Least energy solutions for a quasilinear Schrödinger equation with potential well

Abstract

In this paper, we consider the existence of least energy solutions for the following quasilinear Schrödinger equation:

with $a(x)\ge 0$ having a potential well, where $N\ge 3$ and $\lambda >0$ is a parameter. Under suitable hypotheses, we obtain the existence of a least energy solution $u_{\lambda }$ of ($E_{\lambda }$) which localizes near the potential well $int{a}^{-1}(0)$ for λ large enough by using the variational method and the concentration compactness method in an Orlicz space.

MSC:35J60, 35B33.

1 Introduction

Let us consider the following quasilinear Schrödinger equation:

for sufficiently large λ, where $N\ge 3$.

Our assumptions on $a(x)$ are as follows:

($a_1$) , the potential well $\mathrm{\Omega }:=int{a}^{-1}(0)$ is a non-empty set and $\overline{\mathrm{\Omega }}=a^{-1}(0)$;

($a_2$) There exists a constant $M_0>0$ such that , where μ denotes the Lebesgue measure on .

Condition ($a_2$) is very weak in dealing with the operator $-\mathrm{\Delta }+(\lambda a(x)+1)I$ on , which was firstly used by Bartsch and Wang [1] in dealing with the semilinear Schrödinger equation.

Remark 1.1 $\mathrm{\Omega }:=int{a}^{-1}(0)$ can be unbounded.

For $f(u)$, we assume that f is continuous and satisfies the following conditions:

($f_1$) ${lim}_{u\to 0^{+}}\frac{f(u)}{u}=0$;

($f_2$) $0\le f(u)\le C(1+u^p)$ for $u\ge 0$, where $C>0$ is a constant and $4<p+1<2\cdot 2^{\ast }$, where $2^{\ast }=\frac{2N}{N-2}$;

($f_3$) There is a number $4<\theta \le p+1$ such that for all $u>0$, we have $f(u)\cdot u\ge \theta F(u)$, where $F(u)={\int }_0^{u}f(t)dt$.

Hypotheses ($a_1$), ($a_2$) and ($f_1$), ($f_2$), ($f_3$) will be maintained throughout this paper.

Solutions of ($E_{\lambda }$) are related to the existence of the standing wave solutions of the following quasilinear Schrödinger equation:

(1.1)

where $V(x)$ is a given potential, k is a real constant and f, h are real functions. We would like to mention that (1.1) appears more naturally in mathematical physics and has been derived as models of several physical phenomena corresponding to various types of h. For instance, the case $h(s)=s$ was used for the superfluid film equation in plasma physics by Kurihara [2] (see also [3]); in the case of $h(s)={(1+s)}^{\frac{1}{2}}$, (1.1) was used as a model of the self-changing of a high-power ultrashort laser in matter (see [4–7] and references therein).

In recent years, much attention has been devoted to the quasilinear Schrödinger equation of the following form:

(1.2)

For example, by using a constrained minimization argument, the existence of positive ground state solution was proved by Poppenberg, Schmitt and Wang [8]. Using a change of variables, Liu, Wang and Wang [9] used an Orlicz space to prove the existence of soliton solution of (1.2) via the mountain pass theorem. Colin and Jeanjean [10] also made use of a change of variables but worked in the Sobolev space , they proved the existence of a positive solution for (1.2) from the classical results given by Berestycki and Lions [11]. By using the Nehari manifold method and the concentration compactness principle (see [12]) in the Orlicz space, Guo and Tang [13] considered the following equation:

(1.3)

with $a(x)\ge 0$ having a potential well and $4<p+1<2\cdot 2^{\ast }$, where $2^{\ast }=\frac{2N}{N-2}$ is the critical Sobolev exponent, and they proved the existence of a ground state solution of (1.3) which localizes near the potential well $int{a}^{-1}(0)$ for λ large enough. In [14], Guo and Tang also considered ground state solutions of the corresponding quasilinear Schrödinger systems for (1.3) by the same methods and obtained similar results. For the stability and instability results for the special case of (1.2), one can also see the paper by Colin, Jeanjean and Squassina [15].

It is worth pointing out that the existence of one-bump or multi-bump bound state solutions for the related semilinear Schrödinger equation (1.3) for $k=0$ has been extensively studied. One can see Bartsch and Wang [1], Ambrosetti, Badiale and Cingolani [16], Ambrosetti, Malchiodi and Secchi [17], Byeon and Wang [18], Cingolani and Lazzo [19], Cingolani and Nolasco [20], Del Pino and Felmer [21, 22], Floer and Weinstein [23], Oh [24, 25] and the references therein.

In this paper, based on the idea from Liu, Wang and Wang [9], we consider the more general equation ($E_{\lambda }$), the existence of least energy solutions for equation ($E_{\lambda }$) with a potential well $int{a}^{-1}(0)$ for λ large is proved under the conditions ($a_1$), ($a_2$) and ($f_1$), ($f_2$), ($f_3$).

The paper is organized as follows. In Section 2, we describe our main result (Theorem 2.1). In Section 3, we give some preliminaries that will be used for the proof of the main result. Finally, Theorem 2.1 will be proved in Section 4.

Throughout this paper, we use the same C to denote different universal constants.

