The Study Of Existence Of Solutions For Two Types Of Non-local Elliptic Problems

In this paper,we will discuss the existence of solutions to two types of non-local elliptic problems.First of all,we study the normalized solutions to the following fractional Choquard-type problem where(-Δ)1/2 denotes the 1/2-Laplacian operator,a>0,λ∈R,Iμ=1/|x|μ with μ∈(0,1),F(u)is the primitive function of f(u),and f is a continuous function with exponential critical growth in the sense of Trudinger-Moser inequality.By using a minimax principle based on the homotopy stable family,we obtain that there is at least one positive ground state solution to the above problem.Next,we consider a class of fractional magnetic Kirchhoff type equation(aε2s+bε4s-N[u]A/ε2)(-Δ)A/εsu+V(x)u=|u|p-1u,in RN,where ε>0 is a small parameter,a,b>0,(-Δ)As is the fractional magnetic Laplacian operator,s∈(0,1),1<p<(N+2s)/(N-2s),2s<N<4s,A(x):RN→RN is the bounded magnetic potential and V(x):RN→R is a continuous potential function.Our approach is based on some decaying estimate and nondegenerate of the solutions of the limit problem,we prove that the above equation has multi-peak solutions concentrating at local minimum points of V(x)by applying Lyapunov-Schmidt reduction method.