Existence of renormalized solutions for some noncoercive elliptic problem in a two-component domain with L1 data

In this work, we will focus on studying a specific class of quasilinear elliptic equations with degenerate coercivity in a two-component domain, which is defined as follows: {-div(a(x,u(1),del u(1)))+K(x,del u(1))+lambda(x)|u(1)|(s-1)u(1)=f(x) in Omega(1), -div(a(x,u(2),del u(2)))+K(x,del u(2))+lambda(x)|u(2)|(s-1)u(2)=f(x) in Omega(2), u(1)=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)|u(1)-u(2)|(p-2)(u(1)-u(2)) on Gamma, where Omega is a bounded open set of R-N (N >= 2), with 2 = 2, U-2 Ur, where 2 is an open set such that 2C with a Lipschitz boundary I and N1 = N\N2, lambda(x) > lambda o, s >= 1, 0 <= 8 < 1 and L'(2). We show the existence of a renormalized solutions for this class of equation, and we will conclude some regularity results.