This article deals with the following semilinear Schrödinger equation with constraint
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
In Bartsch and Wang , it is shown that the problem (1.2) possesses infinitely many solutions when is odd with respect to u. Liu  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  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  and .  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:
(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.
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
Note that by (A2) for any , there is a constant such that
Assumption (A3) implies that given , there exists a constant such that
By Zeidler , we have
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  I satisfies (PC)−.
Let G be the Nemytskii operator induced by f, the mapping may be written as
where and . Note that
In other words, KG is uniquely determined by the relation
is globally Lipschitz continuous in H applying (A2) .
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
Suppose Q is a p.g.v.f for on , and consider the initial value problem in
According to the theory of ordinary differential equations in Banach space , (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  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
Applying the theory of ordinary differential equations, we obtain:
Lemma 3.1  There exists a unique solution of (3.1) defined on a maximal interval with . The flow is continuous, where . For , has the expression
Lemma 3.2  If is finite, then as .
In our case, I is bounded and so it follows from Lemma 3.2 that for .
Lemma 3.3  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
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  ).
Lemma 3.4  Suppose E is a real Banach space, D is a closed subset of E, is locally Lipschitz continuous and
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:
2) If , where and S is a closed subset of some finite dimensional subspace W of H, then there is a constant such that
Proof. 1) Obviously.
2) If we define
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
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
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
For any , we can choose small enough such that
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
Set , then , and is strictly increasing. Applying (4.6), we have
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
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 . 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 
Lemma 4.1  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  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
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
holds following from Proposition 4.2 and Lemma 4.1.
Proposition 4.3 If , then . In particular there holds
Proof. First, from (A2) and (A3) we have
Since by (3.3) we infer that
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
Applying Proposition 4.3, we obtain
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).
This article is supported by the Science and Technology Project of Fujian Provincial Department of Education (JAT191148).
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