Positive ground states for a class of superlinear $(p,q)$-Laplacian coupled systems involving Schr\"odinger equations

We study the existence of positive solutions for the following class of $(p,q)$-Laplacian coupled systems $$\{\begin{array}{lrr}-{\mathrm{\Delta }}_{p}u+a(x)\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u=f(u)+\alpha \lambda (x)\mathrm{\mid }u{\mathrm{\mid }}^{\alpha -2}u\mathrm{\mid }v{\mathrm{\mid }}^{\beta },& x\in {\mathbb{R}}^N,-{\mathrm{\Delta }}_{q}v+b(x)\mathrm{\mid }v{\mathrm{\mid }}^{q-2}v=g(v)+\beta \lambda (x)\mathrm{\mid }v{\mathrm{\mid }}^{\beta -2}v\mathrm{\mid }u{\mathrm{\mid }}^{\alpha },& x\in {\mathbb{R}}^N,\end{array}$$ where $N\ge 3$ and $1\le p\le q<N$. Here the coefficient $\lambda (x)$ of the coupling term is related with the potentials by the condition $\mathrm{\mid }\lambda (x)\mathrm{\mid }\le \delta a(x{)}^{\alpha \mathrm{/}p}b(x{)}^{\beta \mathrm{/}q}$ where $\delta \in (0,1)$ and $\alpha \mathrm{/}p+\beta \mathrm{/}q=1$. We deal with periodic and asymptotically periodic potentials. The nonlinear terms $f(s),g(s)$ are “superlinear” at $0$ and at $\mathrm{\infty }$ and are assumed without the well known Ambrosetti-Rabinowitz condition at infinity. Thus, we have established the existence of positive ground states solutions for a large class of nonlinear terms and potentials. Our approach is variational and based on minimization technique over the Nehari manifold.