Existence of positive and sign-changing solutions for a Choquard equation involving mixed local and nonlocal operators

We study the Choquard equation involving mixed local and nonlocal operators $$-\mathrm{\Delta }u+(-\mathrm{\Delta }{)}^{s}u+V(x)u=(\frac{1}{\mathrm{\mid }x{\mathrm{\mid }}^{\mu }}\ast F(u))f(u)\text{in }{\mathbb{R}}^2,$$ where $s\in (0,1)$, $\mu \in (0,2)$, $F(t)={\int }_0^{t}f(\tau )d\tau$, and $f$ has subcritical exponential growth of Trudinger–Moser type. Under suitable assumptions on the potential $V$ and the nonlinearity $f$, we prove the existence of a least energy positive solution by a Nehari manifold approach. We also establish the existence of a sign-changing solution by means of invariant sets of descending flow. If, in addition, the nonlinearity is odd, then the problem admits infinitely many sign-changing solutions.