Ground state solutions for generalized quasilinear Schrödinger equations

Abstract

In this paper we consider the generalized quasilinear Schrödinger equations $\begin{matrix} - div (g^{2} (u) \nabla u) + g (u) g^{'} (u) | \nabla u |^{2} + V (x) u = h (x, u), x \in R^{N}, \end{matrix}$ where V and h are periodic in $x_{i}$ , $1 ⩽ i ⩽ N$ . By using variational methods, we prove the existence of ground state solutions, i.e., nontrivial solutions with least possible energy.

Keywords

Quasilinear Schrödinger equation ground state solution asymptotically linear Nehari manifold

1. Introduction

In this paper, we are concerned with the existence of solutions for the generalized quasilinear Schrödinger equations $\begin{matrix} (1.1) & - div (g^{2} (u) \nabla u) + g (u) g^{'} (u) | \nabla u |^{2} + V (x) u = h (x, u), x \in R^{N} \end{matrix}$ where $N ⩾ 3$ , V and h are periodic in $x_{i}$ , $1 ⩽ i ⩽ N$ , $g \in C^{1} (R)$ is an even positive function and $g^{'} (t) ⩾ 0$ for $t ⩾ 0$ .

These equations are related to the existence of solitary wave solutions for several nonlinear or quasilinear equations in mathematical physics. The most well-known examples are the nonlinear Schrödinger equations $\begin{matrix} i z_{t} = - Δ z + W (x) z + f (z), \end{matrix}$ and the nonlinear Klein–Gordon equations $\begin{matrix} z_{t t} - Δ z + W (x) z + f (z) = 0, \end{matrix}$ where z is a complex wave function, W is a real potential function. If we want to find their standing wave solutions $z (t, x) = exp (- i E t) u (x)$ , we get the nonlinear Schrödinger equation like this $\begin{matrix} (1.2) & - Δ u + V (x) u = h (x, u), x \in R^{N}, \end{matrix}$ which is a special case of (1.1) as $g (u) = 1$ . Equations (1.2) has been investigated widely by many researchers. We only mention the papers [1,15,22] and recall the main result in [11] for comparisons. We first list some assumptions, where and in the following $H (x, t) = \int_{0}^{t} h (x, τ) d τ$ .

V is continuous, 1-periodic in $x_{i}$ , $1 ⩽ i ⩽ N$ , and $V (x) ⩾ V_{0} > 0$ for all $x \in R^{N}$ .

$h (x, t) \in C^{1}$ , 1-periodic in $x_{i}$ , $1 ⩽ i ⩽ N$ , $h_{t}$ is a Caratheodory function and there exists $c > 0$ such that $| h (x, t) | ⩽ c (1 + | t |^{p - 1})$ for some $2 < p < 2^{*}$ , where $2^{*} : = 2 N / (N - 2)$ if $N ⩾ 3$ , $2^{*} : = \infty$ if $N = 1$ or 2.

${lim}_{t \to 0} \frac{h (x, t)}{t} = 0$ , uniformly in $x \in R^{N}$ .

${lim}_{t \to \infty} \frac{H (x, t)}{t^{2}} = \infty$ , uniformly in $x \in R^{N}$ .

$t \mapsto \frac{h (x, t)}{t}$ is strictly increasing on $(0, \infty)$ and strictly decreasing on $(- \infty, 0)$ .

Theorem 1.1 ([11]).

Suppose that $(V)$ and the above four conditions on $h (x, t)$ hold. Then equation ( 1.2 ) has a ground state solution.

The other examples include the equations $\begin{matrix} (1.3) & i z_{t} = - Δ z + W (x) z - h (| z |^{2}) z - Δ (l (| z |^{2})) l^{'} (| z |^{2}) z, \end{matrix}$ where z is a complex wave function, W is a real potential function and l, h are some real functions. They appear naturally in mathematical physics, for example in the superfluid film equation in plasma physics ([9]), in plasma physics and fluid mechanics, in the theory of Heisenberg ferromagnetism and magnons, in dissipative quantum mechanics and in condensed matter theory. For more details and references, see [12]. If we try to find the standing wave solutions $z (t, x) = exp (- i E t) u (x)$ of (1.3), we get $\begin{matrix} (1.4) & - Δ u + V (x) u - Δ (l (u^{2})) l^{'} (u^{2}) u = h (x, u), x \in R^{N}, \end{matrix}$ which is a special case of (1.1) as $g^{2} (u) = 1 + \frac{{[{(l (u^{2}))}^{'}]}^{2}}{2}$ , where l is a real function.

As $g^{2} (u) : = 1 + 2 u^{2}$ , i.e., $l (s) = s$ , (1.4) changes to the superfluid film equation in plasma physics $\begin{matrix} (1.5) & - Δ u + V (x) u - Δ (u^{2}) u = h (x, u), x \in R^{N} . \end{matrix}$ As $g^{2} (u) : = 1 + \frac{u^{2}}{2 (1 + u^{2})}$ , i.e., $l (s) = {(1 + s)}^{1 / 2}$ , (1.4) changes to the equation $\begin{matrix} (1.6) & - Δ u + V (x) u - [Δ {(1 + u^{2})}^{1 / 2}] \frac{u}{2 {(1 + u^{2})}^{1 / 2}} = h (x, u), x \in R^{N} \end{matrix}$ which models the self-channeling of a high-power ultrashort laser in matter.

As far as we know, the first existence result for (1.5) was obtained in [14] by using variational techniques. Then a constrained minimization argument was used in [13] to prove the existence of a positive solution. In [12], the quasilinear equation (1.5) was transformed to a semilinear one by a change of variables and an Orlicz space framework was used. Subsequently, a slightly different change of variables was given in [2], where the usual Sobolev space framework was used as the working space. Since then, many researches have investigated equation (1.5) by various techniques in variational methods, e.g. see [6,8,16,21]. Particularly, Fang and Szulkin([7]) obtained the following result.

Theorem 1.2 ([7]).

Suppose that $(V)$ , $(h_{2})$ and the following conditions are satisfied.

