Title: Liouville-type theorems with finite Morse index for Δ_λ-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.