Uniqueness of the critical points of solutions to two kinds of semilinear elliptic equations in higher dimensional domains

Abstract

In this paper, we provide an affirmative answer to the {\it conjecture A} for bounded simple rotationally symmetric domains $\Omega \subset {\mathbb{R}}^n(n\ge 3)$ along $x_n$ axis. Precisely, we use a new simple argument to study the symmetry of positive stable solutions for two kinds of semilinear elliptic equations. To do this, when $f(\cdot ,s)$ is convex with respect to $s$, we show that the positivity of the first eigenvalue of the corresponding linearized operator in somehow symmetric domains is a sufficient condition for the symmetry of $u$. Moreover, we prove the uniqueness of critical points of a positive stable solution to semilinear elliptic equation $-△u=f(\cdot ,u)$ with zero Dirichlet boundary condition for simple rotationally symmetric domains in ${\mathbb{R}}^n$ by continuity method and a variety of maximum principles.

18 pages