Stable solutions and finite Morse index solutions of nonlinear elliptic equations with Hardy potential

Abstract

We are concerned with Liouville-type results of stable solutions and finite Morse index solutions for the following nonlinear elliptic equation with Hardy potential: Delta mu + mu/vertical bar x vertical bar(2)u + vertical bar x vertical bar(l)vertical bar u vertical bar(p-1)u = 0 in Omega, where Omega = R-N,N-R\{0} for N >= 3, p > 1, l > -2 and mu < (N - 2)(2)/4. Our results depend crucially on a new critical exponent p = p(c)(l, mu) and the parameter mu. in the Hardy term. We prove that there exist no nontrivial stable solution and finite Morse index solution for 1 < p < p(c)(l, mu). We also observe a range of the exponent p larger than p(c)(l, mu) satisfying that our equation admits a positive radial stable solution. (C) 2013 Elsevier Ltd. All rights reserved.