Quasilinear Riccati Type Equations with Super-Critical Exponents

Abstract

We establish explicit criteria of solvability for the quasilinear Riccati type equation − Δ p u = |∇u| q + ω in a bounded 𝒞1 domain Ω ⊂ ℝ n , n ≥ 2. Here Δ p , p > 1, is the p-Laplacian, q is in the supper critical range q > p, and the datum ω is a measure. Our existence criteria are given in the form of potential theoretic or geometric estimates that are sharp when ω is nonnegative and compactly supported in Ω. Our existence results are new even in the case dω =f dx where f belongs to the weak Lebesgue space . Moreover, our methods allow the treatment of more general equations where the principal operators may have discontinuous coefficients. As a consequence of the solvability results, a characterization of removable singularities for the corresponding homogeneous equation is also obtained.