Existence And Concentration Of Semiclassical State Solutions For Two Kinds Of Fractional Equations

In the present paper,by using the variational method and some analysis tech-niques,we study the existence and concentration of semiclassical state solutions for two kinds of fractional equations.Firstly,we research the following fractional Schr(?)dinger equation with critical growth?2s(-?)su+V(x)u=P(x)f(u)+Q(x)|u|2s*-2u,x?RN,where ?>0 is a small parameter,s ?(0,1),2s*=2N/N-2s,n>2s.Under a local condition imposed on the potential function,combining the penalization method and the concentration-compactness principle,we prove the existence of a semiclassical state solution for ? sufficiently small.Secondly,we study the existence of semiclassical state solution for the fractional Kirchhoff equation(?2sa+?4s-3b ? R3|(-?)s/2u|2dx)(-?)su+V(x)u=?(x)f(u),x ? R3,where ?>0 is a small parameter,a,b>0,s ?(3/4,1),the potential function V(x)satisfies a local condition.By using the penalization method,we prove the existence of a semiclassical state solution for ? sufficiently small.At last,we consider the following fractional Kirchhoff equation with critical growth(?2sa+?4s-3b ? R3|(-?)s/2u|2)(-?)su+V(x)u=K(x)f(u)+Q(x)|u|2*s-2u,x ? R3,where ?>0 is a small parameter,a,b 0,s ?(3/4,1),2s*=6/3-2s.Under some suitable assumptions on the positive continuous functions V(x),K(x)and Q(x),combining the concentration-compactness principle.We prove the existence of a positive ground state solution u? with polynomial growth for ? sufficiently small,Moreover,u? concentrates on a concrete set related to V(x),K(x)and Q(x).