Schrödinger–Hardy system without the Ambrosetti–Rabinowitz condition on Carnot groups

Abstract

In this paper, we study the following Schrödinger–Hardy system \begin{equation*} \begin{cases} -\Delta_{\mathbb{G}}u-\mu\frac{\psi^2}{r(\xi)^2}u=F_u(\xi,u,v)\ &{\rm in}\ \Omega, \\ -\Delta_{\mathbb{G}}v-\nu\frac{\psi^2 }{r(\xi)^2}v=F_v(\xi,u,v)\ &{\rm in}\ \Omega, \\ u=v=0 \ \ & {\rm on}\ \partial\Omega, \end{cases} \end{equation*} where $\Omega $ is a smooth bounded domain on Carnot groups $\mathbb{G}$, whose homogeneous dimension is $Q\geq 3$, $\Delta_{\mathbb{G}}$ denotes the sub-Laplacian operator on $\mathbb{G}$, $\mu$ and $\nu$ are real parameters, $r(\xi)$ is the natural gauge associated with fundamental solution of $-\Delta_{\mathbb{G}}$ on $\mathbb{G}$, $\psi$ is the geometrical function defined as $\psi=|\nabla_{\mathbb{G}}r|$, and $\nabla_{\mathbb{G}}$ is the horizontal gradient associated with $\Delta_{\mathbb{G}}$. The difficulty is not only the nonlinearities $F_u$ and $F_v$ without Ambrosetti–Rabinowitz condition, but also the Hardy terms and the structure on Carnot groups. We obtain the existence of nonnegative solution for this system by mountain pass theorem in a new framework.

Keywords

• schrödinger–hardy system
• without ambrosetti–rabinowitz condition
• carnot groups
• mountain pass theorem