Minimization problems for the exterior domain (Q1364186)

This paper contains a discussion on the existence of minimizers for \[ V(u)={1\over 2}\int_{\Omega } | \text{grad } u(x)| ^2dx +\int_{\Omega }F(u(x))dx \] subject to \[ I(u)=\int_{\Omega }G(u(x))dx =\lambda >0, \] where \(F(u)\) and \(G(u)\) are given functions and \(\Omega \) is the complement of a bounded domain in \({\mathbb{R}}^N\). Some applications to the existence and stability of solitary waves for certain evolution equations have been indicated.