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The Dirichlet problem for singular elliptic equations with general nonlinearities
- Source :
- Calculus of Variations and PDEs, 58(4), (2019)
- Publication Year :
- 2018
-
Abstract
- In this paper, under very general assumptions, we prove existence and regularity of distributional solutions to homogeneous Dirichlet problems of the form $$\begin{cases} \displaystyle - \Delta_{1} u = h(u)f & \text{in}\, \Omega,\newline u\geq 0& \text{in}\ \Omega, \newline u=0 & \text{on}\ \partial \Omega, \end{cases} $$ where, $\Delta_{1} $ is the $1$-laplace operator, $\Omega$ is a bounded open subset of $\mathbb{R}^N$ with Lipschitz boundary, $h(s)$ is a continuous function which may become singular at $s=0^{+}$, and $f$ is a nonnegative datum in $L^{N,\infty}(\Omega)$ with suitable small norm. Uniqueness of solutions is also shown provided $h$ is decreasing and $f>0$. As a by-product of our method a general theory for the same problem involving the $p$-laplacian as principal part, which is missed in the literature, is established. The main assumptions we use are also further discussed in order to show their optimality.
- Subjects :
- Mathematics - Analysis of PDEs
35J60, 35J75, 34B16, 35R99, 35A02
Subjects
Details
- Database :
- arXiv
- Journal :
- Calculus of Variations and PDEs, 58(4), (2019)
- Publication Type :
- Report
- Accession number :
- edsarx.1801.03444
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1007/s00526-019-1582-4