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: June 30, 2018

Revised: July 31, 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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References

[1]	X. Cabré and A. Capella, On the stability of radial solutions of semilinear elliptic equations in all of $\Bbb R^n$, C. R. Math. Acad. Sci. Paris, 338 (2004), 769-774. doi: 10.1016/j.crma.2004.03.013.
[2]	X. Cabré, A. Capella and M. Sanchón, Regularity of radial minimizers of reaction equations involving the $p$-Laplacian, Calc. Var. Partial Differential Equations, 34 (2009), 475-494. doi: 10.1007/s00526-008-0192-3.
[3]	X. Cabré and M. Sanchón, Semi-stable and extremal solutions of reaction equations involving the $p$-Laplacian, Commun. Pure Appl. Anal., 6 (2007), 43-67. doi: 10.3934/cpaa.2007.6.43.
[4]	D. Castorina, P. Esposito and B. Sciunzi, Low dimensional instability for semilinear and quasilinear problems in $\Bbb R^N$, Commun. Pure Appl. Anal., 8 (2009), 1779-1793. doi: 10.3934/cpaa.2009.8.1779.
[5]	L. Damascelli, A. Farina, B. Sciunzi and E. Valdinoci, Liouville results for $m$-Laplace equations of Lane-Emden-Fowler type, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1099-1119. doi: 10.1016/j.anihpc.2008.06.001.
[6]	L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations, Monographs and Surveys in Pure and Applied Mathematics, 143, Chapman and Hall/CRC, 2011. doi: 10.1201/b10802.
[7]	A. Farina, Liouville-type results for solutions of $-\Delta u = \vert u\vert ^{p-1}u$ on unbounded domains of $\mathbb{R}^N$, C. R. Math. Acad. Sci. Paris, 341 (2005), 415-418. doi: 10.1016/j.crma.2005.07.006.
[8]	A. Farina, Liouville-type theorems for elliptic problems, Handbook of Differential Equations: Stationary Partial Differential Equations, 4 (2007), 61-116. doi: 10.1016/S1874-5733(07)80005-2.
[9]	A. Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of $\mathbb{R}^N$, J. Math. Pures Appl. (9), 87 (2007), 537-561. doi: 10.1016/j.matpur.2007.03.001.
[10]	A. Farina, Stable solutions of $-\Delta u = e^u$ on $\mathbb{R}^N$, C. R. Math. Acad. Sci. Paris, 345 (2007), 63-66. doi: 10.1016/j.crma.2007.05.021.
[11]	A. Farina and L. Dupaigne, Stable solutions of $ -\Delta u = f(u)$ in $ \mathbb R^N$, Journal of the European Mathematical Society, 12 (2010), 855-882. doi: 10.4171/JEMS/217.
[12]	A. Farina, B. Sciunzi and E. Valdinoci, Bernstein and De Giorgi type problems: New results via a geometric approach, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 7 (2008), 741-791.
[13]	N. Ghoussoub and A. Moradifam, Bessel pairs and optimal Hardy and Hardy-Rellich inequalities, Math. Ann., 349 (2011), 1-57. doi: 10.1007/s00208-010-0510-x.
[14]	E. Giusti, Minimal Surfaces and Functions of Bounded Variation, , Monographs in Mathematics, 80. Birkhauser Verlag, Basel, 1984. doi: 10.1007/978-1-4684-9486-0.
[15]	M. A. Navarro and S. Villegas, Semi-stable radial solutions of $p$-Laplace equations in $\Bbb{R}^N$, Nonlinear Anal., 149 (2017), 111-116. doi: 10.1016/j.na.2016.10.004.
[16]	M. A. Navarro and S. Villegas, Sharp estimates of radial minimizers of $p$–Laplace equations, Proc. Amer. Math. Soc., 145 (2017), 2931-2941. doi: 10.1090/proc/13454.
[17]	S. Villegas, Asymptotic behavior of stable radial solutions of semilinear elliptic equations in $\mathbb{R}^N$, J. Math. Pures Appl. (9), 88 (2007), 241-250. doi: 10.1016/j.matpur.2007.06.004.