Radial solutions to semilinear elliptic equations via linearized operators

Abstract

Let $u$ be a classical solution of semilinear elliptic equations in a ball or an annulus in $\mathbb{R}^N$ with zero Dirichlet boundary condition where the nonlinearity has a convex first derivative. In this note, we prove that if the $N$-th eigenvalue of the linearized operator at $u$ is positive, then $u$ must be radially symmetric.

Keywords

• semilinear elliptic equation
• nonconvex domain
• radial solution
• symmetry