Uniqueness results for equations of the p(x)-Laplacian type

Abstract

We study uniqueness of weak solutions of elliptic equations of the type
- div (a(x,u,∇u))+b(x,u)=f(x)
in a bounded domain Ω⊂ℝ n with Lipschitz boundary γ=∂Ω. We consider in particular mixed boundary conditions - i.e. Dirichlet condition on one part of the boundary and Neumann condition on the other part. A model equation could be
- div (a(x,u)|∇u|p(x)-2∇u)+b(x,u)=f(x),x∈Ω·
We establish uniqueness results for these equations. We also indicate conditions which guarantee existence of solution.