Singular elliptic problems: existence, non-existence and boundary behavior
The authors study the existence, uniqueness and boundary behaviour of solutions of the (possibly) singular semilinear elliptic problem \[ -\Delta u= \rho a(x) g(u)+ \lambda b(x) f(u)\quad\text{in }\Omega, \] \[ u> 0\quad\text{in }\Omega,\qquad u= 0\quad\text{on }\partial\Omega, \] where \(\Omega\subset\mathbb{R}^d\) is a bounded domain with smooth boundary \(\partial\Omega\), \(p\), \(\lambda\geq 0\), \(a,b:\overline\Omega\to [0,\infty)\) are Hölder continuous with exponent \(\nu\in (0,1)\). The main feature here is that \(g(t)\), \(f(t)\to \infty\) as \(t\to 0\) (at least one of them). To this end the authors use a variational method.
scientific article; zbMATH DE number 5144876
scientific article