Nonsymmetric ground states of symmetric variational problems

In this paper we study a minimization problem which is invariant by rotation. The corresponding Euler‐Lagrange equations are semilinear elliptic equations in an exterior domain with Neumann boundary conditions. We prove that this minimization problem has at least one solution. Yet all its solutions are shown not to be rotationally invariant. Furthermore we describe how the radial symmetry is broken.