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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

M. Del Pino, M. Kowalczyk and J. Wei, On De Giorgi conjecture in dimension $N\geq 9$,, to appear in Ann. of Math., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

M. Del Pino, M. Kowalczyk and J. Wei, On De Giorgi conjecture in dimension $N\geq 9$,, to appear in Ann. of Math., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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