In this article, we use Palais-Smale approaches to describe the achieved and nonachieved domains. We characterizes the achieved domain by the existence of a ground state solution for the energy functional $J$ in $\Omega$.
"A Palais-Smale approach to Sobolev subcritical operators." Topol. Methods Nonlinear Anal. 20 (2) 393 - 407, 2002.