$h (x, t) \in C^{1}$ , 1-periodic in $x_{i}$ , $1 ⩽ i ⩽ N$ and there exists $c > 0$ such that $| h (x, t) | ⩽ c (1 + | t |^{p - 1})$ for some $4 < p < 2 \cdot 2^{*}$ , where $2^{*} : = 2 N / (N - 2)$ if $N ⩾ 3$ , $2^{*} : = \infty$ if $N = 1$ or 2.

${lim}_{t \to \infty} \frac{H (x, t)}{t^{4}} = \infty$ , uniformly in $x \in R^{N}$ .

$t \mapsto \frac{h (x, t)}{t^{3}}$ is increasing on $(0, \infty)$ and decreasing on $(- \infty, 0)$ .

Then equation ( 1.5 ) has a ground state solution.

There are a few results in the literature for equation (1.6), e.g. see [18] where the existence of a positive solution was obtained for the asymptotically linear case. In [26], the following result was proved.

Theorem 1.3.
Assume that $N ⩾ 3$ and $12 - 4 \sqrt{6} < r + 1 < 2^{}$ . Then equation (* 1.6 ) where $h (x, u) = | u |^{r - 1} u$ has a ground state solution.

Recently, by introducing a new variable replacement,the authors in [17] proved the existence of a nontrivial solution for equation (1.1) by the mountain pass theorem, where a general Ambrosetti–Rabinowitz condition was used. Subsequently, the existence of a positive solution for (1.1) with a critical growth was shown in [4] (see also [5,19]). Nodal radial solutions were constructed in [3] for (1.1) with a radially symmetric potential. For potentials vanishing at infinity, the weighted Sobolev space was used in [20]. For results about some variations of equation (1.1), see [10,27] and the references therein. To the best of our knowledge, there is still no result for equations (1.1) with periodic potentials and subcritical nonlinearities.

Motivated by the arguments in [7,23,24], we consider the problem $\begin{matrix} (1.7) & - div (g^{2} (u) \nabla u) + g (u) g^{'} (u) | \nabla u |^{2} + V (x) u = h (x, u), u \in H^{1} (R^{N}), \end{matrix}$ where the potential $V (x)$ is periodic and the nonlinear term $h (x, u)$ is subcritical for u.

We list the following assumptions on g and h, where $G (t) : = \int_{0}^{t} g (τ) d τ$ .

$g \in C^{1} (R)$ is an even function, $g^{'} (t) ⩾ 0$ for $t ⩾ 0$ and $g (0) = 1$ .

$h (x, t)$ is continuous, 1-periodic in $x_{i}$ , $1 ⩽ i ⩽ N$ and $| h (x, t) | ⩽ c (1 + g (t) | G (t) |^{p - 1})$ for some $c > 0$ and $2 < p < 2^{}$ , where $2^{} : = 2 N / (N - 2)$ if $N ⩾ 3$ , $2^{} : = \infty$ if $N = 1$ or 2.

${lim}_{t \to \infty} \frac{H (x, t)}{| G (t) |^{2}} = \infty$ , uniformly in $x \in R^{N}$ .

$t \mapsto \frac{h (x, t)}{g (t) G (t)}$ is strictly increasing on $(0, \infty)$ and strictly decreasing on $(- \infty, 0)$ .

We obtain the following result.
Theorem 1.4.
Suppose that* $(V)$ , $(g)$ and $(h_{1})$ – $(h_{4})$ hold. Then problem ( 1.7 ) has a ground state solution. Moreover, if h is odd in t, then it admits infinitely many solutions.
Remark 1.1.
Theorem 1.4 as $g^{2} (u) : = 1$ or $g^{2} (u) : = 1 + 2 u^{2}$ contains the above Theorem 1.1 or Theorem 1.2 respectively. We check the second claim and in fact only indicate that $\frac{h (x, u)}{g (u) G (u)}$ is strictly increasing on $(0, \infty)$ under condition $(h_{4}^{″})$ . By a direct computation, we get $G (u) ⩾ \frac{1}{2} u \sqrt{1 + 2 u^{2}}$ and hence ${(\frac{u^{2}}{G (u)})}^{'} ⩾ 0$ for $u ⩾ 0$ . Therefore, $\frac{u^{3}}{g (u) G (u)}$ is strictly increasing on $(0, \infty)$ . This completes the check. Theorem 1.4 as $g^{2} (u) : = 1 + \frac{u^{2}}{2 (1 + u^{2})}$ also contains Theorem 1.3 since $\frac{t^{r}}{\sqrt{\frac{2 + 3 t^{2}}{2 + 2 t^{2}}} \int_{0}^{t} \sqrt{\frac{2 + 3 s^{2}}{2 + 2 s^{2}}} d s}$ is strictly increasing on $(0, \infty)$ for $11 - 4 \sqrt{6} ⩽ r < 2^{} - 1$ (r can equal to the left number). In fact, by a direct computation $\frac{2 t^{ε} + 2 t^{2 + ε}}{2 + 3 t^{2}}$ is strictly increasing on $(0, \infty)$ for $ε ⩾ 4 \sqrt{6} + 10$ and $\frac{t^{r - ε / 2}}{\int_{0}^{t} \sqrt{\frac{2 + 3 s^{2}}{2 + 2 s^{2}}} d s}$ is positive and increasing on $(0, \infty)$ for $r ⩾ \frac{ε}{2} + 6 - 2 \sqrt{6} = 11 - 4 \sqrt{6}$ .

We also consider the case where the nonlinearity is asymptotically linear at infinity and obtain the following result.
${lim}_{t \to \infty} \frac{h (x, t)}{g (t) G (t)} = q (x)$ , where $q (x)$ is continuous, 1-periodic in $x_{i}$ , $1 ⩽ i ⩽ N$ , and $q (x) > \frac{V (x)}{g^{2} (\infty)}$ for all $x \in R^{N}$ , where $g (\infty) : = {lim}_{t \to \infty} g (t)$ and $\frac{1}{g (\infty)} : = 0$ if g is unbounded.
Theorem 1.5.
Suppose that* $(V)$ , $(g)$ , $(h_{1})$ , $(h_{2})$ , $(h_{3} )$ and* $(h_{4})$ hold. Then problem ( 1.7 ) has a ground state solution. Moreover, if h is odd in t, then it admits infinitely many solutions.
Remark 1.2.
As far as we know, there is still no result for problem (1.7) when the nonlinear term is asymptotically linear at infinity.
Notation.
$C, C_{1}, C_{2}, \dots$ will denote different positive constants whose exact value is inessential. $B_{ρ} (y) : = {x \in R^{N} : | x - y | < ρ}$ . E denotes the Sobolev space $H^{1} (R^{N})$ with the norm $\begin{matrix} ‖ u ‖ : = {(\int_{R^{N}} | \nabla u |^{2} + V (x) u^{2})}^{1 / 2} \end{matrix}$ and S is the unit sphere in E. For $y \in Z^{N}$ , let $y * u$ for the action of $Z^{N}$ on E given by $(y * u) (x) : = u (x - y)$ .

