Bounded solutions for non-parametric mean curvature problems with nonlinear terms
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866913380704452608 |
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| author | Giachetti, Daniela Oliva, Francescantonio Petitta, Francesco |
| author_facet | Giachetti, Daniela Oliva, Francescantonio Petitta, Francesco |
| contents | In this paper we prove existence of nonnegative bounded solutions for the non-autonomous prescribed mean curvature problem in non-parametric form on an open bounded domain $Ω$ of $\mathbb{R}^N$. The mean curvature, that depends on the location of the solution $u$ itself, is asked to be of the form $f(x)h(u)$, where $f$ is a nonnegative function in $L^{N,\infty}(Ω)$ and $h:\mathbb{R}^+\mapsto \mathbb{R}^+$ is merely continuous and possibly unbounded near zero. As a preparatory tool for our analysis we propose a purely PDE approach to the prescribed mean curvature problem not depending on the solution, i.e. $h\equiv 1$. This part, which has its own independent interest, aims to represent a modern and up-to-date account on the subject. Uniqueness is also handled in presence of a decreasing nonlinearity. The sharpness of the results is highlighted by mean of explicit examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_13611 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Bounded solutions for non-parametric mean curvature problems with nonlinear terms Giachetti, Daniela Oliva, Francescantonio Petitta, Francesco Analysis of PDEs In this paper we prove existence of nonnegative bounded solutions for the non-autonomous prescribed mean curvature problem in non-parametric form on an open bounded domain $Ω$ of $\mathbb{R}^N$. The mean curvature, that depends on the location of the solution $u$ itself, is asked to be of the form $f(x)h(u)$, where $f$ is a nonnegative function in $L^{N,\infty}(Ω)$ and $h:\mathbb{R}^+\mapsto \mathbb{R}^+$ is merely continuous and possibly unbounded near zero. As a preparatory tool for our analysis we propose a purely PDE approach to the prescribed mean curvature problem not depending on the solution, i.e. $h\equiv 1$. This part, which has its own independent interest, aims to represent a modern and up-to-date account on the subject. Uniqueness is also handled in presence of a decreasing nonlinearity. The sharpness of the results is highlighted by mean of explicit examples. |
| title | Bounded solutions for non-parametric mean curvature problems with nonlinear terms |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2304.13611 |