Studies on nonlinear partial differential equations by potential analysis

在有界光滑区域中，我们建立了满足一个非线性不等式的正超调和函数的一个合适的边界增长估计。此外，我们还证明了与泊松核相当的半线性椭圆型方程正解的存在性。给出了奇异半线性椭圆型方程Dirichlet边值问题正解存在的一个充分条件。在位势理论方面，我们研究了调和测度的倍增性质，格林函数与Martin核的乘积的估计，以及Martin核的边界性质。

In a bounded smooth domain, we established a suitable boundary growth estimate for positive superharmonic functions satisfying a nonlinear inequality. Moreover, we proved the existence of positive solutions of semilinear elliptic equations that are comparable to the Poisson kernel. Also, we gave a sufficient condition for Dirichlet boundary data to guarantee the existence of positive solutions of singular semilinear elliptic equations. In regard to potential theory, we studied a doubling property of harmonic measure, an estimate for the product of the Green function and the Martin kernel, and the boundary behavior of Martin kernels.