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

We study the Choquard equation involving mixed local and nonlocal operators $$-{\epsilon }^2\mathrm{\Delta }u+{\epsilon }^{2s}(-\mathrm{\Delta }{)}^{s}u+V(x)u={\epsilon }^{\mu -2}(\frac{1}{\mathrm{\mid }x{\mathrm{\mid }}^{\mu }}\ast F(u))f(u)\text{in }{\mathbb{R}}^2,$$ where $\epsilon >0$, $s\in (0,1)$, $0<\mu <2$, $f$ 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 $\epsilon \to 0^{+}$.