<p>This paper is concerned with the following nonlinear elliptic problem with critical exponent <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(P_\varepsilon )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mi>ε</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>: <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Delta ^{2} u= (1+\varepsilon K(x))|u|^{(8/(n-4))}u$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi mathvariant="normal">Δ</mi> <mn>2</mn> </msup> <mi>u</mi> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ε</mi> <mi>K</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <msup> <mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>8</mn> <mo stretchy="false">/</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>-</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </msup> <mi>u</mi> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ \Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Delta u=u= 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>=</mo> <mi>u</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\partial \Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> is a bounded smooth domain in <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$n\ge 5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\varepsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation> is a small positive parameter. We prove the existence of sign-changing solution in higher dimensions: to this aim we develop a general finite-dimensional reduction procedure for perturbed variational functionals.</p>

Sign-changing solutions for a perturbed biharmonic equation with critical exponent

期刊论文
发表日期：

2025年10月10日

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Rabeh Ghoudi,
Moufida Lahrach

摘要

This paper is concerned with the following nonlinear elliptic problem with critical exponent $(P_\varepsilon )$ $(P_{ε})$ : $\Delta ^{2} u= (1+\varepsilon K(x))|u|^{(8/(n-4))}u$ $Δ^{2} u = (1 + ε K (x)) {| u |}^{(8 / (n - 4))} u$ in $ \Omega $ $Ω$ , $\Delta u=u= 0$ $Δ u = u = 0$ on $\partial \Omega $ $\partial Ω$ , where $\Omega $ $Ω$ is a bounded smooth domain in $\mathbb {R}^n$ $R^{n}$ , $n\ge 5$ $n \geq 5$ , $\varepsilon $ $ε$ is a small positive parameter. We prove the existence of sign-changing solution in higher dimensions: to this aim we develop a general finite-dimensional reduction procedure for perturbed variational functionals.