Existence and multiplicity of solutions to a p(x) -Laplacian equation with nonlinear boundary condition on unbounded domain

摘要

We study the existence and multiplicity of positive solutions for the nonlinear bound- ary value problems involving the p(x) -Laplacian of the form . div(a(x)| . u| p(x).2 . u)+b(x)|u| p(x).2 u = f (x,u) in Ω . R N , a(x)| . u| p(x).2 . u .ν = g(x,u) on Γ = .Ω , where Ω . R N is an unbounded domain with non-compact, smooth boundary Γ = . Ω , p ∈ C 0,1 ( Ω ) and 1 < p . p(x) p + < N , a,b are suitable weights. By using the variational methods, we prove that there exist multiple solutions provided f and g are given appropriate assumptions