Nonlinear elliptic equations and intrinsic potentials of Wolff type

Abstract

Abstract. We give necessary and sufficient conditions for the existence of weak solutions to the model equationin the case 0 < q < p − 1, where σ ≥ 0 is an arbitrary locally integrable function, or measure, and ∆pu = div(∇u|∇u| 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 equationwhere 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, ∇u), fractional Laplacians (−∆) α , or fully nonlinear k-Hessian operators.

Pointwise Estimates of Solutions and Existence Criteria for Sublinear Elliptic Equations

“…In this section, we discuss recent results established in [8] for entire solutions to quasilinear elliptic equations of the type…”

“…This approach is applicable to general quasilinear -Laplace operators of divergence type div ( , ∇ ) under standard boundedness and structural assumptions on , as well as to fully nonlinear -Hessian operators, and the fractional Laplacian equations (see [8]). …”

“…T. 20. № 3 23 As we emphasize in [8], such problems are governed by the important integral inequality…”

“…For entire solutions on Ω = R and = −σ ≥ 0, where σ is a locally finite positive Borel measure in R , there are more complicated matching upper and lower bounds given in terms of certain nonlinear potentials of Th. Wolff type [8][9][10]. Earlier pointwise estimates for bounded positive solutions are due to Brezis and Kamin [6].…”

“…The latter class includes all positive solutions 0 < < +∞ σ-a.e., and can be characterized either using certain localization techniques developed in [8] in the case = (−Δ) α 2 and Ω = R , or integral inequalities with weights as in [23] in the general case. These results are discussed in Sec.…”