Exact number of solutions of a prescribed mean curvature equation

We study the exact number of positive solutions of the Dirichlet problem for the one-dimensional prescribed mean curvature equation − ( u ′ 1 + u ′ 2 ) ′ = λ ( u p + u q ) , u ( 0 ) = u ( 1 ) = 0 , where λ > 0 is a parameter and p, q satisfy either 1 < p < q < + ∞ or 0 < p < q < 1 .