Multiplicity and concentration behaviour of solutions for a fractional Choquard equation with critical growth

In this paper, we study the singularly perturbed fractional Choquard equation epsilon(2s)(-Delta)(s)u+ V(x)u = epsilon(mu-3)(integral(3)(R) vertical bar u(y)vertical bar(2)mu,s* + F(u(y))/vertical bar x-y vertical bar(mu) dy)(vertical bar u vertical bar(2)(mu,s)*(-2)(u) + 1/2(mu,s)* f(u)) in R-3, where epsilon > 0 is a small parameter, (-Delta)(s) denotes the fractional Laplacian of order s is an element of (0, 1), 0 < mu < 3, 2(mu,s)* = 6-mu/3-2s the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and fractional Laplace operator. F is the primitive of f which is a continuous subcritical term. Under a local condition imposed on the potential V, we investigate the relation between the number of positive solutions and the topology of the set where the potential attains its minimum values. In the proofs we apply variational methods, penalization techniques and Ljusternik-Schnirelmann theory.

Automated summary

This paper studies the properties of solutions to the singularly perturbed fractional Choquard equation in space R-3, and investigates the relationship between the positivity of solutions and the topology of the potential function's minimum values. The results are proved using variational methods, penalization techniques, and Ljusternik-Schnirelmann theory.