Abstract
In this paper we study solutions, possibly unbounded and sign-changing, of the following equation
where \(n\ge 1\), \(p>1\), \(a \ge 0\) and \(\Delta _{\lambda }\) is a strongly degenerate elliptic operator, the functions \(\lambda =(\lambda _1, \ldots , \lambda _k) : \; {\mathbb {R}}^n \rightarrow {\mathbb {R}}^k\) satisfies some certain conditions, and \(|.|_{\lambda }\) the homogeneous norm associated to the \(\Delta _{\lambda }\)-Laplacian. We prove various Liouville-type theorems for smooth solutions under the assumption that they are stable or stable outside a compact set of \({\mathbb {R}}^n\). 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.
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Rahal, B. Liouville-type theorems with finite Morse index for semilinear \({\varvec{\Delta }}_{{\varvec{\lambda }}}\)-Laplace operators. Nonlinear Differ. Equ. Appl. 25, 21 (2018). https://doi.org/10.1007/s00030-018-0512-z
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DOI: https://doi.org/10.1007/s00030-018-0512-z
Keywords
- Liouville-type theorems
- \(\Delta _{\lambda }\)-Laplace operator
- Stable solutions
- Stability outside a compact set
- Pohozaev identity