Fractional Hardy-Sobolev equations with nonhomogeneous terms

Abstract

This paper deals with existence and multiplicity of positive solutions to the following class of non-local equations with critical nonlinearity: {(-Delta)(s)u-gamma u/vertical bar x vertical bar(2s) = K(x)vertical bar u vertical bar 2*s(t)-2u/vertical bar x vertical bar t +f(x) in R-N,u is an element of (H)/Over dots (RN), where N > 2s, s 2 (0, 1), 0 is an element of t < 2 s < N and 2 * s (t) := 2( N-t) N-2 s. Here 0 < < N, s and N,s is the best Hardy constant in the fractional Hardy inequality. The coefficient K is a positive continuous function on RN, with K(0) = 1 = lim jxj!1 K(x). The perturbation f is a nonnegative nontrivial functional in the dual space. H s( RN) 0 of. H s( RN). We establish the prole decomposition of the Palais-Smale sequence associated with the functional. Further, if K >= 1 and kf k(. H s) 0 is small enough (but f 6 0), we establish existence of at least two positive solutions to the above equation.