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:
For simplicity and without loss of generality, we set , and , then equation (1.2) is equivalent to
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 .
In the sequel, we restrict ourselves to the critical case in which . More specifically, we study the following problem:
where if , and if . satisfies
(f1) for each ;
(f3) there exists such that
(f4) there exists such that
(f5) the function is strictly increasing in for any .
Our main results are the following theorem.
Theorem 1.1Let . Suppose that f satisfies (f1)-(f5), Vis a continuous function in and satisfies . Then whenεis sufficiently small, the problem (1.4) has at least distinct nontrivial solutions.
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
We will also prove that if u is a critical point of satisfying , then u cannot change sign. Hence we obtain at least nontrivial critical points of .
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
is the usual Sobolev space of real-valued functions defined by
with the normal
Let be the subspace of a Hilbert space with respect to the norm
We denote by S the Sobolev constant for the embedding , namely
where is the usual Sobolev space of real-valued functions defined by
We say that a function is a weak solution of the problem (1.4) if
In view of (f2) and (f3), we have that the associated functional given by
is well defined. Moreover, with the following derivative:
Hence, the weak solutions of (1.4) are exactly the critical points of .
Let us recall some known facts about the limiting problem, namely the problem
here acts as a parameter instead of an independent variable. Solutions of (2.2) will be sought in the Sobolev space as critical points of the functional
The least positive critical value can be characterized as
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
Moreover, there exist and such that
For any , we denote . We need to estimate the super bound of . In order to do this, we estimate . We shall use a family of radial function defined by
It is known  that
Moreover, we have
Set , where is a cut-off function satisfying if , if and . After a detailed calculation, we have the following estimates:
Since , from (2.5)-(2.7), we conclude
then the maximum value of the right-hand side is achieved at
Hence we have
We denote the Nehari manifold of by
3 Compactness result
Proposition 3.1Let as . Assume that satisfies as . Then uniformly in , there exist a subsequence of (still denoted by ), and such that . Furthermore, converges strongly in tow, the positive ground state solution of equation (2.2).
Proof Let be such that . Then, by a change of variable , we have
This implies that is bounded in . Noting that
since . Now we prove that there exists a sequence and constants such that
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
as . Therefore
and consequently (3.2) yields
However, recall the definition of S in (2.1),
We now consider such that (see (2.3)). By a change of variable , it follows that
Hence , from which it follows that in .
Since and are bounded in and in , the sequence is bounded. Thus, up to a subsequence, . If , then , which does not occur. Hence , and therefore the sequence satisfies
For fixed , define
By the Hölder inequality,
Hence , the dual space of . Consequently, as , in implies , i.e.,
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
Noting that in , . Therefore we have
By (3.9)-(3.12), we derive that
For any let us consider the measure sequence defined by
By the concentration-compactness lemma , there exists a subsequence of (denoted in the same way) satisfying one of the three following possibilities.
Compactness: There exists a sequence such that for any there is a radius with the property that
Vanishing: For all ,
Dichotomy: There exists a number , , such that for any there is a number and a sequence with the following property: Given there are non-negative measures , such that
(ii) , ,
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,
Hence , contradicting .
Now for the harder part. Let η be a smooth nonincreasing cut-off function, defined in , such that if ; if ; and . Also, let . We define
a nondecreasing function on . Denote by . We show now that dichotomy does not occur. Otherwise there exists such that for some and the function splits into and with the following properties:
If we denote
Denote by , then
Using Dichotomy (iii), we get
Now we observe that , therefore
where as .
Recall that (see (2.3)), which implies
Then using and in place of , respectively, we get
There exists such that , i.e.,
By (f2) and (f3), , we see cannot go zero, that is, . If , by (3.21), (3.22) and (f4), we get
since (f5). By (3.15),
a contradiction. Thus . Assume that , we will show . By (3.21) and (3.22), we have
Hence by the Lebesgue dominated convergence theorem, we get
By (f5), we have . Similarly, . Using this together with (3.16), (3.17), (3.18) and (3.19), we obtain
Contradiction! Thus dichotomy does not occur.
With vanishing and dichotomy ruled out, we obtain the compactness of a sequence , i.e., there exist and for each , there exists such that
Then must be bounded, for otherwise (3.25) would imply, in the limit ,
for some positive constants , independent of δ, which implies , contrary to (3.8).
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
where denotes a Dirac measure, . Take in the support of the singular part of , . We consider such that
Choosing the test function , from , we have
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
When n is large enough, recall (see (2.11)), together with (3.30) and (3.31), we obtain
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
Since is a uniformly convex Banach space, hence
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
Proposition 4.1Supposefsatisfies (f2)-(f4). Then satisfies the -condition for all , that is, every sequence in such that , , as , possesses a convergent subsequence.
Proof Suppose that is a sequence in such that , , as . Using (f4), by a change of variable , we obtain that
This implies that is bounded in . Therefore we may assume in and a.e. Let . Then
and by the Brezis-Lieb lemma ,
For convenience, we denote by
It is clear that
It is easy to verify that . Hence we have
since by (f2) and (f3). If , then , hence as , and we can obtain the desired conclusion. Hence it remains to show that . By a change of variable, from
By the Sobolev inequalities,
Letting , we get , so either which contradicts (4.5) or . □
Let be fixed. Let η be a smooth nonincreasing cut-off function, defined in , such that if ; if ; and for some . For any , let
where w is the positive ground state of (2.2). We may assume that is the unique positive number such that
Let be any positive function tending to 0 as , we define the sublevel
By Lemma 4.2 below, is not empty for ε sufficiently small. By noticing that , we can define as
Lemma 4.2Uniformly in , we have
Proof Let . Computing directly, we have
By a change of variable , we obtain
uniformly for .
By the exponential decay of ω, we get
uniformly for . Therefore, in the limit that ε is very small, thanks to (4.8) (4.9) and (4.11), we find
which contradicts (4.12). Thus, up to a subsequence, .
Since f has subcritical growth and , it follows that . Thus, we can take the limit in (4.13) to obtain
from which it follows that . Since w also belongs to , we conclude that . This and Lebesgue’s theorem imply that
uniformly for . Noting , from (4.12), (4.16) and (4.17), we have
Thus (4.7) is proved. □
Let be the center of mass of in terms of the norm:
Lemma 4.3Let as . Then for ,
uniformly for .
Proof By change of variable , we have
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 ).
By Lemma 4.3, we can assume that for any