2. Preliminary results

Equation (1.7) is formally the Euler–Lagrange equation associated with the natural functional $\begin{matrix} I (u) = \frac{1}{2} \int_{R^{N}} g^{2} (u) | \nabla u |^{2} + V (x) u^{2} - \int_{R^{N}} H (x, u) \end{matrix}$ which may be not well defined in E. So we make a change of variables introduced by [17] $\begin{matrix} v = G (u) = \int_{0}^{u} g (t) d t . \end{matrix}$ Then we have $\begin{matrix} (2.1) & J (v) = \frac{1}{2} \int_{R^{N}} | \nabla v |^{2} + V (x) {| G^{- 1} (v) |}^{2} - \int_{R^{N}} H (x, G^{- 1} (v)) . \end{matrix}$ It follows from $(h_{1})$ , $(h_{2})$ and Lemma 2.1 below that $J \in C^{1} (E, R)$ . According to $(V)$ and $(h_{1})$ , J is invariant with respect to the action of $Z^{N}$ . Obviously, we have $\begin{matrix} (2.2) & ⟨ J^{'} (v), φ ⟩ = \int_{R^{N}} \nabla v \nabla φ + V (x) \frac{G^{- 1} (v)}{g (G^{- 1} (v))} φ - \int_{R^{N}} \frac{h (x, G^{- 1} (v))}{g (G^{- 1} (v))} φ \end{matrix}$ for all $v, φ \in E$ and the critical point of J is the weak solution of the equation $\begin{matrix} (2.3) & - Δ v + V (x) \frac{G^{- 1} (v)}{g (G^{- 1} (v))} = \frac{h (x, G^{- 1} (v))}{g (G^{- 1} (v))}, v \in H^{1} (R^{N}) . \end{matrix}$ It was shown in [17] that $u \in E$ is a solution of (1.7) if and only if $v = G (u) \in E$ is a solution of (2.3).

Let $M : = {v \in E ∖ {0} : ⟨ J^{'} (v), v ⟩ = 0}$ and note that $M$ is called the Nehari manifold.

According to assumption $(g)$ , it is easy to have the following properties whose proofs can be found in [17] and Lemma 2.1 in [4].

Lemma 2.1.
The functions g and G satisfy the following properties:
$G (t)$ and $G^{- 1} (s)$ are odd;

$G (t) ⩽ g (t) t$ , $G^{- 1} (s) ⩽ s$ , for all $t, s ⩾ 0$ ;

$\frac{G^{- 1} (s)}{s}$ is nonincreasing for $s ⩾ 0$ ;

${lim}_{s \to 0} \frac{G^{- 1} (s)}{s} = 1$ , ${lim}_{s \to \infty} \frac{G^{- 1} (s)}{s} = \frac{1}{g (\infty)}$ .

3. Proof of Theorem 1.4

Lemma 3.1.

$H (x, t) ⩾ 0$ , $2 g (t) H (x, t) ⩽ G (t) h (x, t)$ for $t \in R$ .

For every $ε > 0$ , there is $C_{ε} > 0$ such that $| h (x, t) | ⩽ ε | t | + C_{ε} g (t) | G (t) |^{p - 1}$ for all $t \in R$ .

Proof.
(1) It follows from $(h_{2})$ and $(h_{4})$ that $\frac{h (x, t)}{g (t) G (t)} ⩾ 0$ for all $t \in R$ . Using $(g)$ and Lemma 2.1-(1), we get that $h (x, t) ⩾ 0$ for $t > 0$ and $h (x, t) ⩽ 0$ for $t < 0$ . So $H (x, t) ⩾ 0$ for $t \in R$ .

By $(h_{4})$ , we have $\begin{matrix} H (x, t) = \int_{0}^{t} \frac{h (x, τ)}{g (τ) G (τ)} \cdot g (τ) G (τ) d τ ⩽ \frac{h (x, t)}{g (t) G (t)} \int_{0}^{t} \frac{1}{2} \frac{d G^{2} (τ)}{d τ} d τ = \frac{h (x, t) G (t)}{2 g (t)}, \end{matrix}$ that is, $2 g (t) H (x, t) ⩽ G (t) h (x, t)$ .

