A nonlinear elliptic operator and its singular values
The boundary value problem \(\Delta u+\lambda u-u^ 3=g\) on \(\Omega\), \(u| \partial \Omega =0\), where \(\Omega \subset {\mathbb{R}}^ n\) (n\(\leq 4)\) is a bounded domain, defines a real analytic map \(A_{\lambda}\) of the Sobolev space \(H=W_ 0^{1,2}(\Omega)\) onto itself. A point \(u\in H\) is a fold point if \(A_{\lambda}\) at u is \(C^{\infty}\) equivalent to \(f\times id: {\mathbb{R}}\times E\to {\mathbb{R}}\times E\), where \(f(t)=t^ 2.\) (1) There is a closed subset \(\Gamma _{\lambda}\subset H\) such that (a) at each point of \(A_{\lambda}^{-1}(H-\Gamma _{\lambda})\) the map \(A_{\lambda}\) is either locally a diffeomorphism or a fold, and (b) for each nonempty connected open subset \(V\subset H\), \(V-\Gamma _{\lambda}\) is nonempty and connected; thus \(\Gamma _{\lambda}\) is nowhere dense in H and does not locally separate H. Suppose that \(n\leq 3\) and the second eigenvalue \(\lambda _ 2\) of -\(\Delta\) u on \(\Omega\) with \(u| \partial \Omega =0\) is simple. Define \(A: H\times {\mathbb{R}}\to H\times {\mathbb{R}}\) by \(A(u,\lambda)=(A_{\lambda}(u),\lambda).\) (2) There is a connected open neighborhood V of \((0,\lambda _ 2)\) in \(H\times {\mathbb{R}}\) such that \(A^{-1}(V)\) has three components \(U_ 0,U_ 1,U_ 2\) with \(A: U_ i\to V\) a diffeomorphism for \(i=1,2\) and \(A| U_ 0: U_ 0\to V\) \(C^{\infty}\) equivalent to \(w\times id: {\mathbb{R}}^ 2\times E\to {\mathbb{R}}^ 2\times E\) defined by \((w\times id)(t,\lambda,\nu)=(t^ 3-\lambda t,\lambda,\nu)\).
scientific article; zbMATH DE number 4036286