Symmetry and concentration behavior of ground state in axially symmetric domains

Abstract

We let $\Omega \left(r\right)$ be the axially symmetric bounded domains which satisfy some suitable conditions, then the ground-state solutions of the semilinear elliptic equation in $\Omega \left(r\right)$ are nonaxially symmetric and concentrative on one side. Furthermore, we prove the necessary and sufficient condition for the symmetry of ground-state solutions.