Nonlinear Analysis 116 (2015) 100–111 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Stability for p-Laplace type equation in a borderline case Fernando Farroni, Luigi Greco, Gioconda Moscariello ∗ Dipartimento di Matematica e Applicazioni ‘‘R. Caccioppoli’’, Università degli Studi di Napoli ‘‘Federico II’’, Via Cintia – 80126 Napoli, Italy article info Article history: Received 17 December 2014 Accepted 23 December 2014 Communicated by Enzo Mitidieri MSC: 35J60 Keywords: p-Laplacian operator Stability of distributional solutions abstract We study the Dirichlet problem for a p-Laplacian type operator in the setting of the Orlicz– Zygmund space L q log −α L  Ω, R N  , q > 1 and α> 0. More precisely, our aim is to es- tablish under which assumptions on α> 0 existence and uniqueness of the solution are assured and to prove continuity and stability of the associated nonlinear operator. © 2014 Elsevier Ltd. All rights reserved. 1. Introduction Let Ω be a bounded Lipschitz domain of R N , N  2. We consider the Dirichlet problem  div A(x, ∇u) = div f in Ω, u = 0 on ∂ Ω, (1.1) where A : Ω × R N → R N is a Carathéodory vector field satisfying the following conditions for a.e. x ∈ Ω and all ξ,η ∈ R N A(x, 0) = 0 (1.2) ⟨A(x,ξ) − A(x, η), ξ − η⟩  a |ξ − η| 2 (|ξ |+|η|) p−2 (1.3) |A(x,ξ) − A(x, η)|  b |ξ − η| (|ξ |+|η|) p−2 (1.4) where p > 1, 0 < a  b. Let f =  f 1 , f 2 ,..., f N  be a vector field of class L s  Ω, R N  ,1  s  q where q is the conjugate exponent to p, i.e. pq = p + q. Definition 1.1. A function u ∈ W 1,r 0 (Ω), max{1, p − 1}  r  p, is a solution of (1.1) if  Ω ⟨A(x, ∇u), ∇ϕ⟩ dx =  Ω ⟨f , ∇ϕ⟩ dx, (1.5) for every ϕ ∈ C ∞ 0 (Ω). ∗ Corresponding author. E-mail addresses: fernando.farroni@unina.it (F. Farroni), luigreco@unina.it (L. Greco), gmoscari@unina.it (G. Moscariello). http://dx.doi.org/10.1016/j.na.2014.12.023 0362-546X/© 2014 Elsevier Ltd. All rights reserved.