Minima of some non convex non coercive problems (Q1327141)

The authors give an existence proof for a solution of the following problem of the calculus of variations: \[ \int^ 1_ 0 f(x,u,u')dx+ \widetilde h\left( \int^{u(1)}_ 0 a(0,s)ds+ \int^ \lambda_{u(1)} a(1,s)ds\right)\to \min_{u\in \overline w_ p}, \] where \(\overline w_ p= \bigl\{u\in w^{1,p}_{\text{loc}}(0,1)\mid u(0)\geq 0,\;u(1)\leq \lambda,\;u'\geq 0\;\text{a.e.}\bigr\}\) and \(\lambda\in \mathbb{R}_ +\), \(p\geq 1\).