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 $s\in (0,1)$, $\mu \in (0,2)$, $F(t)={\int }_0^{t}f(\tau )d\tau$, 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.

We thank the reviewers and the editor for their suggestions and comments, based on which we have corrected some minor errors. This paper has been accepted by Journal of Pseudo-Differential Operators and Applications

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• Should I stay or should I go? Zero-size jumps in random walks for Lévy flights
• Choquard equation involving mixed local and nonlocal operators
• Normalized solutions to a Choquard equation involving mixed local and nonlocal operators