Nonresonant nonlinear singular problems in the limit circle case
The existence of a solution is established for a nonlinear boundary value problem \({1 \over p(t)} (p(t)y'(t))' + \mu q(t) y(t) = f(t,y(t))\) a.e. on \([0,1]\), \(\lim_{t \to 0^+} p(t) y'(t) = y(1) = 0\), where \(p \in C[0,1] \cap C^1 (0,1)\), \(p > 0\) on \((0,1)\), \(q > 0\) a.e. on \([0,1]\), \(pq \in {\mathcal L}^1 [0,1]\), the endpoint 1 is regular, the endpoint 0 is of limit circle type, \(f : [0,1] \times \mathbb{R} \to \mathbb{R}\) is a so-called Carathéodory function, and \(\mu\) is not an eigenvalue of the corresponding linear boundary value problem. Some inequalities are assumed to hold on a subset of \([0,1]\) of positive measure. They concern the decomposition of \(f, \mu\), and the next eigenvalues of the linear boundary value problem. If \(p^{-1} \in {\mathcal L}^1 [0,1]\), the boundary conditions \(y(0) = y(1) = 0\) also are considered.
scientific article; zbMATH DE number 902467