A system of equations involving the fractional p-Laplacian and doubly critical nonlinearities

Abstract

This article deals with existence of solutions to the following fractional p-Laplacian system of equations:{(-Delta(p))(s) u = vertical bar u vertical bar(ps*-2)u + gamma alpha/p(s)* vertical bar u vertical bar(alpha-2) u vertical bar v vertical bar(beta) in Omega, (-Delta(p))(s) v = vertical bar v vertical bar(ps*-2)v + gamma beta/p(s)* vertical bar u vertical bar(beta-2) v vertical bar u vertical bar(alpha) in Omegawhere s is an element of(0, 1), p is an element of (1, infinity) with N > sp, alpha, beta > 1 such that alpha + beta = p(s)* := Np/N-sp and Omega = R-N or smooth bounded domains in R-N. When Omega = R-N and gamma = 1, we show that any ground state solution of the aforementioned system has the form (lambda U, tau lambda V) for certain tau > 0 and U and V are two positive ground state solutions of (-Delta(p))(s) u = vertical bar u vertical bar(ps*-2)u in R-N. For all gamma > 0, we establish existence of a positive radial solution to the aforementioned system in balls. When Omega = R-N, we also establish existence of positive radial solutions to the aforementioned system in various ranges of gamma.