Ground state solutions of mixed local-nonlolcal equations with Hartree type nonlinearities

TL;DR

This paper proves the existence, regularity, symmetry, and Pohožaev identity for ground state solutions of mixed local-nonlocal equations with Hartree nonlinearities, combining classical and fractional Laplacians.

Contribution

It introduces new existence and symmetry results for solutions to mixed local-nonlocal equations with Hartree nonlinearities, including regularity and Pohožaev identities.

Findings

Existence of ground state solutions established.

Regularity and symmetry properties proven.

Pohožaev-type identity derived for solutions.

Abstract

We study a class of mixed local-nonlocal equations with Hartree-type nonlinearities of the form \begin{equation}\label{meqnab} -\Delta u + (-\Delta)^s u + u = (I_\alpha * F(u))\,F'(u) \quad \text{in } \mathbb{R}^N, \end{equation} where $N \geq 3$, $s \in (0,1)$, and $F \in C^1(\mathbb{R},\mathbb{R})$ satisfies Berestycki-Lions type assumptions. The equation combines the classical Laplacian with the fractional Laplacian, while the Hartree-type nonlinearity is given by a nonlocal convolution term involving the Riesz potential $I_\alpha$, with $\alpha \in (0,N)$. We prove the existence of ground state solutions. To this end, we establish regularity properties and derive a Poho\v{z}aev-type identity for general weak solutions. Moreover, we obtain symmetry properties of ground state solutions via polarization methods.