Existence and Uniqueness for a Class of Quasilinear Elliptic Systems with Kato-Type Lower-Order Terms

We study a quasilinear elliptic system λ u | u | p − 2 − div ( a ( x , ∇ u ) ) + b ( x , u ) = f ( x ) , in a bounded domain Ω ⊂ R l ( l > 2 , 2 ≤ p < l ), with homogeneous Dirichlet boundary conditions. The principal part a depends only on the gradient and satisfies strong monotonicity and growth conditions, while the lower-order term b depends only on u and is monotone; its growth is controlled by a coefficient in the classical Kato class. Using the Minty–Browder theorem for monotone operators, we prove existence and uniqueness of a weak solution in W 0 1 , p ( Ω ) N . The results cover the p -Laplacian system and allow singular potentials.