Exact number of solutions for a class of two-point boundary value problems with one-dimensional \(p\)-Laplacian (Q2473938)

The authors study positive solutions to a first-order scalar nonliner Dirichlet problem of the form \[ -(\phi_p(u'(x)))'=\lambda g(u(x))+ h(u(x)),\quad a< x< b; \] \[ u(x)> 0,\quad a< x< b;\quad u(a)= u(b)= 0. \] Here, \(p> 1\), \(\phi_p(y)= |y|^{p-2} y\), and \(\lambda> 0\) is a parameter. The functions \(g\), \(h\) are positive and strictly increasing. It is shown under some assumptions that there exists a number \(\lambda^*\) such that the considered problem has exactly two nontrivial solutions for \(0<\lambda< \lambda^*\), exactly one nontrivial solution for \(\lambda= \lambda^*\), and no solutions for \(\lambda> \lambda^*\).