Nonexistence of nonconstant global minimizers with limit at \infty of semilinear elliptic equations in all of R^N

Previous Article
A sixth order Cahn-Hilliard type equation arising in oil-water-surfactant mixtures
CPAA Home
This Issue
Next Article
A unified treatment using critical point methods of the existence of multiple solutions for superlinear and sublinear Neumann problems

November 2011, 10(6): 1817-1821. doi: 10.3934/cpaa.2011.10.1817

Nonexistence of nonconstant global minimizers with limit at $\infty$ of semilinear elliptic equations in all of $R^N$

Salvador Villegas ^1,

1.	Departamento de Análisis Matemático, Universidad de Granada, 18071, Granada

Received April 2010 Revised March 2011 Published May 2011

Abstract
Related Papers

We prove nonexistence of nonconstant global minimizers with limit at infinity of the semilinear elliptic equation $-\Delta u=f(u)$ in the whole $R^N$, where $f\in C^1(R)$ is a general nonlinearity and $N\geq 1$ is any dimension. As a corollary of this result, we establish nonexistence of nonconstant bounded radial global minimizers of the previous equation.

Keywords: semilinear elliptic equations., stability, Global minimizers.

Mathematics Subject Classification: Primary: 35J60, 26D10; Secondary: 35B35, 35J2.

Citation: Salvador Villegas. Nonexistence of nonconstant global minimizers with limit at $\infty$ of semilinear elliptic equations in all of $R^N$. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1817-1821. doi: 10.3934/cpaa.2011.10.1817

References:

[1]	G. Alberti, L. Ambrosio and X. Cabré, On a long-standing conjecture of E. De Giorgi: symmetry in 3D for general nonlinearities and a local minimality property, Acta Appl. Math., 65 (2001), 9-33. doi: 10.1023/A:1010602715526.
[2]	G. Ambrosio and X. Cabré, Entire solutions of semilinear elliptic equations in $R^3$ and a conjecture of De Giorgi, J. Amer. Math. Soc., 13 (2000), 725-739. doi: 10.1090/S0894-0347-00-00345-3.
[3]	E. Bombieri, E. De Giorgi and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math., 7 (1969), 243-268. doi: 10.1007/BF01404309.
[4]	X. Cabré and A. Capella, On the stability of radial solutions of semilinear elliptic equations in all of $R^n$, C. R. Math. Acad. Sci. Paris, 338 (2004), 769-774. doi: 10.1016/j.crma.2004.03.013.
[5]	X. Cabré and J. Terra, Saddle-shaped solutions of bistable diffusion equation in all of $R^{2m}$, J. Eur. Math. Soc. (JEMS), 11 (2009), 819-843. doi: 10.4171/JEMS/168.
[6]	L. Caffarelli, N. Garofalo and F. Segàla, A gradient bound for entire solutions of quasi-linear equations and its consequences, Comm. Pure Appl. Math., 47 (1994), 1457-1473. doi: 10.1002/cpa.3160471103.
[7]	E. De Giorgi, Convergence problems for functionals and operators, in "Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978)," Pitagora, Bologna, (1979), 131-138.
[8]	M. Del Pino, M. Kowalczyk and J. Wei, On De Giorgi conjecture in dimension $N\geq 9$, to appear in Ann. of Math., arXiv:0806.3141.
[9]	N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems, Math. Ann., 311 (1998), 481-491. doi: 10.1007/s002080050196.
[10]	D. Jerison and D. Monneau, Towards a counter-example to a conjecture of De Giorgi in high dimensions, Ann. Mat. Pura Appl., 183 (2004), 439-467. doi: 10.1007/s10231-002-0068-7.
[11]	L. Modica, A gradient bound and a Liouville theorem for nonlinear Poisson equations, Comm. Pure Appl. Math., 38 (1985), 679-684. doi: 10.1002/cpa.3160380515.
[12]	O. Savin, Regularity of flat level sets in phase transitions, Ann. of Math., 169 (2009), 41-78. doi: 10.4007/annals.2009.169.41.
[13]	S. Villegas, Asymptotic behavior of stable radial solutions of semilinear elliptic equations in $R^N$, J. Math. Pures Appl., 88 (2007), 241-250. doi: 10.1016/j.matpur.2007.06.004.

show all references

References:

