On the Wolff potential and quasilinear elliptic equations involving measures. (English) Zbl 0860.35041

Annales Academiæ Scientiarum Fennicæ. Mathematica. Dissertationes. 104. Helsinki: Suomalainen Tiedeakatemia. 71 p. (1996).
The author studies weighted degenerated elliptic equations (1) \(-\text{div} A(x, \nabla u) = \mu\), where \(\mu\) is a nonnegative Radon measure, \(A(x,\xi) \cdot \xi \approx w(x) |\xi |^p\), \(1<p< \infty\), and the weight function is \(p\)-admissible in the terminology of J. Heinonen, T. Kilpeläinen, and O. Martio [Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford (1993; Zbl 0780.31001)] (for instance, Muckenhoupt’s \(A_p\) weights will do).
The main result is a substitute for the lack of Riesz representation theorem: if \(u\) is a nonnegative solution of (1) in the ball \(B(x,3r)\), pointwise properly defined, then the following estimate holds \[ c_1 W(x,r) \leq u(x) \leq c_2 \inf_{B(x,r)} u+ c_3 W(x,2r); \] here \(c_j\)’s are constants not depending on \(u\), \(\mu\) or \(r\), and \[ W(x,r) = \int_0^r \left(t^p {\mu\bigl(B(x,t)\bigr) \over w \bigl(B(x,t) \bigr)} \right)^{1/(p-1)} {dt \over t} \] is the weighted version of the Wolff potential. This estimate generalizes that of T. Kilpeläinen and J. Malý [Acta Math. 172, No. 1, 137-161 (1994; Zbl 0820.35063)], and it is used to prove the necessity of Wiener’s criterion for the regularity of boundary points for the Dirichlet problem. Other applications of the potential estimate are also given. Moreover, uniqueness and weaker solutions of (1) are also discussed.


35J70 Degenerate elliptic equations
31C15 Potentials and capacities on other spaces
31C45 Other generalizations (nonlinear potential theory, etc.)
35J67 Boundary values of solutions to elliptic equations and elliptic systems