Liouville-type theorems with finite Morse index for semilinear \({\varvec{\Delta }}_{{\varvec{\lambda }}}\)-Laplace operators



In this paper we study solutions, possibly unbounded and sign-changing, of the following equation
$$\begin{aligned} -\Delta _{\lambda } u=|x|_{\lambda }^a |u|^{p-1}u \quad \text{ in }\; {\mathbb {R}}^n, \end{aligned}$$
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.


Liouville-type theorems \(\Delta _{\lambda }\)-Laplace operator Stable solutions Stability outside a compact set Pohozaev identity 

Mathematics Subject Classification

Primary 35J55 35J65 Secondary 35B65 


Compliance with ethical standards

Conflict of interest

The author declare that he has no competing interests.


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Authors and Affiliations

  1. 1.Faculté des Sciences, Département de MathématiquesUniversité de SfaxSfaxTunisia

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