Sign Changing Solution of a Semilinear Schrödinger Equation with Constraint

The purpose of this paper is to study a semilinear Schrödinger equation with constraint in ( ) 1 N H R , and prove the existence of sign changing solution. Under suitable conditions, we obtain a negative solution, a positive solution and a sign changing solution by using variational methods.


Introduction
This article deals with the following semilinear Schrödinger equation with constraint ( ) Given 0 r > , we try to find ( ) , f x u 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: : N f × → R R R is locally Lipschitz continuous, and there are constants Our main result is the following theorem. Theorem 1.1 Suppose (A 1 )-(A 5 ) hold. Then problem (1.1) has at least three nontrivial solutions , u u + − , and u , where u + is positive, and u − is negative and u 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.

Preliminaries
We first fix some notations.
We will give some relevant definitions below. Definition 2.2 A locally Lipschitz continuous mapping : According to the theory of ordinary differential equations in Banach space [9],

Definition 2.3 A nonempty subset M of E is called an invariant set of descending flow for
Definition 2.4 Let M and D be invariant sets of descending flow for , then D is called a complete invariant set of descending flow relative to M.

Invariant Subsets of the Descending Flow
In this section we shall recall some results about the flow generated by I ′ . We refer to Mawhin and Willem [10] for details.
It is clear that I ′ is globally Lipschitz continuous, and I ′ 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: In our case, I is bounded and so it follows from Lemma 3.2 that ( ) It is easy to verify that ( ) 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] Next we will discuss the convex cones

Three Solutions with One Changing Sign
In this section, we will give some proposition for finding three solutions with one changing sign.
Proposition 4.2 There exists 0 0 ε > such that for is the solution of (3.1), then it will hold that ( ) If finally 0 u ≠ , we conclude ( ) 0 u x < for all x by the maximum principle [12]. Hence, every nontrivial solution r u P S ε − ∈  of (1.1) is negative. Similarly, every nontrivial solution r u P S ε + ∈  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 , where u + is a positive solution of (1.1), u − is a negative solution of (1.1) and u is a sign changing solution of (1.1).

Founding
This article is supported by the Science and Technology Project of Fujian Provincial Department of Education (JAT191148).