Localization of positive critical points in Banach spaces and applications

Radu Precup, Csaba Varga


Two critical point theorems of M. Schechter in a ball of a Hilbert space are extended to uniformly convex Banach spaces by exploiting the properties of the duality mapping. Moreover, the critical points are sought in the intersection of a ball with a wedge, in particular with a cone, making possible applications to positive solutions of variational problems. The extension from Hilbert to Banach spaces not only requires a major refining of reasoning, but also a different statement by adding a third possibility to the original two alternatives from Schechter's results. The theory is applied to positive solutions for $p$-Laplace equations.


Critical point; mountain pass lemma; positive solution; p-Laplace equation

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