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Ground state solution and multiple solutions to asymptotically linear Schrödinger equations
Boundary Value Problems volume 2014, Article number: 216 (2014)
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
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 PalaisSmale 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 $F(x,u):={\int}_{0}^{u}f(x,s)\phantom{\rule{0.2em}{0ex}}ds$, we suppose that V and f satisfy the following assumptions:
(V): V is continuous, 1periodic 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 $x\in {\mathbb{R}}^{N}$.
(f_{1}): f is continuous, 1periodic in ${x}_{i}$, $1\le i\le N$.
(f_{2}): $f(x,u)=o(u)$ as $u\to 0$, uniformly in x.
(f_{3}): 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, 1periodic in ${x}_{i}$, $1\le i\le N$.
(f_{4}): $u\mapsto f(x,u)/u$ 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
It follows from (V) and (f_{1}) 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
$\mathcal{O}({u}_{0})$ 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), (f_{1})(f_{4}) 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
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
2 Preliminary results
Consider the functional
Then I is well defined on E and $I\in {C}^{1}(E,\mathbb{R})$ under the hypotheses (V), (f_{1})(f_{3}). Note also that (V), (f_{1}) imply I is invariant with respect to the action of ${\mathbb{Z}}^{N}$ given by (1.2). It is easy to see that
for all $u,v\in E$.
Let
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 (f_{1})(f_{4}) are satisfied from now on. First, (f_{2}) and (f_{3}) imply that for each $\epsilon >0$ there is ${C}_{\epsilon}>0$ such that
where $2<p<{2}^{\ast}$, ${2}^{\ast}:=2N/(N2)$ if $N\ge 3$, ${2}^{\ast}:=\mathrm{\infty}$ if $N=1\text{or}2$.
For $t>0$, let
Let
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 (f_{2}) and (f_{4}).
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 (f_{2}), (f_{3}), 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
Hence h has a positive maximum. The condition ${h}^{\prime}(t)=0$ is equivalent to
By (f_{4}), the first conclusion holds. The second conclusion follows from ${h}^{\prime}(t)={t}^{1}\u3008{I}^{\prime}(tu),tu\u3009$.

(2)
If $tu\in \mathcal{M}$ for some $t>0$, then $\u3008{I}^{\prime}(tu),u\u3009=0$ and therefore using (f_{3}) and (f_{4})
$${\parallel u\parallel}^{2}={\int}_{u\ne 0}\frac{f(x,tu)}{tu}{u}^{2}<{\int}_{{\mathbb{R}}^{N}}q(x){u}^{2}.$$
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 PalaisSmale 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
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
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
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 $\u3008{I}^{\prime}({u}_{n}),\phi \u3009\to 0$ and hence
By the Lebesgue dominated convergence theorem we therefore have
So $v\ne 0$ and $\mathrm{\Delta}v+V(x)v=q(x)v$. This is impossible because $\mathrm{\Delta}+Vq$ 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
□
Let $U:=\mathcal{E}\cap S$ and define the mapping $m:U\to \mathcal{M}$ by setting
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
Note that by (f_{4}), we have for large enough s, there is $\delta >0$ such that
(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 ${u}_{n}\to {u}_{0}$,
and hence $I({t}_{n}{u}_{n})\to \mathrm{\infty}$. □
The following lemmas are taken from [13], [15].
Below we shall use the notations
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
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
Lemma 2.9

(1)
$\mathrm{\Psi}\in {C}^{1}(U,\mathbb{R})$ and
$$\u3008{\mathrm{\Psi}}^{\prime}(w),z\u3009=\parallel m(w)\parallel \u3008{I}^{\prime}(m(w)),z\u3009\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}z\in {T}_{w}(U).$$

(2)
If $({w}_{n})$ is a PalaisSmale sequence for Ψ, then $(m({w}_{n}))$ is a PalaisSmale sequence for I. If $({u}_{n})\subset \mathcal{M}$ is a bounded PalaisSmale sequence for I, then $({m}^{1}({u}_{n}))$ is a PalaisSmale 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 PalaisSmale 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 PalaisSmale sequences.
It is well known that Ψ admits a pseudogradient 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
where
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 PalaisSmale 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 $\mathrm{\Delta}+Vq$ in ${L}^{2}({\mathbb{R}}^{N})$ is absolutely continuous, $\mathcal{E}\cup \{0\}$ contains an infinitedimensional 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.
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Acknowledgements
The authors thank the referee for providing some of the references and suggestions. The authors thank Professor Szulkin for his encouragement. This work was supported by NSFC 11171047. The first author was supported the Fundamental Research Funds for the Central Universities.
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Fang, XD., Han, ZQ. Ground state solution and multiple solutions to asymptotically linear Schrödinger equations. Bound Value Probl 2014, 216 (2014). https://doi.org/10.1186/s1366101402161
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Keywords
 Schrödinger equation
 ground state solution
 multiplicity of solutions
 asymptotically linear