Abstract
Let Omega be a bounded domain in R-4 with smooth boundary and let x(1), x(2),..., x(m) be m-points in Omega. We are concerned with the problem
Delta(2)u - H(x, u, D(k)u) = rho(4)Pi(n)vertical bar x - p(i)vertical bar(4 alpha i) f(x)g(u),
where the principal term is the bi-Laplacian operator, H(x, u, D(k)u) is a functional which grows with respect to Du at most like vertical bar Du vertical bar(q), 1 <= q <= 4, f : Omega ->[0, +infinity[ is a smooth function satisfying f(p(i)) > 0 for any i = 1,..., n, alpha(i) are positives numbers and g : R -> [0,+infinity[satisfy vertical bar g(u)vertical bar <= ce(u). In this paper, we give sufficient conditions for existence of a family of positive weak solutions (u(rho))(rho>0) in Omega under Navier boundary conditions u = Delta u = 0 on partial derivative Omega. The solutions we constructed are singular as the parameters rho tends to 0, when the set of concentration S = {x(1),..., x(m)} subset of Omega and the set Lambda := {p(1),..., p(n)} subset of Omega are not necessarily disjoint. The proof is mainly based on nonlinear domain decomposition method.