Pseudo-Index Theory for a Schrödinger Equation with Competing Potentials ()
1. Introduction
In this paper, we are interested in the nonlinear Schrödinger equation
(1.1)
where
, i is the imaginary unit,
is the Planck constant,
,
,
,
,
and
are continuous bounded positive functions. An important issue concerning the above nonlinear evolution equation is to study its standing wave solutions of the form
. For small
, these standing wave solutions are referred to as semi-classical states. Byeon and Wang [1] are concerned with the existence and qualitative property of standing waves
for the following Schrödinger equation
where
with E being a critical frequency. It is easy to see that
solves Equation (1.1) if and only if
solves
(1.2)
Research on concentration phenomenon began many years ago, Ambrosetti, Badiale and Cingolani [2] considered
where
and v is a real-valued function,
, V has a possibly degenerate local minimum or maximum at
. Up to translations, they assumed that
and
, then obtained the solution
concentrates near
as
. Wang and Zeng [3] studied the nonlinear elliptic equation with competing poentials
(1.3)
where
, and they proved the ground state concentrates at a global minimum point of ground energy function by the concentration-compactness lemma. Ding and Liu [4] considered the existence, convergence and concentration phenomena of the ground state solution by using Mountain pass technique for
where
, V and W are bounded positive functions. For other convengence and concentration results on nonlinear elliptic equation, we can refer to [5] [6] [7] .
In the past few decades, the research on the multiplicity of solutions has been widely concerned. For example, Cingolani and Lazzo [8] improved the existence result for Equation (1.3) in [3] , and they studied the multiple positive solutions by the topology of the global minima set for energy function. Sun [9] studied the existence and multiplicity for a class of the quasilinear elliptic equations by Morse theory and the minimax method. Bartolo and Bisci [10] proved the existence and multiplicity of solutions to a fractional equation whose nonlinearity is subcritical and asymptotically linear at infinity by using a pseudo-index theory related to the genus. Papageorgiou, Rădulescu and Repovš [11] studied the existence and multiplicity to a class of double-phase Robin problems by the Morse theory, and using the notion of homological local linking. Wu, Tahar, Rafik, Rahmoune and Yang [12] established the existence of infinitely many solutions for the sublinear Schrödinger equations by using the linking theorem and the variant fountain theorem. Wang, Cheng and Wang [13] proved the multiplicity of positive solutions for the fractional Kirchhoff-Choquard equation with magnetic fields by using the penalization method and the Ljusternik-Schnirelmann theory. In [14] , Guo and Li considered the multiplicity of nontrivial solutions by using a global compactness result and Krasnoselskii’s genus theory for the following fractional Schrödinger equation in an open bounded domain of
,
where
,
, V is a sign-changing function. For the multiplicity of solutions to the nonlinear Schrödinger equation, we can refer to [15] [16] [17] .
Recently, Ding and Wei [18] considered the nonlinear Schrödinger equation
where
,
,
,
are bounded positive functions, and studied the existence, concentration phenomena of the positive ground state and multiplicity of semi-classical solutions by Benci pseudo-index theory and Nehari method. Liu and Tang [19] studied the following Choquard equation
where
,
,
is the Riesz potential with order
,
,
and
, they established the multiplicity of semi-classical solutions by Benci pseudo-index theory and the existence of sign-changing solutions by minimizing the energy on Nehari nodal set, they also studied the concentration phenomenon, convergence, decay estimate of ground state solutions. Similar studies appear in [20] [21] .
Motivated by the above works, in this paper, we consider the multiplicity of solutions and the existence, concentration, convergence and decay estimates of the ground state solution for Equation (1.2). There appear the combined nonlinearities in our equation, which make more difficulties in our arguments. Finally, we use the Benci pesudo-index theory to obtain the multiplicity of the semi-classical solutions for Equation (1.2), and we get the sign-changing solutions by resorting to the method. We extend the research in [18] and develop the method in [4] [19] [20] .
Our basic assumptions and the main results are the following.
(P1):
are bounded,
attains a global minimum on
with
, and
attains a global maximum on
with
,
.
To describe our results, for
, we denote by
(P2):
.
We continue to denote by
For vector
, we set
and let
,
,
. For
, we use
to signify
, and use
to signify
and
.
(P3): 1)
, and there is
such that
,
for
;
2)
, and there is
such that
for
. If (P3) - (1) holds, we set
. If (P3) - (2) holds, we set
. In the following,
stands for
in the case (P3) - (1), and
in the case (P3) - (2). Clearly,
is bounded. Furthermore,
, if
is not empty.
