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

Abstract

We study the Choquard equation involving mixed local and nonlocal operators \[ -Δu + (-Δ)^{s}u + V(x)u = \left(\frac{1}{|x|^μ} * F(u)\right) f(u) \quad \text{in } \mathbb{R}^{2}, \] where $s\in(0,1)$, $μ\in(0,2)$, $F(t)=\int_{0}^{t} f(τ)\,dτ$, 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.