Existence criteria for singular boundary value problems with sign changing nonlinearities
In this interesting paper, a general theory on the existence of solutions to the singular Dirichlet problem \[ y''(t)+q(t)f(t,y)=0, \quad 0<t<1,\qquad y(0)=y(1)=0,\tag \(*\) \] is presented. The nonlinearity \(f(t,y)\) might be singular with respect to the variable \(y\) at 0 and it may change sign. The authors show the existence of a solution \(y\in C[0,1]\cap{C^2(0,1)}\) to \((*)\) such that \(y(t)>0,\) for \(0<t<1.\) To do this, they approximate problem \((*)\) with an appropriate sequence of (regular) Dirichlet problems \((*_n)\), say, and by making use of an earlier existence result [the authors, J. Differ. Equations 175, 393-414 (2001; Zbl 0999.34018)], they get the existence of a solution \((y_n)\) to \((*_n).\) The basic assumption is the existence of a lower solution \(\alpha_n\) (generated from a positive given function \(\alpha\) which it turns to be a lower solution to \((*)\)) and of an upper solution \(\beta_n\). Then, the result follows by applying the classical Ascoli theorem. Several comments on the assumptions are presented and, in particular, some general criteria are given which help to verify whether a particular equation has a lower solution. An application of the result is given. On the other hand, two other theorems give existence results by using conditions which do not require any knowledge of upper and lower solutions.
singular nonlinear boundary value problems
scientific article; zbMATH DE number 1836801
scientific article