Theorem 1.1. Assume that (P1) holds and
,
. Then there exists
such that for the maximal integer
with
, Equation (1.2) has at least m pairs of solutions for small
. Moreover, among the solutions, at least one is positive, one is negative and two change sign if
.
Theorem 1.2. Assume that (P1) - (P2) hold and
,
. Then there exists
such that for the maximal integer
with
, Equation (1.2) has at least m pairs of solutions for small
. Moreover, among the solutions, at least one is positive, one is negative and two change sign if
.
Theorem 1.3. Assume that (P1) - (P3) hold. Then for
large small, Equation (1.2) has a positive ground state solution
. If
additionally and
are bounded,
, then
satisfies that
1) There is a maximum point
of
such that
;
2) There are
such that
for all
;
3) Setting
, then for any sequence
as
, there holds
in
as
, where u is a ground state solution of
(1.4)
If particularly
is not empty, then
, and up to a sequence,
in
as
, where u is a ground state solution of
(1.5)
Now we give some preliminary lemmas which will be useful for our arguments.
Lemma 1.4. ( [22] ) For every
and
, there are
and
such that
Lemma 1.5. ( [22] ) The embedding
↪
is continuous for
and the embedding
↪
is compact for
. Furthermore,
↪
is compact for
.
Lemma 1.6. ( [23] ) Let
and
. If
is bounded in
and
as
, then
in
for
as
.
For simplicity, we denote by
And we shall use different patterns of C to denote any positive constant, whose values may change from line to line, and
to denote the quantities that tend to 0 as
or
.
This paper is organized as follows: In Section 2, we give some preliminary results which are proved by Nehari method and play a key role in the arguments of main theorems. In Section 3, we prove the multiplicity of semi-classical solutions by using Benci pseudo-index theory and show the existence of sign-changing solutions. In order to get more detailed and accurate characterization of the properties of solutions, we also study the convergence, concentration phenomenon, and exponential decay estimates of the positive ground state solution.
2. Preliminary Results
2.1. Constant Coefficient Equation
We first consider the following equation
(2.1)
where
,
,
,
,
.
For each
, the energy functional associated to Equation (2.1) is
The weak solutions of Equation (2.1) are critical points of
. We denote the least energy by
, where
is the Nehari manifold. The set of least energy solutions can be denoted by
. In particular, we set
,
and
.
Lemma 2.1. The functional
satisfies that
1) There exist
and
such that
for all
;
2) For
,
as
.
Similar to the proof of Lemma 2.4 in [20] , we have the following result.
Lemma 2.2. Let
, then
Lemma 2.3.
is achieved and
is compact in
.
Proof. For any
, we choose the equivalent norm
. Clearly,
is not empty. Let
with
and
as
, By Lemma 1.4, there is
with
such that
,
,
. Observe that
. If
, then
. If
, then there exists
such that
and
as
. Hence
as
. Define
, then
,
, and
as
.
Clearly,
is bounded. We assume
in
as
up to a subsequence if necessary. By Lemma 1.5 and
, we have
, which ensures that
by letting
. Due to the weakly lower semi-continuity of norm, we obtain
. Thus we have
.
In the end, we can obtain
, where
is positive and radially symmetric. With similar arguments as aboves,
is compact in
.
Lemma 2.4. Let
,
,
.
1) If
, then
;
2) If
and
, then
.
Lemma 2.5. If u is a ground state solution of
(2.2)
with the energy
. Setting
, then z is a ground state solution of
(2.3)
with the energy
.
Proof. Observe that u is a ground state solution of Equation (2.2) if and only if z is a ground state solution of Equation (2.3). Indeed,
Furthermore,
if and only if
. Hence
.
Lemma 2.6. Assume that
,
. Then
.
Proof. Note that if
satisfies
, we have
. By the definition of
, we can find two cases:
(2.4)
or
(2.5)
If (2.4) holds, we choose
, then
. Thus we get
.
If (2.5) holds, we choose
, then
. Thus we get
.
Lemma 2.7. If
, then
and
. If
, then
and
.
Proof. Choose
in Equation (2.1), Equations (2.3), (2.4) and (2.5), respectively. By the definition of
, we have
. By Lemma 2.6, we have
.
Similarly, we choose
in Equation (2.1), Equations (2.3), (2.4) and (2.5), respectively. By the definition of
, we have
. If (2.4) holds, we choose
, then
by Lemma 2.4 and Lemma 2.5. If
, then
by Lemma 2.5. If
, then
by Lemma 2.4. Hence
. If (2.5) holds, we choose
, then
. If
, then
. If
, then
. Thus
. Lemma 2.8. There exist constants
such that for every
,
for all
.
