In 2000, Cingolani and Lazzo (J. Differ. Equ. 160:118-138, 2000) studied nonlinear Schrödinger equations with competing potential functions and considered only the subcritical growth. They related the number of solutions with the topology of the global minima set of a suitable ground energy function. In the present paper, we establish these results in the critical case. In particular, we remove the condition , which is a key condition in their paper. In the proofs we apply variational methods and Ljusternik-Schnirelmann theory.
MSC: 35J60, 35Q55.
Keywords:nonlinear Schrödinger equations; critical growth; variational methods
1 Introduction and main result
We investigate the following nonlinear Schrödinger equation:
which arises in quantum mechanics and provides a description of the dynamics of the particle in a non-relativistic setting. ħ is the Planck’s constant, denotes the mass of the particle, is the electric potential, g is the nonlinear coupling, and ψ is the wave function representing the state of the particle. A standing wave solution of equation (1.1) is a solution of the form . It is clear that solves (1.1) if and only if solves the following stationary equation:
A considerable amount of work has been devoted to investigating solutions of (1.3). The existence, multiplicity and qualitative property of such solutions have been extensively studied. For single interior spikes solutions in the whole space , please see [1-9]etc. For multiple interior spikes, please see [10,11]etc. For single boundary spike solutions with Neumann boundary condition, please see [6,12-15]etc. For multiple boundary spikes, please see [16-18]etc. In particular, Wang and Zeng  studied the existence and concentration behavior of solutions for NLS with competing potential functions. Cingolani and Lazzo in  obtained the multiple solutions for the similar equation. In those papers only the subcritical growth was considered. In the present paper, we complete these studies by considering a class of nonlinearities with the critical growth. In particular, we remove the condition , which is a key condition in .
Our main results are the following theorem.
Here denotes the Ljusternik-Schnirelmann category of Σ in . By definition (e.g., ), the category of A with respect to M, denoted by , is the least integer k such that , with () closed and contractible in M. We set and if there are no integers with the above property. We will use the notation for .
To prove Theorem 1.1, we mainly use the idea of [15,19,21]. More precisely, we can show that the -condition holds in the subset (see (4.6)). Hence the standard Ljusternik-Schnirelmann category theory can be applied in to yield the existence of at least critical points of . And then we construct two continuous mappings
Then a topological argument asserts that
The paper is organized as follows. In Section 2, we collect some notations and preliminaries. A compactness result is given in Section 3, which is a key step in our proof. Finally, in Section 4, we prove Theorem 1.1.
2 Notations and preliminaries
with the normal
Let us recall some known facts about the limiting problem, namely the problem
An associated critical point w actually solves equation (2.2) and is called a ground state solution or the least energy solution, i.e., w satisfies
It is known  that
Moreover, we have
then the maximum value of the right-hand side is achieved at
Hence we have
3 Compactness result
Proposition 3.1Letas. Assume thatsatisfiesas. Then uniformly in, there exist a subsequence of (still denoted by), andsuch that. Furthermore, converges strongly intow, the positive ground state solution of equation (2.2).
Indeed, if this is not true, then the boundedness of in and a lemma due to Lions [, Lemma I.1] imply that in for all . Given , we can use (f2), (f3) and to get
and consequently (3.2) yields
However, recall the definition of S in (2.1),
By the Hölder inequality,
Since converges weakly to w in , is bounded in . Thus is bounded in . It then follows that there is a subsequence of , still denoted by , such that converges weakly to some in . Next we will show . Choose a sequence of open relatively compact subsets, with regular boundaries, of covering , i.e., . It is easy to see that, by compact embedding, in for any . Hence a.e. on . Hence a.e. on . By the Brezis and Lieb lemma , we conclude that strongly in . Thus a.e. on each , and then the diagonal rule implies a.e. on . Hence
Similarly, we have
By (f2) and (f3),
Hence when R is large enough, we get
By (3.9)-(3.12), we derive that
By the concentration-compactness lemma , there exists a subsequence of (denoted in the same way) satisfying one of the three following possibilities.
We are going to rule out the last two possibilities so that compactness holds. Our first goal is to show that vanishing cannot occur. Otherwise,
If we denote
Using Dichotomy (iii), we get
since (f5). By (3.15),
Hence by the Lebesgue dominated convergence theorem, we get
Contradiction! Thus dichotomy does not occur.
