Non-Nehari manifold method for asymptotically periodic Schrodinger equations  

作　　者：TANG XianHua 

机构地区：[1]School of Mathematics and Statistics, Central South University

出　　处：《Science China Mathematics》2015年第4期715-728,共14页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China(Grant No.11171351);Specialized Research Fund for the Doctoral Program of Higher Education of China(Grant No.20120162110021)

摘　　要：We consider the semilinear Schrdinger equation-△u + V(x)u = f(x, u), x ∈ RN,u ∈ H 1(RN),where f is a superlinear, subcritical nonlinearity. We mainly study the case where V(x) = V0(x) + V1(x),V0∈ C(RN), V0(x) is 1-periodic in each of x1, x2,..., x N and sup[σ(-△ + V0) ∩(-∞, 0)] < 0 < inf[σ(-△ +V0)∩(0, ∞)], V1∈ C(RN) and lim|x|→∞V1(x) = 0. Inspired by previous work of Li et al.(2006), Pankov(2005)and Szulkin and Weth(2009), we develop a more direct approach to generalize the main result of Szulkin and Weth(2009) by removing the "strictly increasing" condition in the Nehari type assumption on f(x, t)/|t|. Unlike the Nahari manifold method, the main idea of our approach lies on finding a minimizing Cerami sequence for the energy functional outside the Nehari-Pankov manifold N0 by using the diagonal method.We consider the semilinear Schrdinger equation-△u ＋ V（x）u = f（x, u）, x ∈ RN,u ∈ H 1（RN）,where f is a superlinear, subcritical nonlinearity. We mainly study the case where V（x） = V0（x） ＋ V1（x）,V0∈ C（RN）, V0（x） is 1-periodic in each of x1, x2,..., x N and sup[σ（-△ ＋ V0） ∩（-∞, 0）] 〈 0 〈 inf[σ（-△ ＋V0）∩（0, ∞）], V1∈ C（RN） and lim｜x｜→∞V1（x） = 0. Inspired by previous work of Li et al.（2006）, Pankov（2005）and Szulkin and Weth（2009）, we develop a more direct approach to generalize the main result of Szulkin and Weth（2009） by removing the ＂strictly increasing＂ condition in the Nehari type assumption on f（x, t）/｜t｜. Unlike the Nahari manifold method, the main idea of our approach lies on finding a minimizing Cerami sequence for the energy functional outside the Nehari-Pankov manifold N0 by using the diagonal method.

关 键 词：Schrodinger equation non-Nehari manifold method asymptotically periodic ground state solutions of Nehari-Pankov type 

分 类 号：O175[理学—数学]