cdot\nabla \varphi+g|u|^{q-2}u\varphi\bigr] \\ &{}+c \int|\nabla u|^{p} \int|\nabla u|^{p-2}\nabla u\cdot\nabla\varphi \\ &{}+d \int|\nabla u|^{q} \int|\nabla u|^{q-2}\nabla u\cdot\nabla\varphi - \int f(u)\varphi. \end{aligned}$$
(1.5)
A critical point of I corresponds to a solution of (1.1). Furthermore, if \(u\in W\) is a solution of (1.1) with \(u^{\pm}\ne 0\), then u is a sign-changing solution of (1.1), where
$$ u^{+}(x)=\max\bigl\{ u(x),0\bigr\} , \qquad u^{-}(x)=\min\bigl\{ u(x),0\bigr\} . $$
Obviously, the energy functional \(I_{0}:W\to\mathbb{R}\) of (1.2) is given by
$$ I_{0}(u)=\frac{1}{p}\|u\|_{1}^{p}+ \frac{1}{q}\|u\|_{2}^{q}- \int F(u), \quad u\in W. $$
For \(u\in W\),
$$ I_{0}(u)=I_{0}\bigl(u^{+}\bigr)+I_{0} \bigl(u^{-}\bigr), \qquad \bigl\langle I'_{0}(u),u^{\pm}\bigr\rangle =\bigl\langle I'_{0}\bigl(u^{\pm}\bigr),u^{\pm}\bigr\rangle . $$
(1.6)
When \(c,d>0\), the nonlocal terms \((\int|\nabla u|^{p})\Delta_{p}u\), \((\int |\nabla u|^{q})\Delta_{q}u\) are involved in equation (1.1), for the functional I given by (1.4) it is apparent that
$$\begin{aligned}& I(u)=I\bigl(u^{+}\bigr)+I\bigl(u^{-}\bigr)+\frac{c}{p} \int\bigl\vert \nabla u^{+}\bigr\vert ^{p} \int\bigl\vert \nabla u^{-}\bigr\vert ^{p}+\frac{d}{q} \int\bigl\vert \nabla u^{+}\bigr\vert ^{q} \int\bigl\vert \nabla u^{-}\bigr\vert ^{q}, \end{aligned}$$
(1.7)
$$\begin{aligned}& \bigl\langle I'(u),u^{\pm}\bigr\rangle = \bigl\langle I'\bigl(u^{\pm}\bigr),u^{\pm}\bigr\rangle +c \int \bigl\vert \nabla u^{+}\bigr\vert ^{p} \int\bigl\vert \nabla u^{-}\bigr\vert ^{p}+d \int\bigl\vert \nabla u^{+}\bigr\vert ^{q} \int\bigl\vert \nabla u^{-}\bigr\vert ^{q}. \end{aligned}$$
(1.8)
Clearly, the functional I does no longer satisfy (1.6), since it contains two nonlocal terms. Hence, there may be some differences in investigating the sign-changing solution of equation (1.1) between \(c,d>0\) and \(c=d=0\).
In order to obtain a sign-changing solution of equation (1.1), we try to seek a minimizer of the functional I over the following constraint:
$$ {\mathcal{M}}= \bigl\{ u\in W:u^{\pm}\ne0,\bigl\langle I'(u),u^{+}\bigr\rangle =\bigl\langle I'(u),u^{-}\bigr\rangle =0 \bigr\} $$
(1.9)
and
$$ m=\inf{ \bigl\{ I(u):u\in\mathcal{M} \bigr\} }. $$
(1.10)
Then we show that the minimizer is indeed a sign-changing solution of (1.1). As we have mentioned before, the functional I no longer satisfies the properties (1.6), so it is more difficult to prove that \({\mathcal{M}}\ne\emptyset\). Actually, we will obtain \({\mathcal{M}}\ne\emptyset\) by using the Brouwer fixed point theorem, which is different from the approach in [23].
In order to get the ground state solution of equation (1.1), let
$$ {\mathcal{N}}=\bigl\{ u\in W\setminus\{0\}:\bigl\langle I'(u),u\bigr\rangle =0\bigr\} , $$
(1.11)
and consider the ground state energy
$$ \tilde{m}=\inf\bigl\{ I(u):u\in\mathcal{N}\bigr\} . $$
(1.12)

Now, we state our main results.

Theorem 1.1

If the assumptions (f1)-(f4) hold, then equation (1.1) has one least energy sign-changing solution.

Theorem 1.2

Suppose the assumptions (f1)-(f4) hold. For any sequence \(\{ (c_{n},d_{n})\}\) with \(c_{n},d_{n}\geqslant0\), as \((c_{n},d_{n})\to0\), there exists a subsequence, still denoted by \(\{(c_{n},d_{n})\}\), such that \(u_{c_{n},d_{n}}\to u_{0}\) in W, and \(u_{0}\) is a least energy sign-changing solution of equation (1.2).

Theorem 1.3

Suppose the assumptions (f1)-(f4) hold.
  1. (i)

    There exists a ground state solution v of equation (1.1).

     
  2. (ii)

    \(m>2\tilde{m}\). In particular, the ground state solution must maintain the sign unchanged.

     

Remark 1.4

The three results above are also valid for \((p,q)\)-Laplacian problems in a bounded domain Ω. Consider the following two problems:
$$ \textstyle\begin{cases} - (a+c\int_{\Omega}|\nabla u|^{p} )\Delta_{p}u- (b+d\int_{\Omega}|\nabla u|^{q} )\Delta _{q}u \\ \quad {}+h(x)|u|^{p-2}u+g(x)|u|^{q-2}u =f(u),&x\in\Omega, \\ u=0, &x\in\partial\Omega \end{cases} $$
(1.13)
and
$$ \textstyle\begin{cases} -a\Delta_{p}u-b\Delta_{q}u+h(x)|u|^{p-2}u+g(x)|u|^{q-2}u=f(u),&x\in \Omega, \\ u=0, &x\in\partial\Omega, \end{cases} $$
where Ω is a bounded domain in \(\mathbb{R}^{N}\), h, g are continuous and non-negative functions, including the case \(h\equiv g\equiv0\). Because the embedding \(W_{0}^{1,m}(\Omega)\hookrightarrow L^{s}(\Omega)\) is continuous if \(s\in[1,m^{*}]\) and compact if \(s\in [1,m^{*})\), we find solutions in the space \(W_{0}^{1,p}(\Omega)\cap W_{0}^{1,q}(\Omega)\), and then can also obtain the same conclusions as Theorems 1.1-1.3 for (1.13).

Both the conclusions of (1.1) and of (1.13) are true when \(p=q\), i.e., these results are true for a single p-Laplacian equation with nonlocal term.

The paper is organized as follows. In Section 2, we prove several lemmas, which are important to prove our main results. In Section 3, we first show that the minimizer of the constrained problem (1.9) is a sign-changing solution. Then we prove the convergence property of solutions of (1.1). Finally, we prove the existence of the ground state solution and give the energy comparison.

Throughout this paper, C and \(C_{k}\) denote various positive constants, which may vary from line to line.

2 Preliminaries

We use constraint minimization on \({\mathcal{M}}\) to seek a critical point of I. We begin this section by doing some preparation work.

Lemma 2.1

Assume that (f1)-(f4) hold. If \(u\in W\) with \(u\ne0\), then
  1. (i)

    \(\lim_{s\to0}\int\frac{f(su)u}{|s|^{q-1}}=0\);

     
  2. (ii)

    \(\lim_{|s|\to\infty}\int\frac {f(su)u}{|s|^{2p-2}s}=\infty\);

     
  3. (iii)

    \(\lim_{|s|\to\infty}\int\frac {F(su)}{|s|^{2p}}=\infty\);

     
  4. (iv)

    moreover, if \(u^{\pm}\ne0\), then \(\lim_{|(s,t)|\to\infty }\int\frac{F(su^{+})+F(tu^{-})}{|s|^{2p}+|t|^{2p}}=\infty\).

     

Proof

(i) By the conditions (f1) and (f2), for any given \(\varepsilon >0\), there exists \(C_{\varepsilon}>0\) such that
$$\begin{aligned}& \bigl\vert f(s)\bigr\vert \leqslant\varepsilon|s|^{q-1}+C_{\varepsilon}|s|^{r-1}, \quad s\in \mathbb{R}, \end{aligned}$$
(2.1)
$$\begin{aligned}& \bigl\vert F(s)\bigr\vert \leqslant\frac{\varepsilon}{q}|s|^{q}+ \frac{C_{\varepsilon}}{r}|s|^{r}, \quad s\in\mathbb{R}. \end{aligned}$$
(2.2)
By the condition (f1), we have, for each \(\eta\in\mathbb{R}\),
$$ \lim_{s\to0}\frac{f(s\eta)}{|s|^{q-1}}=0. $$
(2.3)
Thus, by (2.1), (2.3) and the Lebesgue dominated convergence theorem, the conclusion (i) holds.
(ii) By the conditions (f4) and (f3), we have
$$ \lim_{|s|\to\infty}\frac{f(s)}{|s|^{2p-2}s}=\infty. $$
(2.4)
It follows from (2.4) that, for any given \(M_{1}>0\), there exists \(R_{1}>0\) such that
$$ \frac{f(s)s}{|s|^{2p}}\geqslant M_{1}, \quad |s|>R_{1}. $$
(2.5)
By the condition (f1), we have
$$ \lim_{s\to0}\frac{f(s)s-M_{1}|s|^{2p}}{|s|^{q}}=0. $$
Then there exists \(C_{M_{1}}>0\) such that
$$ \frac{f(s)s-M_{1}|s|^{2p}}{|s|^{q}}\geqslant-C_{M_{1}},\quad |s| \in(0,R_{1}]. $$
(2.6)
It follows from (2.5) and (2.6) that
$$ f(s)s\geqslant M_{1}|s|^{2p}-C_{M_{1}}{|s|^{q}}, \quad s\in\mathbb{R}. $$
(2.7)
It follows from (2.7) that
$$ \liminf_{|s|\to\infty} \int\frac{f(su)u}{|s|^{2p-2}s}\geqslant M_{1} \int|u|^{2p}-\lim_{|s|\to\infty}\frac{C_{M_{1}}}{|s|^{2p-q}} \int {|u|^{q}}=M_{1} \int|u|^{2p}. $$
Then, by the arbitrariness of \(M_{1}\), the conclusion (ii) is true.
(iii) By the condition (f3), for any given \(M_{2}>0\), there exists \(R_{2}>0\) such that
$$ \frac{F(s)}{|s|^{2p}}\geqslant M_{2}, \quad |s|>R_{2}. $$
(2.8)
By the condition (f1), we have
$$ \lim_{s\to0}\frac{F(s)-M_{2}|s|^{2p}}{|s|^{q}}=0. $$
Then there exists \(C_{M_{2}}>0\) such that
$$ \frac{F(s)-M_{2}|s|^{2p}}{|s|^{q}}\geqslant-C_{M_{2}}, \quad |s| \in(0,R_{2}]. $$
(2.9)
It follows from (2.8) and (2.9) that
$$ F(s)\geqslant M_{2}|s|^{2p}-C_{M_{2}}|s|^{q}, \quad s\in\mathbb{R}. $$
(2.10)
Then it follows from (2.10) that
$$ \liminf_{|s|\to\infty} \int\frac{F(su)}{|s|^{2p}}\geqslant M_{2} \int |u|^{2p}-\lim_{|s|\to\infty}\frac{C_{M_{2}}}{|s|^{2p-q}} \int |u|^{q}=M_{2} \int|u|^{2p}. $$
Thus, by the arbitrariness of \(M_{2}\), the conclusion (iii) is also true.
(iv) For convenience, we denote functions \(\psi_{1}(s)=\int F(su^{+})\) and \(\psi_{2}(s)=\int F(su^{-})\) for all \(s\in\mathbb{R}\). Then, by (iii), we have
$$ \psi_{1}(s)\to\infty,\qquad \psi_{2}(s)\to \infty, \quad |s|\to\infty. $$
(2.11)
Because of (2.11) and the continuity of \(\psi_{1}\), \(\psi_{2}\), there exists \(C>0\) such that
$$ \psi_{1}(s)\geqslant-C,\qquad \psi_{2}(s) \geqslant-C, \quad s\in\mathbb{R}. $$
(2.12)
By (iii), for any given \(M>0\), there exists \(R>0\) such that
$$ \frac{\psi_{1}(s)+C}{|s|^{2p}}\geqslant2M, \qquad \frac{\psi _{2}(s)+C}{|s|^{2p}} \geqslant2M, \quad |s|\geqslant R. $$
(2.13)
When \(|(s,t)|=\sqrt{s^{2}+t^{2}}\geqslant\sqrt{2}R\), by the inequality
$$ \sqrt{s^{2}+t^{2}}\leqslant\sqrt{2}\max\bigl\{ |s|,|t|\bigr\} , $$
\(\max\{|s|,|t|\}\geqslant R\). We may suppose that \(|s|\geqslant|t|\), so that \(|s|\geqslant R\). Combining with (2.12) and (2.13), we have
$$ \frac{\psi_{1}(s)+C+\psi_{2}(t)+C}{|s|^{2p}+|t|^{2p}}\geqslant\frac {\psi_{1}(s)+C}{2|s|^{2p}}\geqslant M. $$
Then we have
$$ \lim_{|(s,t)|\to\infty}\frac{\psi_{1}(s)+C+\psi _{2}(t)+C}{|s|^{2p}+|t|^{2p}}=\infty. $$
(2.14)
Therefore, it follows from (2.14) that (iv) holds. □

Remark 2.2

By the condition (f4), for each \(\eta \in\mathbb{R}\setminus\{0\}\), we see that \(f(s\eta)\eta /|s|^{2p-1}\) is increasing on \((-\infty,0)\) and \((0,\infty)\), respectively. Therefore, for each \(u\in W\) with \(u\ne0\), we see that \(\int\frac{f(su)u}{s^{2p-1}}\) is increasing on \((0,\infty)\).

