Some remarks for one-dimensional mean curvature problems through a local minimization principle

In this paper we deal with a bifurcation result for the following parametric one-dimensional mean curvature problem: -(u'/root 1+u'(2))' = lambda f(t, u) in (0,1), u(0) = u(1) = 0, where lambda is an element of R and f : [0,1] x R -> R is a Caratheodory function vanishing at zero. More precisely, a critical point theorem ( local minimum result) for differentiable functionals is exploited in order to prove that the above problem admits at least one nontrivial and nonnegative weak solution under an asymptotical behaviour of the nonlinear datum at zero. A concrete example of an application is then presented.