Ground state solution and multiple solutions to asymptotically linear Schrödinger equations

Abstract

In this paper, we consider the Schrödinger equation $-\mathrm{\Delta }u+V(x)u=f(x,u)$, $x\in {\mathbb{R}}^N$, where V and f are periodic in $x_1,\dots ,x_N$, asymptotically linear and satisfies a monotonicity condition. We use the generalized Nehari manifold methods to obtain a ground state solution and infinitely many geometrically distinct solutions when f is odd in u.

1 Introduction

We consider the problem

$$-\mathrm{\Delta }u+V(x)u=f(x,u),u\in H^1\left({\mathbb{R}}^N\right),$$

(1.1)

where f and V are periodic in $x_1,\dots ,x_N$, asymptotically linear and satisfies a monotonicity condition. In the case that the nonlinear term is asymptotically linear at infinity, there are some results in the literature [1]–[12] and the references therein, where multiplicity results are considered in [1]–[3], [9], [10], [12]. As far as we know, there are only a few papers concerned with the existence of infinitely many solutions for the asymptotical linear case when f and V are also periodic in $x_1,\dots ,x_N$; e.g. see [2]. Except for [5], there seem to be few results on the existence of a ground state solution in the asymptotically linear case. Motivated by [13], this paper is to present a different approach involving the critical point theory with the discreteness property of the Palais-Smale in search for a ground state solution and multiple solutions for the asymptotically linear Schrödinger equations. It should be pointed out that in [2], they cannot make sure the existence of a ground state solution. Our results can be regarded as complements or different attempts of the results in [2], [5].

Setting $F(x,u):={\int }_0^{u}f(x,s)ds$, we suppose that V and f satisfy the following assumptions:

(V): V is continuous, 1-periodic in $x_i$, $1\le i\le N$, and there exists a constant $a_0>0$ such that $V(x)\ge a_0$ for all $x\in {\mathbb{R}}^N$.

(f₁): f is continuous, 1-periodic in $x_i$, $1\le i\le N$.

(f₂): $f(x,u)=o(u)$ as $u\to 0$, uniformly in x.

(f₃): There is $q(x)>V(x)$, $\mathrm{\forall }x\in {\mathbb{R}}^N$, such that $f(x,u)/u\to q(x)$, as $|u|\to \mathrm{\infty }$, where q is continuous, 1-periodic in $x_i$, $1\le i\le N$.

(f₄): $u\mapsto f(x,u)/|u|$ is strictly increasing on $(-\mathrm{\infty },0)$ and $(0,\mathrm{\infty })$.

Let ∗ denote the action of ${\mathbb{Z}}^N$ on $H^1({\mathbb{R}}^N)$ given by

$$(k\ast u)(x):=u(x-k),k\in {\mathbb{Z}}^N.$$

(1.2)

It follows from (V) and (f₁) that if $u_0$ is a solution of (1.1), then so is $k\ast u_0$ for all $k\in {\mathbb{Z}}^N$. Set

$$\mathcal{O}(u_0):=\{k\ast u_0:k\in {\mathbb{Z}}^N\}.$$

$\mathcal{O}(u_0)$ is called the orbit of $u_0$ with respect to the action of ${\mathbb{Z}}^N$, and it is called a critical orbit for a functional F if $u_0$ is a critical point of F and F is ${\mathbb{Z}}^N$-invariant, i.e., $F(k\ast u)=F(u)$ for all $k\in {\mathbb{Z}}^N$ and all u (then of course all points of $\mathcal{O}(u_0)$ are critical). Two solutions $u_1$, $u_2$ of (1.1) are said to be geometrically distinct if $\mathcal{O}(u_1)\ne \mathcal{O}(u_2)$.

Theorem 1.1

Suppose that (V), (f₁)-(f₄) are satisfied. Then (1.1) has a ground state solution. In addition, if f is odd in u, then (1.1) admits infinitely many pairs ±u of geometrically distinct solutions.

