Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Inﬁnitely Many Solutions for a Semilinear Elliptic Equation with Sign-Changing Potential | Hindawi Publishing Corporation Boundary Value Problems Volume 2009 Article ID 532546 7 pages doi 2009 532546 Research Article Infinitely Many Solutions for a Semilinear Elliptic Equation with Sign-Changing Potential Chen Yu and Li Yongqing School of Mathematics and Computer Sciences Fujian Normal University Fuzhou 350007 China Correspondence should be addressed to Chen Yu chenyusx@ Received 23 March 2009 Accepted 10 June 2009 Recommended by Martin Schechter We consider a similinear elliptic equation with sign-changing potential -Am - V x u f x u u e H1 RN where V x is a function possibly changing sign in RN. Under certain assumptions on f we prove that the equation has infinitely many solutions. Copyright 2009 C. Yu and L. Yongqing. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction In this paper the existence of solutions of the following elliptic equation - Au - V x u f x u u e H 1 RN P is studied where V x is a function possibly changing sign f is a continuous function on Rn X R. Problem P arises in various branches of applied mathematics and has been studied extensively in recent years. For example Rabinowitz 1 has studied the existence of a nontrivial solution of this kind of equation on a bounded domain. Lien et al. 2 studied the existence of positive solutions of problem P with V x A A is a positive constant and f x u u p-2u. And Grossi et al. 3 established some existence results for -Am Au a x g u where a x is a function possibly changing sign gfu has superlinear growth and A is a positive real parameter he discussed both the cases of subcritical and critical growth for gfu and proved the existence of linking type solutions. Cerami et al. 4 prove that the problem P has infinitely many solutions where a x is a regular function such that liminf x afx ar 0 and some .