Criteria for the existence of a potential well

We consider a critical point u0 of a functional f∈C1(H,R), where H is a real Hilbert space, and formulate criteria ensuring that u0 lies in a potential well of f without supposing that f′ is Fréchet differentiable at u0. The derivative is required to be Gâteaux differentiable at u0, but positive definiteness of f″(u0) does not even ensure that f has a local minimum at u0 when f′ is not Fréchet differentiable at u0. This issue is also discussed in the context of the energy functional for a parameter dependent nonlinear eigenvalue problem and then for a particular case involving a degenerate elliptic Dirichlet problem on a bounded domain in RN.