Uniqueness of Positive Solutions for m-Laplacian Equations with Polynomial Non-linearity

Abstract

The author considers the uniqueness of the following positive solutions of m-Laplacian equation:

*

\{\begin{array}{ll}-{\mathrm{\Delta }}_{m}u=\lambda u^{m-1}+u^{p-1}& \mathrm{i}\mathrm{n}\mathrm{\Omega },\\ u=0& \mathrm{o}\mathrm{n}\mathrm{\partial }\mathrm{\Omega },\end{array}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{cases}-\Delta_{m}u=\lambda u^{m-1}+u^{p-1} & {\rm in} \; \Omega, \\u=0 & {\rm on} \; \partial\Omega,\end{cases}$$\end{document}

where m > 1 is a constant. When p → m, the uniqueness of positive solutions of (*) is shown which is based on the essential uniqueness of first eigenfunction for m-Laplacian equation. Futhermore, it is proved that the uniqueness results when (*) is a perturbation of Laplacian equation.

Keywords

m-Laplacian equation / Uniqueness

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