Existence and uniqueness for nonlinear elliptic equations with lower-order terms

We study the nonlinear Dirichlet problem for the elliptic equation div(A(x,∇u)+b(x,u))=divF in a regular domain Ω⊂RN, N>2N>2, u=0u=0 on ∂Ω. In hypothesis of Lipschitz continuity and strong monotonicity of AA, we assume that the lower order term b(x,s)b(x,s) verify |b(x,s)−b(x,t)|⩽E(x)|s−t||b(x,s)−b(x,t)|⩽E(x)|s−t| for a.e. x∈Ω and for any s,t∈Rs,t∈R, where EE is a non negative function in the Lorentz space LN,q(Ω), N⩽q⩽+∞N⩽q⩽+∞. Without any control on the norm of EE, with F∈LpF∈Lp, p>1p>1 and q<∞q<∞, we obtain existence and uniqueness result for distributional solutions u∈W1,p(Ω) whenever pp is close to two. For q=∞q=∞, uniqueness results are obtained.The main difficulty to solve the problem is due to noncoercivity of the vector field A(x,s,ξ)=A(x,ξ)+b(x,s)A(x,s,ξ)=A(x,ξ)+b(x,s). Moreover, no classical structure conditions are satisfied.