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Ground state solution and multiple solutions to asymptotically linear Schrödinger equations
Boundary Value Problems volume 2014, Article number: 216 (2014)
In this paper, we consider the Schrödinger equation , , where V and f are periodic in , 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.
We consider the problem
where f and V are periodic in , 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 – and the references therein, where multiplicity results are considered in –, , , . 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 ; e.g. see . Except for , there seem to be few results on the existence of a ground state solution in the asymptotically linear case. Motivated by , 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 , 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 , .
Setting , we suppose that V and f satisfy the following assumptions:
(V): V is continuous, 1-periodic in , , and there exists a constant such that for all .
(f1): f is continuous, 1-periodic in , .
(f2): as , uniformly in x.
(f3): There is , , such that , as , where q is continuous, 1-periodic in , .
(f4): is strictly increasing on and .
Let ∗ denote the action of on given by
It follows from (V) and (f1) that if is a solution of (1.1), then so is for all . Set
is called the orbit of with respect to the action of , and it is called a critical orbit for a functional F if is a critical point of F and F is -invariant, i.e., for all and all u (then of course all points of are critical). Two solutions , of (1.1) are said to be geometrically distinct if .
will denote different positive constants whose exact value is inessential. The usual norm in the Lebesgue space is denoted by , and by if . E denotes the Sobolev space 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 , we put
2 Preliminary results
Consider the functional
Then I is well defined on E and under the hypotheses (V), (f1)-(f3). Note also that (V), (f1) imply I is invariant with respect to the action of given by (1.2). It is easy to see that
for all .
Recall that ℳ is called the Nehari manifold. We do not know whether ℳ is of class under our assumptions and therefore we cannot use minimax theory directly on ℳ. To overcome this difficulty, we employ the arguments developed in , , .
We assume that (V) and (f1)-(f4) are satisfied from now on. First, (f2) and (f3) imply that for each there is such that
where , if , if .
For , let
It follows from , , that .
This follows immediately from (f2) and (f4).
For each there is a unique such that for and for . Moreover, if and only if .
If , then for any .
For each , due to the Lebesgue dominated convergence theorem and (f2), (f3), we get
Hence h has a positive maximum. The condition is equivalent to
By (f4), the first conclusion holds. The second conclusion follows from .
If for some , then and therefore using (f3) and (f4)
Hence . □
There exists such that .
for all .
Using (2.4) and the Sobolev inequality we have if ρ is small enough. The inequality is a consequence of Lemma 2.2 since for every there is such that (and ).
For , by Lemma 2.1 we have
We do not know whether I is coercive on ℳ. However, we can prove the following.
All Palais-Smale sequencesare bounded.
Arguing by contradiction, suppose there exists a sequence such that and for some . Let . Then and a.e. in after passing to a subsequence. Choose so that
Since I and ℳ are invariant with respect to the action of given by (1.2), we may assume that is bounded in . If
Taking a sufficiently large s, we get a contradiction. Hence (2.6) cannot hold and, since in , . Hence if .
Let . Then and hence
By the Lebesgue dominated convergence theorem we therefore have
So and . This is impossible because has only an absolutely continuous spectrum. The proof is complete. □
Ifis a compact subset of ℰ, then there existssuch thaton.
We may assume without loss of generality that . Arguing by contradiction, suppose there exist and , where , and . We have
Let and define the mapping by setting
where is as in Lemma 2.2.
U is an open subset of S.
Obvious because ℰ is open in E. □
Assume, , and, then.
Since , . Using this, we have
Note that by (f4), we have for large enough s, there is such that
(see , Remark 1.5). So , as (we have used Fatou’s lemma). Given , choose such that . Since ,
and hence . □
Below we shall use the notations
Since f is odd in u, we can choose a subset ℱ of K such that and each orbit has a unique representative in ℱ. We must show that the set ℱ is infinite. Arguing indirectly, assume
The mapping m is a homeomorphism between U and ℳ, and the inverse of m is given by.
We consider the functional given by
If is a Palais-Smale sequence for Ψ, then is a Palais-Smale sequence for I. If is a bounded Palais-Smale sequence for I, then is a Palais-Smale sequence for Ψ.
w is a critical point of Ψ if and only if is a nontrivial critical point of I. Moreover, the corresponding values of Ψ and I coincide and .
Ψ is even (because I is).
By (2.4), the following lemma also holds.
Let. Ifare two Palais-Smale sequences for Ψ, then eitherasor, wheredepends on d but not on the particular choice of Palais-Smale sequences.
It is well known that Ψ admits a pseudo-gradient vector field (see e.g., p.86). Moreover, since Ψ is even, we may assume H is odd. Let be the flow defined by
and is the maximal existence time for the trajectory . Note that η is odd in w because H is and is strictly decreasing by the properties of a pseudogradient.
Let , and define .
Let. Then for everythere existssuch that
Part (a) is an immediate consequence of (2.7) and (b) has been proved in ; 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 , 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 sequence belonging to U.
For the multiplicity the argument is the same as in Theorem 1.2 (cf.). However, there are details which need to be clarified.
Finally, we need to show that U contains sets of arbitrarily large genus. Since the spectrum of in is absolutely continuous, contains an infinite-dimensional subspace . Hence and . □
There is a small gap in the proof of Theorem 1.2 in . Lemma 4.6 as stated there does not exclude the possibility of approaching the boundary as (because we only know that goes to infinity). But it is easy to prove that goes to infinity as well in . In Lemma 2.7 of this paper we make some proper modifications which also apply to  and were proposed by Andrzej Szulkin.
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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.
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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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/s13661-014-0216-1
- Schrödinger equation
- ground state solution
- multiplicity of solutions
- asymptotically linear