(2) It follows immediately from $(h_{1})$ and $(h_{2})$ . □

Set $h_{v} (t) : = J (t v)$ , for $t > 0$ and $v \in E$ .
Lemma 3.2.
There exists a unique $t_{v} > 0$ such that $h_{v}^{'} (t) > 0$ for $0 < t < t_{v}$ and $h_{v}^{'} (t) < 0$ for $t > t_{v}$ , where $v \neq 0$ . Moreover, $t v \in M$ if and only if $t = t_{v}$ .
Proof.
Using $(h_{2})$ , we get that ${lim}_{t \to 0} \frac{H (x, t)}{| G (t) |^{2}} = 0$ , uniformly in $x \in R^{N}$ . Note that $\begin{matrix} \frac{h_{v} (t)}{t^{2}} = \frac{1}{2} \int_{R^{N}} | \nabla v |^{2} + \frac{1}{2} \int_{v \neq 0} V (x) \frac{| G^{- 1} (t v) |^{2}}{t^{2} v^{2}} v^{2} - \int_{v \neq 0} \frac{H (x, G^{- 1} (t v))}{t^{2} v^{2}} v^{2} . \end{matrix}$ By Lemma 2.1-(4), we have $\begin{matrix} lim_{t \to 0} \frac{h_{v} (t)}{t^{2}} = \frac{1}{2} \int_{R^{N}} | \nabla v |^{2} + V (x) v^{2} > 0 . \end{matrix}$ It follows from Lemma 2.1-(2) and $(h_{3})$ that $\begin{matrix} \frac{h_{v} (t)}{t^{2}} ⩽ \frac{1}{2} \int_{R^{N}} | \nabla v |^{2} + V (x) v^{2} - \int_{v \neq 0} \frac{H (x, G^{- 1} (t v))}{t^{2} v^{2}} v^{2} \to - \infty, \end{matrix}$ as $t \to \infty$ . So $h_{v} (t)$ has a positive maximum. Setting $h_{v}^{'} (t) = ⟨ J^{'} (t v), v ⟩ = 0$ , we have $\begin{matrix} \int_{R^{N}} | \nabla v |^{2} + \int_{v \neq 0} V (x) \frac{G^{- 1} (t v)}{g (G^{- 1} (t v)) t v} v^{2} = \int_{v \neq 0} \frac{h (x, G^{- 1} (t v))}{g (G^{- 1} (t v)) t v} v^{2} . \end{matrix}$ According to $(g)$ and Lemma 2.1-(3), $\begin{matrix} \frac{G^{- 1} (t v)}{g (G^{- 1} (t v)) t v} = \frac{G^{- 1} (t v)}{t v} \cdot \frac{1}{g (G^{- 1} (t v))} \end{matrix}$ is strictly decreasing with respect to $t > 0$ . Using $(h_{4})$ , we obtain that $\frac{h (x, G^{- 1} (t v))}{g (G^{- 1} (t v)) t v}$ is strictly increasing for $t > 0$ . Hence the first conclusion holds. The second conclusion follows since $h_{v}^{'} (t) = t^{- 1} ⟨ J^{'} (t v), t v ⟩$ . □
Lemma 3.3.

There exist $ρ_{0}, a_{0} > 0$ such that $c : = {inf}_{M} J ⩾ {inf}_{S_{ρ_{0}}} J ⩾ a_{0}$ , where $S_{ρ_{0}} : = {v \in E : ‖ v ‖ = ρ_{0}}$ .

$‖ v ‖^{2} ⩾ 2 c$ for all $v \in M$ .

Proof.
(1) It is easy to prove that ${inf}_{S_{ρ_{0}}} J ⩾ a_{0}$ , for some constant $a_{0} > 0$ (see Lemma 2.1 in [17]), so we omit the proof. For every $v \in M$ , there exists $t_{0} > 0$ such that $t_{0} v \in S_{ρ_{0}}$ . By Lemma 3.2, $J (v) ⩾ J (t_{0} v)$ . Hence the first inequality follows.

(2) Using Lemma 2.1-(2) and Lemma 3.1-(1), we have $\begin{aligned} c & ⩽ \frac{1}{2} \int_{R^{N}} | \nabla v |^{2} + V (x) {| G^{- 1} (v) |}^{2} - \int_{R^{N}} H (x, G^{- 1} (v)) \\ ⩽ \frac{1}{2} \int_{R^{N}} | \nabla v |^{2} + V (x) v^{2} . \end{aligned}$ □
Lemma 3.4.
If $V \subset E ∖ {0}$ is a compact subset, there exists $R > 0$ such that $J (v) ⩽ 0$ on $(R^{+} V) ∖ B_{R} (0)$ .
Proof.
Arguing by contradiction, without loss of generality, assume that $V \subset S$ and there exist $v_{n} \in V$ and $t_{n} \to \infty$ such that $J (t_{n} v_{n}) ⩾ 0$ for every n. Going if necessary to a subsequence, assume that $v_{n} \to v$ in E. We have by Lemma 2.1-(2) and $(h_{3})$ , $\begin{matrix} 0 ⩽ \frac{J (t_{n} v_{n})}{t_{n}^{2}} ⩽ \frac{1}{2} - \int_{v_{n} \neq 0} \frac{H (x, G^{- 1} (t_{n} v_{n}))}{t_{n}^{2} v_{n}^{2}} v_{n}^{2} \to - \infty, \end{matrix}$ a contradiction. □
Lemma 3.5.
$J (v) \to \infty$ as $‖ v ‖ \to \infty$ with $v \in M$ .
Proof.
Arguing by contradiction, suppose that $‖ v_{n} ‖ \to \infty$ with $v_{n} \in M$ such that $J (v_{n}) ⩽ d$ . Set $w_{n} : = \frac{v_{n}}{‖ v_{n} ‖}$ . After passing to a subsequence, we have $w_{n} ⇀ w$ in E and $w_{n} \to w$ a.e. in $R^{N}$ .

If $\begin{matrix} lim_{n \to \infty} sup_{y \in R^{N}} \int_{B_{1} (y)} w_{n}^{2} \to 0, \end{matrix}$ then it follows from P.L. Lions’ lemma (see Lemma 1.21 in [25]) that $w_{n} \to 0$ in $L^{p} (R^{N})$ for $2 < p < 2^{}$ . By Lemma 3.1-(2) and Lemma 2.1-(2), we have $\int_{R^{N}} H (x, G^{- 1} (t w_{n})) \to 0$ for every $t \in R$ . Using Lemma 2.1-(4), $\begin{matrix} lim_{s \to 0} \frac{s^{2} - | G^{- 1} (s) |^{2}}{s^{2}} = 0 and lim_{s \to \infty} \frac{s^{2} - | G^{- 1} (s) |^{2}}{s^{p}} = 0 . \end{matrix}$ Then for every $ε > 0$ , there is $C_{ε} > 0$ such that $| s^{2} - | G^{- 1} (s) |^{2} | ⩽ ε s^{2} + C_{ε} s^{p}$ . Hence $\int_{R^{N}} V (x) (t^{2} w_{n}^{2} - | G^{- 1} (t w_{n}) |^{2}) \to 0$ by $(V)$ . Note that $J (t w_{n}) ⩽ J (v_{n}) ⩽ d$ by Lemma 3.2 and $\begin{aligned} J (t w_{n}) & = \frac{t^{2}}{2} \int_{R^{N}} | \nabla w_{n} |^{2} + \frac{1}{2} \int_{R^{N}} V (x) {| G^{- 1} (t w_{n}) |}^{2} - \int_{R^{N}} H (x, G^{- 1} (t w_{n})) \\ = \frac{t^{2}}{2} ‖ w_{n} ‖^{2} - \frac{1}{2} \int_{R^{N}} V (x) (t^{2} w_{n}^{2} - {| G^{- 1} (t w_{n}) |}^{2}) - \int_{R^{N}} H (x, G^{- 1} (t w_{n})) . \end{aligned}$ Taking t large enough, it has a contradiction.

