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

TL;DR

This paper proves the existence of positive ground state solutions for a class of superlinear $(p,q)$-Laplacian coupled systems involving Schrödinger equations with periodic potentials, using variational methods without the Ambrosetti-Rabinowitz condition.

Contribution

It establishes positive ground state solutions for superlinear coupled $(p,q)$-Laplacian systems with periodic potentials, without relying on the Ambrosetti-Rabinowitz condition.

Findings

Existence of positive ground states for superlinear systems.

Applicable to both periodic and asymptotically periodic potentials.

Uses variational methods over the Nehari manifold.

Abstract

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.