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

Estimates on the Green’s Function and Existence of Positive Solutions of Nonlinear Singular... 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 $$K^{\infty}(\mathbb{R}^{n}_{+})$$ . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

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

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Publisher
Kluwer Academic Publishers
Copyright
Copyright © 2005 by Springer
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-003-2713-9
Publisher site
See Article on Publisher Site

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 $$K^{\infty}(\mathbb{R}^{n}_{+})$$ .

Journal

PositivitySpringer Journals

Published: Feb 12, 2003

References

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