On the existence and numerical computation of classical and non-classical solutions for a family of elliptic boundary value problems (Q5957796)

The authors study the existence, pointwise behaviour and numerical computation of positive classical solutions of NEWLINE\[NEWLINE - \Delta u = \lambda u -a(x)u^{p+1} \quad\text{in}\quad \Omega, \qquad u = 0 \quad\text{on}\quad \partial\Omega, NEWLINE\]NEWLINE where \(\Omega\) is a bounded domain in \({\mathbb R}^N\) with \(\partial\Omega\in C^2\), \(\lambda\in{\mathbb R}\), \(p>0\) and \(a\in L^\infty(\Omega)\), \(a\not\equiv 0\), is nonnegative. Several technical assumptions are made regarding the function \(a\) and the sets \(\Omega_+ = \{x\in\Omega:a(x)>0\}\) and \(\Omega\backslash\overline\Omega_+\). The authors also introduce a class of nonclassical solutions and consider similar questions for these solutions.