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Measure solutions to a system of continuity equations driven by Newtonian nonlocal interactions
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
| 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 |
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 $ |
| $ N \ge 2 $ |
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 and Continuous Dynamical Systems,
2020, 40
(2)
: 1233-1256.
doi: 10.3934/dcds.2020076
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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. |
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. |
| [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. |
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