Some Liouville-type results for stable solutions involving the mean curvature operator: The radial case

Alberto Farina; Miguel Angel Navarro

doi:10.3934/dcds.2020076

Article Contents

2020, Volume 40, Issue 2: 1233-1256. Doi: 10.3934/dcds.2020076

This issue Previous Article Measure solutions to a system of continuity equations driven by Newtonian nonlocal interactions Next Article

Some Liouville-type results for stable solutions involving the mean curvature operator: The radial case

Alberto Farina^1, , and
Miguel Angel Navarro^2,

1.
LAMFA, CNRS UMR 7352, Université de Picardie Jules Verne, 33 rue Saint-Leu, 80039 Amiens, France
2.
Departamento de Estatística, Análise Matemática e Optimización, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain

^* Corresponding author: Alberto Farina
^* Corresponding author: Alberto Farina

Received: July 2018

Revised: August 2019

Published: November 2019

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Abstract

We prove some new Liouville-type theorems for stable radial solutions of

$ - {{\rm{div}}}{\left(\frac{\nabla u}{\sqrt{1+\left\vert{\nabla u}\right\vert^2}}\right)} = f(u)\mbox{ in } \mathbb R^N, $

where $ f $ is a smooth nonlinearity and $ N \ge 2 $. Also, the sharpness of our results is discussed by means of some examples.

Keywords:

Stability,
radial solutions,
Liouville-type theorems,
asymptotic behavior of solutions,
quasilinear elliptic equations with mean curvature operator.

Mathematics Subject Classification: Primary: 35B35, 35B53, 35J93, 35B07; Secondary: 35B40.

Citation:

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Figure 1. $ N_{\alpha} $, for $ \left\vert{\alpha\neq0}\right\vert\leq 1 $

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Figure 2. $ \lceil N_{\alpha}\rceil $, $ \overline{N}_{1-\alpha} $, for $ \alpha\neq0<1 $

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