In this paper we study the existence and multiplicity of solutions for the semilinear elliptic equation $-\Delta u = Q(x)f'(u)$ in an exterior domain with Neumann boundary conditions. We prove the existence of a positive ground state as well as a sign-changing solution under a double power growth condition on the nonlinearity.
"Multiple solutions for a null mass Neumann problem in exterior domains." Adv. Differential Equations 15 (1/2) 181 - 199, January/February 2010.