- Open Access
Ground state solution and multiple solutions to asymptotically linear Schrödinger equations
© Fang and Han; licensee Springer 2014
- Received: 5 June 2014
- Accepted: 11 September 2014
- Published: 25 September 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.
- Schrödinger equation
- ground state solution
- multiplicity of solutions
- asymptotically linear
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 .
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 .
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 , , .
where , if , if .
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 .
- (1)For each , due to the Lebesgue dominated convergence theorem and (f2), (f3), we get
- (2)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 ).
- (2)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.
Taking a sufficiently large s, we get a contradiction. Hence (2.6) cannot hold and, since in , . Hence if .
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.
where is as in Lemma 2.2.
U is an open subset of S.
Obvious because ℰ is open in E. □
Assume, , and, then.
and hence . □
The mapping m is a homeomorphism between U and ℳ, and the inverse of m is 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.
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 .
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.
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.
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.
- Costa DG, Tehrani H:On a class of asymptotically linear elliptic problem in . J. Differ. Equ. 2001, 173: 470-494. 10.1006/jdeq.2000.3944MathSciNetView ArticleGoogle Scholar
- Ding YH, Lee C: Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms. J. Differ. Equ. 2006, 222: 137-163. 10.1016/j.jde.2005.03.011MathSciNetView ArticleGoogle Scholar
- Ding YH, Luan SX: Multiple solutions for a class of nonlinear Schrödinger equations. J. Differ. Equ. 2004, 207: 423-457. 10.1016/j.jde.2004.07.030MathSciNetView ArticleGoogle Scholar
- Jeanjean L:On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on . Proc. R. Soc. Edinb., Sect. A 1999, 129: 787-809. 10.1017/S0308210500013147MathSciNetView ArticleGoogle Scholar
- Liu CY, Wang ZP, Zhou HS: Asymptotically linear Schrödinger equation with potential vanishing at infinity. J. Differ. Equ. 2008, 245: 201-222. 10.1016/j.jde.2008.01.006MathSciNetView ArticleGoogle Scholar
- Liu ZL, Wang ZQ:Existence of a positive solution of an elliptic equation on . Proc. R. Soc. Edinb., Sect. A 2004, 134(1):191-200. 10.1017/S0308210500003152View ArticleGoogle Scholar
- Polidoro S, Ragusa MA: Harnack inequality for hypoelliptic ultraparabolic equations with a singular lower order term. Rev. Mat. Iberoam. 2008, 24: 1011-1046. 10.4171/RMI/565MathSciNetView ArticleGoogle Scholar
- Stuart CA, Zhou HS:Applying the mountain pass theorem to an asymptotically linear elliptic equation on . Commun. Partial Differ. Equ. 1999, 24(9-10):1731-1758. 10.1080/03605309908821481MathSciNetView ArticleGoogle Scholar
- van Heerden FA, Francois A: Homoclinic solutions for a semilinear elliptic equation with an asymptotically linear nonlinearity. Calc. Var. Partial Differ. Equ. 2004, 20: 431-455. 10.1007/s00526-003-0242-9View ArticleGoogle Scholar
- van Heerden FA, Wang ZQ: Schrödinger type equations with asymptotically linear nonlinearities. Differ. Integral Equ. 2003, 16: 257-280.MathSciNetGoogle Scholar
- Zhou HS: Positive solution for a semilinear elliptic equation which is almost linear at infinity. Z. Angew. Math. Phys. 1998, 49: 896-906. 10.1007/s000330050128MathSciNetView ArticleGoogle Scholar
- Zhao FK, Zhao LG, Ding YH: Existence and multiplicity of solutions for a non-periodic Schrödinger equation. Nonlinear Anal. 2008, 69: 3671-3678. 10.1016/j.na.2007.10.024MathSciNetView ArticleGoogle Scholar
- Fang XD, Szulkin A: Multiple solutions for a quasilinear Schrödinger equation. J. Differ. Equ. 2013, 254: 2015-2032. 10.1016/j.jde.2012.11.017MathSciNetView ArticleGoogle Scholar
- Rabinowitz PH: Minimax Methods in Critical Point Theory with Applications to Differential Equations. Am. Math. Soc., Providence; 1986.Google Scholar
- Szulkin A, Weth T: Ground state solutions for some indefinite variational problems. J. Funct. Anal. 2009, 257: 3802-3822. 10.1016/j.jfa.2009.09.013MathSciNetView ArticleGoogle Scholar
- Szulkin A, Weth T: The method of Nehari manifold. In Handbook of Nonconvex Analysis and Applications. Edited by: Gao DY, Motreanu D. International Press, Boston; 2010:597-632.Google Scholar
- Willem M: Minimax Theorems. Birkhäuser, Boston; 1996.View ArticleGoogle Scholar
- Struwe M: Variational Methods. Springer, Berlin; 1996.View ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd.Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.