Now we start to check that the set \({\mathcal{M}}\) is nonempty.

For each \(u\in W\) with \(u^{\pm}\ne0\), for convenience, we denote the positive numbers \(A_{1,u}=(\int|\nabla u^{+}|^{p})^{2}\), \(A_{2,u}=(\int|\nabla u^{-}|^{p})^{2}\), \(A_{3,u}=\int|\nabla u^{+}|^{p}\int|\nabla u^{-}|^{p}\); \(B_{1,u}=(\int |\nabla u^{+}|^{q})^{2}\), \(B_{2,u}=(\int|\nabla u^{-}|^{q})^{2}\), \(B_{3,u}=\int|\nabla u^{+}|^{q}\int|\nabla u^{-}|^{q}\).

Lemma 2.3

Assume that (f1)-(f4) hold. If \(u\in W\) with \(u^{\pm}\ne0\), then there is a unique pair \((s_{u},t_{u})\) of positive numbers such that \(s_{u}u^{+}+t_{u}u^{-}\in{\mathcal{M}}\).

Proof

For any given \(u\in W\) with \(u^{\pm}\ne0\), we define a function \(\Psi _{u}:\mathbb{R}_{+}\times\mathbb{R}_{+}\to\mathbb{R}\) by \(\Psi _{u}(s,t)=I(su^{+}+tu^{-})\), where \(\mathbb{R}_{+}=[0,\infty)\), that is,
$$\begin{aligned} \Psi_{u}(s,t) =&\frac{1}{p}s^{p}\bigl\Vert u^{+}\bigr\Vert _{1}^{p}+\frac{1}{q}s^{q} \bigl\Vert u^{+}\bigr\Vert _{2}^{q}+\frac{c}{2p}A_{1,u}s^{2p}+ \frac{c}{p}A_{3,u}s^{p}t^{p} \\ &{}+ \frac{d}{2q}B_{1,u}s^{2q}+\frac{d}{q}B_{3,u}s^{q}t^{q}- \int F\bigl(su^{+}\bigr) \\ &{}+\frac{1}{p}t^{p}\bigl\Vert u^{-}\bigr\Vert _{1}^{p}+\frac{1}{q}t^{q}\bigl\Vert u^{-} \bigr\Vert _{2}^{q}+\frac{c}{2p}A_{2,u}t^{2p} \\ &{}+ \frac{d}{2q}B_{2,u}t^{2q}- \int F\bigl(tu^{-}\bigr). \end{aligned}$$
(2.15)
For \(s,t>0\), since
$$\begin{aligned} \nabla\Psi_{u}(s,t) =& \biggl(\frac{\partial\Psi_{u}}{\partial s}(s,t), \frac{\partial\Psi_{u}}{\partial t}(s,t) \biggr) \\ =& \bigl(\bigl\langle I'\bigl(su^{+}+tu^{-}\bigr),u^{+}\bigr\rangle , \bigl\langle I'\bigl(su^{+}+tu^{-}\bigr),u^{-}\bigr\rangle \bigr) \\ =& \biggl(\frac{1}{s}\bigl\langle I'\bigl(su^{+}+tu^{-} \bigr),su^{+}\bigr\rangle ,\frac{1}{t}\bigl\langle I' \bigl(su^{+}+tu^{-}\bigr),tu^{-}\bigr\rangle \biggr), \end{aligned}$$
we have \(su^{+}+tu^{-}\in{\mathcal{M}}\) if and only if \((s,t)\) is a critical point of \(\Psi_{u}\). Next we will prove the existence of a critical point of \(\Psi_{u}\).
For any given \(t_{0}\in\mathbb{R}_{+}\), we have, for \(s>0\),
$$\begin{aligned}& \frac{\partial}{\partial s}\Psi_{u}(s,t_{0}) \\& \quad = s^{p-1}\bigl\Vert u^{+}\bigr\Vert _{1}^{p}+s^{q-1} \bigl\Vert u^{+}\bigr\Vert _{2}^{q}+cA_{1,u}s^{2p-1}+cA_{3,u}s^{p-1}t_{0}^{p} \\& \qquad {}+dB_{1,u}s^{2q-1}+dB_{3,u}s^{q-1}t_{0}^{q}- \int f\bigl(su^{+}\bigr)u^{+} \\& \quad = s^{q-1} \biggl[s^{p-q}\bigl\Vert u^{+} \bigr\Vert _{1}^{p}+\bigl\Vert u^{+}\bigr\Vert _{2}^{q}+cA_{1,u}s^{2p-q}+cA_{3,u}s^{p-q}t_{0}^{p} \\& \qquad {}+dB_{1,u}s^{q}+dB_{3,u}t_{0}^{q}- \int \frac{f(su^{+})u^{+}}{s^{q-1}} \biggr] \end{aligned}$$
(2.16)
$$\begin{aligned}& \quad = s^{2p-1} \biggl[\frac{1}{s^{p}}\bigl\Vert u^{+}\bigr\Vert _{1}^{p}+\frac{1}{s^{2p-q}}\bigl\Vert u^{+} \bigr\Vert _{2}^{q}+cA_{1,u}+\frac{t_{0}^{p}}{s^{p}}cA_{3,u} \\& \qquad {}+ \frac{dB_{1,u}}{s^{2p-2q}}+ \frac{t_{0}^{q} dB_{3,u}}{s^{2p-q}}- \int\frac{f(su^{+})u^{+}}{s^{2p-1}} \biggr]. \end{aligned}$$
(2.17)
Since \(u^{+}\ne0\), it follows from (2.16) and Lemma 2.1(i) that \(\frac{\partial}{\partial s}\Psi_{u}(s,t_{0})>0\) for \(s>0\) small. It follows from (2.17) and Lemma 2.1(ii) that \(\frac{\partial}{\partial s}\Psi_{u}(s,t_{0})<0\) for \(s>0\) large. Thus there exists \(s_{0}>0\) such that \(\frac{\partial}{\partial s}\Psi_{u}(s_{0},t_{0})=0\).
Suppose that there exist \(s_{1}\), \(s_{2}\) with \(0< s_{1}< s_{2}\) such that \(\frac {\partial}{\partial s}\Psi_{u}(s_{1},t_{0})=\frac{\partial}{\partial s}\Psi_{u}(s_{2},t_{0})=0\). Then (2.17) implies that
$$\begin{aligned}& \frac{1}{s_{i}^{p}}\bigl\Vert u^{+}\bigr\Vert _{1}^{p}+ \frac{1}{s_{i}^{2p-q}}\bigl\Vert u^{+}\bigr\Vert _{2}^{q}+cA_{1,u}+ \frac {t_{0}^{p}}{s_{i}^{p}}cA_{3,u}+\frac{dB_{1,u}}{s_{i}^{2p-2q}}+ \frac{t_{0}^{q} dB_{3,u}}{s_{i}^{2p-q}} \\& \quad = \int\frac {f(s_{i}u^{+})u^{+}}{s_{i}^{2p-1}}, \quad i=1,2. \end{aligned}$$
Hence
$$\begin{aligned}& \biggl(\frac{1}{s_{1}^{p}}-\frac{1}{s_{2}^{p}} \biggr)\bigl\| u^{+}\bigr\| _{1}^{p}+ \biggl(\frac{1}{s_{1}^{2p-q}}-\frac{1}{s_{2}^{2p-q}} \biggr)\bigl\| u^{+}\bigr\| _{2}^{q}+ \biggl(\frac{1}{s_{1}^{p}}-\frac{1}{s_{2}^{p}} \biggr)t_{0}^{p}cA_{3,u} \\& \qquad {}+ \biggl(\frac{1}{s_{1}^{2p-2q}}-\frac{1}{s_{2}^{2p-2q}} \biggr) dB_{1,u}+ \biggl(\frac{1}{s_{1}^{2p-q}}-\frac{1}{s_{2}^{2p-q}} \biggr)t_{0}^{q} dB_{3,u} \\& \quad = \int \biggl[\frac{f(s_{1}u^{+})u^{+}}{s_{1}^{2p-1}}-\frac {f(s_{2}u^{+})u^{+}}{s_{2}^{2p-1}} \biggr]. \end{aligned}$$
(2.18)
But according to Remark 2.2, the right side of (2.18) is negative and (2.18) is absurd. Therefore there exists a unique \(s_{0}=s_{0}(t_{0})>0\) such that \(\frac{\partial}{\partial s}\Psi _{u}(s_{0},t_{0})=0\).
Now we can define a map \(\varphi_{1}:\mathbb{R}_{+}\to(0,\infty)\) by \(\varphi_{1}(t)=s(t)\), where \(s(t)\) satisfies the properties just mentioned previously, with t in the place of \(t_{0}\). By definition, we have
$$ \frac{\partial\Psi_{u}}{\partial s}\bigl(\varphi_{1}(t),t\bigr)=0, \quad t\in \mathbb{R}_{+}, $$
that is, for \(t\geqslant0\),
$$\begin{aligned} \begin{aligned}[b] &\varphi_{1}^{p-1}(t)\bigl\Vert u^{+}\bigr\Vert _{1}^{p}+\varphi_{1}^{q-1}(t)\bigl\Vert u^{+}\bigr\Vert _{2}^{q}+cA_{1,u} \varphi_{1}^{2p-1}(t)+cA_{3,u}\varphi _{1}^{p-1}(t)t^{p} \\ &\qquad {}+dB_{1,u} \varphi_{1}^{2q-1}(t) +dB_{3,u}\varphi_{1}^{q-1}(t)t^{q} \\ &\quad = \int f\bigl(\varphi_{1}(t)u^{+}\bigr)u^{+}. \end{aligned} \end{aligned}$$
(2.19)
We will prove some properties of the function \(\varphi_{1}\).

(a1) \(\varphi_{1}\) has a positive lower bound.

In fact, suppose there exists \(\{t_{n}\}\subset\mathbb{R}_{+}\) such that \(\varphi_{1}(t_{n})\to0\). Then, by (2.19) and Lemma 2.1(i), we have
$$ \bigl\Vert u^{+}\bigr\Vert _{2}^{q}\leqslant\lim _{n\to\infty} \int\frac{f(\varphi _{1}(t_{n})u^{+})u^{+}}{\varphi_{1}^{q-1}(t_{n})}=0. $$
This is absurd. Thus there exists \(C>0\) such that \(\varphi _{1}(s)\geqslant C\) for all \(s\in\mathbb{R}_{+}\).

(a2) \(\varphi_{1}\) is continuous.

In fact, let \(t_{n}\to t_{0}\) in \(\mathbb{R}_{+}\). We firstly prove that \(\{ \varphi_{1}(t_{n})\}\) is bounded. Suppose, by contradiction, that there is a subsequence \(\{t_{n_{k}}\}\) of \(\{t_{n}\}\) such that \(\varphi _{1}(t_{n_{k}})\to\infty\). It follows from (2.19) that
$$\begin{aligned}& \frac{1}{\varphi_{1}^{p}(t_{n_{k}})}\bigl\Vert u^{+}\bigr\Vert _{1}^{p}+\frac{1}{\varphi _{1}^{2p-q}(t_{n_{k}})}\bigl\Vert u^{+}\bigr\Vert _{2}^{q}+cA_{1,u}+\frac{t_{n_{k}}^{p}}{\varphi _{1}^{p}(t_{n_{k}})}cA_{3,u} +\frac{dB_{1,u}}{\varphi_{1}^{2p-2q}(t_{n_{k}})}+\frac {t_{n_{k}}^{q} dB_{3,u}}{\varphi_{1}^{2p-q}(t_{n_{k}})} \\& \quad = \int\frac{f(\varphi_{1}(t_{n_{k}})u^{+})}{\varphi_{1}^{2p-1}(t_{n_{k}})}u^{+}. \end{aligned}$$
(2.20)
Letting \(k\to\infty\) in (2.20), according to Lemma 2.1(ii), we have a contradiction \(cA_{1,u}=\infty\). Thus, \(\{\varphi _{1}(t_{n})\}\) is bounded. For any subsequence \(\{\varphi_{1}(t_{n}')\}\) of \(\{ \varphi_{1}(t_{n})\}\), since \(\{\varphi_{1}(t_{n}')\}\) is bounded, there exists a subsequence \(\{\varphi_{1}(t_{n}'')\}\) of \(\{\varphi_{1}(t_{n}')\}\) such that \(\varphi_{1}(t_{n}'')\to s_{0}\) and it follows from (a1) that \(s_{0}>0\). Passing to the limit as \(n\to\infty\) in (2.19) with \(t=t_{n}''\), we get
$$\begin{aligned}& s_{0}^{p-1}\bigl\Vert u^{+}\bigr\Vert _{1}^{p}+s_{0}^{q-1}\bigl\Vert u^{+}\bigr\Vert _{2}^{q}+cA_{1,u}s_{0}^{2p-1}+cA_{3,u}s_{0}^{p-1}t_{0}^{p}+dB_{1,u}s_{0}^{2q-1} +dB_{3,u}s_{0}^{q-1}t_{0}^{q} \\& \quad = \int f\bigl(s_{0}u^{+}\bigr)u^{+}. \end{aligned}$$
(2.21)
Thus (2.17) and (2.21) imply
$$ \frac{\partial\Psi_{u}}{\partial s}(s_{0},t_{0})=0. $$
Consequently, by the uniqueness, \(s_{0}=\varphi_{1}(t_{0})\). Therefore \(\varphi_{1}\) is continuous.

