Existence of solutions to elliptic problems with fractional p-Laplacian and multiple critical nonlinearities in the entire space [formula omitted

Abstract

Using the Mountain-Pass Theorem of Ambrosetti and Rabinowitz we prove that the following fractional p-Laplacian equation with double critical nonlinearities (−Δp)su=|u|ps∗−2u+|u|ps∗(α)−2u|x|αinRN,admits a positive solution in the class Ẇs,p(RN). In the above, (−Δp)s is the fractional p-Laplacian, s∈(0,1), p>1, 0<α<ps<N, ps∗=NpN−ps is the critical fractional Sobolev exponent and ps∗(α)=p(N−α)N−ps is the critical Hardy–Sobolev exponent, Ẇs,p(RN) denotes the completion of C0∞(RN) with respect to Gagliardo norm [u]s,pp≔∫RN∫RN|u(x)−u(y)|p|x−y|N+psdxdy.Our method is based on the existence of extremals of some fractional Hardy–Sobolev type inequalities, and coupled with some intricate estimates for the nonlocal (s,p)-gradient. Moreover, we also establish the existence of a nontrivial solution to an elliptic system which involves fractional p-Laplacian and critical Hardy–Sobolev exponents in RN.