Positive solutions for the one-dimensional \(p\)-Laplacian (Q5915525)

The author discusses the boundary value problem \[ (b(t)\phi(x'))'+ c(t) f(x)= 0,\quad x(0)= x(T)= 0, \] where \(\phi(u)=|u|^{p-1}u\) and \(p\) is a positive constant. He proves the existence of positive solutions to this problem under the following assumptions: \(b\in C^1([0,+\infty),(0, +\infty))\), \(f\in C^0(\mathbb{R})\), \(f(x)> 0\) for \(x> 0\) and \(f(0)= 0\), \(c\in C^0[0,\infty)\), \(c(t)\geq 0\) on \([0,\infty)\) and is not identically zero on any subinterval of \([0,\infty)\), \[ \lim_{x\to 0} {f(x)\over \phi(x)}= 0\quad\text{and}\quad \lim_{x\to\infty} {f(x)\over \phi(x)}= \infty. \] The forward shooting method combined with a generalized form of the Sturm comparison theorem is used.