Symmetry results for solutions of semilinear elliptic equations with convex nonlinearities
This paper is devoted to the symmetry properties of classical \(C^2(\Omega) \cap C(\overline \Omega)\) solutions of elliptic problems fo the type \[ \begin{cases} -\Delta u=f(x,u) \quad & \text{in }\Omega,\\ u=g(x)\quad & \text{on } \partial\Omega,\end{cases} \tag{1} \] where \(\Omega\) is a bounded, somehow symmetric domain in \(\mathbb{R}^N\), \(N\geq 2\), \(f\) and \(g\) are given functions having some symmetry in \(x\). The author shows that all solutions of (1) of index one are axially symmetric when \(\Omega\) is an annulus or a ball, \(g\equiv 0\) and \(f\) is strictly convex in \(u\). To this end, the author uses spectral properties of the linearized operator.
spectral properties of the linearized operator
scientific article; zbMATH DE number 1797600