<p>In this paper we consider a doubly critical nonlinear elliptic problem with Neumann boundary conditions. The existence of blow-up solutions for this problem is related to the blow-up analysis of the classical geometric problem of prescribing negative scalar curvature <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1042_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="61" /> </InlineMediaObject> <EquationSource Format="TEX">$K=-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo>=</mo> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> on a domain of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1042_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> and mean curvature <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1042_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="154" /> </InlineMediaObject> <EquationSource Format="TEX">$H=D(n(n-1))^{-1/2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>H</mi> <mo>=</mo> <mi>D</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>, for some constant <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1042_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">$D>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>D</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, on its boundary, via a conformal change of the metric. Assuming that <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1042_Article_IEq5.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$n\ge 6$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>6</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1042_Article_IEq6.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="166" /> </InlineMediaObject> <EquationSource Format="TEX">$D>\sqrt{(n+1)/(n-1)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>D</mi> <mo>></mo> <msqrt> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msqrt> </mrow> </math></EquationSource> </InlineEquation>, we establish the existence of a positive solution which concentrates around an elliptic boundary point which is a stable critical point of the original mean curvature.</p>

The role of the boundary in the existence of blow-up solutions for a doubly critical elliptic problem

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发表日期：

2025年3月13日

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Sergio Cruz Blázquez

摘要

In this paper we consider a doubly critical nonlinear elliptic problem with Neumann boundary conditions. The existence of blow-up solutions for this problem is related to the blow-up analysis of the classical geometric problem of prescribing negative scalar curvature $K=-1$ $K = - 1$ on a domain of $\mathbb {R}^n$ $R^{n}$ and mean curvature $H=D(n(n-1))^{-1/2}$ $H = D {(n (n - 1))}^{- 1 / 2}$ , for some constant $D>1$ $D > 1$ , on its boundary, via a conformal change of the metric. Assuming that $n\ge 6$ $n \geq 6$ and $D>\sqrt{(n+1)/(n-1)}$ $D > \sqrt{(n + 1) / (n - 1)}$ , we establish the existence of a positive solution which concentrates around an elliptic boundary point which is a stable critical point of the original mean curvature.