Positive Solutions for a Class of Fractional Choquard Equation in Exterior Domain

This work concerns with the existence of positive solutions for the following class of fractional elliptic problems, { (-Delta)(s)u + u = (integral(Omega)vertical bar u(y)vertical bar(p)/vertical bar x-y vertical bar(N-alpha) dy) vertical bar u vertical bar(p-2)u, in Omega (0.1) u = 0, R-N\Omega where s is an element of (0, 1), N > 2s, alpha is an element of (0, N), Omega subset of R-N is an exterior domain with smooth boundary partial derivative Omega not equal phi and p is an element of (2, 2(s)*). The main feature from problem (0.1) is the lack of compactness due to the unboundedness of the domain and the lack of the uniqueness of solution of the limit problem. To overcome the loss these difficulties we use splitting lemma combined with careful investigation of limit profiles of ground states of limit problem.

Automated summary

This work deals with the existence of positive solutions for a certain class of fractional elliptic problems. The main challenge lies in the lack of compactness due to the unboundedness of the domain and the lack of uniqueness in the solution of the limit problem. To overcome these difficulties, a splitting lemma is used along with a careful investigation of the limit profiles of the ground states of the limit problem.