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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 $R^N$

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 & 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

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##### 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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