Symmetry of solutions of a semilinear elliptic equation with unbounded coefficients

Abstract

We study the equation $- \Delta u + |x|^a |u|^{q-2} u = |x|^b |u|^{p-2} u$ with Dirichlet boundary condition on $B(0,1)$ or on $\mathbb R^N$. We study the radial solutions of this equation on~$\mathbb R^N$ and the symmetry breaking for ground states for $q=2$ on $\mathbb R^N$. Estimates of the transition are also given when $p$ is close to $2$ or $2^*$ on $B(0,1)$.