Proof. Let
, we obtain
. For R large enough, we get
for
. Define
or
, where
,
. Choose C1 large enough such that
for
. Since
, we choose
such that
for
. Therefore,
By comparison principle,
for all
, then
for all
. For C1 large enough, we get that
for all
. Thus
for all
.
2.2. Auxiliary Equation
In this subsection, we consider the following equation for
and
,
(2.6)
where
,
,
and
,
.
For each
, the energy functional associated to Equation (2.6) is
The weak solutions of Equation (2.6) are critical points of
. We denote the least energy by
, where
is the Nehari manifold. The set of least energy solutions was denoted by
. In particular, we set
Lemma 2.9. The functional
satisfies that
1) There exist
and
both dependent on
and independent of
such that
for all
;
2) For
,
as
.
Lemma 2.10. Let
then
Lemma 2.11. If
has a
sequence, then either
or
. Furthermore,
.
Proof. Let
is a
sequence of
, then
and
in
as
. We will show that
when
. Since
is bounded, we may assume
in
as
. Hence
. Set
. By Lemma 1.32 in [23] , we obtain
(2.7)
For all
, we have
(2.8)
For any
, there is
such that
and
. By mean value theorem and Hölder inequality, we obtain
. Furthermore, by Hölder inequality again, we get that
. Thus
(2.9)
Similarly,
(2.10)
By Lemma 1.5, we obtain
in
as
with
, respectively. Hence
(2.11)
Similarly,
(2.12)
By (2.8) - (2.12), we get that
in
as
.
For all
, if
, there is
such that
. Thus
(2.13)
and
. By
, then we obtain
(2.14)
Moreover,
. If
and
as
, we can get
in
as
and
. If
or
in (2.14), we obtain
as
. Hence by (2.7), we have
as
. By (2.13), we get that
. If there is
, then
and
. Hence
.
Observe that
for all
. According to Lemma 2.2 and Lemma 2.10, we obtain
.
Similar to the proof of Lemma 2.11, we also have the following result.
Lemma 2.12. If
has a
sequence, then either
or
.
Lemma 2.13. For all
,
satisfies
condition.
Proof. Let
is a
sequence of
, then
and
in
as
. We assume
in
as
. Hence
. Set
. Due to the proof of Lemma 2.11, we obtain
(2.15)
Next, we will show that
and
in
as
. By definition, we get that for all
, there is
such that for all
,
(2.16)
Thus by (2.16), we have
which together with Lemma 1.5 and (2.15) imply that
(2.17)
Similarly, by Lemma 1.5 and (2.15) again, we have
(2.18)
By (2.17) and (2.18), we obtain
is a
sequence of
. By Lemma 2.11, either
or
. The latter contradicts our assumption
. Hence
and
(2.19)
Now we prove that
in
as
. Since
, we get
. By
, we obtain
. By Lemma 1.6, we assume there exist
and
such that
for some
. Moreover,
which is impossible. By (2.19), we conclude that
as
. Thus
in
as
.
Lemma 2.14.
, where
,
,
,
. Meanwhile, if
,
,
, then
.
Proof. Setting
and
,
, we have
(2.20)
Furthermore,
(2.21)
Due to Lemma 2.3, there is
satisfying
for
. Let
such that
, we obtain
(2.22)
Observe that
, there is
such that
(2.23)
Combining (2.22) with (2.23), we have
. Let
as
. By applying (2.20) - (2.22) and the Lebesgue dominated convergence theorem, we obtain
as
. Hence
.
In the end,
and
,
when
and
,
, namely, for all
, we obtain
,
,
. By Lemma 2.2, Lemma 2.10 and (2.21), we have
. According to
, we obtain
.
Lemma 2.15. If
or
, then there is
such that
is achieved at
for all
.
Proof. By Lemma 2.7, we have
, where
,
,
. By Lemma 2.11 and Lemma 2.14, there is
such that
for all
. By Lemma 2.13,
satisfies the
condition for all
, which combined Lemma 2.9 with Lemma 2.10, we have
is achieved at
. We set
is a ground state solution of Equation (2.6). If
, by
implies that
. Thus
,
which is impossible. Hence
does not change the sign. Then we may assume
. By the elliptic regularity theory,
. By strong maximum principle, we have
.
3. Proofs of the Main Results
Setting
, Equation (1.2) is a solution of
(3.1)
If
is a solution of Equation (3.1), then
is a solution of Equation (1.2).
Since
,
,
, we denote by
3.1. Proof of Theorem 1.1
Without loss of generality, we assume
. Then
,
,
.
