Radial solutions to semilinear elliptic equations via linearized operators

摘要

Let u be a classical solution of semilinear elliptic equations in a ball or anannulus in RN with zero Dirichlet boundary condition where the nonlinearity has aconvex 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.