(a3) \(\varphi_{1}(t)\leqslant t\) for t large.

In fact, if there exists a sequence \(\{t_{n}\}\) with \(t_{n}\to\infty\) such that \(\varphi_{1}(t_{n})>t_{n}\) for all \(n\in\mathbb{N}\), then \(\varphi_{1}(t_{n})\to\infty\) and it follows from (2.20) that \(\infty\leqslant cA_{1,u}+cA_{3,u}\). This is a contradiction. Thus \(\varphi_{1}(t)\leqslant t\) for t large.

Similarly, for each \(s\in\mathbb{R}_{+}\), we consider the function \(\Psi_{u}(s,\cdot)\) and consequently, we can define a map \(\varphi _{2}:\mathbb{R}_{+}\to(0,\infty)\) which satisfies
$$ \frac{\partial\Psi_{u}}{\partial t}\bigl(s,\varphi_{2}(s)\bigr)=0, \quad s\in \mathbb{R}_{+}, $$
that is, for \(s\geqslant0\),
$$\begin{aligned}& \varphi_{2}^{p-1}(s)\bigl\Vert u^{-}\bigr\Vert _{1}^{p}+\varphi_{2}^{q-1}(s)\bigl\Vert u^{-}\bigr\Vert _{2}^{q}+cA_{2,u} \varphi_{2}^{2p-1}(s)+cA_{3,u}\varphi _{2}^{p-1}(s)s^{p} \\& \qquad {}+dB_{2,u} \varphi_{2}^{2q-1}(s) +dB_{3,u}\varphi_{2}^{q-1}(s)s^{q} \\& \quad = \int f\bigl(\varphi_{2}(s)u^{-}\bigr)u^{-}, \end{aligned}$$
(2.22)
and it also satisfies (a1), (a2), and (a3) above.
Now we prove the existence of a critical point of \(\Psi_{u}\) by the Brouwer fixed point theorem. By (a3), there exists \(C_{1}>0\) such that \(\varphi_{1}(t)\leqslant t\) for all \(t>C_{1}\) and \(\varphi_{2}(s)\leqslant s\) for all \(s>C_{1}\). Let
$$ C_{2}=\max \Bigl\{ \max_{t\in[0,C_{1}]}\varphi_{1}(t), \max_{s\in [0,C_{1}]}\varphi_{2}(s) \Bigr\} . $$
Let \(\xi=\max\{C_{1},C_{2}\}\). We define \(T:[0,\xi]\to\mathbb{R}_{+}\) as \(T(s)=\varphi_{1}(\varphi_{2}(s))\). Now we show \(T(s)\in[0,\xi]\) for all \(s\in[0,\xi]\). In fact, let \(0\leqslant s\leqslant\xi=\max\{C_{1},C_{2}\}\). If \(t=\varphi_{2}(s)>C_{1}\), then
$$ T(s)=\varphi_{1}(t)\leqslant t=\varphi_{2}(s)\leqslant \textstyle\begin{cases} s,&s>C_{1}, \\ \max_{s\in[0,C_{1}]}\varphi_{2}(s),&s\leqslant C_{1}, \end{cases} $$
so
$$T(s)\leqslant\max\{C_{1},C_{2}\}. $$
If \(t=\varphi_{2}(s)\leqslant C_{1}\), then
$$ T(s)=\varphi_{1}(t)\leqslant\max_{t\in[0,C_{1}]} \varphi_{1}(t)\leqslant C_{2}. $$
Note that T is continuous. Then, by the Brouwer fixed point theorem, there exists \(s_{u}\in[0,\xi]\) such that \(\varphi_{1}(\varphi _{2}(s_{u}))=s_{u}\). Let \(t_{u}=\varphi_{2}(s_{u})\). Then we have
$$ s_{u}=\varphi_{1}(t_{u}), \qquad t_{u}=\varphi_{2}(s_{u}). $$
(2.23)
Since \(\varphi_{i}>0\), (2.23) implies \(s_{u},t_{u}>0\). By the definition we have
$$ \frac{\partial\Psi_{u}}{\partial s}(s_{u},t_{u})=\frac{\partial\Psi _{u}}{\partial t}(s_{u},t_{u})=0. $$
Thus, \((s_{u},t_{u})\) is a critical point of \(\Psi_{u}\).
Now we prove the uniqueness of \((s_{u},t_{u})\). In fact, considering \(w\in \mathcal{M}\) we have
$$\begin{aligned} \nabla\Psi_{w}(1,1) =& \biggl(\frac{\partial\Psi_{w}}{\partial s}(1,1), \frac{\partial\Psi_{w}}{\partial t}(1,1) \biggr) \\ =& \bigl(\bigl\langle I'\bigl(w^{+}+w^{-}\bigr),w^{+}\bigr\rangle ,\bigl\langle I'\bigl(w^{+}+w^{-}\bigr),w^{-}\bigr\rangle \bigr)=(0,0), \end{aligned}$$
which implies that \((1,1)\) is a critical point of \(\Psi_{w}\). Now we prove that \((1,1)\) is the unique critical point of \(\Psi_{w}\) with positive coordinates. In fact, we may suppose that \((s_{0},t_{0})\) is also a critical point of \(\Psi_{w}\) with \(0< t_{0}\leqslant s_{0}\). Then it follows from (2.19) and (2.22) that
$$\begin{aligned}& s_{0}^{p}\bigl\Vert w^{+}\bigr\Vert _{1}^{p}+s_{0}^{q}\bigl\Vert w^{+}\bigr\Vert _{2}^{q}+cA_{1,w}s_{0}^{2p}+cA_{3,w}s_{0}^{p}t_{0}^{p}+dB_{1,w}s_{0}^{2q}+dB_{3,w}s_{0}^{q}t_{0}^{q} \\& \quad = \int f\bigl(s_{0}w^{+}\bigr)s_{0}w^{+}, \end{aligned}$$
(2.24)
$$\begin{aligned}& t_{0}^{p}\bigl\Vert w^{-}\bigr\Vert _{1}^{p}+t_{0}^{q}\bigl\Vert w^{-}\bigr\Vert _{2}^{q}+cA_{2,w}t_{0}^{2p}+cA_{3,w}s_{0}^{p}t_{0}^{p}+dB_{2,w}t_{0}^{2q}+dB_{3,w}s_{0}^{q}t_{0}^{q} \\& \quad = \int f\bigl(t_{0}w^{-}\bigr)t_{0}w^{-}. \end{aligned}$$
(2.25)
From (2.24) and \(t_{0}\leqslant s_{0}\), we have
$$ s_{0}^{p}\bigl\Vert w^{+}\bigr\Vert _{1}^{p}+s_{0}^{q}\bigl\Vert w^{+}\bigr\Vert _{2}^{q}+c(A_{1,w}+A_{3,w})s_{0}^{2p}+d(B_{1,w}+B_{3,w})s_{0}^{2q} \geqslant \int f\bigl(s_{0}w^{+}\bigr)s_{0}w^{+}. $$
(2.26)
On the other hand, since \(w\in\mathcal{M}\), we have
$$ \bigl\Vert w^{+}\bigr\Vert _{1}^{p}+\bigl\Vert w^{+}\bigr\Vert _{2}^{q}+cA_{1,w}+cA_{3,w}+dB_{1,w}+dB_{3,w}= \int f\bigl(w^{+}\bigr)w^{+}. $$
(2.27)
Hence, from (2.26) and (2.27), we get
$$\begin{aligned}& \biggl(1-\frac{1}{s_{0}^{p}} \biggr)\bigl\Vert w^{+}\bigr\Vert _{1}^{p}+ \biggl(1-\frac{1}{s_{0}^{2p-q}} \biggr)\bigl\Vert w^{+} \bigr\Vert _{2}^{q}+ \biggl(1-\frac{1}{s_{0}^{2p-2q}} \biggr) d(B_{1,w}+B_{3,w}) \\& \quad \leqslant \int \biggl[f\bigl(w^{+}\bigr)w^{+}-\frac {f(s_{0}w^{+})w^{+}}{s_{0}^{2p-1}} \biggr]. \end{aligned}$$
From the above inequality and Remark 2.2 we conclude that \(s_{0}\leqslant1\) and then \(0< t_{0}\leqslant s_{0}\leqslant1\).
Now we prove that \(t_{0}\geqslant1\). In fact, from (2.25) and \(0< t_{0}\leqslant s_{0}\), we have
$$ t_{0}^{p}\bigl\Vert w^{-}\bigr\Vert _{1}^{p}+t_{0}^{q}\bigl\Vert w^{-}\bigr\Vert _{2}^{q}+c(A_{2,w}+A_{3,w})t_{0}^{2p}+d(B_{2,w}+B_{3,w})t_{0}^{2q} \leqslant \int f\bigl(t_{0}w^{-}\bigr)t_{0}w^{-}. $$
(2.28)
On the other hand, since \(w\in\mathcal{M}\), we get
$$ \bigl\Vert w^{-}\bigr\Vert _{1}^{p}+\bigl\Vert w^{-}\bigr\Vert _{2}^{q}+cA_{2,w}+cA_{3,w}+dB_{2,w}+dB_{3,w}= \int f\bigl(w^{-}\bigr)w^{-}. $$
(2.29)
Now from (2.28) and (2.29), we obtain
$$\begin{aligned}& \biggl(1-\frac{1}{t_{0}^{p}} \biggr)\bigl\Vert w^{-}\bigr\Vert _{1}^{p}+ \biggl(1-\frac{1}{t_{0}^{2p-q}} \biggr)\bigl\Vert w^{-} \bigr\Vert _{2}^{q}+ \biggl(1-\frac{1}{t_{0}^{2p-2q}} \biggr)d(B_{2,w}+B_{3,w}) \\& \quad \geqslant \int \biggl[f\bigl(w^{-}\bigr)w^{-}-\frac {f(t_{0}w^{-})w^{-}}{t_{0}^{2p-1}} \biggr]. \end{aligned}$$
By Remark 2.2, we conclude that \(t_{0}\geqslant1\). Consequently, \(t_{0}=s_{0}=1\), this shows that \((1,1)\) is the unique critical point of \(\Psi_{w}\) with positive coordinates.
Now we assume that \(u\in W\) with \(u^{\pm}\ne0\) and \((s_{1},t_{1})\), \((s_{2},t_{2})\) are both critical points with positive coordinates for the map \(\Psi_{u}\). Then
$$ w_{1}=s_{1}u^{+}+t_{1}u^{-}\in{\mathcal{M}},\qquad w_{2}=s_{2}u^{+}+t_{2}u^{-}\in\mathcal{M}. $$
Therefore,
$$ w_{2}= \biggl(\frac{s_{2}}{s_{1}} \biggr)s_{1}u^{+}+ \biggl( \frac {t_{2}}{t_{1}} \biggr)t_{1}u^{-} = \biggl(\frac{s_{2}}{s_{1}} \biggr)w_{1}^{+}+ \biggl( \frac{t_{2}}{t_{1}} \biggr)w_{1}^{-}\in\mathcal{M}. $$
Since \(w_{1}\in\mathcal{M}\) and \((\frac{s_{2}}{s_{1}},\frac {t_{2}}{t_{1}} )\) is a critical point of the map \(\Psi_{w_{1}}\) with positive coordinates, by the uniqueness we have
$$ \frac{s_{2}}{s_{1}}=\frac{t_{2}}{t_{1}}=1, $$
which implies that \((s_{1},t_{1})=(s_{2},t_{2})\). □

Lemma 2.4

For a fixed \(u\in W\) with \(u^{\pm}\ne0\), the vector \((s_{u},t_{u})\), which was obtained in Lemma  2.3, is the unique maximum point of the function \(\Psi_{u}(s,t)\).