Notation

$C,C_1,C_2,\dots$ will denote different positive constants whose exact value is inessential. The usual norm in the Lebesgue space $L^p(\mathrm{\Omega })$ is denoted by ${\parallel u\parallel }_{p,\mathrm{\Omega }}$, and by ${\parallel u\parallel }_p$ if $\mathrm{\Omega }={\mathbb{R}}^N$. E denotes the Sobolev space $H^1({\mathbb{R}}^N)$ and S is the unit sphere in E. It follows from (V) that

$$\parallel u\parallel :={\left({\int }_{{\mathbb{R}}^N}({|\mathrm{\nabla }u|}^2+V(x)u^2)\right)}^{1/2}$$

is an equivalent norm in E. It is more convenient for our purposes than the standard one and will be used henceforth. For a functional I, as in [14], we put

$$I^d:=\{u:I(u)\le d\},I_c:=\{u:I(u)\ge c\},I_c^d:=\{u:c\le I(u)\le d\}.$$

2 Preliminary results

Consider the functional

$$I(u):=\frac{1}{2}{\int }_{{\mathbb{R}}^N}{|\mathrm{\nabla }u|}^2+\frac{1}{2}{\int }_{{\mathbb{R}}^N}V(x)u^2-{\int }_{{\mathbb{R}}^N}F(x,u).$$

(2.1)

Then I is well defined on E and $I\in C^1(E,\mathbb{R})$ under the hypotheses (V), (f₁)-(f₃). Note also that (V), (f₁) imply I is invariant with respect to the action of ${\mathbb{Z}}^N$ given by (1.2). It is easy to see that

$$〈I^{\prime }(u),v〉={\int }_{{\mathbb{R}}^N}\mathrm{\nabla }u\mathrm{\nabla }v+{\int }_{{\mathbb{R}}^N}V(x)uv-{\int }_{{\mathbb{R}}^N}f(x,u)v$$

(2.2)

for all $u,v\in E$.

Let

$$\mathcal{M}:=\{u\in E\setminus \{0\}:〈I^{\prime }(u),u〉=0\}.$$

(2.3)

Recall that ℳ is called the Nehari manifold. We do not know whether ℳ is of class $C^1$ under our assumptions and therefore we cannot use minimax theory directly on ℳ. To overcome this difficulty, we employ the arguments developed in [13], [15], [16].

We assume that (V) and (f₁)-(f₄) are satisfied from now on. First, (f₂) and (f₃) imply that for each $\epsilon >0$ there is $C_{\epsilon }>0$ such that

$$|f(x,u)|\le \epsilon |u|+C_{\epsilon }{|u|}^{p-1}\text{for all }u\in \mathbb{R},$$

(2.4)

where $2<p<2^{\ast }$, $2^{\ast }:=2N/(N-2)$ if $N\ge 3$, $2^{\ast }:=\mathrm{\infty }$ if $N=1\text{ or }2$.

For $t>0$, let

$$h(t):=I(tu)=\frac{t^2}{2}{\int }_{{\mathbb{R}}^N}{|\mathrm{\nabla }u|}^2+V(x)u^2-{\int }_{{\mathbb{R}}^N}F(x,tu).$$

Let

$$\mathcal{E}:=\{u\in E:{\int }_{{\mathbb{R}}^N}{|\mathrm{\nabla }u|}^2+V(x)u^2<{\int }_{{\mathbb{R}}^N}q(x)u^2\}.$$

It follows from $q(x)-V(x)>0$, $\mathrm{\forall }x\in {\mathbb{R}}^N$, that $\mathcal{E}\ne \mathrm{\varnothing }$.

Lemma 2.1

$F(x,u)>0$and$\frac{1}{2}f(x,u)u>F(x,u)$if$u\ne 0$.

This follows immediately from (f₂) and (f₄).

Lemma 2.2

• (1) For each $u\in \mathcal{E}$ there is a unique $t_u>0$ such that $h^{\prime }(t)>0$ for $0<t<t_u$ and $h^{\prime }(t)<0$ for $t>t_u$. Moreover, $tu\in \mathcal{M}$ if and only if $t=t_u$.

