Liouville results for elliptic equations in strips with finite Morse index

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Abstract

Consider the strip \(\varOmega =\mathbb {R}^{n-k}\times \omega \) where \(n\ge 3,\;\;k\ge 1\) and \(\omega \) is a smooth bounded domain of \(\mathbb {R}^k\). We are concerned with the following superlinear elliptic equations:
$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta ^{2} u= |u|^{p-1}u &{}\quad \text {in} \; \varOmega \\ u =|\nabla u| = 0 &{}\quad \text {on} \; \partial \varOmega =\mathbb {R}^{n-k}\times \partial \omega \end{array}\right. } \end{aligned}$$
where \(p>1\) and \(u \in C^4(\overline{\varOmega })\). We prove Liouville-type theorems for stable solutions or solutions which are stable outside a compact set of \(\varOmega \). We first provide an integral estimate from stability which combined with Pohozaev-type identity to obtain nonexistence results for \(p_\mathrm{s}(n,4)< p \le p_\mathrm{s}(n-k,4)\), where \(p_\mathrm{s}(z,4):=\frac{z+4}{z-4}\) is the Sobolev exponent of \(\mathbb {R}^z\). Also, we establish monotonicity formula to prove the nonexistence of nontrivial stable solution for all \(p>1\) and solution which is stable outside a compact set for \(p > p_\mathrm{s}(n-k,4)\). Our classification is close to a sharp result since in the subcritical case [i.e., \(1<p <p_\mathrm{s}(n,4) \)] we prove the existence of a mountain-pass solution with Morse index equal to 1 in the subspace of \(H_0^2(\varOmega )\) having cylindrical symmetry.

Keywords

Morse index Liouville-type theorems Pohozaev identity Monotonicity formula 

Mathematics Subject Classification

Primary: 35J55 35J65 Secondary: 35B65 

Notes

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Compliance with ethical standards

Competing interest

The authors declare that they have no competing interests.

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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Département de Mathématiques, Faculté des SciencesUniversité de SfaxSfaxTunisia

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