Infinitely Many Solutions For The Schr(?)dinger Equations With Sublinear Nonlinearities

First,we investigate the following Schr(?)dinger equation-?u + V(x)u = f(x,u),x?RN,(1)where N?3,is an indefinite potential.By using variant symmetric mountain lemma,we obtain infinitely many nontrivial solutions for problem(1).The main results as follows,Theorem 1 Suppose V and f satisfy the following conditions,(V0)V?LN/2(RN)and |V-|N/2<S,where S is the best Sobolev constant,S = inf {?u?2|u?D1,2(RN),?|u|2*dx=1};(f1)f?C(RN×R,R)and there exist constant ??(1,2)and function h ?L2*/2*-?(RN)such that|f(x,t)|??|h(x)||t|?-1,(?)(x,t)?RN×R;(f2)There exist x0?RN and constant r0>0 such that lim sup(?)=+?,lim inf(?)>-?,where F(x,t)=f0t f(x,s)ds;(f3)f(x,t)is odd with respect to t.Then problem(1)possesses infinitely many nontrivial solutions.Theorem 2 Suppose(V0),(f2)and(f3)hold and f satisfies the following con-dition,(f1')??C(RN×[-?,?],R)with ?>0 and there exist constants ??(1,2),p?(N/2,+?)and function h?L 2*/2*-?(RN)? LP(RN)such that|?(x,t)|??|h(x)||t|?-1(?)(x,t)?RN×[-?,?].Then problem(1)possesses infinitely many nontrivial solutions.And then we study the following fractional Schr(?)dinger equation(??)su + V(x)u = f(x,u),x?RN(2)where N?3,V is an indefinite potential and f ? C(RN × R,R)is nonlinear term.Existence of infinitely many solutions is obtained by using variant symmetric mountain lemma.Theorem 3 Suppose(f2),(f3)hold and V,f satisfy the following conditions,(Vi)V?L N/2(RN)and |V-|N/2<Ss,where Ss is the fractional Sobolev constant,Ss =inf{?RN?RN|u(x)-u(y)|2/|x-y|N+2s dxdy|u?Ds,2(RN),?RN|u|2s*dx=1},where 2s*=2N/N-2s;(f4)f ? C(RN×R,R)and there exist positive functions a ?L N/2s(RN),b?L2s*/2s*-?(RN)such that|f(x,t)|?a(x)|t|+b(x)|t|?-1,(?)(x,t)?RN×R,where ??(1,2),|a|N/2s<Ss/4Then problem(2)possesses infinitely many nontrivial solutions.