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February  2020, 40(2): 1233-1256. doi: 10.3934/dcds.2020076

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

Received  July 2018 Revised  August 2019 Published  November 2019

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: Alberto Farina, Miguel Angel Navarro. Some Liouville-type results for stable solutions involving the mean curvature operator: The radial case. Discrete & Continuous Dynamical Systems, 2020, 40 (2) : 1233-1256. doi: 10.3934/dcds.2020076
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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