A class of quasilinear and noncoercive elliptic problem in a two-component domain with L1 data

In this paper, we will study the following class of non-coercive quasilinear elliptic equations {-div(a(x, u(1), del u(1))) + g(x, u(1), del u(1)) = f (x) in Omega(1), -div(a(x, u(2), del u(2))) + g(x, u(2), del u(2)) = f (x) in Omega(2), u = 0 on partial derivative Omega, a(x, u(1), del u(1)) center dot nu(1) = a(x, u(2), del u(2)) center dot nu(1) on Gamma, a(x, u(1), del u(1)) center dot nu(1) = -h(x)vertical bar u(1) - u(2)vertical bar(p-2)(u(1) - u(2)) on Gamma, where Omega is a connected bounded open set of R-N (N = 2), and can be decomposed as Omega = Omega(1) boolean OR Omega(2) (boolean OR Gamma) under bar, where Omega(2) is an open subset included in Omega, and Omega(1) = Omega/((Omega(2)) over bar boolean AND Omega) and Gamma = Omega boolean AND Omega(1) boolean AND Omega(2), also we assume that f epsilon L-1(Omega). We will show the existence of renormalized solutions for this class of equation and we conclude some regularity results.