The existence of solutions for some classes of nonlinear elliptic equations and systems with sub-natural growth terms

Abstract

Global pointwise estimates are obtained for quasilinear Lane-Emden-type systems involv- ing measures in the “sublinear growth” rate. Necessary and sufficient conditions are presented for the existence of positive solutions to a class of systems of quasilinear elliptic equations involving measures in the “sublinear growth” rate expressed in terms of Wolff’s potential. Our approach is based on recent advances due to T. Kilpeläinen and J. Malý in the potential theory. Also, we are interested in a class of k-Hessian Lane-Emden type systems with measure data in the “sublinear growth” rate. We give global pointwise estimates of the so-called Brezis–Kamin type in terms of Wolff potentials, which allows us to obtain necessary and sufficient condi- tions for the existence of positive solutions. This method enables us to treat several kinds of problems, such as equations involving general quasilinear operators and fractional Laplacian, or fully nonlinear k-Hessian operators. Further, we present a sufficient condition in terms of Wolff potentials for the existence of a finite energy solution to measure data (p, q)-Laplacian equation in the “sublinear growth” rate. We prove that such a solution is minimal. Besides, we show a necessary condition in terms of a suitably generalized potential of the Wolff-type for the eventual solutions, not necessarily of finite energy. Our main tools are integral inequalities closely associated with (p, q)-Laplacian equations with measure data, and pointwise potential estimates which allow us to obtain bounds of solutions. This method enables us to treat other nonlinear elliptic problems involving general quasilinear operators.