On a class of singular Hamiltonian Choquard-type elliptic systems with critical exponential growth

TL;DR

This paper investigates the existence of solutions for a class of singular Hamiltonian Choquard-type elliptic systems with critical exponential growth in two dimensions, employing variational methods and linking theorem.

Contribution

It introduces a novel approach to handle singular weights and exponential growth in Hamiltonian Choquard systems using variational techniques.

Findings

Established existence of solutions for the system.

Handled singular weights with critical exponential growth.

Applied linking theorem in a new context.

Abstract

In this paper, we study the following Hamiltonian Choquard-type elliptic systems involving singular weights \begin{eqnarray*} \begin{aligned}\displaystyle \left\{ \arraycolsep=1.5pt \begin{array}{ll} -\Delta u + V(x)u = \Big(I_{\mu_{1}}\ast \frac{G(v)}{|x|^{\alpha}}\Big)\frac{g(v)}{|x|^{\alpha}} \ \ \ & \mbox{in} \ \mathbb{R}^{2},\\[2mm] -\Delta v + V(x)v = \Big(I_{\mu_{2}}\ast \frac{F(u)}{|x|^{\beta}}\Big)\frac{f(u)}{|x|^{\beta}} \ \ \ & \mbox{in} \ \mathbb{R}^{2}, \end{array} \right. \end{aligned} \end{eqnarray*} where $\mu_{1},\mu_{2}\in(0,2)$, $0<\alpha \leq \frac{\mu_{1}}{2}$, $0<\beta \leq \frac{\mu_{2}}{2}$, $V(x)$ is a continuous positive potential, $I_{\mu_{1}}$ and $I_{\mu_{2}}$ denote the Riesz potential, $\ast$ indicates the convolution operator, $F(s),G(s)$ are the primitive of $f(s),g(s)$ with $f(s),g(s)$ have exponential growth in $\mathbb{R}^{2}$. Using the linking theorem and variational methods, we establish the existence of solutions to the above problem.