Nonlinear Analysis 194 (2020) 111377 Contents lists available at ScienceDirect Nonlinear Analysis www.elsevier.com/locate/na Maximum principles for some quasilinear elliptic systems Salvatore Leonardi a,∗ , Francesco Leonetti b , Cristina Pignotti b , Eugénio Rocha c , Vasile Staicu c a DMI — Department of Mathematics and Informatics, University of Catania, Viale A. Doria 6, 95125 Catania, Italy b DISIM — Department of Information Engineering, Computer Science and Mathematics, University of L’Aquila, 67100 L’Aquila, Italy c CIDMA — Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal article info Article history: Received 1 November 2018 Accepted 6 November 2018 Communicated by Enzo Mitidieri MSC: 35B50 35J47 35J62 49N60 Keywords: Elliptic system Maximum principle r-staircase support abstract We give maximum principles for solutions u : Ω → R N to a class of quasilinear elliptic systems whose prototype is − n ∑ i=1 ∂ ∂x i ( N ∑ β=1 n ∑ j=1 a α,β i,j (x, u(x)) ∂u β ∂x j (x) ⎜ =0, x ∈ Ω , where α ∈¶1,...,N ♢ is the equation index and Ω is an open, bounded subset of R n . We assume that coefficients a α,β i,j (x, y) are measurable with respect to x, continuous with respect to y ∈ R N , bounded and elliptic. In vectorial problems, when trying to bound the solution by means of the boundary data, we need to bypass De Giorgi’s counterexample by means of some additional structure assumptions on the coefficients a α,β i,j (x, y). In this paper, we assume that off-diagonal coefficients a α,β i,j , α ̸= β, have support in some staircase set along the diagonal in the y α ,y β plane. © 2018 Elsevier Ltd. All rights reserved. 1. Introduction We consider the system of N equations − n ∑ i=1 ∂ ∂x i ∏ ∐ N ∑ β=1 n ∑ j=1 a α,β i,j (x, u(x)) ∂u β ∂x j (x) ∫ ˆ =0, x ∈ Ω , (1.1) * Corresponding author. E-mail addresses: leonardi@dmi.unict.it (S. Leonardi), leonetti@univaq.it (F. Leonetti), pignotti@univaq.it (C. Pignotti), eugenio@ua.pt (E. Rocha), vasile@ua.pt (V. Staicu). https://doi.org/10.1016/j.na.2018.11.004 0362-546X/© 2018 Elsevier Ltd. All rights reserved.