Liouville-type theorems with finite Morse index for \Delta_{\lambda}-Laplace operator

Abstract

In this paper we study solutions, possibly unbounded and sign-changing, of the following problem: -\D_{\lambda} u=|x|_{\lambda}^a |u|^{p-1}u, in R^n,\;n\geq 1,\; p>1, and a \geq 0, where \D_{\lambda} is a strongly degenerate elliptic operator, the functions \lambda=(\lambda_1, ..., \lambda_k) : R^n \rightarrow R^k, satisfies some certain conditions, and |.|_{\lambda} the homogeneous norm associated to the \D_{\lambda}-Laplacian. We prove various Liouville-type theorems for smooth solutions under the assumption that they are stable or stable outside a compact set of R^n. First, we establish the standard integralestimates via stability property to derive the nonexistence results for stable solutions. Next, by mean of the Pohozaev identity, we deduce the Liouville-type theorem for solutions stable outside a compact set.