A FRACTIONAL HARDY-SOBOLEV TYPE INEQUALITY WITH APPLICATIONS TO NONLINEAR ELLIPTIC EQUATIONS WITH CRITICAL EXPONENT AND HARDY POTENTIAL

In this paper we consider the attainability of the optimal constant S(N, p, s, mu) associated to the fractional Hardy -Sob olev type embedding which is defined as S(N, p, s, mu) := inf(u is an element of) W(center dot)s,p(RN)\{0}ff|u(x)-u(y)|p|x-y|N+psdxdy - mu f|u|p RN RN RN |x|ps dx ( fRN |u|p(s)& lowast; dx p p & lowast;s , where s is an element of (0, 1), p > 1 and N > ps, 0 <= mu < <mu>H, the latter being the best constant in the fractional Hardy inequality on RN, p(s)& lowast; = Np N-p(s) is the fractional critical Sobolev exponent. The technique that we use to prove the existence of extremals for S(N, p, s, mu) is based on blow-up analysis argument combined with a variational method. Further, as an application of the inequality, we prove an existence result for the critical fractional p -Laplacian equation with Hardy potential and involving continuous nonlinearities having quasicritical growth.