The three-solutions theorem for \(p\)-Laplacian boundary value problems (Q651161)

The author proves the existence of three solutions of the following \(p\)-Laplacian problem \[ (w(t)\, \varphi_p(u'(t)))'+\lambda\, f(t,u(t))=0 \,\, \text{for a.e.}\,\, t\in (0,1), \; u(0)=u(1)=0. \] Here, \(\varphi_p(s)=|s|^{p-2}s\), \(p>1\), is the usual \(p\)-Laplacian operator, \(\lambda>0\) is a real parameter, \(f:(0,1) \times \mathbb R \to \mathbb R\) is a \(L^1\)-Carathédory function and \(w:[0,1] \to [0, \infty)\) is a continuous function such that \(w>0\) on \((0,1)\) and \(\varphi_p^{-1}(1/w) \in L^1(0,1)\). The result follows from the assumption of two pairs of lower and upper solutions and by using the degree theory. Some particular problems are considered in which the existence result holds for some values of \(\lambda\).