Concentrated solutions to the Schrödinger--Bopp--Podolsky system with a positive potential
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866910525566222336 |
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| author | Ramos, Gustavo de Paula |
| author_facet | Ramos, Gustavo de Paula |
| contents | Consider the Schrödinger--Bopp--Podolsky system \[ \begin{cases}
-ε^2Δu+(V+Kϕ)u=u|u|^{p-1};\newline
Δ^2ϕ-Δϕ=4πK u^2 \end{cases} ~\text{in}~\mathbb{R}^3 \] for sufficiently small $ε>0$, where $V,K\colon\mathbb{R}^3\to [0,\infty[$; $p\in ]1,5[$ are fixed and we want to solve for $u,ϕ\colon\mathbb{R}^3\to\mathbb{R}$. Under certain hypotheses, we estimate the multiplicity of solutions in function of a critical manifold of $V$ and we establish the existence of solutions concentrated around critical points of $V$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_01841 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Concentrated solutions to the Schrödinger--Bopp--Podolsky system with a positive potential Ramos, Gustavo de Paula Analysis of PDEs 35J48, 35B25, 35Q60 Consider the Schrödinger--Bopp--Podolsky system \[ \begin{cases} -ε^2Δu+(V+Kϕ)u=u|u|^{p-1};\newline Δ^2ϕ-Δϕ=4πK u^2 \end{cases} ~\text{in}~\mathbb{R}^3 \] for sufficiently small $ε>0$, where $V,K\colon\mathbb{R}^3\to [0,\infty[$; $p\in ]1,5[$ are fixed and we want to solve for $u,ϕ\colon\mathbb{R}^3\to\mathbb{R}$. Under certain hypotheses, we estimate the multiplicity of solutions in function of a critical manifold of $V$ and we establish the existence of solutions concentrated around critical points of $V$. |
| title | Concentrated solutions to the Schrödinger--Bopp--Podolsky system with a positive potential |
| topic | Analysis of PDEs 35J48, 35B25, 35Q60 |
| url | https://arxiv.org/abs/2309.01841 |