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

Abstract

We study the existence of positive solutions for the following class of $(p,q)$-Laplacian coupled systems \[ \left\{ \begin{array}{lr} -Δ_{p} u+a(x)|u|^{p-2}u=f(u)+ αλ(x)|u|^{α-2}u|v|^β, & x\in\mathbb{R}^{N}, -Δ_{q} v+b(x)|v|^{q-2}v=g(v)+ βλ(x)|v|^{β-2}v|u|^α, & x\in\mathbb{R}^{N}, \end{array} \right. \] where $N\geq3$ and $1\leq p\leq q<N$. Here the coefficient $λ(x)$ of the coupling term is related with the potentials by the condition $|λ(x)|\leqδa(x)^{α/p}b(x)^{β/q}$ where $δ\in(0,1)$ and $α/p+β/q=1$. 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.