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One-dimensional symmetry of bounded entire solutions of some elliptic equations. (English) Zbl 0954.35056
The authors deal with the solutions of the semilinear elliptic equation $$\Delta u+f(u)=0$$ in $${\mathbb{R}}^n$$ which satisfy $$|u|\leq 1, \quad u(x',x_n)\rightarrow_{x_n\to \pm\infty} \pm 1\;\text{ uniformly\;in\;} x'=(x_1,\ldots, x_{n-1}).$$
The function $$f=f(u)$$ is supposed to be Lipschitz-continuous in $$[-1,1],$$ moreover, there exists $$\delta>0$$ such that $$f\;\text{ is\;nonincreasing\;on\;} [-1,-1+\delta] \text{ and\;on\;} [1-\delta,1];\;f(\pm 1)=0.$$
The authors prove that if $$u$$ solves the above problem, then $$u(x',x_n)=u_0(x_n),$$ where $$u_0$$ is a solution of $u_0''+f(u_0)=0\quad \text{ in} {\mathbb{R}},\quad u_0(\pm \infty)=\pm 1.$ The proof uses a sliding method and a version of the maximum principle in unbounded domains.

##### MSC:
 35J60 Nonlinear elliptic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35B50 Maximum principles in context of PDEs 35B40 Asymptotic behavior of solutions to PDEs
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