Sign Changing Solution of a Semilinear Schrödinger Equation with Constraint ()
1. Introduction
This article deals with the following semilinear Schrödinger equation with constraint
(1.1)
Given
, we try to find
to satisfy the Equation (1.1). We say
a positive solution if u is positive, a negative solution if u is negative, and a sign changing solution if u is sign changing.
Several authors have considered a Schrödinger equation of the form
(1.2)
In Bartsch and Wang [1], it is shown that the problem (1.2) possesses infinitely many solutions when
is odd with respect to u. Liu [2] obtains a positive solution and a negative solution of the problem (1.2) under the assumption that
and
are periodic with respect to the x-variables. Bartsch, Liu and Weth [3] prove the existence of sign changing solutions to the problem (1.2) and estimate the number of nodal domain.
Some papers concern with the problem (1.1). Under some conditions, a positive and a negative solution can be found in [4] and [5]. [6] gives some results on the existence of sign changing and multiple solutions of the problem (1.1) with different conditions.
In order to state our results, we require the following assumptions:
(A1)
.
(A2)
is locally Lipschitz continuous, and there are constants
and
such that
where
for
and
for
. Moreover,
as
uniformly in x.
(A3) There is a constant
such that
where
for
.
(A4) There is an open subset
such that
for
and
sufficiently large.
(A5)
for every
.
Our main result is the following theorem.
Theorem 1.1 Suppose (A1)-(A5) hold. Then problem (1.1) has at least three nontrivial solutions
, and
, where
is positive, and
is negative and
changes sign.
The key point is to construct certain invariant sets of the gradient flow associated with the energy functional of the elliptic problem. All positive and negative solutions are contained in these invariant sets. And minimax procedures can be used to construct sign changing critical point of the energy functional outside these invariant sets.
2. Preliminaries
We first fix some notations. Denote the usual Sobolev space by
, and set
. Consider the Hilbert space
We introduce the inner product in H by the formula
and the corresponding norm
According to (A1), there is a continuous embedding
↪
, hence
↪
(2.1)
Note that by (A2) for any
, there is a constant
such that
(2.2)
Assumption (A3) implies that given
, there exists a constant
such that
(2.3)
Denote
(2.4)
(2.5)
(2.6)
By Zeidler [7], we have
(2.7)
where
for
. It is easy to see from (2.7) that the critical points of I correspond to the solutions of problem (1.1) with
. And I is bounded.
Definition 2.1 Suppose E is a real Banach space. For
, we say
satisfies the Palais-Smale condition (denoted by (PS)) if any sequence
for which
is bounded and
possesses a convergent subsequence. We say
satisfies (PS)c for a fixed
if any sequence
for which
and
possesses a convergent subsequence. We say
satisfies (PS)+ if
satisfies (PS)c for all
;
satisfies (PS)− if I satisfies (PS)c for all
.
Lemma 2.1 [8] I satisfies (PC)−.
Let G be the Nemytskii operator induced by f, the mapping
may be written as
(2.8)
where
and
. Note that
In other words, KG is uniquely determined by the relation
(2.9)
is globally Lipschitz continuous in H applying (A2) [6].
Let E be a real Banach space,
and
. We will give some relevant definitions below.
Definition 2.2 A locally Lipschitz continuous mapping
is called a pseudo-gradient vector field (denoted by p.g.v.f) for
on
if it satisfies the following conditions
1)
;
2)
.
Suppose Q is a p.g.v.f for
on
, and consider the initial value problem in
(2.10)
According to the theory of ordinary differential equations in Banach space [9], (2.10) has a unique solution in
, denoted by
, with right maximal interval of existence
. Note that
may be either a positive number or
. Note also that
is monotonically decreasing on
and therefore
is called a descending flow curve.
Definition 2.3 A nonempty subset M of E is called an invariant set of descending flow for
determined by Q if
for all
.
Definition 2.4 Let M and D be invariant sets of descending flow for
. Denote
If
, then D is called a complete invariant set of descending flow relative to M.
3. Invariant Subsets of the Descending Flow
In this section we shall recall some results about the flow generated by
. We refer to Mawhin and Willem [10] for details.
It is clear that
is globally Lipschitz continuous, and
is a p.g.v.f of I. In the following we consider the initial value problem
(3.1)
Applying the theory of ordinary differential equations, we obtain:
Lemma 3.1 [10] There exists a unique solution
of (3.1) defined on a maximal interval
with
. The flow
is continuous, where
. For
,
has the expression
(3.2)
Lemma 3.2 [10] If
is finite, then
as
.
In our case, I is bounded and so it follows from Lemma 3.2 that
for
.
Lemma 3.3 [10] Suppose
, for any
, either there exists a unique
such that
or there is a critical point v of I in
, such that
as
.
