In this paper we develop the p-thinness and the p-fine topology for the asymptotic behavior of p-superharmonic functions at singular points. We consider these as extensions of earlier works on superharmonic functions in dimension 2 in [3], on the Riesz and Log potentials in higher dimensions in [23, Section 2.5 and 2.6], and on p-harmonic functions in [13]. It is remarkable that, contrary to the above cases, the p-thinness for the singular behavior (cf. Definition 2.3.1) differs from the p-thinness for continuity by the Wiener criterion for p-superharmonic functions. As applications of asymptotic estimates of p-superharmonic functions, we also obtain asymptotic estimates of solutions to a class of fully nonlinear elliptic equations, which go beyond [17, Theorem 3.6]. This paper grows out of our recent papers on the potential theory in conformal geometry [19–22].
