The Robin eigenvalue problem for the p(x)-Laplacian as p
- Farhod Abdullayev, Marian Bocea
Abstract. We study the Dirichlet problem − div(|∇u|∇u) = 0 in Ω, with u = f on ∂Ω and p(x) = ∞ in D, a subdomain of the reference domain Ω. The main issue is to give a proper sense to what a solution is. To this end, we consider the limit as n → ∞ of the solutions un to the corresponding problem when pn(x) = p(x)∧ n, in particular, with p = n in D. Under suitable assumptions on the data, we find that such a limit exists and that it can be characterized as the unique solution of a variational minimization problem. Moreover, we examine this limit in the viscosity sense and find an equation it satisfies.