It is easy to verify that
, that is
(3.3)
In our further proof, we shall need the following Lemma which is derived by Brézis and extended by Martin to infinite dimensional space (cf. Theorem 1.6.3 in [11] ).
Lemma 3.4 [11] Suppose E is a real Banach space, D is a closed subset of E,
is locally Lipschitz continuous and
(3.4)
where
is the distance on E. If
and
with
is the solution of the initial value problem
then
for all
.
Next we will discuss the convex cones
, and
. Moreover, for
we denote that
and
. Note that
implies
. Consider the sets
as well as
for
. Note that
and
are open convex subsets of H, whereas
is a closed and symmetric subset of H. Moreover,
contains only sign changing functions.
Note that
is a p.g.v.f for I, we can obtain a flow
satisfying (3.1) for all
, where
is the maximal existence time for the trajectory
. We call
the descending flow associated with
. A subset
is invariant for the
if
If M is an invariant subset of H, we also consider
and in addition we put
Note that
is open.
4. Three Solutions with One Changing Sign
In this section, we will give some proposition for finding three solutions with one changing sign.
Proposition 4.1 Suppose W is a finite dimensional subspace of H, there holds:
1)
;
2) If
, where
and S is a closed subset of some finite dimensional subspace W of H, then there is a constant
such that
(4.1)
Proof. 1) Obviously.
2) If we define
then
Inequality (2.3) implies that for any
there exist constants
, such that
Hence for
we have
Thus (4.1) hold.
Using (2.7), we can note that
(4.2)
where
.
Proposition 4.2 There exists
such that for
, there holds
1) If
and
, then
;
2) Every nontrivial solution
of (1.1) is negative, and every nontrivial solution
of (1.1) is positive.
Proof. 1) Let
,
and
, then
Similarly, using (2.1) we find for every
, there is a constant
with
(4.3)
Since
, we have
It follows from (2.2), (2.9) and (4.3) that
with a constant
. Hence
So there exists
such that
for every
with
. Thus
(4.4)
For any
, we can choose
small enough such that
Thus
(4.5)
It follows from Lemma 3.4 that if
is the solution of (3.1), then it will hold that
for all
. So we can obtain from (4.4) that
(4.6)
Set
, then
, and
is strictly increasing. Applying (4.6), we have
(4.7)
If we define
, and
, then
is a compact set of H. According to (4.7),
and hence
, where
is the closed convex hull of
in H. Note that
From (3.2) we get
(4.8)
Denote
, then F is also a compact set of H. Using by (4.6) and (4.8), we obtain that
Hence
for
and
. And
for
and
can be proved analogously.
(2) Put
, it follows from (2.9) that
Any
,
. By (2.2) and (4.3), for
with a constant
, we get
with a constant
. So
with a constant
. Thus
Hence, for
small enough
for every
with
. In particular we have
. If moreover
satisfies
, then
. If finally
, we conclude
for all x by the maximum principle [12]. Hence, every nontrivial solution
of (1.1) is negative. Similarly, every nontrivial solution
of (1.1) is positive.
In view of Proposition 4.2, the next proposition just follows from Liu and Sun [9]
Lemma 4.1 [9] Let E be a Hilbert space. Assume that
,
for
,
, and
. Then there is a p.g.v.f Q for
which enables
and
to be invariant sets of descending flow and
.
Lemma 4.2 [9] Let E be a Hilbert space. Suppose
satisfies (PS) and
has the expression
for
. Assume that
and
are open convex subset of E with the properties that
and
. If there exists a path
such that
and
Then
has at least four critical points, one in
, one in
, one in
, and one in
.
Note that
is a p.g.v.f of I such that
and
are invariant for the associated descending flow. Moreover
(4.9)
holds following from Proposition 4.2 and Lemma 4.1.
Proposition 4.3 If
, then
. In particular there holds
(4.10)
Proof. First, from (A2) and (A3) we have
Since by (3.3) we infer that
Hence, for
Next we recall that
contains no critical points of I by Proposition 4.2. Thus by (4.9), (4.10) and the invariance of
we find that
for every
as claimed.
As a consequence of the preceding discussion, the existence of three solutions with one changing sign follows from Lemma 4.2.
Proof of Theorem 1.1 Choose
, and set
It follow from Proposition 1, there exists
such that
By the choose of r, we have
, now we define the path
then
hence
Applying Proposition 4.3, we obtain
Thus
So we have
Since I satisfies (PS)−,
for
,
and
are open convex subsets of H,
. And by Proposition 4.2,
. It follows from Lemma 4.2 that I has a critical points
,
,
, where
is a positive solution of (1.1),
is a negative solution of (1.1) and
is a sign changing solution of (1.1).
Funding
This article is supported by the Science and Technology Project of Fujian Provincial Department of Education (JAT191148).