Existence and concentration of ground state solutions for an exponentially critical Choquard equation involving mixed local-nonlocal operators

TL;DR

This paper proves the existence and describes the concentration of ground state solutions for a critical exponential growth Choquard equation with mixed local and nonlocal operators in \\(\\mathbb{R}^2\\), using variational methods and advanced inequalities.

Contribution

It introduces new variational techniques to handle the exponential critical growth and nonlocal interactions in a mixed operator Choquard equation.

Findings

Existence of ground state solutions established.

Solutions concentrate as \\varepsilon \\to 0^+.

Application of Trudinger--Moser inequality in a nonlocal context.

Abstract

We study the Choquard equation involving mixed local and nonlocal operators \[-\varepsilon^{2}\Delta u+\varepsilon^{2s}(-\Delta)^{s}u+V(x)u=\varepsilon^{\mu-2}\left(\frac{1}{|x|^{\mu}}*F(u)\right)f(u)\quad \text{in }\R^{2},\] where \(\varepsilon>0\), \(s\in(0,1)\), \(0<\mu<2\), \(f\) has Trudinger--Moser critical exponential growth, and \(F(t)=\int_{0}^{t}f(\tau)\,d\tau\). By variational methods, combined with the Trudinger--Moser inequality and compactness arguments adapted to the critical growth and the nonlocal interaction term, we prove the existence of ground state solutions and describe their concentration behavior as \(\varepsilon\to0^{+}\).