Ground state solution for a periodic p&q‐Laplacian equation involving critical growth without the Ambrosetti–Rabinowitz condition

We study the ground state solutions for the following p&q‐Laplacian equation −Δpu−Δqu+V(x)(|u|p−2u+|u|q−2u)=λK(x)f(u)+|u|q∗−2u,x∈ℝN,u∈W1,p(ℝN)∩W1,q(ℝN),$$ \left\{\begin{array}{l}-{\Delta}_pu-{\Delta}_qu+V(x)\left({\left|u\right|}^{p-2}u+{\left|u\right|}^{q-2}u\right)=\lambda K(x)f(u)+{\left|u\right|}^{q^{\ast }-2}u,x\in {\mathbb{R}}^N,\\ {}u\in {W}^{1,p}\left({\mathbb{R}}^N\right)\cap {W}^{1,q}\left({\mathbb{R}}^N\right),\end{array}\right. $$ where λ>0$$ \lambda >0 $$ is a parameter large enough, Δru=div(|∇u|r−2∇u)$$ {\Delta}_ru=\operatorname{div}\left({\left|\nabla u\right|}^{r-2}\nabla u\right) $$ with r∈{p,q}$$ r\in \left\{p,q\right\} $$ denotes the r$$ r $$‐Laplacian operator, 1