Proof

From the proof of Lemma 2.3, \((s_{u},t_{u})\) is the unique critical point of \(\Psi_{u}\) in \((0,\infty)\times(0,\infty)\). By (2.15), we have
$$\begin{aligned} \Psi_{u}(s,t) =&\bigl(s^{2p}+t^{2p}\bigr) \biggl[ \frac{1}{p}\frac{s^{p}}{s^{2p}+t^{2p}}\bigl\Vert u^{+}\bigr\Vert _{1}^{p}+\frac{1}{q}\frac{s^{q}}{s^{2p}+t^{2p}}\bigl\Vert u^{+} \bigr\Vert _{2}^{q} +\frac{1}{p}\frac{t^{p}}{s^{2p}+t^{2p}} \bigl\Vert u^{-}\bigr\Vert _{1}^{p} \\ &{}+\frac{1}{q} \frac {t^{q}}{s^{2p}+t^{2p}}\bigl\Vert u^{-}\bigr\Vert _{2}^{q} \biggr] \\ &{}+\bigl(s^{2p}+t^{2p}\bigr) \biggl[\frac{d}{2q}B_{1,u} \frac {s^{2q}}{s^{2p}+t^{2p}}+\frac{d}{q}B_{3,u}\frac {s^{q}t^{q}}{s^{2p}+t^{2p}}+ \frac{d}{2q}B_{2,u}\frac {t^{2q}}{s^{2p}+t^{2p}} \biggr] \\ &{}+\bigl(s^{2p}+t^{2p}\bigr) \biggl[\frac{c}{2p}A_{1,u} \frac {s^{2p}}{s^{2p}+t^{2p}}+\frac{c}{p}A_{3,u}\frac {s^{p}t^{p}}{s^{2p}+t^{2p}}+ \frac{c}{2p}A_{2,u}\frac {t^{2p}}{s^{2p}+t^{2p}} \biggr] \\ &{}-\bigl(s^{2p}+t^{2p}\bigr) \int\frac{F(su^{+})+F(tu^{-})}{s^{2p}+t^{2p}} \\ :=&\bigl(s^{2p}+t^{2p}\bigr) \biggl[\Psi_{1}(s,t)+ \Psi_{2}(s,t)+\Psi_{3}(s,t)- \int \frac{F(su^{+})+F(tu^{-})}{s^{2p}+t^{2p}} \biggr]. \end{aligned}$$
It is clear that \(\Psi_{1}(s,t),\Psi_{2}(s,t)\to0\) as \(|(s,t)|\to\infty \) and \(\Psi_{3}(s,t)\) is bounded. Then, by Lemma 2.1(iv), we deduce that \(\Psi_{u}(s,t)\to-\infty\) as \(|(s,t)|\to\infty\). So it is sufficient to check that a maximum point cannot be obtained on the boundary of \(\mathbb{R}_{+}\times\mathbb{R}_{+}\). Without loss of generality, we may assume that \((0,\bar{t})\) is a maximum point of \(\Psi_{u}\). Similar to (2.16), we can get \(\frac{\partial }{\partial s}\Psi_{u}(s,\bar{t})>0\) for s small. Then \(\Psi_{u}(s,\bar {t})\) is an increasing function with respect to s if s is small enough, the pair \((0,\bar{t})\) is not a maximum point of \(\Psi_{u}\) in \(\mathbb{R}_{+}\times\mathbb{R}_{+}\). □

Lemma 2.5

Let (f1)-(f4) hold. Suppose that \(u\in W\) with \(u^{\pm}\ne0\) such that \(\langle I'(u),u^{+}\rangle\leqslant0\), \(\langle I'(u),u^{-}\rangle \leqslant0\). Then the unique pair \((s_{u},t_{u})\) of positive numbers obtained in Lemma  2.3 satisfies \(0< s_{u},t_{u}\leqslant1\).

Proof

We may suppose that \(s_{u}\geqslant t_{u}>0\). Since \(s_{u}u^{+}+t_{u}u^{-}\in \mathcal{M}\),
$$\begin{aligned} \begin{aligned}[b] &s_{u}^{p}\bigl\Vert u^{+}\bigr\Vert _{1}^{p}+s_{u}^{q}\bigl\Vert u^{+}\bigr\Vert _{2}^{q}+cA_{1,u}s_{u}^{2p}+cA_{3,u}s_{u}^{2p}+dB_{1,u}s_{u}^{2q}+dB_{3,u}s_{u}^{2q} \\ &\quad \geqslant s_{u}^{p}\bigl\Vert u^{+}\bigr\Vert _{1}^{p}+s_{u}^{q}\bigl\Vert u^{+}\bigr\Vert _{2}^{q}+cA_{1,u}s_{u}^{2p}+cA_{3,u}s_{u}^{p}t_{u}^{p}+dB_{1,u}s_{u}^{2q}+dB_{3,u}s_{u}^{q}t_{u}^{q} \\ &\quad = \int f\bigl(s_{u}u^{+}\bigr)s_{u}u^{+}. \end{aligned} \end{aligned}$$
(2.30)
The assumption \(\langle I'(u),u^{+}\rangle\leqslant0\) gives
$$ \bigl\Vert u^{+}\bigr\Vert _{1}^{p}+\bigl\Vert u^{+}\bigr\Vert _{2}^{q}+cA_{1,u}+cA_{3,u}+dB_{1,u}+dB_{3,u} \leqslant \int f\bigl(u^{+}\bigr)u^{+}. $$
(2.31)
Combining (2.30) and (2.31), we get
$$\begin{aligned}& \biggl(\frac{1}{s_{u}^{p}}-1 \biggr)\bigl\Vert u^{+}\bigr\Vert _{1}^{p}+ \biggl(\frac{1}{s_{u}^{2p-q}}-1 \biggr)\bigl\Vert u^{+} \bigr\Vert _{2}^{q}+ \biggl(\frac{1}{s_{u}^{2p-2q}}-1 \biggr) (dB_{1,u}+dB_{3,u}) \\& \quad \geqslant \int \biggl[\frac {f(s_{u}u^{+})u^{+}}{s_{u}^{2p-1}}-f\bigl(u^{+}\bigr)u^{+} \biggr]. \end{aligned}$$
If \(s_{u}>1\), then the left side of this inequality is negative. But by Remark 2.2, the right side is positive, thus we must have \(s_{u}\leqslant1\). Then the proof is completed. □

From Lemma 2.3 and \(\mathcal{M}\subset\mathcal{N}\), we know that \(\mathcal{N}\) is nonempty and m, is well defined. Now we prove the following lemma.

Lemma 2.6

Assume that (f1)-(f4) hold. If \(v\in W\) with \(v\ne0\), then there is a unique \(s_{v}\in(0,\infty)\) such that \(s_{v}v\in{\mathcal{N}}\). Moreover, if \(\langle I'(v),v\rangle\leqslant0\), then \(s_{v}\in(0,1]\).

Proof

For fixed \(v\in W\) with \(v\ne0\) and \(s\in(0,\infty)\), \(sv\in \mathcal{N}\) if and only if \(\langle I'(sv),sv\rangle=0\), where
$$\begin{aligned} \bigl\langle I'(sv),sv\bigr\rangle =& s^{q} \biggl[s^{p-q}\|v \|_{1}^{p}+\|v\|_{2}^{q}+cs^{2p-q} \biggl( \int|\nabla v|^{p} \biggr)^{2} \\ &{}+ds^{q} \biggl( \int|\nabla v|^{q} \biggr)^{2}- \int\frac {f(sv)v}{s^{q-1}} \biggr] \end{aligned}$$
(2.32)
$$\begin{aligned} =& s^{2p} \biggl[\frac{1}{s^{p}}\|v \|_{1}^{p}+\frac{1}{s^{2p-q}}\|v\|_{2}^{q}+c \biggl( \int|\nabla v|^{p} \biggr)^{2} \\ &{}+\frac{d}{s^{2p-2q}} \biggl( \int|\nabla v|^{q} \biggr)^{2}- \int\frac{f(sv)v}{s^{2p-1}} \biggr]. \end{aligned}$$
(2.33)
Since \(v\ne0\), it follows from (2.32) and Lemma 2.1 (i) that \(\langle I'(sv),sv\rangle>0\) for \(s>0\) small. On the other hand, it follows from (2.33) and Lemma 2.1(ii) that \(\langle I'(sv),sv\rangle<0\) for \(s>0\) large. Thus there exists \(s_{v}>0\) such that \(\langle I'(s_{v}v),s_{v}v\rangle=0\).
Now we prove the uniqueness of \(s_{v}\). Suppose that there exist \(s_{1}\), \(s_{2}\) with \(0< s_{1}< s_{2}\) such that \(\langle I'(s_{1}v),s_{1}v\rangle =\langle I'(s_{2}v),s_{2}v\rangle=0\). Then (2.33) implies that
$$ \frac{1}{s_{i}^{p}}\|v\|_{1}^{p}+\frac{1}{s_{i}^{2p-q}}\|v \|_{2}^{q}+c \biggl( \int |\nabla v|^{p} \biggr)^{2}+\frac{d}{s_{i}^{2p-2q}} \biggl( \int|\nabla v|^{q} \biggr)^{2}= \int\frac{f(s_{i}v)v}{s_{i}^{2p-1}}, \quad i=1,2. $$
Hence,
$$\begin{aligned}& \biggl(\frac{1}{s_{1}^{p}}-\frac{1}{s_{2}^{p}} \biggr)\|v\|_{1}^{p}+ \biggl(\frac{1}{s_{1}^{2p-q}}-\frac{1}{s_{2}^{2p-q}} \biggr)\|v\|_{2}^{q}+ \biggl(\frac{d}{s_{1}^{2p-2q}}-\frac{d}{s_{2}^{2p-2q}} \biggr) \biggl( \int|\nabla v|^{q} \biggr)^{2} \\& \quad = \int \biggl[\frac{f(s_{1}v)v}{s_{1}^{2p-1}}-\frac {f(s_{2}v)v}{s_{2}^{2p-1}} \biggr]. \end{aligned}$$
Similar to (2.18), it is absurd in view of Remark 2.2.
Now we claim that \(s_{v}\in(0,1]\). It follows from (2.33) that
$$ \frac{1}{s_{v}^{p}}\|v\|_{1}^{p}+ \frac{1}{s_{v}^{2p-q}}\|v\|_{2}^{q}+c \biggl( \int |\nabla v|^{p} \biggr)^{2}+\frac{d}{s_{v}^{2p-2q}} \biggl( \int|\nabla v|^{q} \biggr)^{2}= \int\frac{f(s_{v}v)v}{s_{v}^{2p-1}}. $$
(2.34)
The assumption \(\langle I'(v),v\rangle\leqslant0\) gives
$$ \|v\|_{1}^{p}+\|v\|_{2}^{q}+c \biggl( \int|\nabla v|^{p} \biggr)^{2}+d \biggl( \int |\nabla v|^{q} \biggr)^{2}\leqslant \int f(v) v. $$
(2.35)
Combining (2.34) and (2.35), we have
$$ \biggl(1-\frac{1}{s_{v}^{p}} \biggr)\|v\|_{1}^{p}+ \biggl(1- \frac{1}{s_{v}^{2p-q}} \biggr)\|v\|_{2}^{q}+ \biggl(1- \frac{1}{{s_{v}}^{2p-2q}} \biggr)d \biggl( \int|\nabla v|^{q} \biggr)^{2} \leqslant \int \biggl[f(v)v-\frac {f(s_{v}v)v}{s_{v}^{2p-1}} \biggr]. $$
According to Remark 2.2, it is absurd if \(s_{v}>1\). Thus \(s_{v}\in(0,1]\). □

Lemma 2.7

Assume that (f1)-(f4) hold. Then \(m\geqslant\tilde{m}>0\), and m, can both be obtained.

Proof

(i) For any given \(\varepsilon>0\), by (2.1), we have
$$ \int f(u)u\leqslant\varepsilon \int|u|^{q}+C_{\varepsilon}\int|u|^{r},\quad u\in W. $$
(2.36)
Hence, for some \(\varepsilon>0\) small, by the continuous embedding of \(W_{p,a,h}\hookrightarrow L^{r}(\mathbb{R}^{N})\) and \(W_{q,b,g}\hookrightarrow L^{q}(\mathbb{R}^{N})\), we get
$$ \int f(u)u\leqslant\frac{1}{2}\|u\|_{2}^{q}+C \|u\|_{1}^{r}, \quad u\in W. $$
(2.37)
For every \(v\in\mathcal{N}\), we have \(\langle I'(v),v\rangle=0\), that is,
$$ \|v\|_{1}^{p}+\|v\|_{2}^{q}+c \biggl( \int|\nabla v|^{p} \biggr)^{2}+d \biggl( \int |\nabla v|^{q} \biggr)^{2}= \int f(v)v. $$
(2.38)
For every \(u\in\mathcal{M}\), we have \(\langle I'(u),u^{\pm}\rangle =0\), that is,
$$ \bigl\Vert u^{\pm}\bigr\Vert _{1}^{p}+ \bigl\Vert u^{\pm}\bigr\Vert _{2}^{q}+c \int \vert \nabla u\vert ^{p} \int\bigl\vert \nabla u^{\pm}\bigr\vert ^{p}+d \int \vert \nabla u\vert ^{q} \int\bigl\vert \nabla u^{\pm}\bigr\vert ^{q}= \int f\bigl(u^{\pm}\bigr)u^{\pm}. $$
(2.39)
Hence, for some \(\varepsilon>0\) small, it follows from (2.38), (2.39), (2.37), and (2.36) that
$$\begin{aligned}& \|w\|_{1}^{p}+\|w\|_{2}^{q} \leqslant\frac{1}{2}\|w\|_{2}^{q}+C\|w \|_{1}^{r}, \quad w=v,u^{\pm}, \end{aligned}$$
(2.40)
$$\begin{aligned}& \|w\|_{1}^{p}+\|w\|_{2}^{q} \leqslant\varepsilon \int|w|^{q}+C_{\varepsilon}\int |w|^{r}, \quad w=v,u^{\pm}. \end{aligned}$$
(2.41)
So, by (2.40), there exists a constant \(\alpha>0\), which is not dependent on c, d, such that
$$ \|w\|_{1}\geqslant\alpha, \quad w=v,u^{\pm}, $$
(2.42)
and then \(\|v\|_{1}\geqslant\alpha\), \(\|u^{\pm}\|_{1}\geqslant\alpha\).
By the condition (f4) and \(f\in C^{1}(\mathbb{R},\mathbb{R})\), we have
$$ f'(s)s^{2}-(2p-1)f(s)s\geqslant0,\quad s \in\mathbb{R}. $$
(2.43)
By (2.43), we have
$$ f(s)s-2pF(s)\geqslant0,\quad s\in\mathbb{R}. $$
(2.44)
Then
$$\begin{aligned} I(u) =&I(u)-\frac{1}{2p}\bigl\langle I'(u),u\bigr\rangle \\ =&\frac{1}{2p}\|u\|_{1}^{p}+ \biggl( \frac{1}{q}-\frac{1}{2p} \biggr)\|u\| _{2}^{q}+ \biggl(\frac{1}{2q}-\frac{1}{2p} \biggr)d \biggl( \int|\nabla u|^{q} \biggr)^{2} \\ &{}+\frac{1}{2p} \int \bigl[f(u)u-2pF(u) \bigr] \\ >&\frac{1}{2p}\|u\|_{1}^{p}+ \biggl( \frac{1}{q}-\frac{1}{2p} \biggr)\|u\| _{2}^{q} \geqslant\frac{1}{2p}\alpha^{p}. \end{aligned}$$
(2.45)
This implies that \(\tilde{m}\geqslant\alpha^{p}/(2p)\).

