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A critical point theorem via the Ekeland variational principle. (English) Zbl 1239.58011
Summary: The aim of this paper is to establish the existence of a local minimum for a continuously Gâteaux differentiable function, possibly unbounded from below, without requiring any weak continuity assumption. Several special cases are also emphasized. Moreover, a novel definition of Palais-Smale condition, which is more general than the usual one, is presented and a mountain pass theorem is pointed out. As a consequence, multiple critical points theorems are then established. Finally, as an example of applications, an elliptic Dirichlet problem with critical exponent is investigated.

MSC:
58E30Variational principles on infinite-dimensional spaces
49J52Nonsmooth analysis (other weak concepts of optimality)
49J50Fréchet and Gateaux differentiability
35J40Higher order elliptic equations, boundary value problems
46E05Lattices of continuous, differentiable or analytic functions
58C20Differentiation theory (Gateaux, Fréchet, etc.) on manifolds
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