## 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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