• (2) If $u\notin \mathcal{E}$, then $tu\notin \mathcal{M}$ for any $t>0$.

Proof

• (1) For each $u\in \mathcal{E}$, due to the Lebesgue dominated convergence theorem and (f₂), (f₃), we get

  $$\begin{array}{rl}\underset{t\to \mathrm{\infty }}{lim}\frac{I(tu)}{t^2}& =\frac{1}{2}{\int }_{{\mathbb{R}}^N}{|\mathrm{\nabla }u|}^2+V(x)u^2-\underset{t\to \mathrm{\infty }}{lim}{\int }_{u\ne 0}\frac{F(x,tu)}{t^2{u}^2}{u}^2\\ =\frac{1}{2}[{\int }_{{\mathbb{R}}^N}{|\mathrm{\nabla }u|}^2+V(x)u^2-{\int }_{{\mathbb{R}}^N}q(x)u^2]<0\end{array}$$

and

$$\begin{array}{rl}\underset{t\to 0}{lim}\frac{I(tu)}{t^2}& =\frac{1}{2}{\int }_{{\mathbb{R}}^N}{|\mathrm{\nabla }u|}^2+V(x)u^2-\underset{t\to 0}{lim}{\int }_{u\ne 0}\frac{F(x,tu)}{t^2{u}^2}{u}^2\\ =\frac{1}{2}{\int }_{{\mathbb{R}}^N}{|\mathrm{\nabla }u|}^2+V(x)u^2>0.\end{array}$$

Hence h has a positive maximum. The condition $h^{\prime }(t)=0$ is equivalent to

$${\parallel u\parallel }^2={\int }_{u\ne 0}\frac{f(x,tu)}{tu}{u}^2.$$

By (f₄), the first conclusion holds. The second conclusion follows from $h^{\prime }(t)=t^{-1}〈I^{\prime }(tu),tu〉$.

• (2) If $tu\in \mathcal{M}$ for some $t>0$, then $〈I^{\prime }(tu),u〉=0$ and therefore using (f₃) and (f₄)

  $${\parallel u\parallel }^2={\int }_{u\ne 0}\frac{f(x,tu)}{tu}{u}^2<{\int }_{{\mathbb{R}}^N}q(x)u^2.$$

Hence $u\in \mathcal{E}$. □

Lemma 2.3

• (1) There exists $\rho >0$ such that $c:={inf}_{\mathcal{M}}I\ge {inf}_{S_{\rho }}I>0$.

• (2) ${\parallel u\parallel }^2\ge 2c$ for all $u\in \mathcal{M}$.

Proof

• (1) Using (2.4) and the Sobolev inequality we have ${inf}_{S_{\rho }}I>0$ if ρ is small enough. The inequality ${inf}_{\mathcal{M}}I\ge {inf}_{S_{\rho }}I$ is a consequence of Lemma 2.2 since for every $u\in \mathcal{M}$ there is $s>0$ such that $su\in S_{\rho }$ (and $I(t_{u}u)\ge I(su)$).

• (2) For $u\in \mathcal{M}$, by Lemma 2.1 we have

  $$c\le \frac{1}{2}{\parallel u\parallel }^2-{\int }_{{\mathbb{R}}^N}F(x,u)\le \frac{1}{2}{\parallel u\parallel }^2.$$

 □

We do not know whether I is coercive on ℳ. However, we can prove the following.

Lemma 2.4

All Palais-Smale sequences$(u_n)\subset \mathcal{M}$are bounded.

