Variational and topological methods for operator equations involving duality mappings on Orlicz-Sobolev spaces

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].