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

Abstract

<p>Let α : ℝ → ℝ be a strictly increasing odd continuous function with lim_t→+∞ 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 ℝ^N, N ≥ 2, T[u, u] a nonnegative quadratic form involving the only generalized derivatives of order m of the function u ∈ W^m₀ E_A(Ω) and g_α : Ω x ℝ → ℝ, |α| < m, be Carathéodory functions.</p> <p>We study the problem</p> <pre>J_αu = ∑_|α|<m (-1)^|α| D^α g_α(x, D^αu) in Ω,<br> D^αu = 0 on ∂Ω, |α| ≤ m - 1,</pre> <p>where J_α is the duality mapping on (W^m₀ E_A(Ω), ∥ ⋅ ∥_m,A), subordinated to the gauge function α (given by (1.5)) and</p> <pre>∥u∥_m,A = ∥√T[u, u]∥_(A),</pre> <p>∥ ⋅ ∥_A being the Luxemburg norm on E_A(Ω).</p> <p>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].</p>