Symmetry breaking for a class of semilinear elliptic problems

Abstract:

We study positive solutions of the Dirichlet problem for $- \Delta u = {u^p} - \lambda$, $p > 1$, $\lambda > 0$, on the unit ball $\Omega$. We show that there exists a positive solution $({u_0}, {\lambda _0})$ of this problem which satisfies in addition $\partial {u_0}/\partial n = 0$ on $\partial \Omega$. We prove also that at $({u_0}, {\lambda _0})$, the symmetry breaks, i.e. asymmetric solutions bifurcate from the positive radial solutions.