Variational and topological methods for operator equations involving duality mappings on Orlicz-Sobolev spaces
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Date
2007-06-21
Authors
Dinca, George
Matei, Pavel
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University-San Marcos, Department of Mathematics
Abstract
Let α : ℝ → ℝ be a strictly increasing odd continuous function with limt→+∞ α(t) = +∞ and A(t) = ∫t0 α(s) ds, t ∈ ℝ, the N-function generated by α. Let Ω be a bounded open subset of ℝN, N ≥ 2, T[u, u] a nonnegative quadratic form involving the only generalized derivatives of order m of the function u ∈ Wm0 EA(Ω) and gα : Ω x ℝ → ℝ, |α| < m, be Carathéodory functions.
We study the problem
Jαu = ∑|α|<m (-1)|α| Dα gα(x, Dαu) in Ω,
Dαu = 0 on ∂Ω, |α| ≤ m - 1,
where Jα is the duality mapping on (Wm0 EA(Ω), ∥ ⋅ ∥m,A), subordinated to the gauge function α (given by (1.5)) and
∥u∥m,A = ∥√T[u, u]∥(A),
∥ ⋅ ∥A being the Luxemburg norm on EA(Ω).
By using the Leray-Schauder topological degree and the mountain pass theorem of Ambrosetti and Rabinowitz, the existence of nontrivial solutions is established. The results of this paper generalize the existence results for Dirichlet problems with p-Laplacian given in [12] and [13].
Description
Keywords
A priori estimate, Critical points, Orlicz-Sobolev spaces, Leray-Schauder topological degree, Duality mapping, Nemytskij operator, Mountain Pass Theorem
Citation
Dinca, G., & Matei, P. (2007). Variational and topological methods for operator equations involving duality mappings on Orlicz-Sobolev spaces. <i>Electronic Journal of Differential Equations, 2007</i>(93), pp. 1-47.
Rights
Attribution 4.0 International