Standing Waves for Quasilinear Schrödinger Equations with Indefinite Nonlinearity ()
Keywords:
1. Introduction
Solutions of semilinear elliptic equations
(1.1)
are standing waves of the corresponding time-dependent Schrödinger. For the existence of solutions of Equation (1.1), one of the important role is the sign of
and
. We say Equation (1.1) is linearly indefinite if
changes sign, and superlinearly indefinite if
changes sign. There are many results of Equation (1.1) for the superlinearly indefinite problem, linearly indefinite or not, we refer to [1] [2] [3]. In this paper, we consider the following modified Schrödinger equations
(1.2)
This kind of equations arise when we are looking for standing waves
for the time-dependent quasilinear Schrödinger equation
which was used for the superfluid film equation in plasma physics by Kurihar [4]. This model also appears in plasma physics and fluid mechanics, dissipative quantum mechanics and condensed matter theory. For more information on the relevance of these models and their deduction, we refer to [5].
To the best of our knowledge, the first mathematical studies of the Equation (1.2) seem to be Poppenberg et al. [6] for the one dimensional case and Liu-Wang [7] for higher dimensional case. The proofs in these papers are based on constrained minimization argument. Formally, Equation (1.2) associates with the Euler functional
Unfortunately, the functional J is not defined for all
, unless
. Therefore, it is difficult to use the standard variational methods to study the functional J. To overcome this difficulty, Jeanjean [8] introduced a transformation f so that if v is a critical point of
(1.3)
where f is defined by
Then
is a solution of (1.2).
Since the publication of [8], Problem (1.2) has been studied extensively. For example, the case that the potential V is
is studied in Silva-Vieira [9]. By Nehari manifold method, Fang-Szulkin [10] studied the case that the nonlinearity is 4-superlinear and the potential has a positive lower bound. For problems with critical nonlinearities, see Silva-Vieira [9].
In all these papers, it is required that the potential V and nonlinearity satisfy the positive condition. With this condition and suitable conditions on the nonlinearity, the mountain pass theorem can be applied to produce a solution of (1.2).
In the literature, there are some existence results which allow the potential V to be negative somewhere. The strategy is to write
with
. Then if
is in some sense small, it can be absorbed and the functional still verifies the mountain pass geometry. We refer the reader to [11]. Recently, by a local linking argument and Morse theory, Liu-Zhou [12] obtains a nontrivial solution for the problem (1.2) with indefinite potential. For linearly indefinite case, we also refer to [13].
However, this is a gap in the high dimensional quasilinear Schrödinger equations with indefinite nonlinearity. The one dimensional case has been partially studied in [14] by critical point theory. The purpose of this paper is to present some results about indefinite quasilinear Schrödinger equations in higher dimensional. More precisely, we present our assumptions on the potential
and
(V1)
;
(V2)
and for each
,
, where
is a constant and
denotes the Lebesgue measure of a measurable set
;
(A1)
and
, where
.
(P1)
where the critical Sobolev exponent
for
and
for
.
Then we have
Theorem 1. Suppose that (V1), (V2), (A1) and (P1) hold. Then Equation (1.2) has at least one nontrivial solutions.
Notation.
will denote different positive constants whose exact value is inessential.
2. Preliminaries
Before prove our results, we shall introduce the appropriate space to find critical points of the Euler functional. Let
with the inner product
and the norm
Then X is a Hilbert space. By Bartsch and Wang [15], we know that the embedding
↪
for is compact for
.
Below we summarize the properties of f in (1.3). Proofs may be found in [8].
Lemma 2.1. The function f has the following properties:
(f1) f is uniquely defined,
and invertible.
(f2)
and
for all
. Moreover,
.
(f3) For all
we have
.
(f4) For all
we have
and
.
(f5) There exists a positive constant
such that
for
,
for
.
By Lemma 2.1, it is easy to see that
, moreover
(2.1)
for all
.
3. Proof of the Theorem 1
Because the principle part of
, denoted by
is not a quadratic form on v, it’s not so obvious to verify that
satisfies the mountain pass geometry. Similar to [12], by taking into account the Taylor expansion of Q at the origin, it is easy to deduce that
is a strict local minimizer of
.
Lemma 3.1. Under the assumptions of Theorem 1, then
(i)
is a strict local minimizer of
.
(ii) There is
with
such that
.
Proof. By the properties of the transformation f, it is easy to see that Q is a C2-functional on X. Since
, we get
. According to the Taylor formula, as
, we have
Therefore, combining this with Lemma 2.1 (f2), there exists
such that
this implies that the zero function 0 is a strict local minimizer of
.
On the other hand, since
and
is continuous in
, we may choose
such that
and
for all
. Then for any
, using Lemma 2.1 (f2),(f5), we deduce
Since
, we know that
for s sufficiently large. Thus the conclusion(ii) follows from choosing
with
large. £
Lemma 3.2. Under the assumptions of Theorem 1. Then the functional
satisfies Cerami condition.
Proof. Let
be a Cerami sequence of
, that is
,
for some
.
First we claim that there exists
such that
(3.1)
Let
. By direct computation, we get
. By (1.3) and (2.1), there exists
such that
Therefore, our claim is true.
Next, we claim that there exists
such that
(3.2)
Indeed, we may assume that
(otherwise the conclusion is trivial). We argue by contradiction and assume that
(3.3)
where
and
. By direct computation, we have
(3.4)
This implies
is strictly increasing. So we get
is positive if
. Combining this with (3.3), we obtain
(3.5)
We claim that for each
, there exists a constant
independent of n such that
, where
. Otherwise, there is an
and a subsequence
of
such that for any positive integer k,
, where
. By the properties of f described in Lemma 2.1 and (V1), there exists a constant
such that
a contradiction. Hence the assertion is true. Then for each
,
may be chosen so that
. Next, keeping
in mind. Let
. By (3.4), as in the proof of the Lemma 3.10 in [10], it is easy to see that as
, there exists
such that
Combining this with (3.3) and the Mean Value Theorem, we have
(3.6)
Since
is uniformly bounded, by the integral absolutely continuity there exists
such that whenever
,
. For this
, we have
This and (3.6) contradict with (3.5). Therefore, this claim is true.
Lastly, together (3.2) and Lemma 2.1(f4) give us
Combining this with (3.1) implies
is bounded in X. Up to a subsequence we may assume
in X. Since embedding
↪
is compact for
, by a standard argument, we can show that
has a convergent subsequence, see [16] (Theorem 2.1, Step 3). We omit it here. This completes the proof.
To prove Theorem 1, we will apply the following Mountain Pass Theorem [17].
Theorem 2. Let X be a Banach space and
be a functional satisfying the Cerami condition. If
and
are such that
then
is a critical value of
with
, where
.
Proof of the Theorem 1
Proof. From Lemma 3.1 and Lemma 3.2, we know
satisfies the conditions of Theorem 2. Hence Equation (1.2) has at least one nontrivial solution under assumptions (V1), (V2), (A1) and (P1).
4. Conclusion
By mountain pass theorem and Taylor expansion, we prove the existence of solutions for the quasilinear Schrödinger equations with indefinite nonlinearity. This indefinite problem had never been considered so far. So our main results can be regarded as complementary work in the literature. On the other hand, our approach seems much simpler than those presented in [9] [16].
Acknowledgements
This project is supported by National Natural Science Foundation of China (Nos.11701114, 11871171).
NOTES
*Corresponding author.