Positive solution of a singular nonlinear elliptic boundary value problem.
The authors consider the following singular elliptic problem \[ \begin{cases} \Delta u+ g(x) u^\alpha+ h(x) u^{-\beta}= 0\quad &\text{in }\Omega,\\ u> 0\quad &\text{in }\Omega,\\ u= 0\quad &\text{on }\partial\Omega,\end{cases}\tag{1} \] where \(\alpha\in (0,1)\), \(\beta> 0\), \(\Omega\) is a bounded domain in \(\mathbb R^N\) with smooth boundary. Using super- and subsolution argument together with a monotonic approaching, the authors establish two theorems regarding, respectively the existence and the uniqueness of solutions for (1), in which the restriction on constant \(\beta\) and the functions \(g\) and \(h\) are relaxed in some sense.
nonlinear elliptic boundary value problem
scientific article; zbMATH DE number 2032664
scientific article