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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. 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. 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. 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. 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. 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. doi: 10.1002/cpa.3160471103. [7] E. De Giorgi, Convergence problems for functionals and operators,, in, (1979), 131. [8] M. Del Pino, M. Kowalczyk and J. Wei, On De Giorgi conjecture in dimension $N\geq 9$,, to appear in Ann. of Math., (). [9] N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems,, Math. Ann., 311 (1998), 481. 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. 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. doi: 10.1002/cpa.3160380515. [12] O. Savin, Regularity of flat level sets in phase transitions,, Ann. of Math., 169 (2009), 41. 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. 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. 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. 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. 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. 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. 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. doi: 10.1002/cpa.3160471103. [7] E. De Giorgi, Convergence problems for functionals and operators,, in, (1979), 131. [8] M. Del Pino, M. Kowalczyk and J. Wei, On De Giorgi conjecture in dimension $N\geq 9$,, to appear in Ann. of Math., (). [9] N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems,, Math. Ann., 311 (1998), 481. 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. 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. doi: 10.1002/cpa.3160380515. [12] O. Savin, Regularity of flat level sets in phase transitions,, Ann. of Math., 169 (2009), 41. 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. doi: 10.1016/j.matpur.2007.06.004.
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