We establish a new 3G-Theorem for the Green’s function for the half space
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\begin{document}$$\mathbb{R}^{n}_{+} := \{x = (x_{1},\ldots,x_{n}) \in \mathbb{R}^{n} : x_{n} > 0\}, (n \geq 3).$$\end{document} We exploit this result to introduce a new class of potentials \documentclass[12pt]{minimal}
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\begin{document}$$K(\mathbb{R}^{n}_{+})$$\end{document} that we characterize by means of the Gauss semigroup on \documentclass[12pt]{minimal}
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\begin{document}$$\mathbb{R}^{n}_{+}$$\end{document}. Next, we define a subclass \documentclass[12pt]{minimal}
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\begin{document}$$K^{\infty}(\mathbb{R}^{n}_{+})$$\end{document} of \documentclass[12pt]{minimal}
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\begin{document}$$K(\mathbb{R}^{n}_{+})$$\end{document} and we study it. In particular, we prove that
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\begin{document}$$K^{\infty}(\mathbb{R}^{n}_{+})$$\end{document} properly contains the classical Kato class
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\begin{document}$$K^\infty_n (\mathbb{R}^{n}_{+})$$\end{document}. Finally, we study the existence of positive continuous solutions in
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\begin{document}$$\mathbb{R}^{n}_{+}$$\end{document} of the following nonlinear elliptic problem
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\begin{document}$$\left\{\begin{array}{ll} \Delta u + h(., u) = 0,\\hbox{in}\mathbb{R}^{n}_{+} \ \hbox{(in the sense of distributions)},&\\ u|_{\partial\mathbb{R}^{n}_{+}} = 0, &\end{array}\right.$$\end{document} where h is a Borel measurable function in
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\begin{document}$$\mathbb{R}^{n}_{+} \times (0,\infty),$$\end{document} satisfying some appropriate conditions related to the class
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\begin{document}$$K^{\infty}(\mathbb{R}^{n}_{+})$$\end{document}.
