A positive ground state for a planar Choquard equation with mixed diffusion and critical exponential growth

We study a two-dimensional Choquard equation driven by the mixed local and nonlocal operator $L:=-\Delta+(-\Delta)^s$, where the nonlinearity has critical exponential growth of Trudinger--Moser type.

Under a coercive assumption on the potential and suitable one-sided assumptions on the nonlinearity, we prove the existence of a least energy positive solution.

The proof combines Nehari manifold minimization, compactness below the critical Trudinger--Moser threshold, local regularity, and a strong maximum principle.