I am reading this paper from Peter Hess: On multiple positive solutions of nonlinear elliptic eigenvalue problems. Comm. Partial Differential Equations 6 (1981), no. 8, 951–961.
In the last page (part (vii)), he finds sub and super solutions to problem $2_k$, and then he uses a theorem that guarantee the existence of a solution $\hat{u}_k$ to problem $2_k$. From here, he implicity assumes that this solution is a minimal of the functional $$\frac{1}{2}\|u\|_H^2-\lambda\int_\Omega F_k(u)$$
Im saying that he assume this, because he uses the step (v) and (vi). Well, I know that this soluiton is a critical point of this functional, but how to show that this solution is a minimum point? I am also assuming that he is using a theorem of sub and super solutions like this (see theorem 11.1)
If you need the paper, please send me an e-mail.