Least Energy Nodal Solution For Nonlinear Schr(o|¨)dinger Equation Without(AR) Condition

In this paper,we are concerned with the existence of least energy nodal solution for the following stationary nonlinear Schrodinger equation-Δu+λV(x)u=f(u),x∈rn,N,N≥3,(P)where λ>0 is a parameter,potential function V(x)has double potential well.The nonlinearity f(u)has asymptotically linear growth or superlinear growth at infinity without the Ambrosetti-Rabinowitz type superlinear condition,((AR)condition for short).By using the constrained variational method and Ekeland’s variational principle,we prove that the above equation(P)has a least energy nodal solution.That is,the sign-changing solution has the least energy in all the sign-changing solutions of(P).The main contents of this article contain the following parts:Chapter 1 In the first chapter,we mainly discuss the research backgrounds and significance,as well as the existing literature results,of the above equation(P).Chapter 2 In the second chapter,we introduce some symbols commonly used,and some definitions and theorems which are used in the proofs.Chapter 3 In the third chapter,we mainly prove some necessary lemmas.While the nonlinear term f(u)is asymptotically linear,it is not possible to directly obtain the non-empty constraint set Mλ.Therefore,we use the condition that the potential function V(x)has double potential wells and prove that the constraint set Mλ is not empty by construction methodChapter 4 In the fourth chapter,based on the Ekeland’s variational principle and the Implicit function theorem,we provide a new method to deal with the existence of the least energy nodal solution for the stationary nonlinear Schrodinger equation when the nonlinear term does not satisfy the(AR)condition.Chapter 5 In the fifth chapter,we summarize the contents of this paper and provide some ideas for the future research.