Opuscula Mathematica
Opuscula Math. 43, no. 1 (2023), 19-46

Singular elliptic problems with Dirichlet or mixed Dirichlet-Neumann non-homogeneous boundary conditions

Tomas Godoy
Abstract. Let \(\Omega\) be a \(C^{2}\) bounded domain in \(\mathbb{R}^{n}\) such that \(\partial\Omega=\Gamma_{1}\cup\Gamma_{2}\), where \(\Gamma_{1}\) and \(\Gamma_{2}\) are disjoint closed subsets of \(\partial\Omega\), and consider the problem\(-\Delta u=g(\cdot,u)\) in \(\Omega\), \(u=\tau\) on \(\Gamma_{1}\), \(\frac{\partial u}{\partial\nu}=\eta\) on \(\Gamma_{2}\), where \(0\leq\tau\in W^{\frac{1}{2},2}(\Gamma_{1})\), \(\eta\in(H_{0,\Gamma_{1}}^{1}(\Omega))^{\prime}\), and \(g:\Omega \times(0,\infty)\rightarrow\mathbb{R}\) is a nonnegative Carathéodory function. Under suitable assumptions on \(g\) and \(\eta\) we prove the existence and uniqueness of a positive weak solution of this problem. Our assumptions allow \(g\) to be singular at \(s=0\) and also at \(x\in S\) for some suitable subsets \(S\subset\overline{\Omega}\). The Dirichlet problem \(-\Delta u=g(\cdot,u)\) in \(\Omega\), \(u=\sigma\) on \(\partial\Omega\) is also studied in the case when \(0\leq\sigma\in W^{\frac{1}{2},2}(\Omega)\).
Keywords: singular elliptic problems, mixed boundary conditions, weak solutions.
Mathematics Subject Classification: 35J75, 35M12, 35D30.
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About the authors ▼
Tomas Godoy
https://orcid.org/0000-0002-8804-9137
Universidad Nacional de Córdoba, Facultad de Matemática, Astronomía, Física y Computación, Av. Medina Allende s/n, Ciudad Universitaria, 5000 Córdoba, Argentina
About the article ▼
Communicated by Vicentiu D. Rădulescu.
Received: 2022-08-07.
Revised: 2022-11-21.
Accepted: 2022-11-25.
Published online: 2022-12-30.
Cite this article as ▼
Tomas Godoy, Singular elliptic problems with Dirichlet or mixed Dirichlet-Neumann non-homogeneous boundary conditions, Opuscula Math. 43, no. 1 (2023), 19-46, https://doi.org/10.7494/OpMath.2023.43.1.19.
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