(ii) Let \(\{v_{n}\}\subset\mathcal{N}\) such that \(I(v_{n})\to\tilde {m}\). Then it follows from (2.45) that \(\{v_{n}\}\) is bounded in W and there exists \(v\in W\) such that \(v_{n}\rightharpoonup v\) in W.

Let \(\{u_{n}\}\subset\mathcal{M}\subset{\mathcal{N}}\) such that \(I(u_{n})\to m\). Then it follows from (2.45) that \(\{u_{n}\}\) is bounded in W and there exists \(u\in W\) such that \(u_{n}^{\pm}\rightharpoonup u^{\pm}\) in W (see Lemma A.1).

Since \(v_{n}\in{\mathcal{N}}\), \(u_{n}\in{\mathcal{M}}\), it follows from (2.41) that
$$ \alpha^{p}\leqslant\|w_{n}\|_{1}^{p} \leqslant\varepsilon \int |w_{n}|^{q}+C_{\varepsilon}\int|w_{n}|^{r},\quad w_{n}=v_{n},u_{n}^{\pm}. $$
Using the boundedness of \(\{w_{n}\}\), there is \(C_{1}>0\) such that
$$ \alpha^{p}\leqslant\varepsilon C_{1}+C_{\varepsilon}\int|w_{n}|^{r}. $$
Choosing \(\varepsilon=\alpha^{p}/(2C_{1})\), we get
$$ \int|w_{n}|^{r}\geqslant\frac{\alpha^{p}}{2C_{2}}, $$
where \(C_{2}=C_{\varepsilon}\). By the compactness of the embedding \(W\hookrightarrow L^{r}(\mathbb{R}^{N})\), we get
$$ \int|w|^{r}\geqslant\frac{\alpha^{p}}{2C_{2}},\quad w=v,u^{\pm}. $$
(2.46)
Thus \(w\ne0\). Equations (2.1) and (2.2) combined with the Lebesgue dominated convergence theorem give
$$ \lim_{n\to\infty} \int f(w_{n})w_{n}= \int f(w)w,\qquad \lim_{n\to\infty } \int F(w_{n})= \int F(w). $$
(2.47)
(iii) By the weak lower semi-continuity of the norm, we have
$$\begin{aligned}& \|v\|_{1}^{p}+\|v\|_{2}^{q}+c \biggl( \int|\nabla v|^{p} \biggr)^{2}+d \biggl( \int |\nabla v|^{q} \biggr)^{2} \\& \quad \leqslant \liminf_{n\to\infty} \biggl\{ \|v_{n} \|_{1}^{p}+\|v_{n}\| _{2}^{q}+c \biggl( \int|\nabla v_{n}|^{p} \biggr)^{2}+d \biggl( \int|\nabla v_{n}|^{q} \biggr)^{2} \biggr\} . \end{aligned}$$
Then from (2.47) we get
$$ \|v\|_{1}^{p}+\|v\|_{2}^{q}+c \biggl( \int|\nabla v|^{p} \biggr)^{2}+d \biggl( \int |\nabla v|^{q} \biggr)^{2}\leqslant \int f(v)v. $$
(2.48)
From (2.48) and Lemma 2.6, there exists \(s_{v}\in (0,1]\) such that \(\bar{v}=s_{v}v\in\mathcal{N}\).
It follows from (2.43) that \(G(s)=f(s)s-2pF(s)\) is a non-negative function, increasing on \([0,\infty)\), and decreasing on \((-\infty,0]\). Then we have
$$\begin{aligned} \tilde{m} \leqslant&I(\bar{v})-\frac{1}{2p}\bigl\langle I'(\bar{v}),\bar{v}\bigr\rangle \\ =&\frac{1}{2p}\|\bar{v}\|_{1}^{p}+ \biggl( \frac{1}{q}-\frac{1}{2p} \biggr)\| \bar{v}\|_{2}^{q}+ \biggl(\frac{1}{2q}-\frac{1}{2p} \biggr)d \biggl( \int|\nabla\bar{v}|^{q} \biggr)^{2} \\ &{}+ \frac{1}{2p} \int\bigl[f(\bar {v})\bar{v}-2pF(\bar{v})\bigr] \\ =&\frac{1}{2p}s_{v}^{p}\|v\|_{1}^{p}+ \biggl(\frac{1}{q}-\frac{1}{2p} \biggr)s_{v}^{q} \| v\|_{2}^{q}+ \biggl(\frac{1}{2q} - \frac{1}{2p} \biggr) ds_{v}^{2q} \biggl( \int|\nabla v|^{q} \biggr)^{2} \\ &{}+\frac{1}{2p} \int \bigl[f(s_{v}v)s_{v}v-2pF(s_{v}v) \bigr] \\ \leqslant&\frac{1}{2p}\|v\|_{1}^{p}+ \biggl( \frac{1}{q}-\frac{1}{2p} \biggr)\| v\|_{2}^{q}+ \biggl(\frac{1}{2q}-\frac{1}{2p} \biggr)d \biggl( \int|\nabla v|^{q} \biggr)^{2} \\ &{}+\frac{1}{2p} \int \bigl[f(v)v-2pF(v) \bigr] \\ \leqslant&\liminf_{n\to\infty} \biggl\{ I (v_{n} )- \frac{1}{2p}\bigl\langle I' (v_{n} ),v_{n}\bigr\rangle \biggr\} =\tilde{m}. \end{aligned}$$
We then deduce that \(s_{v}=1\). Thus, \(\bar{v}=v\) and \(I(v)=\tilde{m}\).
(iv) By the weak lower semi-continuity of the norm, we have
$$\begin{aligned}& \bigl\Vert u^{\pm}\bigr\Vert _{1}^{p}+\bigl\Vert u^{\pm}\bigr\Vert _{2}^{q}+c \int|\nabla u|^{p} \int\bigl\vert \nabla u^{\pm}\bigr\vert ^{p}+d \int|\nabla u|^{q} \int\bigl\vert \nabla u^{\pm}\bigr\vert ^{q} \\& \quad \leqslant \liminf_{n\to\infty} \biggl\{ \bigl\Vert u_{n}^{\pm}\bigr\Vert _{1}^{p}+\bigl\Vert u_{n}^{\pm}\bigr\Vert _{2}^{q}+c \int|\nabla u_{n}|^{p} \int\bigl\vert \nabla u_{n}^{\pm}\bigr\vert ^{p}+d \int|\nabla u_{n}|^{q} \int\bigl\vert \nabla u_{n}^{\pm}\bigr\vert ^{q} \biggr\} . \end{aligned}$$
Then from (2.47) we get
$$ \bigl\Vert u^{\pm}\bigr\Vert _{1}^{p}+ \bigl\Vert u^{\pm}\bigr\Vert _{2}^{q}+c \int \vert \nabla u\vert ^{p} \int\bigl\vert \nabla u^{\pm}\bigr\vert ^{p}+d \int \vert \nabla u\vert ^{q} \int\bigl\vert \nabla u^{\pm}\bigr\vert ^{q} \leqslant \int f\bigl(u^{\pm}\bigr)u^{\pm}. $$
(2.49)
From (2.49) and Lemma 2.5, there exists \((s_{u},t_{u})\in (0,1]\times(0,1]\) such that
$$ \bar{u}=s_{u}u^{+}+t_{u}u^{-}\in\mathcal{M}. $$
Note that \(G(s)=f(s)s-2pF(s)\) is a non-negative function, increasing on \([0,\infty)\) and decreasing on \((-\infty,0]\). Then we have
$$\begin{aligned} m \leqslant&I(\bar{u})-\frac{1}{2p}\bigl\langle I'( \bar{u}),\bar{u}\bigr\rangle \\ =&\frac{1}{2p}\Vert \bar{u}\Vert _{1}^{p}+ \biggl(\frac{1}{q}-\frac{1}{2p} \biggr)\Vert \bar{u}\Vert _{2}^{q}+ \biggl(\frac{1}{2q}-\frac{1}{2p} \biggr)d \biggl( \int|\nabla\bar{u}|^{q} \biggr)^{2}+ \frac{1}{2p} \int\bigl[f(\bar {u})\bar{u}-2pF(\bar{u})\bigr] \\ =&\frac{1}{2p}s_{u}^{p}\bigl\Vert u^{+}\bigr\Vert _{1}^{p}+ \biggl(\frac{1}{q}-\frac{1}{2p} \biggr)s_{u}^{q}\bigl\Vert u^{+}\bigr\Vert _{2}^{q}+ \biggl(\frac{1}{2q} -\frac{1}{2p} \biggr) ds_{u}^{2q} \biggl( \int\bigl\vert \nabla u^{+}\bigr\vert ^{q} \biggr)^{2} \\ &+\frac{1}{2p} \int \bigl[f\bigl(s_{u}u^{+}\bigr)s_{u}u^{+}-2pF \bigl(s_{u}u^{+}\bigr) \bigr]+ \biggl(\frac{1}{q}-\frac{1}{p} \biggr) ds_{u}^{q}t_{u}^{q} \int\bigl\vert \nabla u^{+}\bigr\vert ^{q} \int\bigl|\nabla u^{-}\bigr|^{q} \\ &+\frac{1}{2p}t_{u}^{p}\bigl\Vert u^{-}\bigr\Vert _{1}^{p}+ \biggl(\frac{1}{q}-\frac{1}{2p} \biggr)t_{u}^{q}\bigl\Vert u^{-}\bigr\Vert _{2}^{q}+ \biggl(\frac{1}{2q} -\frac{1}{2p} \biggr) dt_{u}^{2q} \biggl( \int\bigl\vert \nabla u^{-}\bigr\vert ^{q} \biggr)^{2} \\ &+\frac{1}{2p} \int \bigl[f\bigl(t_{u}u^{-}\bigr)t_{u}u^{-}-2pF \bigl(t_{u}u^{-}\bigr) \bigr] \\ \leqslant&\frac{1}{2p}\Vert u\Vert _{1}^{p}+ \biggl(\frac{1}{q}-\frac{1}{2p} \biggr)\Vert u\Vert _{2}^{q}+ \biggl(\frac{1}{2q}-\frac{1}{2p} \biggr)d \biggl( \int|\nabla u|^{q} \biggr)^{2}+\frac{1}{2p} \int \bigl[f(u)u-2pF(u) \bigr] \\ \leqslant&\liminf_{n\to\infty} \biggl\{ I (u_{n} )- \frac{1}{2p}\bigl\langle I' (u_{n} ),u_{n}\bigr\rangle \biggr\} =m. \end{aligned}$$
We then deduce that \(s_{u}=t_{u}=1\). Thus, \(\bar{u}=u\) and \(I(u)=m\). □

3 Proof of the main results

The purpose of this section is to prove our main results. We start to prove that the minimizer u for the minimization problem (1.10) is indeed a sign-changing solution of (1.1), that is, \(I'(u)=0\).

Proof of Theorem 1.1

Using the quantitative deformation lemma and topological degree theory, we prove that \(I'(u)=0\).

It is clear that \(\langle I'(u),u^{+}\rangle=\langle I'(u),u^{-}\rangle =0\). It follows from Lemma 2.4 that, for \((s,t)\in\mathbb {R}_{+}\times\mathbb{R}_{+}\) and \((s,t)\ne(1,1)\),
$$ I\bigl(su^{+}+tu^{-}\bigr)< I\bigl(u^{+}+u^{-}\bigr)=m. $$
(3.1)
It follows from (2.46) that \(\int|u^{\pm}|^{r}\geqslant\alpha ^{p}/(2C_{2}):=\tau^{r}\). Then \(|u^{\pm}|_{r}\geqslant\tau\). We denote by \(\gamma_{r}\) the embedding constant of \(W\hookrightarrow L^{r}(\mathbb{R}^{N})\).
If \(I'(u)\ne0\), then there exist \(r_{0},\rho>0\) such that
$$ \bigl\Vert I'(v)\bigr\Vert \geqslant\rho, \qquad \Vert v-u\Vert \leqslant r_{0}. $$
(3.2)
Let \(\delta\in(0,\min\{\tau/(2\gamma_{r}),r_{0}/3\})\) and let \(\sigma \in(0,\min\{1/2,\delta/(2\|u\|_{1}),\delta/{(2\|u\|_{2})}\})\). Let \(D=(1-\sigma,1+\sigma)\times(1-\sigma,1+\sigma)\) and \(\varphi (s,t)=su^{+}+tu^{-}\) for all \((s,t)\in\overline{D}\). It follows from (3.1) that
$$ \bar{m}=\max_{(s,t)\in\partial D}I\bigl(\varphi(s,t)\bigr)< m. $$
(3.3)
Let \(\varepsilon=\min\{(m-\bar{m})/2,\rho\delta/8\}\) and \(S=B(u,\delta)\). Then it follows from (3.2) that
$$ \bigl\Vert I'(v)\bigr\Vert \geqslant8 \varepsilon/\delta,\quad v\in I^{-1}\bigl([m-2\varepsilon,m+2 \varepsilon]\bigr)\cap S_{2\delta}. $$
(3.4)
Applying (3.4) and Lemma 2.3 in [27], p.38, there exists a deformation \(\eta\in C([0,1]\times W,W)\) such that
(b1): 

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

(b2): 

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

(b3): 

\(\|\eta(1,v)-v\|\leqslant\delta\) for all \(v\in W\).

