Weighted Robin eigenvalue problems and nonlinear elliptic equations with general growth in the gradient
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915692517785600 |
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| author | Della Pietra, Francesco di Blasio, Giuseppina Riey, Giuseppe |
| author_facet | Della Pietra, Francesco di Blasio, Giuseppina Riey, Giuseppe |
| contents | We prove an existence result for Robin boundary value problems modeled on \[
\begin{cases}
Δu + |\nabla u|^2 + λf(x) = 0 & \text{in } Ω
\\ \frac{\partial u}{\partial ν} + βu = 0 & \text{on } \partialΩ\end{cases} \] where $Ω$ is a bounded, sufficiently smooth open set in $\mathbb R^N$, $f(x)$ belongs to the Marcinkiewicz space $M^{\frac N2}$ and {$β>0$}, under a smallness assumption on the datum $λ$. In order to study such problem, we will show several properties of the weighted, singular Robin eigenvalue problem \[ λ_{1,f,γ}(Ω)= \inf_{ψ\in H^{1},\;\int_Ωfψ^{2}=1}\left\{\int_Ω|\nabla ψ|^{2}dx+γ\int_{\partialΩ}ψ^{2}\right\}. \] |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_20192 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weighted Robin eigenvalue problems and nonlinear elliptic equations with general growth in the gradient Della Pietra, Francesco di Blasio, Giuseppina Riey, Giuseppe Analysis of PDEs 35J66, 35J20, 35P15 We prove an existence result for Robin boundary value problems modeled on \[ \begin{cases} Δu + |\nabla u|^2 + λf(x) = 0 & \text{in } Ω \\ \frac{\partial u}{\partial ν} + βu = 0 & \text{on } \partialΩ\end{cases} \] where $Ω$ is a bounded, sufficiently smooth open set in $\mathbb R^N$, $f(x)$ belongs to the Marcinkiewicz space $M^{\frac N2}$ and {$β>0$}, under a smallness assumption on the datum $λ$. In order to study such problem, we will show several properties of the weighted, singular Robin eigenvalue problem \[ λ_{1,f,γ}(Ω)= \inf_{ψ\in H^{1},\;\int_Ωfψ^{2}=1}\left\{\int_Ω|\nabla ψ|^{2}dx+γ\int_{\partialΩ}ψ^{2}\right\}. \] |
| title | Weighted Robin eigenvalue problems and nonlinear elliptic equations with general growth in the gradient |
| topic | Analysis of PDEs 35J66, 35J20, 35P15 |
| url | https://arxiv.org/abs/2512.20192 |