Estimates on the Green’s Function and Existence of Positive Solutions of Nonlinear Singular Elliptic Equations in the Half Space

Abstract

We establish a new 3G-Theorem for the Green’s function for the half space \(\mathbb{R}^{n}_{+} := \{x = (x_{1},\ldots,x_{n}) \in \mathbb{R}^{n} : x_{n} > 0\}, (n \geq 3).\) We exploit this result to introduce a new class of potentials \(K(\mathbb{R}^{n}_{+})\) that we characterize by means of the Gauss semigroup on \(\mathbb{R}^{n}_{+}\). Next, we define a subclass \(K^{\infty}(\mathbb{R}^{n}_{+})\) of \(K(\mathbb{R}^{n}_{+})\) and we study it. In particular, we prove that \(K^{\infty}(\mathbb{R}^{n}_{+})\) properly contains the classical Kato class \(K^\infty_n (\mathbb{R}^{n}_{+})\). Finally, we study the existence of positive continuous solutions in \(\mathbb{R}^{n}_{+}\) of the following nonlinear elliptic problem $$\left\{\begin{array}{ll} \Delta u + h(., u) = 0,\\hbox{in}\mathbb{R}^{n}_{+} \ \hbox{(in the sense of distributions)},&\\ u|_{\partial\mathbb{R}^{n}_{+}} = 0, &\end{array}\right.$$ where h is a Borel measurable function in \(\mathbb{R}^{n}_{+} \times (0,\infty),\) satisfying some appropriate conditions related to the class \(K^{\infty}(\mathbb{R}^{n}_{+})\).