Proof

Arguing by contradiction, suppose there exists a sequence $(u_n)\subset \mathcal{M}$ such that $\parallel u_n\parallel \to \mathrm{\infty }$ and $I(u_n)\le d$ for some $d\in [c,\mathrm{\infty })$. Let $v_n:=u_n/\parallel u_n\parallel$. Then $v_n\rightharpoonup v$ and $v_n(x)\to v(x)$ a.e. in ${\mathbb{R}}^N$ after passing to a subsequence. Choose $y_n\in {\mathbb{R}}^N$ so that

$${\int }_{B_1(y_n)}{v}_n^2=\underset{y\in {\mathbb{R}}^N}{max}{\int }_{B_1(y)}{v}_n^2.$$

(2.5)

Since I and ℳ are invariant with respect to the action of ${\mathbb{Z}}^N$ given by (1.2), we may assume that $(y_n)$ is bounded in ${\mathbb{R}}^N$. If

$${\int }_{B_1(y_n)}{v}_n^2\to 0\text{as }n\to \mathrm{\infty },$$

(2.6)

then it follows that $v_n\to 0$ in $L^r({\mathbb{R}}^N)$ for $2<r<2^{\ast }$ by Lions’ lemma (cf.[17], Lemma 1.21), and therefore (2.4) implies that ${\int }_{{\mathbb{R}}^N}F(x,s{v}_n)\to 0$ for every $s\in \mathbb{R}$. Lemma 2.2 implies that

$$d\ge I(u_n)\ge I(s{v}_n)=\frac{s^2}{2}-{\int }_{{\mathbb{R}}^N}F(x,s{v}_n)\to \frac{s^2}{2}.$$

Taking a sufficiently large s, we get a contradiction. Hence (2.6) cannot hold and, since $v_n\to v$ in $L_{\mathrm{loc}}^2({\mathbb{R}}^N)$, $v\ne 0$. Hence $|u_n(x)|\to \mathrm{\infty }$ if $v(x)\ne 0$.

Let $\phi \in C_0^{\mathrm{\infty }}({\mathbb{R}}^N)$. Then $〈I^{\prime }(u_n),\phi 〉\to 0$ and hence

$${\int }_{{\mathbb{R}}^N}\mathrm{\nabla }{v}_n\mathrm{\nabla }\phi +V(x)v_n\phi -{\int }_{{\mathbb{R}}^N}\frac{f(x,u_n)}{u_n}{v}_n\phi \to 0.$$

By the Lebesgue dominated convergence theorem we therefore have

$${\int }_{{\mathbb{R}}^N}\mathrm{\nabla }v\mathrm{\nabla }\phi +V(x)v\phi ={\int }_{{\mathbb{R}}^N}q(x)v\phi .$$

So $v\ne 0$ and $-\mathrm{\Delta }v+V(x)v=q(x)v$. This is impossible because $-\mathrm{\Delta }+V-q$ has only an absolutely continuous spectrum. The proof is complete. □

Lemma 2.5

If$\mathcal{V}$is a compact subset of ℰ, then there exists$R>0$such that$I\le 0$on$({\mathbb{R}}^{+}\mathcal{V})\setminus B_R(0)$.

Proof

We may assume without loss of generality that $\mathcal{V}\subset S$. Arguing by contradiction, suppose there exist $u_n\in \mathcal{V}$ and $w_n=t_n{u}_n$, where $u_n\to u$, $t_n\to \mathrm{\infty }$ and $I(w_n)\ge 0$. We have

$$\begin{array}{rl}0& \le \frac{I(t_n{u}_n)}{t_n^2}=\frac{1}{2}{\int }_{{\mathbb{R}}^N}{|\mathrm{\nabla }{u}_n|}^2+V(x)u_n^2-{\int }_{u_n\ne 0}\frac{F(x,t_n{u}_n)}{t_n^2{u}_n^2}{u}_n^2\\ \to \frac{1}{2}{\int }_{{\mathbb{R}}^N}{|\mathrm{\nabla }u|}^2+V(x)u^2-\frac{1}{2}{\int }_{{\mathbb{R}}^N}q(x)u^2<0.\end{array}$$

 □

Let $U:=\mathcal{E}\cap S$ and define the mapping $m:U\to \mathcal{M}$ by setting

$$m(w):=t_{w}w,$$

where $t_w$ is as in Lemma 2.2.

Lemma 2.6

U is an open subset of S.

Proof

Obvious because ℰ is open in E. □

Lemma 2.7

Assume$u_n\in U$, $u_n\to u_0\in \partial U$, and$t_n{u}_n\in \mathcal{M}$, then$I(t_n{u}_n)\to \mathrm{\infty }$.

