Ground state solutions of mixed local-nonlolcal equations with Hartree type nonlinearities
Gurdev Chand Anthal, Prashanta Garain, Nidhi Nidhi
TL;DR
This work establishes the existence and symmetry of ground state solutions for a mixed local-nonlocal equation with a Hartree-type nonlinearity on $\mathbb{R}^N$, combining the classical and fractional Laplacians. The authors overcome non-scale-invariance via Jeanjean's scaling to obtain a nontrivial weak solution, and then demonstrate it is a ground state by deriving a Pohožaev identity and employing a concentration-compactness argument. Regularity results place the solution in $L^q$, $W^{2,q}_{\text{loc}}$, and $C^{1,\delta}_{\text{loc}}$, enabling the Pohožaev identity and facilitating symmetry via polarization. The results extend Choquard-type analyses to mixed local-nonlocal operators and provide rigorous symmetry conclusions for positive ground states.
Abstract
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.