Extremal positive solutions for a class of singular and discontinuous second order problems
Using an iterative scheme, the author proves the existence of positive extremal solutions to the initial value problem \[ (k(u(x)) u'(x))'= f(x, u(x)) u'(x)+ g(x, u(x)) u(x)\quad\text{for a.e. }x\in I= [0,\alpha], \] \[ u(0)= 0,\quad k(u(0)) u'(0)= 0, \] under the following assumptions. The function \(k\in L^1_{\text{loc}}([0,\infty))\) is such that \(k(y)> 0\) for a.e. \(y\in [0,\infty)\), \(\int^1_0 {k(s)\over s} ds< \infty\), \(\int^\infty_1 {k(s)\over s} ds= \infty\). The function \(g: I\times [0,\infty)\to \mathbb{R}\) is such that \(g(x,y)y\) is nondecreasing with respect to \(y\in [0,\infty)\) for a.e. \(x\in I\), \(g(\cdot,h(\cdot))\) is measurable on \(I\), whenever \(h: I\to [0,\infty)\) is continuous, and there exists \(\psi\in L^1(I)\) such that \(0\leq g(x,y)\leq \psi(x)\), \(y\in [0,\infty)\), for a.e. \(x\in I\). Finally, \(f: I\times [0,\infty)\to [0,\infty)\) is such that \(f(\cdot,y)\) is nonincreasing on \(I\) for a.e. \(y\in [0,\infty)\), \(f(h(\cdot),\cdot)\) is measurable on \([0,\infty)\), whenever \(h: [0,\infty)\to I\) is continuous and nondecreasing, and there exist constants \(M\), \(m> 0\) such that \(f(x,y)\leq M\), \(x\in I\), for a.e. \(y\in [0,\infty)\) and \(m\leq f(x,y)\), \(x\in I\), for a.e. \(y\in [0,r_0]\), where \(r_0\) is a suitable constant.
scientific article; zbMATH DE number 1836630