An approximation approach to eigenvalue intervals for singular boundary value problems with sign changing and superlinear nonlinearities (Q2380984)

The authors consider the eigenvalue interval for the singular boundary value problem \[ \begin{gathered} -u''= g(t,u)+\lambda h(t,u),\quad t\in (0,1),\\ u(0)= 0= u(1),\end{gathered}\tag{1} \] where \(g: (0,1)\times (0,\infty)\to \mathbb{R}\); \(h: (0,1)\times [0,\infty)\to (0,\infty)\) are continuous; \(g+ h\) may be singular at \(u= 0\), \(t= 0,1\). By using an approximation method together with the upper and lower solutions method, the authors obtains the results: Theorem 1.1: There exists \(\lambda^*_1> 0\) such that for every \(\lambda\geq\lambda_1\) problem (1) has at least one positive solution \(u\in C[0,1]\cap C^1(0,1)\) and \(u> 0\) for \(t\in (0,1)\). Theorem 1.2 (1.4): There exists \(\lambda^*_2> 0\) such that {\parindent7mm \begin{itemize}\item[(i)] for \(0<\lambda<\lambda^*_2\) problem (1) has at least one solution \(u\in C[0,1]\cap \mathbb C^1(0,1)\) and \(u> 0\) for \(t\in (0,1)\); \item[(ii)] for \(\lambda>\lambda^*_2\) problem (1) has no solution. \end{itemize}}