Multiplicity And Concentration Of Solutions For Some Fractional Equations

This paper is mainly in four aspects devoted to studying the existence,multiplicity and concentration of solutions for several fractional equations.First,we consider the multiplicity and concentration of solutions for the fractional Choquard equationε^2s（-Δ）^su+V（x）u=ε^α-N(|x|^-α*F（u）)f（u）,x∈R^N,where N>2s,0<α<min{N,4s},the potential V:R^N→R satisfies Rabinowitz type global minimum condition,f is subcritical growth and does not satisfies the Ambrosetti-Rabinowitz condition,F is the primitive function of f.Then,we investigate the existence of positive ground state solution,the multiplicity and concentration of positive solutions for the fractional Kirchhoff equation(aε^2s+bε^4s-3[u]²)（-Δ）^su+V（x）u=f（u）,x∈R³,where a,b,>0 are positive constants,ε>0 is a small parameter,s ∈（3/4,1）,[u]²=∫∫_R⁶（|u|（x）-u（y）|²）/(|x-y|^3+2s)dxdy,（-Δ）^s is the fractional Laplacian,the potential V:R³→R satisfies Rabinowitz type global minimum condition,f is subcritical growth but does not satisfies the usual Ambrosetti-Rabinowitz contition.We obtain a new method to verify the compactness of Palais-Smale sequence to the corresponding variational functional.Later,we investigate the existence of positive ground state solution,the multiplicity and concentration of positive solutions for the fractional Kirchhoff-Choquard equation (aε^2s+bε^4s-3[u]²)（-Δ）^su+V（x）u=ε^α-3∫_R³（|u（y）|²_α^*+F（u（y）））/（|x-y|^α）dy(|u|^2_α*-2u+1/2_α^*f（u）),x∈R³where a,b are positive constants,ε is a small positive parameter,s ∈（0,1）,0<α<min{3,4s},2_α^*=（6-α）/（3-2s） is the critical exponent in the sense of HardyLittlewood-Sobolev inequality,the potential V:R³→R satisfies Rabinowitz type global minimum condition.f is subcritical growth and F is the primitive function of f.Last,we study the multiplicity and concentration of nontrivial solutions to the fractional Kirchhoff equation with magnetic potential(aε^2s+bε^4s-3[u]_A/ε²)（-Δ）_A/ε^su+V（x）u=f（|u|²）u,x ∈R³,where ε is a small parameter,a,b are positive constants,s ∈（3/4,1）,the magnetic potential A:R³→R³ is C0,α（R³,R³）continuous(α∈（0,1]）,（-Δ）_A^s is the fractional magenetic Laplacian,V:R³→R satisfies Rabinowitz type global minimum condition,f is subcritical growth and does not satisfies the Ambrosetti-Rabinowitz condition.The main methods of this paper are based on the variational method and Ljusternik-Schnirelmann category theory...