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 growthε2s(-△)su+V(x)u=P(x)f(u)+Q(x)|u|2s*-2u,x∈RN,where ε>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 ε sufficiently small.Secondly,we study the existence of semiclassical state solution for the fractional Kirchhoff equation(ε2sa+ε4s-3b ∫ R3|(-Δ)s/2u|2dx)(-Δ)su+V(x)u=Γ(x)f(u),x ∈ R3,where ε>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 ε sufficiently small.At last,we consider the following fractional Kirchhoff equation with critical growth(ε2sa+ε4s-3b ∫ R3|(-Δ)s/2u|2)(-Δ)su+V(x)u=K(x)f(u)+Q(x)|u|2*s-2u,x ∈ R3,where ε>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 uε with polynomial growth for ε sufficiently small,Moreover,uε concentrates on a concrete set related to V(x),K(x)and Q(x).