Exact number of solutions of a one-dimensional prescribed mean curvature equation with concave–convex nonlinearities

In this paper, an exact number of solutions is obtained for the one-dimensional prescribed mean curvature equation with concave–convex nonlinearities in the form of { − ( u ′ 1 + u ′ 2 ) ′ = λ ( u p + u q ) , u ( x ) > 0 , 0 < x < 1 , u ( 0 ) = u ( 1 ) = 0 , where λ > 0 is a parameter and p , q satisfy 0 < p < 1 < q < + ∞ . The arguments are based upon a time-map method.