Some Results on Non-Coercive Variational Problems and Applications

Abstract

We give a necessary and sufficient condition to ensure the existence of solutions of three problems of the calculus of variations with non-coercive integrands. The solutions $u$ we consider are lipschitz functions, i.e. $u\in W^{1,\infty }(0,1)$. The three problems depends on the same functionals but are different in the constraints. We consider respectively a problem without constraint, a problem with $u^{\prime }\ge 0$ and finally a problem with $u\ge 0$. These problems can be related to optimal foraging models in behavioural ecology.