Then there is a sequence $(y_{n}) \subset R^{N}$ and $δ > 0$ such that $\begin{matrix} \underset{n \to \infty}{lim inf} \int_{B_{1} (y_{n})} w_{n}^{2} ⩾ δ . \end{matrix}$ Since J and $M$ are invariant under translations of the form $w \mapsto w (\cdot - k)$ with $k \in Z^{N}$ , we can assume that $(y_{n})$ is bounded in $R^{N}$ . Hence $w \neq 0$ by the local compactness of the Sobolev embedding theorem. By $v_{n} (x) \to \infty$ if $w (x) \neq 0$ , we have $\begin{matrix} 0 ⩽ \frac{J (v_{n})}{‖ v_{n} ‖} ⩽ \frac{1}{2} - \int_{v_{n} \neq 0} \frac{H (x, G^{- 1} (v_{n}))}{v_{n}^{2}} w_{n}^{2} \to - \infty, \end{matrix}$ in view of Lemma 2.1-(2) and $(h_{3})$ . We get a contradiction. □

Define $m : S \mapsto M$ by $m (w) : = t_{w} w$ , where $t_{w}$ is in Lemma 3.2. According to Proposition 8 and Corollary 10 in [24], we have the following two lemmas.
Lemma 3.6.
The map m is a homeomorphism with the inverse given by* $m^{- 1} (v) = \frac{v}{‖ v ‖}$ .

Consider the functional $Ψ : S \to R$ given by $Ψ (w) : = J (m (w))$ . Lemma 3.7.

$Ψ \in C^{1} (S, R)$ and $\begin{matrix} ⟨ Ψ^{'} (w), z ⟩ = ‖ m (w) ‖ ⟨ J^{'} (m (w)), z ⟩ for all z \in T_{w} (S) . \end{matrix}$

If $(w_{n})$ is a Palais–Smale sequence for Ψ, then $(m (w_{n}))$ is a Palais–Smale sequence for J. If $(v_{n}) \subset M$ is a bounded Palais–Smale sequence for J, then $(m^{- 1} (v_{n}))$ is a Palais–Smale sequence for Ψ.

w is a critical point of Ψ if and only if $m (w)$ is a nontrivial critical point of J. Moreover, the corresponding values of Ψ and J coincide and ${inf}_{S} Ψ = {inf}_{M} J$ .

Proof of Theorem 1.4.
By Ekeland’s variational principle, there is a sequence $(w_{n}) \subset S$ such that $Ψ (w_{n}) \to c$ and $Ψ^{'} (w_{n}) \to 0$ , where c is as in Lemma 3.3-(1). Set $v_{n} = m (w_{n}) \in M$ . Then $J (v_{n}) \to c$ and $J^{'} (v_{n}) \to 0$ . By Lemma 3.5, $(v_{n})$ is bounded in E. Passing to a subsequence, $v_{n} ⇀ v$ in E and $v_{n} \to v$ a.e. in $R^{N}$ .

If $\begin{matrix} lim_{n \to \infty} sup_{y \in R^{N}} \int_{B_{1} (y)} v_{n}^{2} \to 0, \end{matrix}$ it follows from P.L. Lions’ lemma that $v_{n} \to 0$ in $L^{p} (R^{N})$ , $2 < p < 2^{*}$ . Using Lemma 3.1-(2) and Lemma 2.1-(2), we have $\begin{matrix} \int_{v_{n} \neq 0} \frac{h (x, G^{- 1} (v_{n}))}{g (G^{- 1} (v_{n}))} v_{n} \to 0 . \end{matrix}$ Since $\begin{matrix} lim_{s \to 0} \frac{s^{2} - \frac{G^{- 1} (s)}{g (G^{- 1} (s))} s}{s^{2}} = 0 and lim_{s \to \infty} \frac{s^{2} - \frac{G^{- 1} (s)}{g (G^{- 1} (s))} s}{s^{p}} = 0 \end{matrix}$ by $(g)$ and Lemma 2.1-(4), for every $ε > 0$ there is $C_{ε} > 0$ such that $\begin{matrix} (3.1) & | s^{2} - \frac{G^{- 1} (s)}{g (G^{- 1} (s))} s | ⩽ ε s^{2} + C_{ε} s^{p} . \end{matrix}$ Then $\begin{matrix} \int_{v_{n} \neq 0} V (x) [v_{n}^{2} - \frac{G^{- 1} (v_{n})}{g (G^{- 1} (v_{n}))} v_{n}] \to 0, \end{matrix}$ as $n \to \infty$ . Note that $\begin{aligned} o (1) = J^{'} (v_{n}) v_{n} & = \int_{v_{n} \neq 0} | \nabla v_{n} |^{2} + V (x) \frac{G^{- 1} (v_{n})}{g (G^{- 1} (v_{n}))} v_{n} - \int_{v_{n} \neq 0} \frac{h (x, G^{- 1} (v_{n}))}{g (G^{- 1} (v_{n}))} v_{n} \\ = ‖ v_{n} ‖^{2} - \int_{v_{n} \neq 0} V (x) [v_{n}^{2} - \frac{G^{- 1} (v_{n})}{g (G^{- 1} (v_{n}))} v_{n}] + o (1) \\ = ‖ v_{n} ‖^{2} + o (1), \end{aligned}$ where $o (1) \to 0$ as $n \to \infty$ . Hence $‖ v_{n} ‖ \to 0$ , a contradiction with $c > 0$ .

