Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : Infinitely many sign-changing solutions for a Schrödinger equation | Qian Advances in Difference Equations 2011 2011 39 http content 2011 1 39 REVIEW o Advances in Difference Equations a SpringerOpen Journal Open Access Infinitely many sign-changing solutions for a Schrodinger equation Aixia Qian Correspondence qaixia@ School of Mathematic Sciences Qufu Normal University Qufu Shandong 273165 P. R. China Abstract We study a superlinear Schrodinger equation in the whole Euclidean space Rn. By using a suitable sign-changing critical point we prove that the problem admits infinitely many sign-changing solutions under weaker conditions. Keywords Schrodinger equation sign-changing critical point w -PS condition 1 Introduction In this paper we consider the following Schrodinger equation Au V x u f x u x e RN u x 0 x TO. In order to overcome the lack of compactness of the problem we assume that the potential V x has a good behavior at infinity in such a way the Schrodinger operator -A V x on L2 RN has a discrete spectrum. More precisely we suppose V1 V e q2oc RN V is bounded from below V2 There exists r0 0 such that for any h 0 meas Br0 y n Vh 0 y TO where meas A denotes the Lebesgue measure of A on RN Br0 y is the ball centered at y with radius r0 and Vh x e RN V x h . Of course V x above can satisfy the condition S1 or S1 S1 in 1 so that the Schrodinger operator could have the same good properties. We denote lj to be the eigenvalues sequence of - A V x see Proposition in Section 2 . Set F x t Jf x s ds F x t f x t t 2F x t . We assume the following conditions. f1 f Rn X R R is a Carathéodory function with a subcritical growth If x t c 1 t s 1 t e R x e RN where s e 2 2 f x t 0 for all x t e RN X R andf x t o t as t 0. f2 lim x TO uniformly for x e RN. t TO t 2 Springer 2011 Qian licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License http licenses by which permits unrestricted use distribution .
