The singular set of a nonlinear elliptic operator (Q913098)

Let \(\Omega\) be a bounded domain in \(\mathbb{R}^ n\) (\(n\leq 4\)) and let \(H\) denote the Sobolev space \(W_ 0^{1,2}(\Omega)\). In the present note the authors continue their study on the singular set of mapping \(A: H\times\mathbb{R} \to H\times\mathbb{R}\) defined by \(A(u,\lambda) = (A_{\lambda}(u),\lambda)\), where \[ \langle A_{\lambda}(u),\phi \rangle_ H = \int_{\Omega} [\nabla u\nabla \phi - \lambda u\phi + u^3\phi] \text{ for all } \phi \in C_0^{\infty}(\Omega). \] The results are applied to investigate the number of weak solutions of the boundary value problem \[ \Delta u + \lambda u-u^3 = g \text{ in } \Omega,\quad u=0 \text{ on } \partial\Omega. \]