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

Opuscula Mathematica

article

Tomas Godoy¹ 

[1]Universidad Nacional de Córdoba, Facultad de Matemática, Astronomía, Física y Computación, Av. Medina Allende s/n, Ciudad Universitaria

关键词: singular elliptic problems;    mixed boundary conditions;    weak solutions.;   

DOI  :  10.7494/OpMath.2023.43.1.19

学科分类：环境科学（综合）

来源: AGH University of Science and Technology Press

【 摘 要 】

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 \(0\leq\sigma\in W^{\frac{1}{2},2}(\Omega)\).

【 授权许可】

CC BY-NC