Non-Coercive Variational Problems with Constraints on the Derivatives

We establish a necessary and sufficient condition for the existence of the minimum of the functional ∫₀¹ f(t, v′(t))dt in the class $\mathcal{W}^p_d = \{v \in W^{1,p}([0,1]) : v(0) = 0, v(1) = d, v'(t) \geq \alpha\}$, in terms of a limitation of the slope d. Some applications to quasi-coercive and non-coercive integrands are also derived.