Nonlinear elliptic equations and intrinsic potentials of Wolff type

We give necessary and sufficient conditions for the existence of weak solutions to the model equation -Delta(p)u = sigma u(q) on R-n in the case 0 < q < p - 1, where sigma >= 0 is an arbitrary locally integrable function, or measure, and Delta(p)u = div(del u vertical bar del u vertical bar(p-2)) is the p-Laplacian. Sharp global pointwise estimates and regularity properties of solutions are obtained as well. As a consequence, we characterize the solvability of the equation -Delta(p)v = b vertical bar del v vertical bar(p)/v + sigma on R-n, where b > 0. These results are new even in the classical case p = 2. Our approach is based on the use of special nonlinear potentials of Wolff type adapted for "sublinear" problems, and related integral inequalities. It allows us to treat simultaneously several problems of this type, such as equations with general quasilinear operators div A(x, del u), fractional Lapincians (-Delta)(alpha), or fully nonlinear k-Hessian operators. (C) 2016 Elsevier Inc. All rights reserved.