2 Main result

Let $V_{\lambda }(x)=\lambda a(x)+1$. Formally, we define the following functional:

(2.1)

for . Note that under our assumptions, the functional $J_{\lambda }$ is not well defined on X.

We follow the idea of [9] and make the following change of variable.

Let $v=h(u)=\frac{1}{2}u\sqrt{1+u^2}+\frac{1}{2}ln(u+\sqrt{1+u^2})$, then $dv=\sqrt{1+u^2}du$. Moreover, $h(u)$ satisfies

$$h(u)\sim \{\begin{array}{cc}u,\hfill & |u|\ll 1,\hfill \\ \frac{1}{2}u|u|,\hfill & |u|\gg 1.\hfill \end{array}$$

Since $h^{\prime }(u)=\sqrt{1+u^2}>0$, $h(u)$ is strictly monotone and hence has an inverse function denoted by $u=g(v)$. Obviously,

$$g(v)\sim \{\begin{array}{cc}v,\hfill & |v|\ll 1,\hfill \\ \sqrt{\frac{2}{|v|}}v,\hfill & |v|\gg 1,\hfill \end{array}{g}^{\prime }(v)=\frac{1}{\sqrt{1+g^2(v)}}.$$

Let $G(v)=g^2(v)$. Then it holds that

$$G(v)=g^2(v)\sim \{\begin{array}{cc}{v}^2,\hfill & |v|\ll 1,\hfill \\ 2|v|,\hfill & |v|\gg 1\hfill \end{array}$$

and $G(v)$ is convex. Moreover, there exists $C_0>0$ such that $G(2v)\le C_{0}G(v)$,

$$G^{\prime }(v)=\frac{2g(v)}{\sqrt{1+g^2(v)}},G^{″}(v)=\frac{2}{{(1+g^2(v))}^2}>0.$$

Now we introduce the Orlicz space (see [26])

equipped with the norm

Then $E_G^{\lambda }$ is a Banach space.

Let

equipped with the norm

$${\parallel v\parallel }_{\lambda }={\parallel \mathrm{\nabla }v\parallel }_{L^2}+{|v|}_G^{\lambda }.$$

Using the change of variable, we define the functional ${\mathrm{\Phi }}_{\lambda }$ on $H_G^{\lambda }$ by

(2.2)

where $v_{+}=max\{v,0\}$ is the positive part of v.

Let

$$N_{\lambda }:=\{v\in H_G^{\lambda }\mathrm{\setminus }\{0\}|〈{\mathrm{\Phi }}_{\lambda }^{\prime }(v),v〉=0\}$$

be the Nehari manifold and let

$$c_{\lambda }:=\underset{v\in N_{\lambda }}{inf}{\mathrm{\Phi }}_{\lambda }(v)$$

be the infimum of ${\mathrm{\Phi }}_{\lambda }$ on the Nehari manifold $N_{\lambda }$, where ${\mathrm{\Phi }}_{\lambda }^{\prime }(v)$ is the Gateaux derivative (see Proposition 3.3).

We say that $u_{\lambda }=g(v_{\lambda })$ is a least energy solution of ($E_{\lambda }$) if $v_{\lambda }\in N_{\lambda }$ such that $c_{\lambda }$ is achieved.

Note that under our assumptions, for λ large enough, the following Dirichlet problem is a kind of a ‘limit’ problem:

where $\mathrm{\Omega }:=int{a}^{-1}(0)$.

Similar to the definition of the least energy solution of ($E_{\lambda }$), we can define the least energy solution of (D) which will be given in Section 4.

Our main result is as follows.

Theorem 2.1 Assume that ($a_1$), ($a_2$) and ($f_1$), ($f_2$), ($f_3$) are satisfied. Then for λ large, $c_{\lambda }$ is achieved by a critical point $v_{\lambda }$ of ${\mathrm{\Phi }}_{\lambda }$ such that $u_{\lambda }=g(v_{\lambda })$ is a least energy solution of ($E_{\lambda }$). Furthermore, for any sequence ${\lambda }_n\to \mathrm{\infty }$, $\{v_{{\lambda }_n}\}$ has a subsequence converging to v such that $u=g(v)$ is a least energy solution of (D).

3 Preliminaries

In order to obtain the compactness of the functional ${\mathrm{\Phi }}_{\lambda }$, we recall the following Lemmas 3.1 and 3.2 which can be found in [13].

Lemma 3.1 There exist two constants $C_1>0$, $C_2>0$ such that

(3.1)

for any $v\in H_G^{\lambda }$.

Lemma 3.2 The map: $v\to g(v)$ from $H_G^{\lambda }$ into is continuous for $2\le q\le 2\cdot 2^{\ast }$.

Now we consider the functional ${\mathrm{\Phi }}_{\lambda }$ defined on $H_G^{\lambda }$ by (2.2), the following Proposition 3.3 is due to [9].

