Ground state solutions of mixed local-nonlolcal equations with Hartree type nonlinearities
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866908805610078208 |
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| author | Anthal, Gurdev Chand Garain, Prashanta Nidhi, Nidhi |
| author_facet | Anthal, Gurdev Chand Garain, Prashanta Nidhi, Nidhi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_02168 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Ground state solutions of mixed local-nonlolcal equations with Hartree type nonlinearities Anthal, Gurdev Chand Garain, Prashanta Nidhi, Nidhi Analysis of PDEs 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. |
| title | Ground state solutions of mixed local-nonlolcal equations with Hartree type nonlinearities |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2602.02168 |