Infinitely Many Solutions For Two Classes Of Fractional Schr(?)dinger Equations

In this thesis,we first investigate the following fractional Schr(?)dinger equation with superlinear growth:(—A?)su+V(x)u=f(x,u),x?RN(1)where N>2s,V:RN?R and f?C(RN×R,R),we assume the following conditions on V and f:(V1)V(x)?C(RN)and infRN V(x)>—?;(V2)For any M>0,there exists r>0 such that lim meas{x?RN:|x-y|?r,V(x)?M}=0;|y|??(f1)lim/t??=f(x,t)/|t|2s*-2t=0uniformly in x?RN;(f2)lim sup/t?0|f(x,t)/t|<+OO uniformly in x?RN;(f3)lim/|t|??F(x,t)/|t|2=+? uniformly in x?N;(f4)There exists D,r0>0,k>max{l,N/2s} and a nonnegative function W(x)?L1(RN)such tha(F(x,t)/t2)?D,F(x,t)+W(x)for|t|>r0 and x?RN,where F(x?t)=tf(x,t)-2F(x?t)?(f5)f(x?t)is odd with respect to t.According to these conditions,infinitely high energy solutions of the equation(1)are obtained by using the symmetric mountain lemma.Secondly,we research the following fractional Schr(?)dinger equation with critical exponent:(—?)su+u=?h(x)|u|q-1u+|u|2s*?2u,x?RN,(2)where?>0 is a parameter,1<q<2,0<s<1,N>2s and 2s*=N-2s/2N is a non-local fractional Sobolev exponent.The weight function h(x)is sign-changing and satisfies the following condition:(H)h?Lq*where q*=2s*-q/2s*and h+=max{h,0}?0.Using Brezis-Lieb lemma to overcome the lack of compactness,and then apply the symmetric Mountain Pass theorem,there exists ?*>0 such that for all ??(0,?*),problem(2)has infinitely many small energy solutions.