Then there is $(y_{n}) \subset R^{N}$ and $δ > 0$ such that ${lim inf}_{n \to \infty} \int_{B_{1} (y_{n})} v_{n}^{2} ⩾ δ$ . Since J and $M$ are invariant under translations of the form $w \mapsto w (\cdot - k)$ with $k \in Z^{N}$ , we may assume that $(y_{n})$ is bounded in $R^{N}$ . Then $v_{n} ⇀ v \neq 0$ and $J^{'} (v) = 0$ (cf: P.199 in [17]).

Setting $G (t) = s$ , for $s > 0$ , we have $s ⩽ g (G^{- 1} (s)) G^{- 1} (s)$ by Lemma 2.1-(2). Using Lemma 2.1-(1) and the definition of G, it is easy to prove that $\begin{matrix} {| G^{- 1} (s) |}^{2} - \frac{G^{- 1} (s)}{g (G^{- 1} (s))} s ⩾ 0 \end{matrix}$ for $s \in R$ . Note that $c + o (1) = J (v_{n}) - \frac{1}{2} J^{'} (v_{n}) v_{n}$ and we have by Lemma 3.1-(1) and Fatou’s lemma, $\begin{aligned} c + o (1) & = \int_{v_{n} \neq 0} V (x) [{| G^{- 1} (v_{n}) |}^{2} - \frac{G^{- 1} (v_{n})}{g (G^{- 1} (v_{n}))} v_{n}] \\ + \int_{v_{n} \neq 0} [\frac{h (x, G^{- 1} (v_{n}))}{2 g (G^{- 1} (v_{n}))} v_{n} - H (x, G^{- 1} (v_{n}))] \\ ⩾ J (v) - \frac{1}{2} J^{'} (v) v + o (1) = J (v) + o (1), \end{aligned}$ where $o (1) \to 0$ as $n \to \infty$ . Since $v \in M$ , $J (v) = c$ .

By (3.1), it is not difficult to prove the discreteness of Palais–Smale sequences (see Lemma 2.14 in [23] and Lemma 3.13 in [7]). If h is odd in t, the existence of infinitely many solutions can be obtained by using the similar arguments as in [7,23]. We omit the details. □

4. Proof of Theorem 1.5

Set $E : = {v \in E : \int_{R^{N}} | \nabla v |^{2} + \int_{R^{N}} \frac{V (x)}{g^{2} (\infty)} v^{2} < \int_{R^{N}} q (x) v^{2}}$ . It follows from $(h_{3} *)$ that $E \neq \emptyset$ .

Lemma 4.1.

For every $v \in E$ , there is a unique $t_{v} > 0$ such that $h_{v}^{'} (t) > 0$ for $0 < t < t_{v}$ and $h_{v}^{'} (t) < 0$ for $t > t_{v}$ . Moreover, $t v \in M$ if and only if $t = t_{v}$ .

If $v \notin E$ , $t v \notin M$ for any $t > 0$ .

Proof.

(1) Note that $\begin{matrix} \frac{h_{v} (t)}{t^{2}} = \frac{1}{2} \int_{R^{N}} | \nabla v |^{2} + \frac{1}{2} \int_{v \neq 0} V (x) \frac{| G^{- 1} (t v) |^{2}}{t^{2} v^{2}} v^{2} - \int_{v \neq 0} \frac{H (x, G^{- 1} (t v))}{t^{2} v^{2}} v^{2} . \end{matrix}$ Using the same arguments as in Lemma 3.2, we have $\begin{matrix} lim_{t \to 0} \frac{h_{v} (t)}{t^{2}} = \frac{1}{2} \int_{R^{N}} | \nabla v |^{2} + V (x) v^{2} > 0 . \end{matrix}$ Using $(h_{3} *)$ , we have ${lim}_{t \to \infty} \frac{H (x, t)}{| G (t) |^{2}} = \frac{1}{2} q (x)$ . Then by Lemma 2.1-(4), $\begin{matrix} lim_{t \to \infty} \frac{h_{v} (t)}{t^{2}} = \frac{1}{2} \int_{R^{N}} | \nabla v |^{2} + \frac{V (x)}{g^{2} (\infty)} v^{2} - \frac{1}{2} \int_{R^{N}} q (x) v^{2} < 0 . \end{matrix}$ Note that $h_{v}^{'} (t) = 0$ is equivalent to $\begin{matrix} \int_{R^{N}} | \nabla v |^{2} + \int_{v \neq 0} V (x) \frac{G^{- 1} (t v)}{g (G^{- 1} (t v)) t v} v^{2} = \int_{v \neq 0} \frac{h (x, G^{- 1} (t v))}{g (G^{- 1} (t v)) t v} v^{2} . \end{matrix}$ By a similar argument as in Lemma 3.2, we finish the proof.

(2) It follows from $(g)$ and Lemma 2.1-(1),(3) that $\frac{t}{g (t) G (t)} = \frac{1}{g (t)} \cdot \frac{t}{G (t)}$ is nonincreasing on $(0, \infty)$ and nondecreasing on $(- \infty, 0)$ . Thus $\begin{matrix} \frac{t}{g (t) G (t)} ⩾ \frac{1}{g^{2} (\infty)}, \end{matrix}$ for every $t \in R$ . If there is $t_{0} > 0$ such that $t_{0} v \in M$ , we have, under conditions $(h_{3} *)$ and $(h_{4})$ , $\begin{aligned} \int_{R^{N}} | \nabla v |^{2} + \frac{V (x)}{g^{2} (\infty)} v^{2} & ⩽ \int_{R^{N}} | \nabla v |^{2} + \int_{v \neq 0} V (x) \frac{G^{- 1} (t_{0} v)}{g (G^{- 1} (t_{0} v)) t_{0} v} v^{2} \\ < \int_{R^{N}} q (x) v^{2} . \end{aligned}$ Hence $v \in E$ , a contradiction. □

The following lemma is similar as Lemma 3.3, so we omit the proof.

Lemma 4.2.

There is $ρ_{0} > 0$ and $a_{0} > 0$ such that $c : = {inf}_{M} J ⩾ {inf}_{S_{ρ_{0}}} J ⩾ a_{0}$ , where $S_{ρ_{0}} : = {v \in E : ‖ v ‖ = ρ_{0}}$ .

$‖ v ‖^{2} ⩾ 2 c$ for all $v \in M$ .

