Publications
2002
Issue 51

MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server end of 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint

51/2002

On a nonlinear elliptic equation arising in a free boundary problem

Guofang Wang and Dong Ye

Abstract

Let $p^{*} = n / (n - 2)$ and $n \geq 3$ . In this paper, we first classify all non-constant solutions of ${\begin{aligned} - Δ u & = u_{+}^{p^{*}} in R^{n}, \\ \int_{R^{n}} u_{+}^{p^{*}} d x & < \infty . \end{aligned}$ We then establish a $sup + inf$ and a Moser-Trudinger type inequalities for the equation $- Δ u = u_{+}^{p^{*}}$ . Our results illustrate that this equation is much closer to the Liouville problem $- Δ u = e^{u}$ in dimension two than the usual critical exponent equation, namely $- Δ u = u^{\frac{n + 2}{n - 2}}$ is.

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Received:: 26.06.02

Published:: 26.06.02

MSC Codes:: 35J60

Keywords:: free boundary problem, liouville equation

Related publications

inJournal

2003 Repository Open Access

Guofang Wang and Dong Ye

On a nonlinear elliptic equation arising in a free boundary problem

In: Mathematische Zeitschrift, 244 (2003) 3, pp. 531-548

BibTex DOI: 10.1007/s00209-003-0493-3