By Lemmas 2.4 and 2.7, 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
$$\begin{aligned} \bigl\Vert \varphi(s,t)-u\bigr\Vert _{1}^{p} =&\bigl\Vert su^{+}+tu^{-}-u^{+}-u^{-}\bigr\Vert _{1}^{p} \\ =&|s-1|^{p}\bigl\Vert u^{+}\bigr\Vert _{1}^{p}+|t-1|^{p} \bigl\Vert u^{-}\bigr\Vert _{1}^{p} \\ \leqslant&\sigma^{p}\Vert u\Vert _{1}^{p} \\ < &(\delta/2)^{p}, \end{aligned}$$
and similarly \(\|\varphi(s,t)-u\|_{2}^{q}<(\delta/2)^{q}\), we have \(\| \varphi(s,t)-u\|=\|\varphi(s,t)-u\|_{1}+\|\varphi(s,t)-u\|_{2}<\delta\). Thus \(\varphi(s,t)\in S\). By (b2), we have \(I(\eta(1,\varphi (s,t)))< m-\varepsilon\). Then it is clear that
$$ \max_{(s,t)\in\overline{D}}I\bigl(\eta\bigl(1,\varphi(s,t)\bigr) \bigr)\leqslant m-\varepsilon< m. $$
(3.5)
We will prove that \(\eta(1,\varphi(D))\cap{\mathcal{M}}\ne \emptyset\), which is a contradiction with (3.5). Therefore, \(I'(u)=0\), that is, u is a sign-changing solution for equation (1.1). In fact, on we also define \(\psi(s,t)=\eta(1,\varphi (s,t))\) and
$$\begin{aligned}& \Phi_{0}(s,t)= \bigl(\bigl\langle I'\bigl(\varphi(s,t) \bigr),su^{+}\bigr\rangle ,\bigl\langle I'\bigl(\varphi(s,t)\bigr),tu^{-} \bigr\rangle \bigr) \\& \hphantom{\Phi_{0}(s,t)} = \bigl(\bigl\langle I'\bigl(su^{+}+tu^{-}\bigr),su^{+} \bigr\rangle ,\bigl\langle I'\bigl(su^{+}+tu^{-}\bigr),tu^{-}\bigr\rangle \bigr), \\& \Phi_{1}(s,t)= \bigl(\bigl\langle I'\bigl(\psi(s,t) \bigr),\psi^{+}(s,t)\bigr\rangle ,\bigl\langle I'\bigl(\psi(s,t) \bigr),\psi^{-}(s,t)\bigr\rangle \bigr). \end{aligned}$$
Let
$$\begin{aligned}& P(s,t) = \bigl\langle I'\bigl(su^{+}+tu^{-}\bigr),su^{+}\bigr\rangle \\& \hphantom{P(s,t)} = s^{p}\bigl\Vert u^{+}\bigr\Vert _{1}^{p}+s^{q}\bigl\Vert u^{+}\bigr\Vert _{2}^{q}+cA_{1,u}s^{2p}+cA_{3,u}s^{p}t^{p} \\& \hphantom{P(s,t) ={}}{}+dB_{1,u}s^{2q} +dB_{3,u}s^{q}t^{q}- \int f\bigl(su^{+}\bigr)su^{+}, \\& Q(s,t) = \bigl\langle I'\bigl(su^{+}+tu^{-}\bigr),tu^{-}\bigr\rangle \\& \hphantom{Q(s,t)} = t^{p}\bigl\Vert u^{-}\bigr\Vert _{1}^{p}+t^{q}\bigl\Vert u^{-}\bigr\Vert _{2}^{q}+cA_{2,u}t^{2p}+cA_{3,u}s^{p}t^{p} \\& \hphantom{Q(s,t) ={}}{}+dB_{2,u}t^{2q} +dB_{3,u}s^{q}t^{q}- \int f\bigl(tu^{-}\bigr)tu^{-}. \end{aligned}$$
By direct calculation, we have
$$\begin{aligned}& \frac{\partial P(s,t)}{\partial s}\bigg|_{(1,1)} = (p-1)\bigl\Vert u^{+}\bigr\Vert _{1}^{p}+(q-1)\bigl\Vert u^{+}\bigr\Vert _{2}^{q} +(2p-1)cA_{1,u}+(p-1)cA_{3,u} \\& \hphantom{\frac{\partial P(s,t)}{\partial s}\bigg|_{(1,1)} ={}}{}+(2q-1) dB_{1,u}+(q-1) dB_{3,u}- \int f'\bigl(u^{+}\bigr)\bigl|u^{+}\bigr|^{2}, \\& \frac{\partial P(s,t)}{\partial t} \bigg|_{(1,1)}=pcA_{3,u}+q dB_{3,u}, \qquad \frac{\partial Q(s,t)}{\partial s}\bigg|_{(1,1)}=pcA_{3,u}+q dB_{3,u}, \\& \frac{\partial Q(s,t)}{\partial t}\bigg|_{(1,1)} = (p-1)\bigl\Vert u^{-}\bigr\Vert _{1}^{p}+(q-1)\bigl\Vert u^{-}\bigr\Vert _{2}^{q} +(2p-1)cA_{2,u}+(p-1)cA_{3,u} \\& \hphantom{\frac{\partial Q(s,t)}{\partial t}\bigg|_{(1,1)} ={}}{}+(2q-1) dB_{2,u}+(q-1) dB_{3,u}- \int f'\bigl(u^{-}\bigr)\bigl|u^{-}\bigr|^{2}. \end{aligned}$$
By (2.43), we get
$$ \frac{\partial P(s,t)}{\partial s}\bigg|_{(1,1)}< -(pcA_{3,u}+q dB_{3,u}) $$
and
$$ \frac{\partial Q(s,t)}{\partial t}\bigg|_{(1,1)}< -(pcA_{3,u}+q dB_{3,u}). $$
Set the matrix
$$ M=\left [ \textstyle\begin{array}{@{}c@{\quad}c@{}} \frac{\partial P(1,1)}{\partial s} & \frac{\partial P(1,1)}{\partial t} \\ \frac{\partial Q(1,1)}{\partial s} & \frac{\partial Q(1,1)}{\partial t} \end{array}\displaystyle \right ]. $$
Then we get
$$ J_{\Phi_{0}}(1,1)=\det M>0. $$
Since \(\Phi_{0}\) is \(C^{1}\), \((1,1)\) is the unique isolated zero of \(\Phi _{0}\), by Lemmas 1.5 and 1.6 in [28], p.112, we have
$$ \deg(\Phi_{0},D,0)=\operatorname{ind}\bigl(\Phi_{0},(1,1) \bigr)=\operatorname{sgn}J_{\Phi _{0}}(1,1)=1. $$
It follows from (3.3), \(\bar{m}< m-2\varepsilon\), and (b1) above that \(\varphi=\psi\) on ∂D. Hence \(\deg(\Phi _{1},D,0)=\deg(\Phi_{0},D,0)=1\). Thus there exists a pair \((s_{0},t_{0})\in D\) such that \(\Phi_{1}(s_{0},t_{0})=0\). Since \(|u^{\pm}|_{r}\geqslant\tau\), \((s_{0},t_{0})\in D\), we have \(|\varphi ^{+}(s_{0},t_{0})|_{r}=s_{0}|u^{+}|_{r}\geqslant\tau/2\) and \(|\varphi ^{-}(s_{0},t_{0})|_{r}=t_{0}|u^{-}|_{r}\geqslant\tau/2\). By (b3), we have
$$ \bigl\vert \psi(s_{0},t_{0})-\varphi(s_{0},t_{0}) \bigr\vert _{r}\leqslant\gamma_{r}\bigl\Vert \psi (s_{0},t_{0})-\varphi(s_{0},t_{0}) \bigr\Vert \leqslant\gamma_{r}\delta. $$
This implies that
$$ \bigl\vert \psi^{\pm}(s_{0},t_{0})- \varphi^{\pm}(s_{0},t_{0})\bigr\vert _{r}\leqslant\bigl\vert \psi (s_{0},t_{0})- \varphi(s_{0},t_{0})\bigr\vert _{r}\leqslant \gamma_{r}\delta. $$
Thus we have
$$ \bigl\vert \psi^{\pm}(s_{0},t_{0})\bigr\vert _{r}\geqslant\bigl\vert \varphi^{\pm}(s_{0},t_{0}) \bigr\vert _{r}-\gamma _{r}\delta\geqslant \frac{\tau}{2}-\gamma_{r}\delta>0. $$
That is, \(\psi^{\pm}(s_{0},t_{0})\ne0\). Then \(\eta(1,\varphi (s_{0},t_{0}))=\psi(s_{0},t_{0})\in{\mathcal{M}}\). □

Now, we are in a situation to prove Theorem 1.2. We first claim that equation (1.2) has one least energy sign-changing solution.

Remark 3.1

The proof of equation (1.1) is still suited to (1.2) where \(c=d=0\) although the proof for equation (1.2) may be easier than (1.1). So, if the assumptions (f1)-(f4) hold, there exists a least energy sign-changing solution of (1.2).

In the following, we regard \(c,d\geqslant0\) as parameters in equation (1.1). Let \(u_{c,d}\in W\) be the least energy sign-changing solution of (1.1) obtained in Theorem 1.1. The relative functional and constraint are denoted by \(I_{c,d}\) and \({\mathcal{M}}_{c,d}\), respectively. We will analyze the convergence property of \(u_{c,d}\) as \((c,d)\to0\).