Proof

Since $u_0\in \partial U$, ${\int }_{{\mathbb{R}}^N}{|\mathrm{\nabla }{u}_0|}^2+V(x)u_0^2={\int }_{{\mathbb{R}}^N}q(x)u_0^2$. Using this, we have

$$\begin{array}{rcl}I(t{u}_0)& =& \frac{1}{2}{t}^2{\int }_{{\mathbb{R}}^N}{|\mathrm{\nabla }{u}_0|}^2+V(x)u_0^2-t^2{\int }_{{\mathbb{R}}^N}\frac{F(x,t{u}_0)}{t^2{u}_0^2}{u}_0^2\\ =& \frac{1}{2}{t}^2{\int }_{{\mathbb{R}}^N}(q(x)-\frac{2F(x,t{u}_0)}{t^2{u}_0^2})u_0^2\\ =& \frac{1}{2}{t}^2{\int }_{{\mathbb{R}}^N}(q(x)-\frac{f(x,t{u}_0)}{t{u}_0})u_0^2\\ +{\int }_{{\mathbb{R}}^N}\frac{1}{2}f(x,t{u}_0)t{u}_0-F(x,t{u}_0).\end{array}$$

Note that by (f₄), we have for large enough s, there is $\delta >0$ such that

$$\frac{1}{2}f(x,s)s-F(x,s)\ge \delta$$

(see [4], Remark 1.5). So $I(t{u}_0)\to \mathrm{\infty }$, as $t\to \mathrm{\infty }$ (we have used Fatou’s lemma). Given $C>0$, choose $t>0$ such that $I(t{u}_0)\ge C$. Since $u_n\to u_0$,

$$\underset{n\to \mathrm{\infty }}{lim}I(t_n{u}_n)\ge \underset{n\to \mathrm{\infty }}{lim}I(t{u}_n)=I(t{u}_0)\ge C$$

and hence $I(t_n{u}_n)\to \mathrm{\infty }$. □

The following lemmas are taken from [13], [15].

Below we shall use the notations

$$\begin{array}{c}K:=\{w\in S:{\mathrm{\Psi }}^{\prime }(w)=0\},\hfill \\ K_d:=\{w\in K:\mathrm{\Psi }(w)=d\}.\hfill \end{array}$$

Since f is odd in u, we can choose a subset ℱ of K such that $\mathcal{F}=-\mathcal{F}$ and each orbit $\mathcal{O}(w)\subset K$ has a unique representative in ℱ. We must show that the set ℱ is infinite. Arguing indirectly, assume

$$\mathcal{F}\text{ is a finite set}.$$

(2.7)

Lemma 2.8

The mapping m is a homeomorphism between U and ℳ, and the inverse of m is given by$m^{-1}(u)=\frac{u}{\parallel u\parallel }$.

We consider the functional $\mathrm{\Psi }:U\to \mathbb{R}$ given by

$$\mathrm{\Psi }(w):=I(m(w)).$$

Lemma 2.9

• (1) $\mathrm{\Psi }\in C^1(U,\mathbb{R})$ and

  $$〈{\mathrm{\Psi }}^{\prime }(w),z〉=\parallel m(w)\parallel 〈I^{\prime }(m(w)),z〉\mathit{\text{for all }}z\in T_w(U).$$

• (2) If $(w_n)$ is a Palais-Smale sequence for Ψ, then $(m(w_n))$ is a Palais-Smale sequence for I. If $(u_n)\subset \mathcal{M}$ is a bounded Palais-Smale sequence for I, then $(m^{-1}(u_n))$ is a Palais-Smale sequence for Ψ.

• (3) w is a critical point of Ψ if and only if $m(w)$ is a nontrivial critical point of I. Moreover, the corresponding values of Ψ and I coincide and ${inf}_U\mathrm{\Psi }={inf}_{\mathcal{M}}I$.

• (4) Ψ is even (because I is).

By (2.4), the following lemma also holds.

