On nontrivial solutions of an asymptotically linear Dirichlet problem
The author adds to the large existing literature dealing with solutions of \[ -\Delta u=f(u)\quad in\quad \Omega,\quad f(0)=0;\quad u\equiv 0\quad on\quad \partial \Omega, \] where \(\Omega\) is a bounded domain in \({\mathbb{R}}^ n\), and \(\partial \Omega\) is smooth. He assumes \(\lim_{\zeta \to \pm \infty} f(\zeta)/\zeta\) exists. However, he does not require differentiability of f(\(\zeta)\), and the interval \([f_-,f_+]\) is allowed to contain an eigenvalue of - \(\Delta\). He proves that if there exists a constant \(\beta\), such that \[ (f(\zeta)-f(\eta))/(\zeta -\eta)\leq \beta <\lambda_{k+1},\quad (k+1\quad eigenvalue) \] \(\forall \zeta\), \(\eta\in {\mathbb{R}}\), \(\zeta\neq \eta\), and \(\exists \delta >0\), such that \(0<| \zeta | <\delta \Rightarrow f(\zeta)/\zeta <\lambda_ k\) then the Dirichlet problem stated above has a solution for negative values of Galluët-Kavakian numbers. Other, related results are also proved.
scientific article; zbMATH DE number 4091985
scientific article