### Abstract

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

**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,
*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

For simplicity and without loss of generality, we set

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
*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 [9] studied the existence and concentration behavior of solutions for NLS with competing
potential functions. Cingolani and Lazzo in [19] 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

In the sequel, we restrict ourselves to the critical case in which

where

(f_{1})

(f_{2})

(f_{3}) there exists

(f_{4}) there exists

where

(f_{5}) the function

Our main results are the following theorem.

**Theorem 1.1***Let*
*Suppose that f satisfies* (f_{1})-(f_{5}), *V**is a continuous function in*
*and satisfies*
*Then when**ε**is sufficiently small*, *the problem* (1.4) *has at least*
*distinct nontrivial solutions*.

Here
*e.g.*, [20]), the category of *A* with respect to *M*, denoted by
*k* such that
*M*. We set

To prove Theorem 1.1, we mainly use the idea of [15,19,21]. More precisely, we can show that the

and

where

Then a topological argument asserts that

We will also prove that if *u* is a critical point of
*u* cannot change sign. Hence we obtain at least

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

We denote by *S* the Sobolev constant for the embedding

where

We say that a function

In view of (f_{2}) and (f_{3}), we have that the associated functional

is well defined. Moreover,

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

The least positive critical value

where

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

For more details, please see [22,23].

Set

For any

It is known [20] that

Moreover, we have

Set

Since

then the maximum value of the right-hand side is achieved at

and

Hence we have

We denote the Nehari manifold of

### 3 Compactness result

**Proposition 3.1***Let*
*as*
*Assume that*
*satisfies*
*as*
*Then uniformly in*
*there exist a subsequence of*
*still denoted by*
*and*
*such that*
*Furthermore*,
*converges strongly in*
*to**w*, *the positive ground state solution of equation* (2.2).

*Proof* Let

This implies that

hence

since

Indeed, if this is not true, then the boundedness of
_{2}), (f_{3}) and

Moreover,

as

and consequently (3.2) yields

*i.e.*,

However, recall the definition of S in (2.1),

equivalent to

We now consider

Hence

Since

For fixed

By the Hölder inequality,

Hence
*i.e.*,

Since
*w* in
*i.e.*,

Similarly, we have

By (f_{2}) and (f_{3}),

Hence when *R* is large enough, we get

Noting that

Hence

By (3.9)-(3.12), we derive that

*i.e.*,

For any

We assume

By the concentration-compactness lemma [24], there exists a subsequence of

Compactness: There exists a sequence

Vanishing: For all

Dichotomy: There exists a number

(i)

(ii)

(iii)

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

Now for the harder part. Let *η* be a smooth nonincreasing cut-off function, defined in

a nondecreasing function on

If we denote

(3.14) becomes

Denote by

Using Dichotomy (iii), we get

which implies

Hence

Now we observe that

and

where

Recall that

Then using

There exists
*i.e.*,

By (f_{2}) and (f_{3}),
_{4}), we get

since (f_{5}). By (3.15),

a contradiction. Thus

Hence by the Lebesgue dominated convergence theorem, we get

By (f_{5}), we have

Contradiction! Thus dichotomy does not occur.

With vanishing and dichotomy ruled out, we obtain the compactness of a sequence
*i.e.*, there exist

Then

for some positive constants
*δ*, which implies

From the foregoing, it follows that there exist bounded nonnegative measures

where

Choosing the test function

This reduces to

hence (3.27)(3) states

*i.e.*,

Consequently,

and hence

which implies that the set *J* is at most finite. Here Card*J* is the cardinal numbers of set *J*. Hence

since

When *n* is large enough, recall

a contradiction. Therefore *J* is empty, that is,

Equation (3.25) and compact embedding theorem imply

This together with (3.13), (3.20) and (3.32) allows us to deduce easily

Since

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 4.1***Suppose**f**satisfies* (f_{2})-(f_{4}). *Then*
*satisfies the*
*condition for all*
*that is*, *every sequence*
*in*
*such that*
*as*
*possesses a convergent subsequence*.

*Proof* Suppose that
_{4}), by a change of variable

This implies that

and by the Brezis-Lieb lemma [26],

For convenience, we denote by

and

It is clear that

It is easy to verify that

and thus

since
_{2}) and (f_{3}). If

and

we get

*i.e.*,

By the Sobolev inequalities,

Letting

Let
*η* be a smooth nonincreasing cut-off function, defined in

where *w* is the positive ground state of (2.2). We may assume that

Let

By Lemma 4.2 below,
*ε* sufficiently small. By noticing that

**Lemma 4.2***Uniformly in*
*we have*

*Proof* Let

By a change of variable

uniformly for

By the exponential decay of *ω*, we get

uniformly for
*ε* is very small, thanks to (4.8) (4.9) and (4.11), we find

On the other hand, following the idea of [21,25], from

which contradicts (4.12). Thus, up to a subsequence,

Since *f* has subcritical growth and

from which it follows that
*w* also belongs to

and

uniformly for

Thus (4.7) is proved. □

Let

**Lemma 4.3***Let*
*as*
*Then for*

*uniformly for*

*Proof* By change of variable

By Proposition 3.1,
*w*, which is a positive ground state solution of equation (2.2). Thanks to the exponential
decay of *w* (see (2.4)), we obtain

*Proof of Theorem 1.1* By Proposition 4.1,

By Lemma 4.3, we can assume that for any