Proposition 3.3

• (i) ${\mathrm{\Phi }}_{\lambda }$ is well defined on $H_G^{\lambda }$;

• (ii) ${\mathrm{\Phi }}_{\lambda }$ is continuous in $H_G^{\lambda }$;

• (iii) ${\mathrm{\Phi }}_{\lambda }$ is Gateaux differentiable, the Gateaux derivative ${\mathrm{\Phi }}_{\lambda }^{\prime }(v)$ for $v\in H_G^{\lambda }$ is a linear functional and ${\mathrm{\Phi }}_{\lambda }^{\prime }(v)$ is continuous in v in the strong-weak topology, that is, if $v_n\to v$ strongly in $H_G^{\lambda }$, then ${\mathrm{\Phi }}_{\lambda }^{\prime }(v_n)\rightharpoonup {\mathrm{\Phi }}_{\lambda }^{\prime }(v)$ weakly. Moreover, the Gateaux derivative ${\mathrm{\Phi }}_{\lambda }^{\prime }(v)$ has the form

  

  (3.2)

Recall that $\{v_n\}\subset H_G^{\lambda }$ is called a Palais-Smale sequence ((PS) _c sequence in short) for ${\mathrm{\Phi }}_{\lambda }$ if ${\mathrm{\Phi }}_{\lambda }(v_n)\to c$ and ${\mathrm{\Phi }}_{\lambda }^{\prime }(v_n)\to 0$ in ${(H_G^{\lambda })}^{\ast }$, the dual space of $H_G^{\lambda }$. We say that the functional ${\mathrm{\Phi }}_{\lambda }$ satisfies the (PS) _c condition if any of (PS) _c sequence (up to a subsequence, if necessary) $\{v_n\}$ converges strongly in $H_G^{\lambda }$.

Lemma 3.4 Any of (PS) _c sequence $\{v_n\}$ for ${\mathrm{\Phi }}_{\lambda }$ is bounded.

Proof Suppose that $\{v_n\}$ is a (PS) _c sequence of ${\mathrm{\Phi }}_{\lambda }$. We have and ${\mathrm{\Phi }}_{\lambda }^{\prime }(v_n)\to 0$ in the space ${(H_G^{\lambda })}^{\ast }$.

Taking $w_n=\frac{g(v_n)}{g^{\prime }(v_n)}$, then $|\mathrm{\nabla }{w}_n|=(1+\frac{g^2(v_n)}{1+g^2(v_n)})|\mathrm{\nabla }{v}_n|\le 2|\mathrm{\nabla }{v}_n|$, we have ${\parallel w_n\parallel }_{\lambda }\le C{\parallel v_n\parallel }_{\lambda }$, thus

(3.3)

and

(3.4)

Taking $\text{(3.4)}-\frac{1}{\theta }\text{(3.3)}$ yields

Note that

$$\begin{array}{c}\frac{1}{2}-\frac{1}{\theta }(1+\frac{g^2(v_n)}{1+g^2(v_n)})>\frac{1}{2}-\frac{2}{\theta }=\frac{\theta -4}{2\theta },\hfill \\ \frac{1}{2}-\frac{1}{\theta }=\frac{\theta -2}{2\theta },f(u)\cdot u>\theta F(u),\theta >4,\hfill \end{array}$$

we have

It follows from Lemma 3.1 that

$$C_{1}min\{{\parallel v\parallel }_{\lambda },{\parallel v\parallel }_{\lambda }^2\}\le \frac{2\theta }{\theta -4}(c+o(1)+o\left({\parallel v_n\parallel }_{\lambda }\right)),$$

(3.5)

thus $\{v_n\}$ is bounded in $H_G^{\lambda }$.

Let $K_{\lambda }=\{v\in H_G^{\lambda }|{\mathrm{\Phi }}_{\lambda }^{\prime }(v)=0\}$ be the critical set of ${\mathrm{\Phi }}_{\lambda }$. Suppose $v\in K_{\lambda }$, then it is easy to check that either $v>0$ or $v\equiv 0$ in by the definition of ${\mathrm{\Phi }}_{\lambda }$ and the strong maximum principle. □

Lemma 3.5 There exists $0<\sigma <1$ which is independent of λ such that ${\parallel v\parallel }_{\lambda }\ge {\parallel v\parallel }_1>\sigma$ for all $v\in K_{\lambda }\mathrm{\setminus }\{0\}$ and $\lambda \ge 1$.

Proof Assume that ${\parallel v\parallel }_{\lambda }\le 1$ for any $v\in K_{\lambda }\mathrm{\setminus }\{0\}$ (otherwise, the conclusion is true). From ($f_1$), ($f_2$), we see that for any $\epsilon >0$, there is a constant $C_{\epsilon }>0$ such that $f(x,u)\le \epsilon u+C_{\epsilon }{u}^p$ for $u>0$. We have

and we can easily deduce the desired result. □

Lemma 3.6 There exists a positive constant $c_0>0$ such that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\parallel v_n\parallel }_{\lambda }\le max\{\frac{2\theta }{(\theta -4)C_1}c,\sqrt{\frac{2\theta }{(\theta -4)C_1}c}\}$$

and either $c\ge c_0$ or $c=0$ if $\{v_n\}$ is a (PS) _c sequence for ${\mathrm{\Phi }}_{\lambda }$, where $C_1$ is the constant in Lemma  3.1.

