<p>We introduce a new minimax principle to prove the existence of multi-peak solutions to the Neumann problem of the <i>p</i>-Laplace equation <Equation ID="Equ80"> <EquationSource Format="TEX">$ -\varepsilon ^p \Delta _p u = u^{q-1} - u^{p-1} \ \ \text {in}\ \Omega , $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mo>-</mo> <msup> <mi>ε</mi> <mi>p</mi> </msup> <msub> <mi mathvariant="normal">Δ</mi> <mi>p</mi> </msub> <mi>u</mi> <mo>=</mo> <msup> <mi>u</mi> <mrow> <mi>q</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>-</mo> <msup> <mi>u</mi> <mrow> <mi>p</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mspace width="4pt" /> <mspace width="4pt" /> <mtext>in</mtext> <mspace width="4pt" /> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> is a bounded domain in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$C^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-boundary, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$1<p<n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$p<q< \frac{np}{n-p}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo><</mo> <mi>q</mi> <mo><</mo> <mfrac> <mrow> <mi mathvariant="italic">np</mi> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mi>p</mi> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation>. The minimax principle will be applied to the set of peak functions, which is a subset of the Sobolev space <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$W^{1,p} (\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>W</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>p</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. The argument is based on a combination of variational methods, topological degree theory, and gradient flow.</p>

A New Minimax Principle and Application to the p-Laplace Equation

期刊论文
发表日期：

2026年5月31日

查看全文

Xu-Jia Wang,
Xinyue Zhang

摘要

We introduce a new minimax principle to prove the existence of multi-peak solutions to the Neumann problem of the p-Laplace equation $ -\varepsilon ^p \Delta _p u = u^{q-1} - u^{p-1} \ \ \text {in}\ \Omega , $ $- ε^{p} Δ_{p} u = u^{q - 1} - u^{p - 1} in Ω,$ where $\Omega $ $Ω$ is a bounded domain in $\mathbb {R}^n$ $R^{n}$ with $C^2$ $C^{2}$ -boundary, $1<p<n$ $1 < p < n$ and $p<q< \frac{np}{n-p}$ $p < q < \frac{np}{n - p}$ . The minimax principle will be applied to the set of peak functions, which is a subset of the Sobolev space $W^{1,p} (\Omega )$ $W^{1, p} (Ω)$ . The argument is based on a combination of variational methods, topological degree theory, and gradient flow.