Generalized Ambrosetti-Prodi conditions for nonlinear two-point boundary value problems
For the following two-point boundary value problem \[ (P)\quad u''+f(x,u)=s(\frac{2}{\pi})^{1/2}\sin (x),\quad u(0)=u(\pi)=0 \] where f is a continuous function satisfying the condition (H) \(\lim_{| u| \to \infty}f(x,u)=+\infty\) it is shown that there exist real numbers \(so\leq so^+\) such that (1) for \(s<so\), (P) has no solution; (2) for \(s\in [so,so^+]\) (P) has at least one solution; (3) for \(s>so^ olution\) u(t), \(t\in [0,\infty [\). Theorem 1 states that Kiguradze- Kvinikadze condition cannot replace the condition \(J_{p\phi}=\int^{\infty}_{0}p(\tau)\phi^{-1}(\tau)\tau^{(n- 1)\lambda}d\tau <\infty,\) where \(\phi\) (t)\(\to \infty\) under \(t\to \infty\). The authors investigate case \(I_{p\phi}=\infty\).
scientific article; zbMATH DE number 4054072
scientific article