Classification of positive solutions for a static Schrödinger–Maxwell equation with fractional Laplacian

In this paper, we study the fractional-order nonlocal static Schrödinger equation (−Δ)α2u=pup−1(|x|α−n∗up),u>0inRn, with n≥3n≥3, α∈(1,n)α∈(1,n) and p>1p>1. It can be viewed as an integral system involving the Riesz potentials {u(x)=p∫Rnup−1(y)v(y)dy|x−y|n−α,u>0inRn,v(x)=p∫Rnup(y)dy|x−y|n−αv>0inRn. First, the fact pp is larger than the Serrin exponent nn−α is a necessary condition for the existence of the positive solution. Based on this result, we investigate the classification of the positive solutions. If the system has solutions in Ln(p−1)α(Rn), then pp must be the critical exponent n+αn−α, and hence all the positive solutions can be classified as u(x)=v(x)=c(tt2+|x−x∗|2)n−α2, where c,tc,t are positive constants, and x∗∈Rnx∗∈Rn.