Lemma 4.3.

If $V \subset E$ is a compact subset, then there is $R > 0$ such that $J (v) ⩽ 0$ on $(R^{+} V) ∖ B_{R} (0)$ .

Proof.

Without loss of generality, we assume $V \subset S$ . Arguing indirectly, there is $v_{n} \in V$ and $t_{n} > 0$ such that $J (t_{n} v_{n}) ⩾ 0$ with $v_{n} \to v \in V$ and $t_{n} \to \infty$ . It follows from Lemma 2.1-(4), $(h_{3} *)$ and the Lebesgue dominated convergence theorem that $\begin{aligned} 0 ⩽ \frac{J (t_{n} v_{n})}{t_{n}^{2}} & = \frac{1}{2} \int_{R^{N}} | \nabla v_{n} |^{2} + \frac{1}{2} \int_{v_{n} \neq 0} V (x) \frac{| G^{- 1} (t_{n} v_{n}) |^{2}}{t_{n}^{2} v_{n}^{2}} v_{n}^{2} \\ - \int_{v_{n} \neq 0} \frac{H (x, G^{- 1} (t_{n} v_{n}))}{t_{n}^{2} v_{n}^{2}} v_{n}^{2} \\ \to \frac{1}{2} \int_{R^{N}} | \nabla v |^{2} + \frac{V (x)}{g^{2} (\infty)} v^{2} - \frac{1}{2} \int_{R^{N}} q (x) v^{2} < 0, \end{aligned}$ a contradiction. □

Lemma 4.4.

All Palais–Smale sequences $(v_{n})$ in $M$ are bounded in E.

Proof.

Arguing by contradiction, assume $‖ v_{n} ‖ \to \infty$ and set $w_{n} : = \frac{v_{n}}{‖ v_{n} ‖}$ . Up to a subsequence, $w_{n} ⇀ w$ in E and $w_{n} \to w$ a.e. in $R^{N}$ . Using a similar argument as in Lemma 3.5, we have $w \neq 0$ . So $v_{n} (x) \to \infty$ if $w (x) \neq 0$ . Since $⟨ \frac{J^{'} (v_{n})}{‖ v_{n} ‖}, φ ⟩ \to 0$ for all $φ \in C_{0}^{\infty} (R^{N})$ , $\begin{matrix} \int_{R^{N}} \nabla w_{n} \nabla φ + \int_{v_{n} \neq 0} V (x) \frac{G^{- 1} (v_{n})}{g (G^{- 1} (v_{n})) v_{n}} w_{n} φ = \int_{v_{n} \neq 0} \frac{h (x, G^{- 1} (v_{n}))}{g (G^{- 1} (v_{n})) v_{n}} w_{n} φ + o (1), \end{matrix}$ where $o (1) \to 0$ as $n \to \infty$ . We get by the Lebesgue dominated convergence theorem, $\begin{matrix} \int_{R^{N}} \nabla w \nabla φ + \frac{V (x)}{g^{2} (\infty)} w φ = \int_{R^{N}} q (x) w φ . \end{matrix}$ Since $- Δ - (q (x) - \frac{V (x)}{g^{2} (\infty)})$ has only absolutely continuous spectrum by $(V)$ and $(h_{3} *)$ , a contradiction. □

Lemma 4.5.

$v \mapsto T (x, v) : = \frac{h (x, v) G (v)}{2 g (v)} - H (x, v)$ is increasing on $(0, \infty)$ and decreasing on $(- \infty, 0)$ . Moreover, there is $δ > 0$ such that $T (x, v) ⩾ δ$ , for large $v \in R$ .

Proof.

We only consider the case $v > 0$ . For $0 < t_{1} < t_{2}$ , we have by $(h_{4})$ $\begin{aligned} T (x, t_{2}) - T (x, t_{1}) & = \frac{h (x, t_{2}) G (t_{2})}{2 g (t_{2})} - \frac{h (x, t_{1}) G (t_{1})}{2 g (t_{1})} - H (x, t_{2}) + H (x, t_{1}) \\ = \int_{0}^{t_{2}} \frac{h (x, t_{2})}{2 g (t_{2}) G (t_{2})} \frac{d G^{2} (τ)}{d τ} d τ - \int_{0}^{t_{1}} \frac{h (x, t_{1})}{2 g (t_{1}) G (t_{1})} \frac{d G^{2} (τ)}{d τ} d τ \\ - \int_{t_{1}}^{t_{2}} h (x, τ) d τ \\ = \int_{0}^{t_{2}} \frac{h (x, t_{2}) g (τ) G (τ)}{g (t_{2}) G (t_{2})} d τ - \int_{0}^{t_{1}} \frac{h (x, t_{1}) g (τ) G (τ)}{g (t_{1}) G (t_{1})} d τ \\ - \int_{t_{1}}^{t_{2}} h (x, τ) d τ \\ = \int_{0}^{t_{1}} [\frac{h (x, t_{2})}{g (t_{2}) G (t_{2})} - \frac{h (x, t_{1})}{g (t_{1}) G (t_{1})}] g (τ) G (τ) d τ \\ + \int_{t_{1}}^{t_{2}} [\frac{h (x, t_{2})}{g (t_{2}) G (t_{2})} - \frac{h (x, τ)}{g (τ) G (τ)}] g (τ) G (τ) d τ \\ > 0 . \end{aligned}$ So there exists $δ > 0$ such that $T (x, v) ⩾ δ$ , for large v. □

Set $U : = E \cap S$ and define the map $m : U \mapsto M$ by $m (w) : = t_{w} w$ , where $t_{w}$ is as in Lemma 4.1-(1). Since $E$ is an open subset in E, the set U is open in S.

Lemma 4.6.

Assume $w_{n} \in U$ , $w_{n} \to w_{0} \in \partial U$ and $t_{n} w_{n} \in M$ , then $t_{n} \to \infty$ and $J (t_{n} w_{n}) \to \infty$ .

Proof.

If $(t_{n})$ is bounded, up to a subsequence, $t_{n} \to t_{0} ⩾ 2 c > 0$ by Lemma 4.2-(2). Since $0 = ⟨ J^{'} (t_{n} w_{n}), w_{n} ⟩ \to ⟨ J^{'} (t_{0} w_{0}), w_{0} ⟩$ , we have $t_{0} w_{0} \in M$ . This is a contradiction to $w_{0} \in \partial U$ .

