On a nonlocal elliptic system of p-Kirchhoff-type under Neumann boundary condition

In this paper we investigate questions of existence of solution for the system {- [M-1 (integral(Omega) vertical bar del u vertical bar(p))](p-1) Delta(p)u = f(u, v) + rho(1)(x) in Omega, - [M-2 (integral(Omega) vertical bar del v vertical bar(p))](p-1) Delta(p)v = g(u, v) + rho(2)(x) in Omega, partial derivative u/partial derivative n = partial derivative v/partial derivative n - 0 on partial derivative Omega. Motivated by a problem in [ D. G. Costa, Topicos em analise funcional nao- linear e aplicaoes as equacoes diferenciais, VIII Escola Latino-Americana de Matematica, Rio de Janeiro, Brazil, 1986. [3]], who studies a single local equation, we study the above problem by using variational methods. Since we will work in the space W-1,W-p (Omega), the functional associated to the above problem will not be coercive. So, we have to consider the Poincare-Wirtinger's inequality in the subspace of W-1,W-p(Omega) formed by the functions with null mean in Omega. In this way, and motivated by physical motivations related to wave equation we consider the conditions (F-1)-(F-2). (C) 2008 Elsevier Ltd. All rights reserved.