Nonlinear elliptic equations with singular nonlinearities (Q2857020)

The author of this paper studies the following problem NEWLINE\[NEWLINE\mathrm{div }a(x, \nabla u)=\frac{f(x)}{u^\gamma} \text{ in } \Omega,\quad u>0 \text{ in } \Omega,\quad u=0 \text{ on } \partial\Omega,NEWLINE\]NEWLINE where \(\Omega\) is a bounded open set of \(\mathbb R^N\), \(N\geq 2\), \(\gamma\) is a positive parameter, \(f\in L^1(\Omega)\) is positive and \(a\) is a Carathéodory function whose model is given by \(a(x,\xi)=|\xi|^{p-2}\xi\) for some \(1<p<N\).NEWLINENEWLINEThe difficulties in treating this problem are related to the presence of the singular term \(u^{-\gamma}\) and to the homogeneous Dirichlet boundary condition. In the main result of the paper, the author proves both the existence of at least one solution for the problem, using an approximation technique, and a regularity result. The regularity of the solution depends on \(\gamma\) and, in some cases, also on \(p\).