Existence of solution for a class of fractional Hamiltonian-type elliptic systems with exponential critical growth in R

TL;DR

This paper proves the existence of at least one positive solution for a class of fractional Hamiltonian elliptic systems with exponential critical growth in the real line, using variational methods and linking theorem.

Contribution

It introduces a new approach to establish solutions for fractional Hamiltonian systems with exponential growth in unbounded domains.

Findings

Existence of at least one positive solution is proven.

The methods handle exponential critical growth in fractional systems.

Variational techniques are effectively applied to nonlocal operators.

Abstract

In this paper, we study the following class of fractional Hamiltonian systems: \begin{eqnarray*} \begin{aligned}\displaystyle \left\{ \arraycolsep=1.5pt \begin{array}{ll} (-\Delta)^{\frac{1}{2}} u + u = \Big(I_{\mu_{1}}\ast G(v)\Big)g(v) \ \ \ & \mbox{in} \ \mathbb{R},\\[2mm] (-\Delta)^{\frac{1}{2}} v + v = \Big(I_{\mu_{2}}\ast F(u)\Big)f(u) \ \ \ & \mbox{in} \ \mathbb{R}, \end{array} \right. \end{aligned} \end{eqnarray*} where $(-\Delta)^{\frac{1}{2}}$ is the square root Laplacian operator, $\mu_{1},\mu_{2}\in(0,1)$, $I_{\mu_{1}},I_{\mu_{2}}$ 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 $\mathbb{R}$. Using the linking theorem and variational methods, we establish the existence of at least one positive solution to the above problem.