Uniqueness results for the one-dimensional \(m\)-Laplacian considering superlinear nonlinearities (Q1400268)

The authors study existence and uniqueness of positive solutions to the boundary value problem for the one-dimensional \(m\)-Laplacian \[ -(|u'|^{m-2}u')'=\lambda f(u) \quad \text{in}\quad (0,1), \qquad u(0)=u(1)=0, \] where \(\lambda\) is a positive parameter, \(m>1,\) and \(f:[0,\infty)\to \mathbb{R}\) is a continuous function which vanishes at most once in \((0,\infty)\) and is superlinear at \(+\infty.\) Furthemore, infinite multiplicity results on nonnegative solutions are established. The well-known shooting method is used.