Existence and concentration of semiclassical bound states for a quasilinear Schrödinger-Poisson system
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866918202549731328 |
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| author | Ramos, Gustavo de Paula Siciliano, Gaetano |
| author_facet | Ramos, Gustavo de Paula Siciliano, Gaetano |
| contents | In the paper we consider the following quasilinear Schrödinger--Poisson system in the whole space $\mathbb R^{3}$ $$ \begin{cases}
- \varepsilon^2 Δu + (V + ϕ) u = u |u|^{p - 1} \newline
- Δϕ- βΔ_4 ϕ= u^2, \end{cases} $$ where $1 < p < 5, β> 0,V :\mathbb R^{3}\to ]0, \infty[$ and look for solutions $u,ϕ:\mathbb R^{3}\to \mathbb R$ in the semiclassical regime, namely when $\varepsilon\to 0.$ By means of the Lyapunov--Schmidt method we estimate the number of solutions by the cup-length of the critical manifold of the external potential $V$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_03161 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Existence and concentration of semiclassical bound states for a quasilinear Schrödinger-Poisson system Ramos, Gustavo de Paula Siciliano, Gaetano Analysis of PDEs 35J10, 35J50, 35Q60 In the paper we consider the following quasilinear Schrödinger--Poisson system in the whole space $\mathbb R^{3}$ $$ \begin{cases} - \varepsilon^2 Δu + (V + ϕ) u = u |u|^{p - 1} \newline - Δϕ- βΔ_4 ϕ= u^2, \end{cases} $$ where $1 < p < 5, β> 0,V :\mathbb R^{3}\to ]0, \infty[$ and look for solutions $u,ϕ:\mathbb R^{3}\to \mathbb R$ in the semiclassical regime, namely when $\varepsilon\to 0.$ By means of the Lyapunov--Schmidt method we estimate the number of solutions by the cup-length of the critical manifold of the external potential $V$. |
| title | Existence and concentration of semiclassical bound states for a quasilinear Schrödinger-Poisson system |
| topic | Analysis of PDEs 35J10, 35J50, 35Q60 |
| url | https://arxiv.org/abs/2312.03161 |