Proof Since $\{v_n\}$ is a (PS) _c sequence, we have

It follows from (3.5) that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\parallel v_n\parallel }_{\lambda }\le max\{\frac{2\theta }{(\theta -4)C_1}c,\sqrt{\frac{2\theta }{(\theta -4)C_1}c}\}.$$

On the other hand, for ${\parallel v_n\parallel }_{\lambda }\le 1$, we have

$$\begin{array}{rcl}o\left({\parallel v_n\parallel }_{\lambda }\right)& =& 〈{\mathrm{\Phi }}_{\lambda }^{\prime }(v_n),\frac{g(v_n)}{g^{\prime }(v_n)}〉\\ \ge & C_{1}min\{{\parallel v_n\parallel }_{\lambda },{\parallel v_n\parallel }_{\lambda }^2\}-C_2{(max\{{\parallel v_n\parallel }_{\lambda },{\parallel v_n\parallel }_{\lambda }^2\})}^{\frac{p+1}{2}}\\ =& C_1{\parallel v_n\parallel }_{\lambda }^2-C_2{\parallel v_n\parallel }_{\lambda }^{\frac{p+1}{2}}.\end{array}$$

(3.6)

Thus, there exists ${\sigma }_1>0$ (${\sigma }_1<1$) such that

$$〈{\mathrm{\Phi }}_{\lambda }^{\prime }(v_n),\frac{g(v_n)}{g^{\prime }(v_n)}〉\ge \frac{1}{4}{C}_1{\parallel v_n\parallel }_{\lambda }^2\text{for }{\parallel v_n\parallel }_{\lambda }\le {\sigma }_1.$$

(3.7)

Taking $c_0=\frac{{\sigma }_1}{max\{\frac{2\theta }{(\theta -4)C_1},\sqrt{\frac{2\theta }{(\theta -4)C_1}}\}}$, then we have

$$max\{\frac{2\theta }{(\theta -4)C_1}c,\sqrt{\frac{2\theta }{(\theta -4)C_1}}c\}<{\sigma }_1<1$$

if $c<c_0$. It follows from (3.6) and (3.7) that

$$\frac{1}{4}{\parallel v_n\parallel }_{\lambda }^2\le o\left({\parallel v_n\parallel }_{\lambda }\right),$$

hence, ${\parallel v_n\parallel }_{\lambda }\to 0$ and $c=0$. Therefore, we have proved that there exists a constant $c_0$ such that either $c\ge c_0$ or $c=0$. □

Proposition 3.7 Let $M>0$ be a constant. Then for any $\epsilon >0$, there exist ${\mathrm{\Lambda }}_{\epsilon }>0$, $R_{\epsilon }>0$ such that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\int }_{B_{R_{\epsilon }}^c}(\frac{1}{2}f\left(g\left({(v_n)}_{+}\right)\right)g\left({(v_n)}_{+}\right)-F\left(g\left({(v_n)}_{+}\right)\right))dx\le \epsilon$$

if $\{v_n\}$ is a (PS) _c sequence of ${\mathrm{\Phi }}_{\lambda }$ with $\lambda >{\mathrm{\Lambda }}_{\epsilon }$, $c\le M$, where .

Proof For all $R>0$, let

We have

$$\begin{array}{rcl}{\int }_{A(R)}{g}^2(v_n)dx& \le & \frac{1}{\lambda M_0+1}{\int }_{A(R)}(\lambda a(x)+1)g^2(v_n)dx\\ \le & \frac{1}{\lambda M_0+1}{\int }_{A(R)}({|\mathrm{\nabla }{v}_n|}^2+(\lambda a(x)+1)g^2(v_n))dx\\ \le & \frac{1}{\lambda M_0+1}(\frac{2\theta }{\theta -4}c+o\left({\parallel v_n\parallel }_{\lambda }\right))\\ =& \frac{1}{\lambda M_0+1}(\frac{2\theta }{\theta -4}c+o(1))\\ \to & 0(\lambda \to \mathrm{\infty }).\end{array}$$

(3.8)

On the other hand, by the Hölder inequality and interpolation inequality, we have

(3.9)

By using the Gagliardo-Nirenberg inequality, we obtain

$$\begin{array}{c}{\int }_{B_{R_{\epsilon }}^c}(\frac{1}{2}f\left(g\left({(v_n)}_{+}\right)\right)g\left({(v_n)}_{+}\right)-F\left(g\left({(v_n)}_{+}\right)\right))dx\hfill \\ \le {\int }_{B_{R_{\epsilon }}^c}(\frac{1}{2}f\left(g\left({(v_n)}_{+}\right)\right)g\left({(v_n)}_{+}\right)-C{\left(g\left({(v_n)}_{+}\right)\right)}^{\theta })dx(\text{by }(f_3))\hfill \\ \le C{\int }_{B_R^c}{|g(v_n)|}^{p+1}dx\hfill \\ \le C{\left({\int }_{B_R^c}{|\mathrm{\nabla }g(v_n)|}^{2}dx\right)}^{\frac{(p+1)\beta }{2}}{({\int }_{B_R^c}{g}^2(v_n)dx)}^{\frac{(p+1)(1-\beta )}{2}}(\beta =\frac{N(p-1)}{2(p+1)})\hfill \\ \le C{\parallel v_n\parallel }_{\lambda }^{(p+1)\beta }{({\int }_{A(R)}{g}^2(v_n)dx+{\int }_{B(R)}{g}^2(v_n)dx)}^{\frac{(p+1)(1-\beta )}{2}}\hfill \\ \le C{({\int }_{A(R)}{g}^2(v_n)dx+{\int }_{B(R)}{g}^2(v_n)dx)}^{\frac{(p+1)(1-\beta )}{2}}\left({\parallel v_n\parallel }_{\lambda }\text{ is bounded}\right).\hfill \end{array}$$

Let λ and R be large enough, from (3.8) and (3.9), we get the desired result. □

Lemma 3.8 $c_{\lambda }={inf}_{N_{\lambda }}{\mathrm{\Phi }}_{\lambda }(v)$ is achieved by some $v>0$.

