Nonhomogeneous boundary value problems in anisotropic Sobolev spaces

We study the nonlinear boundary value problem − ∑ i = 1 N ( | u x i | p i ( x ) − 2 u x i ) x i = λ | u | q ( x ) − 2 u in Ω , u = 0 on ∂ Ω , where Ω ⊂ R N ( N ⩾ 3 ) is a bounded domain with smooth boundary, λ is a positive real number, and the continuous functions p i and q satisfy 2 ⩽ p i ( x ) < N and q ( x ) > 1 for any x ∈ Ω ¯ and any i ∈ { 1 , … , N } . By analyzing the growth of the functions p i and q we prove in this Note several existence results in Sobolev spaces with variable exponents.