<div>We study the ground state solutions for the following
p\&amp;q-Laplacain equation \[
\left\{
\begin{array}{ll}
-\Delta_pu-\Delta_qu+V(x)
(|u|^{p-2}u+|u|^{q-2}u)=\lambda
K(x)f(u)+|u|^{q^*-2}u,~x\in\R^N,
\\ u\in
W^{1,p}(\R^N)\cap
W^{1,q}(\R^N), \end{array}
\right. \] where
$\lambda&gt;0$ is a parameter large enough,
$\Delta_ru =
\text{div}(|\nabla
u|^{r-2}\nabla u)$ with
$r\in\{p,q\}$ denotes
the $r$ Laplacian operator, $1</div>