On the summability of weak solutions for a singular Dirichlet problem in bounded domains (Q2450330)

In this paper the authors consider a class of semilinear Dirichlet problems in the form \[ -\Delta u = f(x,u)+\lambda h(x,u) \text{ in } \Omega, \;\;u = 0 \text{ on } \partial \Omega, \] where \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^N\) with \(N\geq 3\) and \(\lambda\geq 0\). The authors show the existence of positive weak solutions to the above equation under growth conditions such as given below: There exist \(p>0\) and \(\psi_0\in L^m(\Omega)\) with \(m\geq 1\) such that \(f(x,s)\neq 0\), \(0\leq f(x,s)\leq \displaystyle{\psi_0(x)\over s^p}\) for \(x\in \Omega\), \(s>0\). There exists \(\mu_0>0\) such that \(\displaystyle \liminf_{s\rightarrow 0^+}{f(x,s)\over s}\geq \mu_0\). There exist \(q\geq 0\) and \(\psi_{\infty}\in L^M(\Omega)\) with \(M>N/2\) such that \(0\leq h(x,s)\leq \psi_{\infty}(x)s^q\) for \(x\in \Omega\) and \(s>0\). For various values of \(p\) and \(q\), the paper provides results on the structure of the set of parameter values (\(\lambda\)) for which the equation has a positive weak solution \(u_{\lambda}\) and the summability of \(u_{\lambda}\) in different spaces.