Proof of Theorem 1.2

(1) We first of all claim that, for any sequence \(\{(c_{n},d_{n})\}\) with \((c_{n},d_{n})\to0\) as \(n\to \infty\), \(\{u_{c_{n},d_{n}}\}:=\{u_{n}\}\) is bounded in W. In fact, choosing a function \(\varphi\in C_{0}^{\infty}(\mathbb{R}^{N})\) with \(\varphi^{\pm}\ne0\), \(\int h|\varphi|^{p},\int g|\varphi|^{q}<\infty\). For any \(c,d\in[0,1]\), since \(c,d\leqslant1\), Lemma 2.1(ii) implies that there exists a pair \((\mu_{1},\mu_{2})\) of positive numbers, which does not depend on c, d, such that
$$ \textstyle\begin{cases} \mu_{1}^{p}\|\varphi^{+}\|_{1}^{p}+\mu_{1}^{q}\|\varphi^{+}\| _{2}^{q}+cA_{1,\varphi}\mu_{1}^{2p} +cA_{3,\varphi}\mu_{1}^{p}\mu_{2}^{p} \\ \quad {}+dB_{1,\varphi}\mu _{1}^{2q}+dB_{3,\varphi}\mu_{1}^{q}\mu_{2}^{q}-\int f(\mu_{1}\varphi^{+})\mu _{1}\varphi^{+}< 0, \\ \mu_{2}^{p}\|\varphi^{-}\|_{1}^{p}+\mu_{2}^{q}\|\varphi^{-}\|_{2}^{q}+cA_{2,\varphi }\mu_{2}^{2p} +cA_{3,\varphi}\mu_{1}^{p}\mu_{2}^{p} \\ \quad {}+dB_{2,\varphi}\mu _{2}^{2q}+dB_{3,\varphi}\mu_{1}^{q}\mu_{2}^{q}-\int f(\mu_{2}\varphi^{-})\mu _{2}\varphi^{-}< 0. \end{cases} $$
In view of Lemmas 2.3 and 2.5, for any \(c,d\in [0,1]\), there is a unique pair \((s_{\varphi}(c,d),t_{\varphi}(c,d))\in (0,1]\times(0,1]\) such that
$$ \bar{\varphi}=s_{\varphi}(c,d)\mu_{1}\varphi^{+}+t_{\varphi}(c,d) \mu _{2}\varphi^{-}\in{\mathcal{M}}_{c,d}. $$
Thus, for any \(c,d\in[0,1]\), combining (2.1) and (2.2), we have
$$\begin{aligned} I_{c,d}(u_{c,d}) \leqslant&I_{c,d}(\bar{\varphi})- \frac{1}{2p}\bigl\langle I'_{c,d}(\bar {\varphi}), \bar{\varphi}\bigr\rangle \\ =&\frac{1}{2p}\Vert \bar{\varphi} \Vert _{1}^{p}+ \biggl(\frac{1}{q}-\frac{1}{2p} \biggr)\Vert \bar{\varphi} \Vert _{2}^{q} + \biggl(\frac{1}{2q}-\frac{1}{2p} \biggr)d \biggl( \int|\nabla\bar{\varphi}|^{q} \biggr)^{2} \\ &{}+ \frac{1}{2p} \int \bigl[f(\bar{\varphi})\bar{\varphi}-2pF(\bar{\varphi} ) \bigr] \\ \leqslant&\frac{1}{2p}\Vert \bar{\varphi} \Vert _{1}^{p}+ \biggl(\frac{1}{q}-\frac{1}{2p} \biggr)\Vert \bar{\varphi} \Vert _{2}^{q} + \biggl(\frac{1}{2q}-\frac{1}{2p} \biggr)d \biggl( \int|\nabla\bar{\varphi}|^{q} \biggr)^{2} \\ &{}+ \frac{1}{2p} \int \bigl[C_{1}|\bar{\varphi}|^{q}+C_{2}| \bar{\varphi}|^{r} \bigr] \\ \leqslant&\frac{1}{2p}\mu_{1}^{p}\bigl\Vert \varphi^{+}\bigr\Vert _{1}^{p}+\frac{1}{2p} \mu_{2}^{p}\bigl\Vert \varphi^{-}\bigr\Vert _{1}^{p} + \biggl(\frac{1}{q}-\frac{1}{2p} \biggr)\mu_{1}^{q}\bigl\Vert \varphi^{+}\bigr\Vert _{2}^{q} + \biggl(\frac{1}{q}-\frac{1}{2p} \biggr)\mu_{2}^{q}\bigl\Vert \varphi^{-}\bigr\Vert _{2}^{q} \\ &{}+ \biggl(\frac{1}{2q}-\frac{1}{2p} \biggr) \bigl[B_{1,\varphi}\mu _{1}^{2q}+2B_{3,\varphi} \mu_{1}^{q}\mu_{2}^{q} +B_{2,\varphi} \mu _{2}^{2q} \bigr] \\ &{}+\frac{C_{1}}{2p} \int \bigl[\mu_{1}^{q}\bigl\vert \varphi^{+}\bigr\vert ^{q}+\mu_{2}^{q}\bigl\vert \varphi ^{-}\bigr\vert ^{q} \bigr] +\frac{C_{2}}{2p} \int \bigl[\mu_{1}^{r}\bigl\vert \varphi^{+}\bigr\vert ^{r}+\mu_{2}^{r}\bigl\vert \varphi ^{-}\bigr\vert ^{r} \bigr]:=C_{0}, \end{aligned}$$
where \(C_{0}\) does not depend on c, d. It follows from (2.45) that, for n large enough,
$$ C_{0}\geqslant I_{n}(u_{n})=I_{n}(u_{n})- \frac{1}{2p}\bigl\langle I'_{n}(u_{n}),u_{n} \bigr\rangle \geqslant\frac{1}{2p}\|u_{n}\|_{1}^{p}+ \biggl(\frac{1}{q}-\frac{1}{2p} \biggr)\| u_{n} \|_{2}^{q}, $$
where \(I_{n}\) denotes \(I_{c_{n},d_{n}}\). Then \(\{u_{n}\}\) is bounded in W.
(2) There exists a subsequence of \(\{(c_{n},d_{n})\}\), still denoted by \(\{ (c_{n},d_{n})\}\), such that \(u_{n}\rightharpoonup u_{0}\) in W. Now we will prove that \(u_{0}\) is a sign-changing solution of (1.2). Indeed, by (2.1), the Hölder inequality and the compactness of the embedding \(W\hookrightarrow L^{s}(\mathbb{R}^{N})\) for \(s=q,r\), we get
$$\begin{aligned}& \int\bigl\vert f(u_{n}) (u_{n}-u_{0}) \bigr\vert \\& \quad \leqslant \int \bigl[C_{1}|u_{n}|^{q-1}|u_{n}-u_{0}|+C_{2}|u_{n}|^{r-1}|u_{n}-u_{0}| \bigr] \\& \quad \leqslant C_{1}|u_{n}|_{q}^{q-1}|u_{n}-u_{0}|_{q}+C_{2}|u_{n}|_{r}^{r-1}|u_{n}-u_{0}|_{r} \to0 \end{aligned}$$
(3.6)
and
$$ \int\bigl\vert f(u_{0}) (u_{n}-u_{0}) \bigr\vert \to0. $$
(3.7)
We have
$$\begin{aligned}& \bigl\langle I'_{n}(u_{n})-I'_{0}(u_{0}),u_{n}-u_{0} \bigr\rangle \\& \quad = \int a\bigl(|\nabla u_{n}|^{p-2}\nabla u_{n}-|\nabla u_{0}|^{p-2}\nabla u_{0} \bigr)\cdot\nabla(u_{n}-u_{0}) \\& \qquad {}+ \int h\bigl(|u_{n}|^{p-2}u_{n}-|u_{0}|^{p-2}u_{0} \bigr) (u_{n}-u_{0}) \\& \qquad {} + \int b\bigl(|\nabla u_{n}|^{q-2}\nabla u_{n}-|\nabla u_{0}|^{q-2}\nabla u_{0} \bigr)\cdot\nabla(u_{n}-u_{0}) \\& \qquad {}+ \int g\bigl(|u_{n}|^{q-2}u_{n}-|u_{0}|^{q-2}u_{0} \bigr) (u_{n}-u_{0}) \\& \qquad {} +c_{n} \int|\nabla u_{n}|^{p} \int|\nabla u_{n}|^{p-2}\nabla u_{n}\cdot \nabla (u_{n}-u_{0}) \\& \qquad {}+d_{n} \int|\nabla u_{n}|^{q} \int|\nabla u_{n}|^{q-2}\nabla u_{n}\cdot \nabla(u_{n}-u_{0}) \\& \qquad {} - \int f(u_{n}) (u_{n}-u_{0})+ \int f(u_{0}) (u_{n}-u_{0}) \\& \quad \geqslant C_{1} \int a\bigl\vert \nabla(u_{n}-u_{0})\bigr\vert ^{p}+C_{2} \int h|u_{n}-u_{0}|^{p} \\& \qquad {}+C_{3} \int b\bigl\vert \nabla(u_{n}-u_{0})\bigr\vert ^{q}+C_{4} \int g|u_{n}-u_{0}|^{q} \\& \qquad {} +c_{n} \int|\nabla u_{n}|^{p} \int|\nabla u_{n}|^{p-2}\nabla u_{n}\cdot \nabla (u_{n}-u_{0}) \\& \qquad {}+d_{n} \int|\nabla u_{n}|^{q} \int|\nabla u_{n}|^{q-2}\nabla u_{n}\cdot \nabla(u_{n}-u_{0}) \\& \qquad {} - \int f(u_{n}) (u_{n}-u_{0})+ \int f(u_{0}) (u_{n}-u_{0}) \\& \quad \geqslant C\bigl(\|u_{n}-u_{0}\|_{1}^{p}+ \|u_{n}-u_{0}\|_{2}^{q} \bigr)+c_{n} \int|\nabla u_{n}|^{p} \int|\nabla u_{n}|^{p-2}\nabla u_{n}\cdot \nabla(u_{n}-u_{0}) \\& \qquad {} +d_{n} \int|\nabla u_{n}|^{q} \int|\nabla u_{n}|^{q-2}\nabla u_{n}\cdot \nabla (u_{n}-u_{0}) \\& \qquad {}- \int f(u_{n}) (u_{n}-u_{0})+ \int f(u_{0}) (u_{n}-u_{0}), \end{aligned}$$
where we have used the inequality \((|\xi|^{s-2}\xi-|\eta|^{s-2}\eta ,\xi-\eta)\geqslant C_{s}|\xi-\eta|^{s}\) for all \(\xi,\eta\in\mathbb {R}^{N}\) and \(s\geqslant2\) (see [29]). By (3.6) and (3.7), it follows that, passing to the limit on \(n\to\infty\), we deduce \(\|u_{n}-u_{0}\|_{1}^{p}+\|u_{n}-u_{0}\|_{2}^{q}\to0\) as \(n\to\infty\). Then \(u_{n}\to u_{0}\) in W. It follows from (2.42) that \(\|u_{n}^{\pm}\| _{1}\geqslant\alpha\), where α is not dependent on \(c_{n}\), \(d_{n}\). By Lemma A.2, we know that \(\|u_{0}^{\pm}\|_{1}\geqslant\alpha\). So \(u_{0}\) changes sign and \(u_{0}\) is a solution of (1.2).
(3) Now we prove that \(u_{0}\) is also a least energy sign-changing solution of (1.2). In fact, suppose that \(v_{0}\) is a least energy sign-changing solution of (1.2). Then by Lemma 2.3, for each \(c_{n},d_{n}\geqslant0\), there is a unique pair \((s_{n},t_{n})\) of positive numbers such that
$$ s_{n}v_{0}^{+}+t_{n}v_{0}^{-}\in{ \mathcal{M}}_{n}, $$
where \({\mathcal{M}}_{n}\) denotes \({\mathcal{M}}_{c_{n},d_{n}}\). Then we have
$$\begin{aligned}& s_{n}^{p}\bigl\| v_{0}^{+} \bigr\| _{1}^{p}+s_{n}^{q}\bigl\| v_{0}^{+} \bigr\| _{2}^{q}+c_{n}s_{n}^{2p} \biggl( \int\bigl\vert \nabla v_{0}^{+}\bigr\vert ^{p} \biggr)^{2}+d_{n}s_{n}^{2q} \biggl( \int\bigl\vert \nabla v_{0}^{+}\bigr\vert ^{q} \biggr)^{2} \\& \quad {}+c_{n}s_{n}^{p}t_{n}^{p} \int\bigl\vert \nabla v_{0}^{+}\bigr\vert ^{p} \int\bigl\vert \nabla v_{0}^{-}\bigr\vert ^{p}+d_{n}s_{n}^{q}t_{n}^{q} \int\bigl\vert \nabla v_{0}^{+}\bigr\vert ^{q} \int\bigl\vert \nabla v_{0}^{-}\bigr\vert ^{q}= \int f\bigl(s_{n}v_{0}^{+}\bigr)s_{n}v_{0}^{+} \end{aligned}$$
(3.8)
and
$$\begin{aligned}& t_{n}^{p}\bigl\| v_{0}^{-} \bigr\| _{1}^{p}+t_{n}^{q}\bigl\| v_{0}^{-} \bigr\| _{2}^{q}+c_{n}t_{n}^{2p} \biggl( \int\bigl\vert \nabla v_{0}^{-}\bigr\vert ^{p} \biggr)^{2}+d_{n}t_{n}^{2q} \biggl( \int\bigl\vert \nabla v_{0}^{-}\bigr\vert ^{q} \biggr)^{2} \\& \quad {}+c_{n}s_{n}^{p}t_{n}^{p} \int\bigl\vert \nabla v_{0}^{+}\bigr\vert ^{p} \int\bigl\vert \nabla v_{0}^{-}\bigr\vert ^{p}+d_{n}s_{n}^{q}t_{n}^{q} \int\bigl\vert \nabla v_{0}^{+}\bigr\vert ^{q} \int\bigl\vert \nabla v_{0}^{-}\bigr\vert ^{q}= \int f\bigl(t_{n}v_{0}^{-}\bigr)t_{n}v_{0}^{-}. \end{aligned}$$
(3.9)
We first claim that \((s_{n},t_{n})\) is bounded. Otherwise, we may assume there is a subsequence such that \(s_{n}\geqslant t_{n}\), \(s_{n}\to\infty\). Thus, by (3.8) and Lemma 2.1(ii), we get \(0=\infty \). This is a contradiction. Therefore, there are \(s_{0},t_{0}\geqslant0\) and a subsequence \((s_{n},t_{n})\) such that \((s_{n},t_{n})\to(s_{0},t_{0})\). If \(s_{0}=0\), by (3.8) and Lemma 2.1(i), we have
$$ \bigl\Vert v_{0}^{+}\bigr\Vert _{2}^{q}=\lim _{n\to\infty} \int\frac{f(s_{n}v_{0}^{+})v_{0}^{+}}{s_{n}^{q-1}}=0. $$
It is absurd in view of \(v_{0}^{+}\ne0\). Thus \(s_{0}\ne0\). We also get \(t_{0}\ne0\) by (3.9) in the same way.
Recall that \(v_{0}^{\pm}\) satisfies
$$ \bigl\Vert v_{0}^{\pm}\bigr\Vert _{1}^{p}+\bigl\Vert v_{0}^{\pm}\bigr\Vert _{2}^{q}= \int f\bigl(v_{0}^{\pm}\bigr)v_{0}^{\pm}. $$
(3.10)
Passing to the limit on \(n\to\infty\) in (3.8), we have
$$ \frac{1}{s_{0}^{p}}\bigl\Vert v_{0}^{+}\bigr\Vert _{1}^{p}+\frac{1}{s_{0}^{2p-q}}\bigl\Vert v_{0}^{+} \bigr\Vert _{2}^{q}= \int\frac {f(s_{0}v_{0}^{+})v_{0}^{+}}{s_{0}^{2p-1}}. $$
(3.11)
Combining (3.10) and (3.11) we get
$$ \biggl(\frac{1}{s_{0}^{p}}-1 \biggr)\bigl\Vert v_{0}^{+} \bigr\Vert _{1}^{p}+ \biggl(\frac{1}{s_{0}^{2p-q}}-1 \biggr) \bigl\Vert v_{0}^{+}\bigr\Vert _{2}^{q}= \int \biggl[\frac {f(s_{0}v_{0}^{+})v_{0}^{+}}{s_{0}^{2p-1}}- f\bigl(v_{0}^{+} \bigr)v_{0}^{+} \biggr]. $$
(3.12)
As follows from (3.12) and Remark 2.2, we just get \(s_{0}=1\). And \(t_{0}=1\) is similar available. That yields
$$ (s_{n},t_{n})\to(1,1), \quad n\to\infty. $$
(3.13)
Now, we can prove \(u_{0}\) is a least energy sign-changing solution of (1.2). In fact, from (3.13), we have
$$ I_{0}(v_{0})\leqslant I_{0}(u_{0})= \lim_{n\to\infty}I_{n}(u_{n})\leqslant\lim _{n\to\infty}I_{n}\bigl(s_{n}v_{0}^{+}+t_{n}v_{0}^{-} \bigr) =I_{0}\bigl(v_{0}^{+}+v_{0}^{-} \bigr)=I_{0}(v_{0}). $$
Thus, \(I_{0}(u_{0})=I_{0}(v_{0})\) and then \(u_{0}\) is a least energy sign-changing solution of (1.2). This completes the proof of Theorem 1.2. □

Proof of Theorem 1.3

(i) Let \(\mathcal{N}\) and be given by (1.11) and (1.12), respectively. We prove that the minimizer v for the minimization problem (1.12) is indeed a ground state solution of (1.1), that is, \(I'(v)=0\).

