We study the existence of positive solutions for the following class of $(p,q)$-Laplacian coupled systems \[ \left\{ \begin{ array}{ lr} -\Delta_{ p} u+a(x)|u|^{ p-2}u=f(u)+ \alpha\lambda(x)|u|^{ \alpha-2}u|v|^{ \beta}, & x\in\mathbb{ R}^{ N}, -\Delta_{ q} v+b(x)|v|^{ q-2}v=g(v)+ \beta\lambda(x)|v|^{ \beta-2}v|u|^{ \alpha}, & x\in\mathbb{ R}^{ N}, \end{ array} \right. \] where $N\geq3$ and $1\leq p\leq q<N$. Here the coefficient $\lambda(x)$ of the coupling term is related with the potentials by the condition $|\lambda(x)|\leq\delta a(x)^{ \alpha/p}b(x)^{ \beta/q}$ where $\delta\in(0,1)$ and $\alpha/p+\beta/q=1$. We deal with periodic and asymptotically periodic potentials. The nonlinear terms $f(s), \; g(s)$ are "superlinear" at $0$ and at $\infty$ 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.
Categories: math.AP