  In this work, we study the of positive ground state solution for the
semilinear elliptic problem $$ \left\{ \begin{array} [c]{ll}% -\Delta
u=u^{p(x)-1},\quad u>0 & \mathrm{in}\,G\subseteq\mathbb{R}^{N}% ,\,N\geq3\\
u\in D_{0}^{1,2}(G), & \end{array} \right. $$ where $G$ is either
$\mathbb{R}^{N}$ or a bounded domain, and $p:G\rightarrow \mathbb{R}$ is a
continuous function assuming critical and subcritical values.
