Upper and lower solution method and a singular boundary value problem for the one-dimensional \(p\)-Laplacian
The authors study the singular boundary value problem for the one-dimensional \(p\)-Laplacian \[ (\Phi (u'))' = - g(t,u) \text{ for all } t \in (0,1), \quad u(0)=u(1)=0, \] with \(\Phi(s) = \mid s \mid^{p-2} s\) and \(p > 1\). The singularity may appear at \(u=0\), \(t =0\) and \(t= 1\) and the function \(g\) may change sign. The authors prove that if there exist a lower solution \(\alpha\) and an upper solution \(\beta\) such that \(\alpha \leq \beta\), then this problem has at least one solution lying between both functions. Such solution is given as the limit of a sequence of solutions to nonsingular truncated problems. Finally, the authors give different existence and uniqueness results derived from the previous existence results.
scientific article; zbMATH DE number 1563084
scientific article