Mountain pass theorem

Abstract

Let us recall some notions of differentiability.

Definition 1.1. Let φ : U → ℝ where U is an open subset of a Banach space X. The functional φ has a Gateaux derivative f ∈ X′ at u ∈ U if, for every h ∈ X,

$$\mathop{{\lim }}\limits_{{t \to 0}} \frac{1}{t}[\varphi (u + th) - \varphi (u) - \langle f,th\rangle ] = 0 $$

.