We are concerned with the problem of existence of solutions to the variational problem [GRAPHICS] with only one fixed endpoint prescribed. The map g : [0, R] x R ---> (R) over bar is a normal integrand, for which neither convexity nor superlinear growth conditions are assumed. As an application, we give an existence result for the radially symmetric variational problem [GRAPHICS] where B-R is the ball of R-n centered at the origin and with radius R > 0, the map f : [0, R] x [0, + infinity [--> (R) over bar is a normal integrand, and a is an element of L-1 (0, R). Again, neither convexity nor superlinear growth conditions are made on f. These kinds of problems, with nonconvex Lagrangians with respect to del(u), arise in different fields of mathematical physics such as optimal design and nonlinear elasticity.