Note that $\begin{matrix} \int_{R^{N}} | \nabla w_{0} |^{2} + \int_{R^{N}} \frac{V (x)}{g^{2} (\infty)} w_{0}^{2} = \int_{R^{N}} q (x) w_{0}^{2} \end{matrix}$ and we have $| G^{- 1} (s) |^{2} ⩾ \frac{1}{g^{2} (\infty)} s^{2}$ by Lemma 2.1-(1),(3),(4). It follows from $(h_{3} *)$ and $(h_{4})$ that $\begin{aligned} J (t w_{0}) & = \frac{t^{2}}{2} \int_{R^{N}} | \nabla w_{0} |^{2} + \frac{1}{2} \int_{R^{N}} V (x) {| G^{- 1} (t w_{0}) |}^{2} - \int_{R^{N}} H (x, G^{- 1} (t w_{0})) \\ = \frac{1}{2} \int_{R^{N}} V (x) [{| G^{- 1} (t w_{0}) |}^{2} - \frac{t^{2} w_{0}^{2}}{g^{2} (\infty)}] + \int_{R^{N}} \frac{1}{2} q (x) t^{2} w_{0}^{2} - H (x, G^{- 1} (t w_{0})) \\ ⩾ \frac{1}{2} \int_{R^{N}} [q (x) - \frac{h (x, G^{- 1} (t w_{0}))}{g (G^{- 1} (t w_{0})) t w_{0}}] t^{2} w_{0}^{2} \\ + \int_{R^{N}} [\frac{h (x, G^{- 1} (t w_{0})) t w_{0}}{2 g (G^{- 1} (t w_{0}))} - H (x, G^{- 1} (t w_{0}))] \\ ⩾ \int_{R^{N}} [\frac{h (x, G^{- 1} (t w_{0})) t w_{0}}{2 g (G^{- 1} (t w_{0}))} - H (x, G^{- 1} (t w_{0}))], \end{aligned}$ for every $t > 0$ . According to Lemma 4.5, given $C > 0$ , there exists t large enough such that $J (t w_{0}) ⩾ C$ . Since $J (t_{n} w_{n}) ⩾ J (t w_{n}) \to J (t w_{0})$ by Lemma 4.1-(1), as $n \to \infty$ , we obtain that $J (t_{n} w_{n}) \to \infty$ . □

If the functional J is coercive on $M$ , it is easy to prove that the map m is continuous by Lemma 2.8 in [23]. Under our assumptions the coerciveness of the functional does not hold, while we can still prove the continuity of the map m.

Lemma 4.7.

The map $m : U \mapsto M$ is continuous.

Proof.

Consider $w_{n} \in U$ with $w_{n} \to w_{0}$ and set $m (w_{n}) = t_{n} w_{n}$ . If $w_{0} \in \partial U$ , we have $t_{n} \to \infty$ and $J (t_{n} w_{n}) \to \infty$ by Lemma 4.6. This is a contradiction to Lemma 4.3.

We claim that $(t_{n})$ is bounded. In fact, if $t_{n} \to \infty$ , by Lemma 2.1-(4), $(h_{3} *)$ and the Lebesgue dominated convergence theorem we have $\begin{aligned} 0 ⩽ \frac{J (t_{n} w_{n})}{t_{n}^{2}} & = \frac{1}{2} \int_{R^{N}} | \nabla w_{n} |^{2} + \frac{1}{2} \int_{w_{n} \neq 0} V (x) \frac{| G^{- 1} (t_{n} w_{n}) |^{2}}{t_{n}^{2} w_{n}^{2}} w_{n}^{2} \\ - \int_{w_{n} \neq 0} \frac{H (x, G^{- 1} (t_{n} w_{n}))}{t_{n}^{2} w_{n}^{2}} w_{n}^{2} \\ \to \frac{1}{2} \int_{R^{N}} | \nabla w_{0} |^{2} + \frac{V (x)}{g^{2} (\infty)} w_{0}^{2} - \frac{1}{2} \int_{R^{N}} q (x) w_{0}^{2} < 0, \end{aligned}$ a contradiction.

Assume $t_{n} \to t_{0}$ . By Lemma 4.2-(2), $t_{0} ⩾ 2 c > 0$ . Hence $\begin{matrix} 0 = ⟨ J^{'} (t_{n} w_{n}), t_{n} w_{n} ⟩ \to ⟨ J^{'} (t_{0} w_{0}), t_{0} w_{0} ⟩, \end{matrix}$ that is, $t_{0} w_{0} = m (w_{0})$ . □

The following two lemmas correspond to Lemmas 3.6 and 3.7.

Lemma 4.8.

The map $m : U \mapsto M$ is a homeomorphism and the inverse of m is given by $m^{- 1} (v) = \frac{v}{‖ v ‖}$ .

Define the functional $Ψ : U \to R$ by $Ψ (w) : = J (m (w))$ .

Lemma 4.9.

$Ψ \in C^{1} (U, R)$ and $\begin{matrix} ⟨ Ψ^{'} (w), z ⟩ = ‖ m (w) ‖ ⟨ J^{'} (m (w)), z ⟩ for all z \in T_{w} (U) . \end{matrix}$

If $(w_{n})$ is a Palais–Smale sequence for Ψ, then $(m (w_{n}))$ is a Palais–Smale sequence for J. If $(v_{n}) \subset M$ is a bounded Palais–Smale sequence for J, then $(m^{- 1} (v_{n}))$ is a Palais–Smale sequence for Ψ.

w is a critical point of Ψ if and only if $m (w)$ is a nontrivial critical point of J. Moreover, the corresponding values of Ψ and J coincide and ${inf}_{U} Ψ = {inf}_{M} J$ .

Proof of Theorem 1.5.

By Lemma 4.6 and Ekeland’s variational principle, there is a Palais–Smale sequence $(w_{n}) \subset U$ . By the same arguments as in Theorem 1.4, the proof is complete. □

Footnotes

Acknowledgements

The authors are grateful to the referee for carefully reading the paper.

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