Church, P. T.; Timourian, J. G. The singular set of a nonlinear elliptic operator. (English) Zbl 0699.35086 Mich. Math. J. 35, No. 2, 197-213 (1988). 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 \[ \Delta u + \lambda u-u^3 = g \text{ in } \Omega,\quad u=0 \text{ on } \partial\Omega. \] Reviewer: V.Mustonen Cited in 4 Documents MSC: 35J65 Nonlinear boundary value problems for linear elliptic equations Keywords:nonlinear elliptic operator; Sobolev space; weak solutions; boundary value problem PDFBibTeX XMLCite \textit{P. T. Church} and \textit{J. G. Timourian}, Mich. Math. J. 35, No. 2, 197--213 (1988; Zbl 0699.35086) Full Text: DOI