Proof By the definition of $c_{\lambda }$ and the Ekeland variational principle, there exists a (PS) _c sequence $\{v_n\}$, by Lemma 3.4, we know that $\{v_n\}$ is bounded. Hence (up to a subsequence) we have $v_n\rightharpoonup v$ in $L^{2^{\ast }}$, $\mathrm{\nabla }{v}_n\rightharpoonup \mathrm{\nabla }v$ in $L^2$, $v_n\to v$ a.e. in , $g(v_n)\rightharpoonup g(v)$ in $L^q$ for $2\le q\le 2\cdot 2^{\ast }$.

It is sufficient to prove that $v\ne 0$ and $v\in N_{\lambda }$. In fact,

(3.10)

it follows that

Let ${\epsilon }_0=\frac{1}{4}c$, since $g(v_n)\to g(v)$ strongly in $L^{p+1}(B_R)$ for $R>0$, by Proposition 3.7, there exist ${\mathrm{\Lambda }}_0>0$, $R_0>0$ such that for $\lambda \ge {\mathrm{\Lambda }}_0$, $R\ge R_0$,

$${\int }_{B_R^c}|(\frac{1}{2}f\left(g\left({(v_n)}_{+}\right)\right)g\left({(v_n)}_{+}\right)-F\left(g\left({(v_n)}_{+}\right)\right))-(\frac{1}{2}f(g(v_{+}))g(v_{+})-F(g(v_{+})))|dx<{\epsilon }_0,$$

thus

$${\int }_{B_R}(\frac{1}{2}f(g(v_{+}))g(v_{+})-F(g(v_{+})))dx\ge \frac{1}{2}c>0.$$

Hence $v\ne 0$.

Now we prove $v\in N_{\lambda }$. Indeed, since $\{v_n\}$ is a (PS) _c sequence, we have

(3.11)

where $o({\parallel v_n\parallel }_{\lambda })\to 0$ as $n\to \mathrm{\infty }$.

Let $l_n:=\frac{g(v_n)}{\sqrt{1+g^2(v_n)}}$, then $\{l_n\}$ is bounded in for $2\le q\le 2\cdot 2^{\ast }$, by the continuity of g, we have, up to a subsequence, $l_n\rightharpoonup l=\frac{g(v)}{\sqrt{1+g^2(v)}}$ in .

Similarly, we have $s_n:=\frac{f(g(v_n))}{\sqrt{1+g^2(v_n)}}\le f(g(v_n))$ is bounded in . Again, by the continuity of g, we have $s_n\rightharpoonup s=\frac{f(g(v))}{\sqrt{1+g^2(v)}}$ in . Passing to the limits in (3.11), we get

which is equivalent to $〈{\mathrm{\Phi }}_{\lambda }^{\prime }(v),v〉=0$, that is, $v\in N_{\lambda }$. □

4 Proof of the main result

Consider the following quasilinear Schrödinger equation in ($N\ge 3$):

We have the same change of variables and the same notation as in the previous sections. Define the corresponding Orlicz space $E_G(\mathrm{\Omega })$ by

$$E_G(\mathrm{\Omega })=\left\{v\right|{\int }_{\mathrm{\Omega }}{g}^2(v)dx<+\mathrm{\infty }\}$$

with the norm

$${|v|}_{G(\mathrm{\Omega })}:=\underset{\xi >0}{inf}\xi (1+{\int }_{\mathrm{\Omega }}G\left({\xi }^{-1}v\right)dx).$$

The space $H_G(\mathrm{\Omega })$ is defined by

$$H_G(\mathrm{\Omega }):=\left\{v\right|{\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }v|}^{2}dx<+\mathrm{\infty },{\int }_{\mathrm{\Omega }}{g}^2(v)dx<+\mathrm{\infty }\}$$

with the norm

$${\parallel v\parallel }_{\mathrm{\Omega }}={\parallel v\parallel }_{L^2}+{|v|}_{G(\mathrm{\Omega })}.$$

The following Lemma 4.1 is a counterpart of Lemma 3.1.

Lemma 4.1 There exist two constants $C_1>0$, $C_2>0$ such that

$$C_{1}min\{{\parallel v\parallel }_{\mathrm{\Omega }},{\parallel v\parallel }_{\mathrm{\Omega }}^2\}\le {\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }v|}^{2}dx+{\int }_{\mathrm{\Omega }}{g}^2(v)dx\le C_{2}max\{{\parallel v\parallel }_{\mathrm{\Omega }},{\parallel v\parallel }_{\mathrm{\Omega }}^2\}$$

for any $v\in H_G(\mathrm{\Omega })$.