From the foregoing, it follows that there exist bounded nonnegative measures , on such that weakly and tightly as . Lemma I.1 in  declares that there exist sequences , such that
This reduces to
hence (3.27)(3) states
which implies that the set J is at most finite. Here CardJ is the cardinal numbers of set J. Hence
a contradiction. Therefore J is empty, that is, as . By the Brezis and Lieb lemma  again, we get
Equation (3.25) and compact embedding theorem imply
This together with (3.13), (3.20) and (3.32) allows us to deduce easily
From (3.32), (3.33) and (3.34), we can obtain
i.e., w is the ground state solution of (2.2) in view of (3.13). The proof of Proposition 3.1 is complete. □
4 Proof of Theorem 1.1
and by the Brezis-Lieb lemma ,
For convenience, we denote by
It is clear that
By the Sobolev inequalities,
By the exponential decay of ω, we get
Thus (4.7) is proved. □
By Proposition 3.1, converges strongly in to w, which is a positive ground state solution of equation (2.2). Thanks to the exponential decay of w (see (2.4)), we obtain as . This completes the proof of Lemma 4.3. □
Proof of Theorem 1.1 By Proposition 4.1, satisfies the -condition for all . Now let us choose a function such that as and such that is not a critical level for . For such , let us introduce the set as in (4.6). Then the standard Ljusternik-Schnirelmann theory implies that has at least critical points on (also see ).
as uniformly for . Hence the map is homotopical equivalence to the inclusion for ε small enough. We denote . It is easy to verify that and (cf. [, Lemma 2.2]). Hence we have
The authors declare that they have no competing interests.
WL carried out the genetic studies, participated in the sequence alignment and drafted the manuscript. PZ checked the references.
This work was partially supported by the National Natural Science Foundation of China (11261052). The authors are grateful to Prof. Guowei Dai for pointing out several mistakes and valuable comments.
Floer, A, Weinstein, A: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal.. 69, 397–408 (1986). Publisher Full Text
Oh, Y-G: Existence of semi-classical bound states of nonlinear Schrödinger equations with potentials of the class . Commun. Partial Differ. Equ.. 13, 1499–1519 (1988). Publisher Full Text
Ambrosetti, A, Badiale, M, Cingolani, S: Semiclassical states of nonlinear Schrödinger equations. Arch. Ration. Mech. Anal.. 140, 285–300 (1997). Publisher Full Text
Del Pino, M, Felmer, P: Local mountain passes for semilinear elliptic problems in unbounded domains. Calc. Var.. 4, 121–137 (1996). Publisher Full Text
Rabinowitz, P: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys.. 43, 270–291 (1992). Publisher Full Text
Wang, X: On concentration of positive bound states of nonlinear Schrödinger equations. Commun. Math. Phys.. 153, 229–244 (1993). Publisher Full Text
Wang, X, Zeng, B: On concentration of positive bound states of nonlinear Schrödinger equation with competing potential functions. SIAM J. Math. Anal.. 28, 633–655 (1997). Publisher Full Text
Del Pino, M, Felmer, P: Multi-peak bound states of nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré, Anal. Non Linéaire. 15(2), 127–149 (1998). Publisher Full Text
Gui, C: Existence of multi-bump solutions for nonlinear Schrödinger equations. Commun. Partial Differ. Equ.. 21, 787–820 (1996). Publisher Full Text
Lin, C, Ni, W-M, Takagi, I: Large amplitude stationary solutions to a chemotaxis systems. J. Differ. Equ.. 72, 1–27 (1988). Publisher Full Text
Ni, WM, Takagi, I: Locating the peaks of least-energy solutions to a semilinear Neumann problem. Duke Math. J.. 70, 247–281 (1993). Publisher Full Text
Wang, ZQ: On the existence of multiple, single peaked solutions for a semilinear Neumann problem. Arch. Ration. Mech. Anal.. 120, 375–399 (1992). Publisher Full Text
Gui, C: Multi-peak solutions for a semilinear Neumann problem. Duke Math. J.. 84, 739–769 (1996). Publisher Full Text
Wei, J, Winter, M: Multiple boundary spike solutions for a wide class of singular perturbation problems. J. Lond. Math. Soc.. 59(2), 585–606 (1999). Publisher Full Text
Cingolani, S, Lazzo, M: Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions. J. Differ. Equ.. 160, 118–138 (2000). Publisher Full Text
Alves, CO, Figueiredo, GM, Furtado, MF: Multiple solutions for a nonlinear Schrödinger equation with magnetic fields. Commun. Partial Differ. Equ.. 36, 1565–1586 (2011). Publisher Full Text