Lemma 2.10

Let$d\ge c$. If$(v_n^1),(v_n^2)\subset {\mathrm{\Psi }}^d$are two Palais-Smale sequences for Ψ, then either$\parallel v_n^1-v_n^2\parallel \to 0$as$n\to \mathrm{\infty }$or${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}\parallel v_n^1-v_n^2\parallel \ge \rho (d)>0$, where$\rho (d)$depends on d but not on the particular choice of Palais-Smale sequences.

It is well known that Ψ admits a pseudo-gradient vector field $H:U\setminus K\to TU$ (see e.g.[18], p.86). Moreover, since Ψ is even, we may assume H is odd. Let $\eta :\mathcal{G}\to U\setminus K$ be the flow defined by

$$\{\begin{array}{c}\frac{d}{dt}\eta (t,w)=-H(\eta (t,w)),\hfill \\ \eta (0,w)=w,\hfill \end{array}$$

(2.8)

where

$$\mathcal{G}:=\{(t,w):w\in U\setminus K,T^{-}(w)<t<T^{+}(w)\}$$

and $(T^{-}(w),T^{+}(w))$ is the maximal existence time for the trajectory $t\mapsto \eta (t,w)$. Note that η is odd in w because H is and $t\mapsto \mathrm{\Psi }(\eta (t,w))$ is strictly decreasing by the properties of a pseudogradient.

Let $P\subset U$, $\delta >0$ and define $U_{\delta }(P):=\{w\in U:dist(w,P)<\delta \}$.

Lemma 2.11

Let$d\ge c$. Then for every$\delta >0$there exists$\epsilon =\epsilon (\delta )>0$such that

• (a) ${\mathrm{\Psi }}_{d-\epsilon }^{d+\epsilon }\cap K=K_d$ and

• (b) ${lim}_{t\to T^{+}(w)}\mathrm{\Psi }(\eta (t,w))<d-\epsilon$ for $w\in {\mathrm{\Psi }}^{d+\epsilon }\setminus U_{\delta }(K_d)$.

Part (a) is an immediate consequence of (2.7) and (b) has been proved in [15]; see Lemmas 2.15 and 2.16 there. The argument is exactly the same except that S should be replaced by U. We point out that an important role in the proof of Lemma 2.11 is played by the discreteness property of the Palais-Smale sequences expressed in Lemma 2.10.

3 Proof of Theorem 1.1

Proof of Theorem 1.1

Taking a similar argument as in the proof of Theorem 1.1 in [15], it is easy to get a ground state solution. Noting that by Lemma 2.7 and Ekeland’s variational principle, it can make sure the existence of a ${(\mathit{PS})}_c$ sequence belonging to U.

For the multiplicity the argument is the same as in Theorem 1.2 (cf.[15]). However, there are details which need to be clarified.

Let η be the flow given by (2.8). If $T^{+}(w)<\mathrm{\infty }$, then ${lim}_{t\to T^{+}(w)}\eta (t,w)$ exists (cf.[15], Lemma 2.15, Case 1) but unlike the situation in [15], this limit may be a point $w_0\in \partial U$. This possibility is ruled out by Lemma 2.7.

Finally, we need to show that U contains sets of arbitrarily large genus. Since the spectrum of $-\mathrm{\Delta }+V-q$ in $L^2({\mathbb{R}}^N)$ is absolutely continuous, $\mathcal{E}\cup \{0\}$ contains an infinite-dimensional subspace $E_0$. Hence $E_0\cap S\subset U$ and $\gamma (E_0\cap S)=\mathrm{\infty }$. □

Remark 3.1

There is a small gap in the proof of Theorem 1.2 in [13]. Lemma 4.6 as stated there does not exclude the possibility of $\eta (t,w)$ approaching the boundary as $t\to T^{+}(w)$ (because we only know that $\eta (t,w)$ goes to infinity). But it is easy to prove that $I(\eta (t,w))$ goes to infinity as well in [13]. In Lemma 2.7 of this paper we make some proper modifications which also apply to [13] and were proposed by Andrzej Szulkin.