The author considers the uniqueness of the following positive solutions of m-Laplacian equation: <Equation ID="Equ1"> <EquationNumber></EquationNumber> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11401_2025_21_Article_Equ1.gif" Format="GIF" Height="54" Rendition="HTML" Resolution="72" Type="Linedraw" Width="245" /> </MediaObject> <EquationSource Format="TEX">$\begin{cases}-\Delta_{m}u=\lambda u^{m-1}+u^{p-1} & {\rm in} \; \Omega, \\u=0 & {\rm on} \; \partial\Omega,\end{cases}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mo>{</mo> <mtable columnalign="left left" columnspacing="1em" displaystyle="false" rowspacing=".2em"> <mtr> <mtd> <mo>−</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow> <mi>m</mi> </mrow> </msub> <mi>u</mi> <mo>=</mo> <mi>λ</mi> <msup> <mi>u</mi> <mrow> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>u</mi> <mrow> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> <mtd> <mrow> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> <mspace width="thickmathspace" /> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>u</mi> <mo>=</mo> <mn>0</mn> </mtd> <mtd> <mrow> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> </mrow> <mspace width="thickmathspace" /> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" /> </mrow> </math></EquationSource> </Equation> 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.

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

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2025年5月1日

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Wei Ke

摘要

The author considers the uniqueness of the following positive solutions of m-Laplacian equation: * $\begin{cases}-\Delta_{m}u=\lambda u^{m-1}+u^{p-1} & {\rm in} \; \Omega, \\u=0 & {\rm on} \; \partial\Omega,\end{cases}$ ${\begin{cases} - Δ_{m} u = λ u^{m - 1} + u^{p - 1} & i n Ω, \\ u = 0 & o n \partial Ω, \end{cases}$ 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.