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In this paper we study solutions, possibly unbounded and signchanging, of the following equationwhere n ≥ 1, p > 1, a ≥ 0 and Δ λ is a strongly degenerate elliptic operator, the functions λ = (λ1, . . . , λ k ) : R n → R k satisfies some certain conditions, and |.| λ the homogeneous norm associated to the Δ λ -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 integral estimates 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.

In this paper we study the nonexistence of solutions, which are stable or stable outside a compact set, possibly unbounded and sign-changing, of some nonlinear elliptic equations with mixed boundary value conditions. The main methods used are the integral estimates and the monotonicity formula.