Variational and Topological Methods for a Class of Nonlinear Equations which Involves a Duality Mapp

The goal of this paper is to demonstrate the results of the existence of the following abstract equation Jpu = Nfu, where Jp is the application of duality on a real reflexive and smooth X Banach space, which corresponds to the function of the gauge φ(t) = tp-1, 1 < p < ∞. We assume that X is compactly embedded in Lq( Ω ), where Ω is an RN-bounded domain, N ≥2, 1 < q < p, p ⁎ is the exponent of Sobolev conjugate. Nf: Lq( Ω ) -> Lq'( Ω ), 1/q + 1/q' = 1, is the operator of Nemytskii that has a Caratheodory function created by a f: Ω× R -> R that satisfies certain conditions of growth. In order to demonstrate the existence of solutions for the equation Jpu = Nfu, we use topological methods (via Leray-Schauder degree), critical points methods (the Mountain Pass theorem) and a direct variation method.

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