Concavity properties for quasilinear equations and optimality remarks
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866908415174901760 |
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| author | Almousa, Nouf M. Assettini, Jacopo Gallo, Marco Squassina, Marco |
| author_facet | Almousa, Nouf M. Assettini, Jacopo Gallo, Marco Squassina, Marco |
| contents | In this paper we study quasiconcavity properties of solutions of Dirichlet problems related to modified nonlinear Schrödinger equations of the type $$-{\rm div}\big(a(u) \nabla u\big) + \frac{a'(u)}{2} |\nabla u|^2 = f(u) \quad \hbox{in $Ω$},$$ where $Ω$ is a convex bounded domain of $\mathbb{R}^N$. In particular, we search for a function $φ:\mathbb{R} \to \mathbb{R}$, modeled on $f\in C^1$ and $a\in C^1$, which makes $φ(u)$ concave. Moreover, we discuss the optimality of the conditions assumed on the source. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_09982 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Concavity properties for quasilinear equations and optimality remarks Almousa, Nouf M. Assettini, Jacopo Gallo, Marco Squassina, Marco Analysis of PDEs 26B25, 35B99, 35E10, 35J60, 35J62 In this paper we study quasiconcavity properties of solutions of Dirichlet problems related to modified nonlinear Schrödinger equations of the type $$-{\rm div}\big(a(u) \nabla u\big) + \frac{a'(u)}{2} |\nabla u|^2 = f(u) \quad \hbox{in $Ω$},$$ where $Ω$ is a convex bounded domain of $\mathbb{R}^N$. In particular, we search for a function $φ:\mathbb{R} \to \mathbb{R}$, modeled on $f\in C^1$ and $a\in C^1$, which makes $φ(u)$ concave. Moreover, we discuss the optimality of the conditions assumed on the source. |
| title | Concavity properties for quasilinear equations and optimality remarks |
| topic | Analysis of PDEs 26B25, 35B99, 35E10, 35J60, 35J62 |
| url | https://arxiv.org/abs/2305.09982 |