We denote by $H_G^0(\mathrm{\Omega })$ the closure of $C_0^{\mathrm{\infty }}(\mathrm{\Omega })$ in $H_G(\mathrm{\Omega })$. We define the functional ${\mathrm{\Phi }}_{\mathrm{\Omega }}$ on $H_G^0(\mathrm{\Omega })$ by

$${\mathrm{\Phi }}_{\mathrm{\Omega }}(v)=\frac{1}{2}{\int }_{\mathrm{\Omega }}({|\mathrm{\nabla }v|}^2+g^2(v))dx-{\int }_{\mathrm{\Omega }}F(g(v_{+}))dx,$$

(4.1)

and we define the Nehari manifold $N_{\mathrm{\Omega }}$ by

$$N_{\mathrm{\Omega }}:=\{v\in H_G(\mathrm{\Omega })\mathrm{\setminus }\{0\}|〈{\mathrm{\Phi }}_{\mathrm{\Omega }}^{\prime }(v),v〉=0\}.$$

Let

$$c(\mathrm{\Omega })=\underset{N_{\mathrm{\Omega }}}{inf}{\mathrm{\Phi }}_{\mathrm{\Omega }}(v).$$

We recall that $u=g(v)$ is a least energy solution of (D) if $v\in N_{\mathrm{\Omega }}$ such that $c(\mathrm{\Omega })$ is achieved.

Lemma 4.2 Suppose $c_{\lambda }={inf}_{N_{\lambda }}{\mathrm{\Phi }}_{\lambda }$. Then ${lim}_{\lambda \to +\mathrm{\infty }}{c}_{\lambda }=c(\mathrm{\Omega })$.

Proof It is easy to see that $c_{\lambda }\le c(\mathrm{\Omega })$ for $\lambda \ge 1$. We claim that $c_{\lambda }$ is monotone increasing with respect to λ. In fact, for ${\lambda }_1<{\lambda }_2$, we assume that $c_{{\lambda }_1}$, $c_{{\lambda }_2}$ are achieved for $v_{{\lambda }_1}>0$, $v_{{\lambda }_2}>0$. Obviously,

(4.2)

We first prove that there exists $0<\alpha <1$ such that $\alpha v_{{\lambda }_2}\in N_{{\lambda }_1}$. This is sufficient to prove that

That is,

Let

Then by ($f_1$), we can obtain ${lim}_{\alpha \to 0}I(\alpha )\le 0$ and

Hence, there exists $0<{\alpha }_0<1$ such that , i.e., ${\alpha }_0{v}_{{\lambda }_2}\in N_{{\lambda }_1}$. Thus

$$c_{{\lambda }_1}\le {\mathrm{\Phi }}_{{\lambda }_1}({\alpha }_0{v}_{{\lambda }_2}).$$

In the following, we will prove that

$${\mathrm{\Phi }}_{{\lambda }_1}({\alpha }_0{v}_{{\lambda }_2})\le {\mathrm{\Phi }}_{{\lambda }_2}(v_{{\lambda }_2})=c_{{\lambda }_2}.$$

In fact, we consider the function $\rho (\alpha )$ defined by

$$\rho (\alpha )=\frac{(f(g(\alpha v_{{\lambda }_2}))-({\lambda }_{2}a+1)g(\alpha v_{{\lambda }_2}))v_{{\lambda }_2}}{\alpha \sqrt{1+g^2(\alpha v_{{\lambda }_2})}}.$$

By $g(t)\sqrt{1+g^2(t)}\le 2t$ for $t\ge 0$, we have $\frac{g(\alpha v_{{\lambda }_2})}{g^{\prime }(\alpha v_{{\lambda }_2})}=g(\alpha v_{{\lambda }_2})\sqrt{1+g^2(\alpha v_{{\lambda }_2})}\le 2\alpha v_{{\lambda }_2}\le (\theta -2)\alpha v_{{\lambda }_2}$. It follows that

$$\frac{\alpha v_{{\lambda }_2}g(\alpha v_{{\lambda }_2})}{g(\alpha v_{{\lambda }_2})/g^{\prime }(\alpha v_{{\lambda }_2})+\alpha v_{{\lambda }_2}}\ge \frac{\alpha v_{{\lambda }_2}g(\alpha v_{{\lambda }_2})}{(\theta -2)\alpha v_{{\lambda }_2}+\alpha v_{{\lambda }_2}}.$$

