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

TL;DR

This paper proves the existence of positive and sign-changing solutions for a nonlinear Choquard equation with mixed local and nonlocal operators in two dimensions, using variational methods and invariant set techniques.

Contribution

It introduces new existence results for solutions to a Choquard equation involving mixed operators and subcritical exponential growth, extending previous work in the field.

Findings

Existence of a least energy positive solution.

Existence of sign-changing solutions.

Infinitely many sign-changing solutions if nonlinearity is odd.

Abstract

We study the Choquard equation involving mixed local and nonlocal operators $$-\Delta u+(-\Delta)^{s}u+V(x)u=(\frac{1}{|x|^{\mu}}* F(u))f(u)\quad\text{in }\R^{2},$$ 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.