Lemma 3.1. Equation (3.1) has at least m pairs of solutions.
Proof. We choose
in Equation (2.1) and by Lemma 2.3 and Lemma 2.8, there are
and
. Let
,
satisfies
if
and
if
with
. Assume
for
. By
as
, we get that
in
as
and
in
for
as
. There is a unique
such that
. Therefore,
(3.2)
Furthermore,
(3.3)
uniformly of x on any bounded set. There is a unique
such that
. Observe that
as
. Hence (3.2) and (3.3) imply that
(3.4)
By Lemma 2.7,
. We choose
. For the maximal integer
with
, we have
. Define
for
, and set
. Clearly,
if
. Hence
. Combining (3.2) with (3.3) again, for all
, we have
where
and
are the unique constants satisfying
and
, respectively, and
as
. Therefore, for all
, there are
and
such that for all
, we get
(3.5)
Let
for any
, where
. According to (3.5), for all
and
, we obtain
. Thus
for all
and
. By Lemma 2.6,
. We choose
, then there exist
and
such that
(3.6)
Next, we shall define constants
and prove that they are critical values of
. Consider the symmetric group
and we denote by
and
For any
, we define a version of Benci pseudo-index of A as follows,
, where
is the Krasnoselskii genus of A, and
is a constant given in Lemma 2.9. Let
,
. Observe that
. For any
and
, we have
, then
is not empty. By Lemma 2.9, it follows from
that
, where
is defined in Lemma 2.9.
Noticing that
satisfies dimension property in [24] , for all
, we have
. Hence
, then we obtain
. Combining (3.6) with Lemma 2.11, we have
(3.7)
Let
,
,
, and
. Clearly,
is an even functional. For all
, we obtain
(3.8)
By using (3.7) and Lemma 2.13, for all
,
satisfies
condition and
(3.9)
Set
, where
for any
, then we choose
, we have there is
such that
(3.10)
Let
, then
. By (3.10), we have
for all
. By Lemma 2.3 in [23] , there is
such that for all
,
is
an odd homeomorphism of
and
. Set
, then
is an odd homeomorphism of
and
(3.11)
For any
and
, it follows from
for all
that
. Hence
and
(3.12)
Moreover,
(3.13)
By applying the Theorem 1.4 in [24] , (3.8), (3.9) and (3.11) - (3.13), we have
are critical values of
, and
, if
with
and
. Since
is even, then
has at least m pairs of critical points being solutions of Equation (3.1).
Lemma 3.2. Equation (3.1) has at least one positive and one negative ground state solutions for
and has at least a pair of sign-changing solutions for
.
Proof. If
,
,
in Equation (2.1), then
,
,
. By Lemma 2.9 and Lemma 2.13,
has a
sequence and satisfies
condition. By Lemma 2.15, there exists
such that
is achieved at
for all
. Thus
and
are positive and negative ground state solutions of Equation (3.1), respectively.
Let
with
. Define
for
, where
is given in Lemma 3.1. Then
in
as
.
Choose
,
with
large enough and
.
Let
such that
and
. Then
and
,
is empty and
. Define
, then we have
. Define
, then
.
Next, we will prove
for
small enough. Due to
, we get
(3.14)
Observe that
as
and
(3.15)
By
and combining Lemma 2.6 with Lemma 2.11, we have
(3.16)
By (3.14) - (3.16), we get that
for
small enough, which implies
satisfies
condition for
small enough.
Now we show that there is a
sequence of
. Since
, then
. We assume
in
with
. There exist
and
such that
,
, we get
. Assume by contradiction that if
is not a sign-changing solution of Equation (3.1), there exists
such that
. We choose
small enough, satisfying
for all
. Let
be a cut off function such that
Then
. Hence
. By a degree theory argument in [25] , we find
such that
and
, which contradits that the defination of
.
In the end, we prove
is achieved at some
. Let
and
as
. By Ekeland vainational principle there is
such that
and
as
, then
as
. Hence
is a
sequence of
. Going of necessary to a subsequence, for
small enough we may assume
in
as
. Hence
and
. Then
, we have
,
. Thus
and
are a pair of sign-changing solutions of
Equation (3.1). Let
, then
are a pair of sign-changing solutions of Equation (1.2).
This completes the proof of Theorem 1.1.
3.2. Proof of Theorem 1.2
We can assume without loss of generality that
. Then
,
,
. Letting
,
,
in Equation (2.1), there is
by Lemma 2.3. Due to Lemma 2.7,
, we choose
For the maximal integer
, then
. By Lemma 2.6 and Lemma 2.7, we have
. The following proof of Theorem 1.2 is similar to that of Theorem 1.1 and so is omitted.