Obviously,

$$f^{\prime }(g(\alpha v_{{\lambda }_2}))g^{\prime }(\alpha v_{{\lambda }_2})v_{{\lambda }_2}^2\alpha g(\alpha v_{{\lambda }_2})-f(g(\alpha v_{{\lambda }_2}))v_{{\lambda }_2}(g(\alpha v_{{\lambda }_2})+\alpha g^{\prime }(\alpha v_{{\lambda }_2})v_{{\lambda }_2})\ge 0$$

and hence it is easy to check that

$$\begin{array}{c}\frac{d}{d\alpha }\left(\frac{f(g(\alpha v_{{\lambda }_2}))v_{{\lambda }_2}}{\alpha \sqrt{1+g^2(\alpha v_{{\lambda }_2})}}\right)\hfill \\ =\frac{d}{d\alpha }\left(\frac{g(\alpha v_{{\lambda }_2})}{\sqrt{1+g^2(\alpha v_{{\lambda }_2})}}\right)\frac{f(g(\alpha v_{{\lambda }_2}))v_{{\lambda }_2}}{\alpha g(\alpha v_{{\lambda }_2})}+\frac{d}{d\alpha }\left(\frac{f(g(\alpha v_{{\lambda }_2}))v_{{\lambda }_2}}{\alpha g(\alpha v_{{\lambda }_2})}\right)\frac{g(\alpha v_{{\lambda }_2})}{\sqrt{1+g^2(\alpha v_{{\lambda }_2})}}\hfill \\ =\frac{v_{{\lambda }_2}^2}{{(1+g^2(\alpha v_{{\lambda }_2}))}^2}\cdot \frac{f(g(\alpha v_{{\lambda }_2}))v_{{\lambda }_2}}{\alpha g(\alpha v_{{\lambda }_2})}\hfill \\ +\frac{f^{\prime }(g(\alpha v_{{\lambda }_2}))g^{\prime }(\alpha v_{{\lambda }_2})v_{{\lambda }_2}^2\alpha g(\alpha v_{{\lambda }_2})-f(g(\alpha v_{{\lambda }_2}))v_{{\lambda }_2}(g(\alpha v_{{\lambda }_2})+\alpha g^{\prime }(\alpha v_{{\lambda }_2})v_{{\lambda }_2})}{{\alpha }^2{g}^2(\alpha v_{{\lambda }_2})}\hfill \\ \times \frac{g(\alpha v_{{\lambda }_2})}{\sqrt{1+g^2(\alpha v_{{\lambda }_2})}}\hfill \\ \ge \frac{f^{\prime }(g(\alpha v_{{\lambda }_2}))g^{\prime }(\alpha v_{{\lambda }_2})v_{{\lambda }_2}^2\alpha g(\alpha v_{{\lambda }_2})-f(g(\alpha v_{{\lambda }_2}))v_{{\lambda }_2}(g(\alpha v_{{\lambda }_2})+\alpha g^{\prime }(\alpha v_{{\lambda }_2})v_{{\lambda }_2})}{{\alpha }^{2}g(\alpha v_{{\lambda }_2})\sqrt{1+g^2(\alpha v_{{\lambda }_2})}}\hfill \\ \ge 0.\hfill \end{array}$$

On the other hand,

$$\begin{array}{c}\frac{d}{d\alpha }(({\lambda }_{2}a+1)\frac{g(\alpha v_{{\lambda }_2})v_{{\lambda }_2}}{\alpha \sqrt{1+g^2(\alpha v_{{\lambda }_2})}})\hfill \\ =({\lambda }_{2}a+1)\frac{\alpha v_{{\lambda }_2}^2-g(\alpha v_{{\lambda }_2})\sqrt{1+g^2(\alpha v_{{\lambda }_2})}(1+g^2(\alpha v_{{\lambda }_2}))}{{\alpha }^2{(1+g^2(\alpha v_{{\lambda }_2}))}^2}{v}_{{\lambda }_2},\hfill \end{array}$$

by $v=\frac{1}{2}g(v)\sqrt{1+g^2(v)}+\frac{1}{2}ln(g(v)+\sqrt{1+g^2(v)})$, it is easy to check that for any $t\ge 0$,

$$t-g(t)\sqrt{1+g^2(t)}(1+g^2(t))\le 0,$$

which implies

$$\frac{d}{d\alpha }(({\lambda }_{2}a+1)\frac{g(\alpha v_{{\lambda }_2})v_{{\lambda }_2}}{\alpha \sqrt{1+g^2(\alpha v_{{\lambda }_2})}})\le 0$$

for any $\alpha >0$, thus we have proved that $\rho (\alpha )$ is monotone increasing for $\alpha >0$.

Now we consider the function $\gamma (\alpha )$ defined by

Then

for $0<\alpha <1$. Therefore, $\gamma (\alpha )$ is monotone increasing with respect to $\alpha <1$. Thus, we deduce that

Assume that ${lim}_{\lambda \to \mathrm{\infty }}{c}_{\lambda }=k\le c(\mathrm{\Omega })$. If $k<c(\mathrm{\Omega })$, then for any sequence $\{{\lambda }_n\}$ (${\lambda }_n\to +\mathrm{\infty }$), we have $c_{{\lambda }_n}\to k<c(\mathrm{\Omega })$.

We assume that $v_n$ is such that $c_{{\lambda }_n}$ is achieved, by Lemma 3.4, $\{v_n\}$ is bounded in $H_G^{{\lambda }_n}$. Since ${\parallel v_n\parallel }_{H_G^0}\le {\parallel v_n\parallel }_{H_G^{{\lambda }_n}}$, $\{v_n\}$ is bounded in $H_G^0$, as a result, we have $\mathrm{\nabla }{v}_n\rightharpoonup \mathrm{\nabla }v$ in , $g(v_n)\to g(v)$ in for $2\le q\le 2\cdot 2^{\ast }$, $v_n\rightharpoonup v$ in for $2\le q\le 2\cdot 2^{\ast }$, $v_n\to v$ a.e. in .

