Comparison estimates for the Green function and the Martin kernel

A comparison estimate for the product of the Green function and the Martin kernel is given in a uniform domain. As its application, we show the equivalence of ordinary thinness and minimal thinness of a set contained in a non-tangential cone. We also give a comparison estimate for the Martin kernels with distinct singularities

Martin boundary points of cones generated by spherical John regions

We study Martin boundary points of cones generated by spherical John regions. In particular, we show that such a cone has a unique (minimal) Martin boundary point at the vertex, and also at infinity. We also study a relation between ordinary thinness and minimal thinness, and the boundary behavior of positive superharmonic functions

The boundary growth of superharmonic functions and positive solutions of nonlinear elliptic equations

We investigate the boundary growth of positive superharmonic functions on a bounded domain satisfying a certain nonlinear elliptic inequality. The result is applied to show the Harnack inequality for such superharmonic functions. Also, we study the existence of positive solutions, with singularity on the boundary, of elliptic equations with nonlinear term conditioned by the generalized Kato class

Sharp estimates for the Green function, 3G inequalities, and nonlinear Schr\"{o}dinger problems in uniform cones

We find and prove sharp estimates for the Green function and 3G inequalities in uniform cones. Estimates are applied to give equivalent conditions for measures to satisfy the generalized Cranston-McConnell inequality, and to show the existence of infinitely many continuous solutions to nonlinear Schr\"{o}dinger problems

DOUBLING CONDITIONS FOR HARMONIC MEASURE IN JOHN DOMAINS

We introduce new classes of domains, i.e., semi-uniform domains and inner emi-uniform domains. Both of them are intermediate between the class of John domains nd the class of uniform domains. Under the capacity density condition, we show that the armonic measure of a John domain D satisfies certain doubling conditions if and only if is a semi-uniform domain or an inner semi-uniform domain

The Dirichlet problem for sublinear elliptic equations with source

We present a necessary and sufficient condition on nonnegative Radon measures $\mu$ and $\nu$ for the existence of a positive continuous solution of the Dirichlet problem for the sublinear elliptic equation $-\Delta u=\mu u^q+\nu$ with prescribed nonnegative continuous boundary data in a general domain. Moreover, two-sided pointwise estimates of Brezis-Kamin type for positive bounded solutions and the uniqueness of a positive continuous $L^q$-solution are investigated.Comment: 14 page