3.3. Proof of Theorem 1.3
In this subsection, we will consider the case (P3) - (1), the other case can be handled similarly. Without loss of generality, we assume
. Then
,
,
.
Lemma 3.3.
as
up to a sequence after translations.
Proof. Let
as
,
with
. By Lemma 2.14, we obtain
, which together with
, implies that
is bounded. By Lemma 2.8, there exist
,
and
such that
(3.17)
Let
,
,
,
. Then
is a solution of
(3.18)
Furthermore,
(3.19)
Since
is bounded, we can assume that
in
as
. Then
in
for
as
. By (3.17),
.
Since V and
,
are bounded, up to a subsequence if necessary, we can assume
(3.20)
and
. For all
, by the boundedness of
, for given arbitrarily
, we obtain
for all
. Hence
as
uniformly on any bounded set of x. Similarly,
,
as
uniformly on any bounded set of x. Similar to the proof of Lemma 2.14, we have
(3.21)
By (3.18), for any
, we obtain
which implies that u is a ground state solution of
(3.22)
with the energy functional
(3.23)
By Fatou’s Lemma,
(3.24)
Combining (3.19) with (3.22) - (3.24), we have
Hence
(3.25)
Set
satisfies
if
and
if
. Define
and
for
. Then
and
in
as
,
in
for
and
in
for
as
,
and
a.e. on
as
. We define
. Now we show that
and
as
. By Remark 1.33 in [23] , we have
(3.26)
For any
, there exists
such that
(3.27)
By choosing
in (3.27), respectively, we obtain
(3.28)
(3.29)
(3.30)
By using the Lebesgue dominated convergence theorem,
(3.31)
(3.32)
(3.33)
Moreover,
(3.34)
Combining (3.25) - (3.34) and (3.18) with (3.22), we have
(3.35)
and
(3.36)
In the end, by (3.35) and (3.36), we have
, which implies that
in
as
. Thus
in
as
.
Lemma 3.4.
as
uniformly in
.
Proof. We use the contradiction method to obtain that there are
for
,
as
such that
. Moreover, there exists
(independent of k) such that
. Thus by the Minkowski inequality, we have
as
, which is impossible.
Lemma 3.5.
is bounded on
.
Proof. Assume by contradiction that there is
as
up to a subsequence. Hence
and
,
. By Lemma 2.4, we have
. According to (3.19), (3.25) and Lemma 2.14,
, which is impossible.
By Lemma 3.5, we may assume
as
. By (3.20), we obtain
and
,
. Applying (3.22), we get that u is a ground state solution of Equation (1.4).
Lemma 3.6.
is bounded, where
is a maximum point of
.
Proof. If the thesis were not true, there were
with
, where
is a maximum point of
. Repeating Lemma 3.3 - Lemma 3.5, we can get that there exists
such that
in
as
,
as
uniformly in
,
is bounded on
. Thus
as
, then
as
. Since
as
, then
as
uniformly in
, which contradicts with
.
Lemma 3.7.
.
Proof. By Lemma 3.5 and Lemma 3.6, there exists
as
with
(3.37)
where
is a maximum point of
. By Lemma 3.3 and Lemma 3.5, there exists
such that
. By Lemma 3.4, we may assume
and
is bounded on
. Hence
and
as
. By (3.32) and (3.34), which imply that
(3.38)
Assume indirectly that
, then
,
,
or
,
,
or
,
,
. By Lemma 2.4,
(3.39)
Combining (3.19), (3.25), (3.38) and (3.39) with Lemma 2.14, we have
which is impossible. Hence
.
By Lemma 3.6, if
is not empty, we assume
, which implies that
Hence u is a groundstate solution of Equation (1.5). This completes the proof of Theorem 1.3.
Similar to the proof of Step 6 in [18] , we have the following result.
Lemma 3.8. There exists
such that for small
,
for all
.
Now we prove Theorem 1.3 by Lemma 3.3 - Lemma 3.8. Set
, then
. By Lemma 3.6,
is a maximum point of
and
is bounded on
. By Lemma 3.7,
. By Lemma 3.3 and Lemma 3.4,
, where
is a maximum point of
with
as
. By Lemma 3.8, we obtain
, where C depends on
.
Consequently, we establish the multiplicity of the semi-classical solutions for Equation (1.2), and we obtain the existence, concentration, convergence, exponential decay estimates of the positive ground state solution. We also prove the existence of sign-changing solutions of Equation (1.2).