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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 \(R> 1\).

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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