Stable Solutions of -Delta u plus lambda u = vertical bar u vertical bar(p-1)u in Strips

摘要

This paper is devoted to study the following equation -Delta u + lambda u=vertical bar u vertical bar(p-1)u in Omega, with homogeneous Dirichlet or Neumann boundary conditions wherep>1, lambda > 0, Omega = Rn-k x Omega, n >= 2, k >= 1, and Omega is a smoothly bounded domain of R-k. We prove Liouville-type theorems for C-2 solutions which are stable or stable outside a compact set of Omega. We first provide an integral estimate using stability argument which combined with Pohozaev-type identity allow to obtain nonexistence results for p is an element of [p(s)(n), p(s)(n - k)], whereps(n) is the classical Sobolev exponent in dimension n. Also, we establish monotonicity formula to prove the nonexistence of nontrivial solution which is stable or stable outside a compact set of Omega for all p > 1.