Sharp decay estimate for solutions of general Choquard equations

Abstract

Consider the general Choquard equation{Δpu+(∫RN|u(y)|pα⁎|x−y|αdy)|u|pα⁎−2u=0,inRN,u∈D1,p(RN), where 1<p<N,0<α<N,p<pα⁎=(N−α2)pN−p and Δpu=div(|∇u|p−2∇u) is the p-Laplacian operator. By using the Wolff potential theory, we prove that for any solution u of the equation, there exists a constant c such that|u(x)|≤c(1+|x|)−N−pp−1, for x∈RN. The decay estimate is sharp in the sense that the positive solution u satisfiesu(x)≥c(1+|x|)−N−pp−1.