Quasilinear Equations with Neumann Boundary Conditions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908696307564544 |
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| author | Canino, Annamaria Mauro, Simone |
| author_facet | Canino, Annamaria Mauro, Simone |
| contents | We prove a multiplicity result for non-constant weak solutions $u \in H^1(Ω)$ for the quasilinear elliptic equation \[ \begin{cases} \displaystyle-\text{div}(A(x,u)\nabla u) + \frac{1}{2} D_sA(x,u)\nabla u \cdot \nabla u = g(x,u) - λu & \text{in } Ω\\ A(x,u)\nabla u \cdot η= 0 & \text{on } \partial Ω\end{cases} \] where $λ\in \mathbb{R}$, $ Ω$ is a bounded lipschitz domain, $ η$ is the outward normal to the boundary $ \partial Ω$, and $g(x,u)$ is a Carathéodory function that satisfies a general subcritical (and superlinear) growth condition. We also prove that any weak solution is bounded under a stronger growth assumption. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_17374 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quasilinear Equations with Neumann Boundary Conditions Canino, Annamaria Mauro, Simone Analysis of PDEs 35A01, 35A15, 35J05, 35J20, 35J25, 35J62 We prove a multiplicity result for non-constant weak solutions $u \in H^1(Ω)$ for the quasilinear elliptic equation \[ \begin{cases} \displaystyle-\text{div}(A(x,u)\nabla u) + \frac{1}{2} D_sA(x,u)\nabla u \cdot \nabla u = g(x,u) - λu & \text{in } Ω\\ A(x,u)\nabla u \cdot η= 0 & \text{on } \partial Ω\end{cases} \] where $λ\in \mathbb{R}$, $ Ω$ is a bounded lipschitz domain, $ η$ is the outward normal to the boundary $ \partial Ω$, and $g(x,u)$ is a Carathéodory function that satisfies a general subcritical (and superlinear) growth condition. We also prove that any weak solution is bounded under a stronger growth assumption. |
| title | Quasilinear Equations with Neumann Boundary Conditions |
| topic | Analysis of PDEs 35A01, 35A15, 35J05, 35J20, 35J25, 35J62 |
| url | https://arxiv.org/abs/2510.17374 |