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

Main Article Content

Jenica Cringanu

Abstract

The purpose of this paper is to show the existence results for the following abstract equation Jpu = Nfu,
where Jp is the duality application on a real reflexive and smooth X Banach space, that corresponds to the gauge function φ(t) = tp-1, 1 < p < ∞. We assume that X is compactly imbedded in Lq(Ω), where Ω is a bounded domain in RN, N ≥ 2, 1 < q < p∗, p∗ is the Sobolev conjugate exponent.
Nf : Lq(Ω) → Lq′(Ω), 1/q + 1/q′ = 1, is the Nemytskii operator that Caratheodory function generated by a f : Ω × R → R which satisfies some growth conditions. We use topological methods (via Leray-Schauder degree), critical points methods (the Mountain Pass theorem) and a direct variational method to prove the existence of the solutions for the equation Jpu = Nfu.

Keywords:
Duality mapping, leray-schauder degree, mountain pass theorem, p-Laplacian.

Article Details

How to Cite
Cringanu, J. (2020). Variational and Topological Methods for a Class of Nonlinear Equations which Involves a Duality Mapping. Asian Research Journal of Mathematics, 16(10), 56-71. https://doi.org/10.9734/arjom/2020/v16i1030230
Section
Original Research Article