Multiplicity and uniqueness for Lane-Emden equations and systems with Hardy potential and measure data

Abstract

Let Omega be a C-2 bounded domain in R-N (N >= 3), delta(x) = dist(x, partial derivative Omega) and C-H(Omega) be the best constant in the Hardy inequality with respect to Q. We investigate positive solutions to a boundary value problem for Lane-Emden equations with Hardy potential of the form -Delta u - mu/delta(2) u = u(p) in Omega, u = rho nu on partial derivative Omega, (P-rho) where 0 < mu < C-H (Q), rho is a positive parameter, nu is a positive Radon measure on partial derivative Omega with norm 1 and 1 < p < N-mu, with N-mu being a critical exponent depending on N and mu. It is known from [22] that there exists a threshold value rho* such that problem (P-rho) admits a positive solution if 0 < rho <= rho*, and no positive solution if rho > rho*. In this paper, we go further in the study of the solution set of (P-rho). We show that the problem admits at least two positive solutions if 0 < rho < rho* and a unique positive solution if rho= rho*. We also prove the existence of at least two positive solutions for Lane-Emden systems {- Delta u - mu/delta(2) u = v(p) in Omega, - Delta v - mu/delta(2) v = u(q) in Omega, u = rho nu, v = sigma tau on Omega, under the smallness condition on the positive parameters rho and sigma. (C) 2021 Published by Elsevier Inc.