We claim that $v{|}_{{\mathrm{\Omega }}^c}=0$, where . Indeed, it is sufficient to prove $g(v){|}_{{\mathrm{\Omega }}^c}=0$. If not, then there exists a compact subset $\mathrm{\Sigma }\subset {\mathrm{\Omega }}^c$ with $dist\{\mathrm{\Sigma },\partial \mathrm{\Omega }\}>0$ such that $g(v){|}_{\mathrm{\Sigma }}\ne 0$ and

$${\int }_{\mathrm{\Sigma }}{g}^2(v_n)dx\to {\int }_{\mathrm{\Sigma }}{g}^2(v)dx>0.$$

Moreover, there exists ${\epsilon }_0>0$ such that $a(x)\ge {\epsilon }_0$ for any $x\in \mathrm{\Sigma }$.

By the choice of $v_n$, we have

hence,

This contradiction shows that $g(v){|}_{{\mathrm{\Omega }}^c}=0$ and so does v.

Now we show that

(4.3)

Suppose that (4.3) is not true, then by the concentration compactness principle of Lions (see [12]), there exist $\delta >0$, $\varrho >0$ and with $|x_n|\to +\mathrm{\infty }$ such that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\int }_{B_{\varrho }(x_n)}{|g(v_n)-g(v)|}^2\ge \delta >0.$$

On the other hand, by the choice of $\{v_n\}$, we have

which shows that $g(v_n)\to g(v)$ in for $2<q<2^{\ast }$. In the above proof, we have used the fact that $\mu \{B_{\varrho }(x_n)\cap \{x|a(x)\le M_0\}\}\to 0$ as $n\to \mathrm{\infty }$ and the $L^2$ bounded property of $g(v_n)$.

Now, since $\{v_n\}$ is bounded, by the Fatou lemma, we obtain

But, by the choice of $v_n$, we have

hence,

(4.4)

In the following, we will prove that

Indeed,

Since $f(g(v_{+}))g^{\prime }({(v_n)}_{+})v_n\rightharpoonup f(g(v_{+}))g^{\prime }(v_{+})v$, one can easily see that $I_2\to 0$ as $n\to \mathrm{\infty }$, and

by using $g(v_n)\to g(v)$ in for $2<q<2^{\ast }$. It follows from (4.4) that

thus, there exists $0<{\alpha }_0\le 1$ such that ${\alpha }_{0}v\in N_{\mathrm{\Omega }}$ and

$${\mathrm{\Phi }}_{\mathrm{\Omega }}({\alpha }_{0}v)\le {\mathrm{\Phi }}_{\mathrm{\Omega }}(v),$$

hence $c(\mathrm{\Omega })\le {\mathrm{\Phi }}_{\mathrm{\Omega }}({\alpha }_{0}v)<{\mathrm{\Phi }}_{\mathrm{\Omega }}(v)\le {lim}_{n\to \mathrm{\infty }}{\mathrm{\Phi }}_{{\lambda }_n}=k<c(\mathrm{\Omega })$. A contradiction. Thus we have proved that ${lim}_{\lambda \to \mathrm{\infty }}{c}_{\lambda }\to c(\mathrm{\Omega })$ as $\lambda \to +\mathrm{\infty }$. □

Proof of Theorem 2.1 Suppose that $\{v_n\}$ is a sequence such that $v_n\in N_{\lambda }$, ${\mathrm{\Phi }}_{{\lambda }_n}(v_n)=c_{{\lambda }_n}$, by the proof of Lemma 3.2, we have $\mathrm{\nabla }{v}_n\rightharpoonup \mathrm{\nabla }v$ in , $g(v_n)\to g(v)$ in for $2<q<2^{\ast }$ and $v{|}_{{\mathrm{\Omega }}^c}=0$. Moreover, ${\mathrm{\Phi }}_{\mathrm{\Omega }}(v)\le {lim}_{n\to \mathrm{\infty }}{\mathrm{\Phi }}_{{\lambda }_n}(v_n)\le c(\mathrm{\Omega })$, and if $v\in N_{\mathrm{\Omega }}$, then ${\mathrm{\Phi }}_{\mathrm{\Omega }}(v)=c(\mathrm{\Omega })$. Hence, in the following, we need only to prove that $v\in N_{\mathrm{\Omega }}$. To do this, it is sufficient to prove that

and

In fact, if one of the above three limits does not hold, by the Fatou lemma, we have

Similar to above, there exists ${\alpha }_0\in (0,1)$ such that ${\alpha }_{0}v\in N_{\mathrm{\Omega }}$ and $c(\mathrm{\Omega })\le {\mathrm{\Phi }}_{\mathrm{\Omega }}({\alpha }_{0}v)<{\mathrm{\Phi }}_{\mathrm{\Omega }}(v)\le {lim}_{n\to \mathrm{\infty }}{\mathrm{\Phi }}_{{\lambda }_n}(v_n)\le c(\mathrm{\Omega })$. A contradiction, and thus we complete the proof of Theorem 2.1. □