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 μ∈(1,2)and function h ∈L2*/2*-μ(RN)such that|f(x,t)|≤μ|h(x)||t|μ-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×[-δ,δ],R)with δ>0 and there exist constants μ∈(1,2),p∈(N/2,+∞)and function h∈L 2*/2*-μ(RN)∩ LP(RN)such that|∫(x,t)|≤μ|h(x)||t|μ-1(?)(x,t)∈RN×[-δ,δ].Then problem(1)possesses infinitely many nontrivial solutions.And then we study the following fractional Schr(?)dinger equation(一△)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*-μ(RN)such that|f(x,t)|≤a(x)|t|+b(x)|t|μ-1,(?)(x,t)∈RN×R,where μ∈(1,2),|a|N/2s<Ss/4Then problem(2)possesses infinitely many nontrivial solutions.