Existence of positive solutions of one-dimensional prescribed mean curvature equation (Q1718770)

Summary: We consider the existence of positive solutions of one-dimensional prescribed mean curvature equation \(-(u' / \sqrt{1 + u^{\prime 2}})' = \lambda f(u)\), \(0 < t < 1\), \(u(t) > 0\), \(t \in(0, 1)\), \(u(0) = u(1) = 0\) where \(\lambda > 0\) is a parameter, and \(f : [0, \infty) \rightarrow [0, \infty)\) is continuous. Further, when \(f\) satisfies \(\max \{u^p, u^q \} \leq f(u) \leq u^p + u^q\), \(0 < p \leq q < + \infty\), we obtain the exact number of positive solutions. The main results are based upon quadrature method.