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Entire solutions of semilinear elliptic equations in \(\mathbb{R}^{3}\) and a conjecture of De Giorgi. (English) Zbl 0968.35041
The paper gives a partial answer to a conjecture of De Giorgi from 1978.
Theorem 1. Let \(F\in C^2(\mathbb{R})\) and let \(u\) be a bounded solution of \(\Delta u- F'\circ u= 0\) in \(\mathbb{R}^3\) satisfying \(\partial_3 u>0\) in \(\mathbb{R}^3\) and suppose
a) \(F\geq \min\{F(-1), F(1)\}\) in \(]-1,1[\) and \(\lim_{x_3\to \pm\infty} u(x',x_3)= \pm 1\) for all \(x'\in \mathbb{R}^2\) or
b) \(F\geq \min\{F(m), F(M)\}\) in \(]m,M[\) for each \(m\), \(M\in\mathbb{R}\) such that \(m< M\), \(F'(m)= F'(M)= 0\), \(F''(m)\geq 0\), \(F''(M)\geq 0\);
then there exist \(a\in\mathbb{R}^3\) and \(g\in C^2(\mathbb{R})\) such that \(u(x)= g(ax)\) for all \(x\in\mathbb{R}^3\).
The key result for the proof is Theorem 2: Let \(F\in C^2(\mathbb{R})\) and let \(u\) be a bounded solution of \(\Delta u- F'\circ u= 0\) in \(\mathbb{R}^n\) satisfying \(\partial_n u>0\) in \(\mathbb{R}^n\) and \(\lim_{x_n\to\infty} u(x', x_n)= 1\) for all \(x'\in \mathbb{R}^{n-1}\); then there exists \(C> 0\) such that \[ \int_{S(0,R)} (1/2|\nabla u|^2+ F\circ u- F(1)) dx\leq CR^{n-1} \] for every \(R> 1\).

MSC:
35J60 Nonlinear elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35B45 A priori estimates in context of PDEs
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References:
[1] G. Alberti, L. Ambrosio and X. Cabré, On a long-standing conjecture of E. De Giorgi: Old and recent results, forthcoming.
[2] Martin T. Barlow, On the Liouville property for divergence form operators, Canad. J. Math. 50 (1998), no. 3, 487 – 496. · Zbl 0912.31004
[3] M. T. Barlow, R. F. Bass and C. Gui, The Liouville property and a conjecture of De Giorgi, preprint. · Zbl 1072.35526
[4] Henri Berestycki, Luis Caffarelli, and Louis Nirenberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no. 1-2, 69 – 94 (1998). Dedicated to Ennio De Giorgi. · Zbl 1079.35513
[5] H. Berestycki, F. Hamel and R. Monneau, One-dimensional symmetry of bounded entire solutions of some elliptic equations, preprint. · Zbl 0954.35056
[6] E. Bombieri, E. De Giorgi, and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math. 7 (1969), 243 – 268. · Zbl 0183.25901
[7] Luis A. Caffarelli and Antonio Córdoba, Uniform convergence of a singular perturbation problem, Comm. Pure Appl. Math. 48 (1995), no. 1, 1 – 12. · Zbl 0829.49013
[8] Luis Caffarelli, Nicola Garofalo, and Fausto Segàla, A gradient bound for entire solutions of quasi-linear equations and its consequences, Comm. Pure Appl. Math. 47 (1994), no. 11, 1457 – 1473. · Zbl 0819.35016
[9] Ennio De Giorgi, Convergence problems for functionals and operators, Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978) Pitagora, Bologna, 1979, pp. 131 – 188. · Zbl 0405.49001
[10] Alberto Farina, Some remarks on a conjecture of De Giorgi, Calc. Var. Partial Differential Equations 8 (1999), no. 3, 233 – 245. · Zbl 0938.35057
[11] A. Farina, Symmetry for solutions of semilinear elliptic equations in \(\mathbb R^{N}\) and related conjectures, Ricerche di Matematica XLVIII (1999), 129-154. · Zbl 0940.35084
[12] A. Farina, forthcoming.
[13] N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems, Math. Ann. 311 (1998), no. 3, 481 – 491. · Zbl 0918.35046
[14] Enrico Giusti, Minimal surfaces and functions of bounded variation, Monographs in Mathematics, vol. 80, Birkhäuser Verlag, Basel, 1984. · Zbl 0545.49018
[15] Stephan Luckhaus and Luciano Modica, The Gibbs-Thompson relation within the gradient theory of phase transitions, Arch. Rational Mech. Anal. 107 (1989), no. 1, 71 – 83. · Zbl 0681.49012
[16] Luciano Modica, A gradient bound and a Liouville theorem for nonlinear Poisson equations, Comm. Pure Appl. Math. 38 (1985), no. 5, 679 – 684. · Zbl 0612.35051
[17] Luciano Modica, Monotonicity of the energy for entire solutions of semilinear elliptic equations, Partial differential equations and the calculus of variations, Vol. II, Progr. Nonlinear Differential Equations Appl., vol. 2, Birkhäuser Boston, Boston, MA, 1989, pp. 843 – 850. · Zbl 0699.35082
[18] Luciano Modica and Stefano Mortola, Un esempio di \Gamma \(^{-}\)-convergenza, Boll. Un. Mat. Ital. B (5) 14 (1977), no. 1, 285 – 299 (Italian, with English summary). · Zbl 0356.49008
[19] Luciano Modica and Stefano Mortola, Some entire solutions in the plane of nonlinear Poisson equations, Boll. Un. Mat. Ital. B (5) 17 (1980), no. 2, 614 – 622 (English, with Italian summary). · Zbl 0448.35044
[20] William P. Ziemer, Weakly differentiable functions, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989. Sobolev spaces and functions of bounded variation. · Zbl 0692.46022
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