We define a functional \(H(u)=\langle I'(u),u\rangle\) for all \(u\in W\), that is,
$$ H(u)=\|u\|_{1}^{p}+\|u\|_{2}^{q}+c \biggl( \int|\nabla u|^{p} \biggr)^{2}+d \biggl( \int|\nabla u|^{q} \biggr)^{2}- \int f(u)u, \quad u\in W. $$
(3.14)
Since \(v\in{\mathcal{N}}\), \(I(v)=\tilde{m}\), there is a Lagrange multiplier \(l\in\mathbb{R}\) such that
$$ I'(v)-lH'(v)=0. $$
(3.15)
Hence, \(l\langle H'(v),v\rangle=\langle I'(v),v\rangle=0\). But it follows from (3.14), (2.43), and \(H(v)=0\) that
$$\begin{aligned} \bigl\langle H'(v),v\bigr\rangle =&p\|v\|_{1}^{p}+q \|v\|_{2}^{q}+2pc \biggl( \int|\nabla v|^{p} \biggr)^{2}+2qd \biggl( \int|\nabla v|^{q} \biggr)^{2}- \int \bigl[f'(v)v^{2}+f(v)v \bigr] \\ < &-p\|v\|_{1}^{p}-(2p-q)\|v\|_{2}^{q}-(2p-2q)d \biggl( \int|\nabla v|^{q} \biggr)^{2}< 0. \end{aligned}$$
Thus we have \(l=0\). Then (3.15) implies that \(I'(v)=0\). Therefore, v is a ground state solution of (1.1).
(ii) From Theorem 1.1, we know that equation (1.1) has a least energy sign-changing solution u. Let \(u=u^{+}+u^{-}\) with \(u^{\pm}\neq0\). Then, combining \(\langle I'(u^{\pm}),u^{\pm}\rangle <\langle I'(u),u^{\pm}\rangle=0\) and Lemma 2.6, there is a unique \(t_{u^{+}},s_{u^{-}}\in(0,1)\) such that \(t_{u^{+}}u^{+},s_{u^{-}}u^{-}\in \mathcal{N}\). Then it follows from the definition of that
$$\begin{aligned} \tilde{m} \leqslant&I\bigl(t_{u^{+}}u^{+}\bigr) \\ =&I\bigl(t_{u^{+}}u^{+}\bigr)-\frac{1}{2p}\bigl\langle I'\bigl(t_{u^{+}}u^{+}\bigr),t_{u^{+}}u^{+}\bigr\rangle \\ =&\frac{1}{2p}t_{u^{+}}^{p}\bigl\Vert u^{+} \bigr\Vert _{1}^{p}+ \biggl(\frac{1}{q}- \frac{1}{2p} \biggr)t_{u^{+}}^{q}\bigl\Vert u^{+}\bigr\Vert _{2}^{q}+ \biggl(\frac{1}{2q}- \frac{1}{2p} \biggr) dt_{u^{+}}^{2q} \biggl( \int\bigl\vert \nabla u^{+}\bigr\vert ^{q} \biggr)^{2} \\ &{}+\frac{1}{2p} \int \bigl[f\bigl(t_{u^{+}}u^{+}\bigr)t_{u^{+}}u^{+}-2pF \bigl(t_{u^{+}}u^{+}\bigr) \bigr] \\ < &\frac{1}{2p}\bigl\Vert u^{+}\bigr\Vert _{1}^{p}+ \biggl(\frac{1}{q}-\frac{1}{2p} \biggr)\bigl\Vert u^{+}\bigr\Vert _{2}^{q}+ \biggl( \frac{1}{2q}-\frac{1}{2p} \biggr)d \biggl( \int\bigl\vert \nabla u^{+}\bigr\vert ^{q} \biggr)^{2} \\ &{}+\frac{1}{2p} \int \bigl[f\bigl(u^{+}\bigr)u^{+}-2pF\bigl(u^{+}\bigr) \bigr]. \end{aligned}$$
(3.16)
Similarly, we have
$$\begin{aligned} \tilde{m} < &\frac{1}{2p}\bigl\Vert u^{-}\bigr\Vert _{1}^{p}+ \biggl(\frac{1}{q}-\frac{1}{2p} \biggr)\bigl\Vert u^{-}\bigr\Vert _{2}^{q}+ \biggl( \frac{1}{2q}-\frac{1}{2p} \biggr)d \biggl( \int\bigl\vert \nabla u^{-}\bigr\vert ^{q} \biggr)^{2} \\ &{}+\frac{1}{2p} \int \bigl[f\bigl(u^{-}\bigr)u^{-}-2pF\bigl(u^{-}\bigr) \bigr]. \end{aligned}$$
(3.17)
Combining (3.16) and (3.17), we have
$$\begin{aligned} 2\tilde{m} < &\frac{1}{2p}\bigl(\bigl\Vert u^{+}\bigr\Vert _{1}^{p}+\bigl\Vert u^{-}\bigr\Vert _{1}^{p} \bigr)+ \biggl(\frac{1}{q}-\frac{1}{2p} \biggr) \bigl(\bigl\Vert u^{+}\bigr\Vert _{2}^{q}+\bigl\Vert u^{-}\bigr\Vert _{2}^{q}\bigr) \\ &{}+ \biggl(\frac{1}{2q}- \frac{1}{2p} \biggr)d \biggl[ \biggl( \int\bigl\vert \nabla u^{+}\bigr\vert ^{q} \biggr)^{2}+ \biggl( \int\bigl\vert \nabla u^{-}\bigr\vert ^{q} \biggr)^{2} \biggr] \\ &{}+\frac{1}{2p} \int \bigl[f\bigl(u^{+}\bigr)u^{+}-2pF\bigl(u^{+}\bigr)+f\bigl(u^{-}\bigr)u^{-}-2pF \bigl(u^{-}\bigr) \bigr] \\ < &\frac{1}{2p}\Vert u\Vert _{1}^{p}+ \biggl( \frac{1}{q}-\frac{1}{2p} \biggr)\Vert u\Vert _{2}^{q}+ \biggl(\frac{1}{2q}-\frac{1}{2p} \biggr)d \biggl( \int|\nabla u|^{q} \biggr)^{2} \\ &{}+\frac{1}{2p} \int \bigl[f(u)u-2pF(u) \bigr] \\ =&I(u)u-\frac{1}{2p}\bigl\langle I'(u),u\bigr\rangle \\ =&I(u)=m. \end{aligned}$$
That is, \(m>2\tilde{m}\). This implies that cannot be obtained by a sign-changing solution. This completes the proof. □

Declarations

Acknowledgements

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

References

  1. Figueiredo, GM: Existence of positive solutions for a class of \(p\, \&\, q\) elliptic problems with critical growth on \(\mathbb{R}^{N}\). J. Math. Anal. Appl. 378(2), 507-518 (2011) MathSciNetView ArticleMATHGoogle Scholar
  2. He, C, Li, G: The regularity of weak solutions to nonlinear scalar field elliptic equations containing \(p\, \&\, q\)-Laplacians. Ann. Acad. Sci. Fenn., Math. 33(2), 337-371 (2008) MathSciNetMATHGoogle Scholar
  3. Li, G, Liang, X: The existence of nontrivial solutions to nonlinear elliptic equation of p-q-Laplacian type on \(\mathbb {R}^{N}\). Nonlinear Anal. 71(5-6), 2316-2334 (2009) MathSciNetView ArticleMATHGoogle Scholar
  4. Wu, M, Yang, Z: A class of p-q-Laplacian type equation with potentials eigenvalue problem in \(\mathbf{R}^{N}\). Bound. Value Probl. 2009, Article ID 185319 (2009) MathSciNetView ArticleMATHGoogle Scholar
  5. Ambrosetti, A, Rabinowitz, PH: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349-381 (1973) MathSciNetView ArticleMATHGoogle Scholar
  6. Barile, S, Figueiredo, GM: Existence of a least energy nodal solution for a class of \(p\, \&\, q\)-quasilinear elliptic equations. Adv. Nonlinear Stud. 14(2), 511-530 (2014) MathSciNetView ArticleMATHGoogle Scholar
  7. Barile, S, Figueiredo, GM: Existence of least energy positive, negative and nodal solutions for a class of \(p\, \&\, q\)-problems with potentials vanishing at infinity. J. Math. Anal. Appl. 427(2), 1205-1233 (2015) MathSciNetView ArticleMATHGoogle Scholar
  8. Bartsch, T, Liu, Z, Weth, T: Sign changing solutions of superlinear Schrödinger equations. Commun. Partial Differ. Equ. 29(1-2), 25-42 (2004) MathSciNetView ArticleMATHGoogle Scholar
  9. Bartsch, T, Weth, T: Three nodal solutions of singularly perturbed elliptic equations on domains without topology. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 22(3), 259-281 (2005) MathSciNetView ArticleMATHGoogle Scholar
  10. Bartsch, T, Weth, T, Willem, M: Partial symmetry of least energy nodal solutions to some variational problems. J. Anal. Math. 96, 1-18 (2005) MathSciNetView ArticleMATHGoogle Scholar
  11. Ben-Naoum, AK, Troestler, C, Willem, M: Extrema problems with critical Sobolev exponents on unbounded domains. Nonlinear Anal. 26(4), 823-833 (1996) MathSciNetView ArticleMATHGoogle Scholar
  12. Brézis, H, Nirenberg, L: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun. Pure Appl. Math. 36(4), 437-477 (1983) MathSciNetView ArticleMATHGoogle Scholar
  13. Costa, DG, Miyagaki, OH: Nontrivial solutions for perturbations of the p-Laplacian on unbounded domains. J. Math. Anal. Appl. 193(3), 737-755 (1995) MathSciNetView ArticleMATHGoogle Scholar
  14. Ding, W, Ni, W-M: On the existence of positive entire solutions of a semilinear elliptic equation. Arch. Ration. Mech. Anal. 91(4), 283-308 (1986) MathSciNetView ArticleMATHGoogle Scholar
  15. Figueiredo, GM, Nascimento, RG: Existence of a nodal solution with minimal energy for a Kirchhoff equation. Math. Nachr. 288(1), 48-60 (2015) MathSciNetView ArticleMATHGoogle Scholar
  16. Li, G, Zhou, H-S: The existence of a positive solution to asymptotically linear scalar field equations. Proc. R. Soc. Edinb., Sect. A 130(1), 81-105 (2000) MathSciNetView ArticleMATHGoogle Scholar
  17. Li, G, Zhu, X: Nontrivial solution of nonlinear scalar field equations with strong nonlinearity. Acta Math. Sci. 8(4), 431-448 (1988) MathSciNetMATHGoogle Scholar
  18. Liu, S: On superlinear problems without the Ambrosetti and Rabinowitz condition. Nonlinear Anal. 73(3), 788-795 (2010) MathSciNetView ArticleMATHGoogle Scholar
  19. Miyagaki, OH, Souto, MAS: Superlinear problems without Ambrosetti and Rabinowitz growth condition. J. Differ. Equ. 245(12), 3628-3638 (2008) MathSciNetView ArticleMATHGoogle Scholar
  20. Chaves, MF, Ercole, G, Miyagaki, OH: Existence of a nontrivial solution for the \((p,q)\)-Laplacian in \(\mathbb{R}^{N}\) without the Ambrosetti-Rabinowitz condition. Nonlinear Anal. 114, 133-141 (2015) MathSciNetView ArticleMATHGoogle Scholar
  21. He, C, Li, G: The existence of a nontrivial solution to the \(p\, \&\, q\)-Laplacian problem with nonlinearity asymptotic to \(u^{p-1}\) at infinity in \(\mathbb {R}^{N}\). Nonlinear Anal. 68(5), 1100-1119 (2008) MathSciNetView ArticleMATHGoogle Scholar
  22. Mugnai, D, Papageorgiou, NS: Wang’s multiplicity result for superlinear \((p,q)\)-equations without the Ambrosetti-Rabinowitz condition. Trans. Am. Math. Soc. 366(9), 4919-4937 (2014) MathSciNetView ArticleMATHGoogle Scholar
  23. Shuai, W: Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains. J. Differ. Equ. 259(4), 1256-1274 (2015) MathSciNetView ArticleMATHGoogle Scholar
  24. Berestycki, H, Lions, P-L: Nonlinear scalar field equations. II. Existence of infinitely many solutions. Arch. Ration. Mech. Anal. 82(4), 347-375 (1983) MathSciNetMATHGoogle Scholar
  25. Strauss, WA: Existence of solitary waves in higher dimensions. Commun. Math. Phys. 55(2), 149-162 (1977) MathSciNetView ArticleMATHGoogle Scholar
  26. Yang, J, Zhu, X: On the existence of nontrivial solution of a quasilinear elliptic boundary value problem for unbounded domains. I. Positive mass case. Acta Math. Sci. 7(3), 341-359 (1987) MathSciNetMATHGoogle Scholar
  27. Willem, M: Minimax Theorems. Progress in Nonlinear Differential Equations and Their Applications, vol. 24. Birkhäuser Boston, Boston (1996) View ArticleMATHGoogle Scholar
  28. Guo, D: Nonlinear Functional Analysis, 2nd edn. Shandong Science and Technology Press, Jinan (2001) Google Scholar
  29. Damascelli, L: Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 15(4), 493-516 (1998) MathSciNetView ArticleMATHGoogle Scholar

Copyright

© Li and Liang 2016