Existence and multiple solutions for a variable exponent system

In this paper, we study the following variable exponent system { -div(vertical bar del u vertical bar(p(x)-2)del u) + a(x) vertical bar u vertical bar(p(x)-2) u = f(x, u, v, lambda) in Omega, -div(vertical bar del v vertical bar(p(x)-2)del v) + b(x) vertical bar v vertical bar(p(x)-2) v = g(x, u, v, lambda) in Omega, B(u,v) = 0 under some various conditions, we present several results for the existence of multiple nontrivial solutions. Crown Copyright (C) 2010 Published by Elsevier Ltd. All rights reserved.