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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 $R^N$

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.
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.

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