[1]	G. Alberti, L. Ambrosio and X. Cabré, On a long-standing conjecture of E. De Giorgi: symmetry in 3D for general nonlinearities and a local minimality property, Acta Appl. Math., 65 (2001), 9-33. doi: 10.1023/A:1010602715526.
[2]	G. Ambrosio and X. Cabré, Entire solutions of semilinear elliptic equations in $R^3$ and a conjecture of De Giorgi, J. Amer. Math. Soc., 13 (2000), 725-739. doi: 10.1090/S0894-0347-00-00345-3.
[3]	E. Bombieri, E. De Giorgi and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math., 7 (1969), 243-268. doi: 10.1007/BF01404309.
[4]	X. Cabré and A. Capella, On the stability of radial solutions of semilinear elliptic equations in all of $R^n$, C. R. Math. Acad. Sci. Paris, 338 (2004), 769-774. doi: 10.1016/j.crma.2004.03.013.
[5]	X. Cabré and J. Terra, Saddle-shaped solutions of bistable diffusion equation in all of $R^{2m}$, J. Eur. Math. Soc. (JEMS), 11 (2009), 819-843. doi: 10.4171/JEMS/168.
[6]	L. Caffarelli, N. Garofalo and F. Segàla, A gradient bound for entire solutions of quasi-linear equations and its consequences, Comm. Pure Appl. Math., 47 (1994), 1457-1473. doi: 10.1002/cpa.3160471103.
[7]	E. De Giorgi, Convergence problems for functionals and operators, in "Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978)," Pitagora, Bologna, (1979), 131-138.
[8]	M. Del Pino, M. Kowalczyk and J. Wei, On De Giorgi conjecture in dimension $N\geq 9$, to appear in Ann. of Math., arXiv:0806.3141.
[9]	N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems, Math. Ann., 311 (1998), 481-491. doi: 10.1007/s002080050196.
[10]	D. Jerison and D. Monneau, Towards a counter-example to a conjecture of De Giorgi in high dimensions, Ann. Mat. Pura Appl., 183 (2004), 439-467. doi: 10.1007/s10231-002-0068-7.
[11]	L. Modica, A gradient bound and a Liouville theorem for nonlinear Poisson equations, Comm. Pure Appl. Math., 38 (1985), 679-684. doi: 10.1002/cpa.3160380515.
[12]	O. Savin, Regularity of flat level sets in phase transitions, Ann. of Math., 169 (2009), 41-78. doi: 10.4007/annals.2009.169.41.
[13]	S. Villegas, Asymptotic behavior of stable radial solutions of semilinear elliptic equations in $R^N$, J. Math. Pures Appl., 88 (2007), 241-250. doi: 10.1016/j.matpur.2007.06.004.

[1]	Hwai-Chiuan Wang. Stability and symmetry breaking of solutions of semilinear elliptic equations. Conference Publications, 2005, 2005 (Special) : 886-894. doi: 10.3934/proc.2005.2005.886
[2]	Shoichi Hasegawa. Stability and separation property of radial solutions to semilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 4127-4136. doi: 10.3934/dcds.2019166
[3]	Shitao Liu, Roberto Triggiani. Determining damping and potential coefficients of an inverse problem for a system of two coupled hyperbolic equations. Part I: Global uniqueness. Conference Publications, 2011, 2011 (Special) : 1001-1014. doi: 10.3934/proc.2011.2011.1001
[4]	Lei Wei, Zhaosheng Feng. Isolated singularity for semilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 3239-3252. doi: 10.3934/dcds.2015.35.3239
[5]	Xavier Fernández-Real, Xavier Ros-Oton. On global solutions to semilinear elliptic equations related to the one-phase free boundary problem. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 6945-6959. doi: 10.3934/dcds.2019238
[6]	Junping Shi, R. Shivaji. Semilinear elliptic equations with generalized cubic nonlinearities. Conference Publications, 2005, 2005 (Special) : 798-805. doi: 10.3934/proc.2005.2005.798
[7]	Xavier Cabré, Manel Sanchón, Joel Spruck. A priori estimates for semistable solutions of semilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 601-609. doi: 10.3934/dcds.2016.36.601
[8]	Hwai-Chiuan Wang. On domains and their indexes with applications to semilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2007, 19 (2) : 447-467. doi: 10.3934/dcds.2007.19.447
[9]	Claudia Anedda, Giovanni Porru. Boundary estimates for solutions of weighted semilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2012, 32 (11) : 3801-3817. doi: 10.3934/dcds.2012.32.3801
[10]	Diane Denny. A unique positive solution to a system of semilinear elliptic equations. Conference Publications, 2013, 2013 (special) : 193-195. doi: 10.3934/proc.2013.2013.193
[11]	Antonio Greco, Marcello Lucia. Gamma-star-shapedness for semilinear elliptic equations. Communications on Pure and Applied Analysis, 2005, 4 (1) : 93-99. doi: 10.3934/cpaa.2005.4.93
[12]	Ying-Chieh Lin, Tsung-Fang Wu. On the semilinear fractional elliptic equations with singular weight functions. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 2067-2084. doi: 10.3934/dcdsb.2020325
[13]	Marco Degiovanni, Michele Scaglia. A variational approach to semilinear elliptic equations with measure data. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1233-1248. doi: 10.3934/dcds.2011.31.1233
[14]	Pierre Baras. A generalization of a criterion for the existence of solutions to semilinear elliptic equations. Discrete and Continuous Dynamical Systems - S, 2021, 14 (2) : 465-504. doi: 10.3934/dcdss.2020439
[15]	Jiabao Su, Zhaoli Liu. A bounded resonance problem for semilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2007, 19 (2) : 431-445. doi: 10.3934/dcds.2007.19.431
[16]	Mousomi Bhakta, Debangana Mukherjee. Semilinear nonlocal elliptic equations with critical and supercritical exponents. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1741-1766. doi: 10.3934/cpaa.2017085
[17]	David L. Finn. Convexity of level curves for solutions to semilinear elliptic equations. Communications on Pure and Applied Analysis, 2008, 7 (6) : 1335-1343. doi: 10.3934/cpaa.2008.7.1335
[18]	Massimo Grossi. On the number of critical points of solutions of semilinear elliptic equations. Electronic Research Archive, 2021, 29 (6) : 4215-4228. doi: 10.3934/era.2021080
[19]	Norimichi Hirano, Wen Se Kim. Multiplicity and stability result for semilinear parabolic equations. Discrete and Continuous Dynamical Systems, 1996, 2 (2) : 271-280. doi: 10.3934/dcds.1996.2.271
[20]	John M. Ball. Global attractors for damped semilinear wave equations. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 31-52. doi: 10.3934/dcds.2004.10.31

2021 Impact Factor: 1.273

Tools

Metrics

Other articles
by authors

on AIMS
Salvador Villegas
on Google Scholar
Salvador Villegas

[Back to Top]