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

We study a class of mixed local-nonlocal equations with Hartree-type nonlinearities of the form \begin{equation}\label{meqnab} -Δu + (-Δ)^s u + u = (I_α* 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 $I_α$, with $α\in (0,N)$. We prove the existence of ground state solutions. To this end, we establish regularity properties and derive a Pohožaev-type identity for general weak solutions. Moreover, we obtain symmetry properties of ground state solutions via polarization methods.

💡 Research Summary

The paper investigates a class of elliptic equations that combine the classical Laplacian with a fractional Laplacian and feature a Hartree‑type nonlocal nonlinearity. The model under consideration is

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