Normalized solutions to an exponential growth Choquard equation driven by mixed local-nonlocal operator in ${\mathbb{R}}^2$

Abstract

In this paper, we study the existence of normalized solutions to the mixed nonlinear Choquard equation with exponential growth \begin{equation} \left\{ \mathcal{L}u+λu = Λ(I_α*F(u))f(u) \text{ in }\mathbb{R}^{2}; \;\; \int_{\mathbb{R}^{2}}|u|^{2}\,dx = a^{2}, \right. \end{equation} where $\mathcal{L}=-\Delta +(-\Delta {)}^s$, $0<s<1$, $a>0$, $I_{\alpha }$ is the Riesz potential of order $\alpha \in (0,2)$, $\Lambda >0$, and $\lambda \in \mathbb{R}$ arises as a Lagrange multiplier. The nonlinearity has critical exponential growth in the sense of the Trudinger-Moser inequality. Using variational methods, the associated Pohožaev manifold, and suitable Moser-type estimates, we establish the existence of a normalized solution of mountain-pass type with associated non zero Lagrange multiplier. Under an additional monotonicity assumption on the nonlinearity, we further show that the obtained solution is a normalized ground state. We also establish the regularity properties required for the proof of the corresponding Pohožaev identity.