We consider the equation <Equation ID="Equ29"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_907_Article_Equ29.gif" Format="GIF" Height="43" Rendition="HTML" Resolution="72" Type="Linedraw" Width="240" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} \left\{ \begin{gathered} \begin{array}{lll} \Delta ^{2}u+c\Delta u=\lambda f(x,u),\ x \in \Omega ,\\ u=\Delta u=0,\ x \in \partial \Omega , \end{array} \end{gathered} \right. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mrow> <msup> <mi mathvariant="normal">Δ</mi> <mn>2</mn> </msup> <mi>u</mi> <mo>+</mo> <mi>c</mi> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>=</mo> <mi>λ</mi> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="4pt" /> <mi>x</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mi>u</mi> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="4pt" /> <mi>x</mi> <mo>∈</mo> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_907_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$\Delta ^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="normal">Δ</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> denotes the biharmonic operator, c is a given constant, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_907_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\Omega$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> is a bounded domain in <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_907_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="73" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {R}^n (n \ge 1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>≥</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with smooth boundary <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_907_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$\partial \Omega$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_907_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda > 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> is a parameter. The nonlinearity f exhibits an oscillatory behavior. We establish the existence of multiple positive solutions, multiple negative solutions, and multiple sign-changing solutions, depending on <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_907_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda$</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>.

On nodal solutions for a class of fourth-order elliptic equations

期刊论文
发表日期：

2025年4月24日

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Abderrazek Benhassine,
Saida Farhani,
Taib Talbi

摘要

We consider the equation $\begin{aligned} \left\{ \begin{gathered} \begin{array}{lll} \Delta ^{2}u+c\Delta u=\lambda f(x,u),\ x \in \Omega ,\\ u=\Delta u=0,\ x \in \partial \Omega , \end{array} \end{gathered} \right. \end{aligned}$ $\begin{matrix} \{\begin{matrix} \begin{matrix} Δ^{2} u + c Δ u = λ f (x, u), x \in Ω, \\ u = Δ u = 0, x \in \partial Ω, \end{matrix} \end{matrix}) \end{matrix}$ where $\Delta ^2$ $Δ^{2}$ denotes the biharmonic operator, c is a given constant, $\Omega$ $Ω$ is a bounded domain in $\mathbb {R}^n (n \ge 1)$ $R^{n} (n \geq 1)$ with smooth boundary $\partial \Omega$ $\partial Ω$ , and $\lambda > 0$ $λ > 0$ is a parameter. The nonlinearity f exhibits an oscillatory behavior. We establish the existence of multiple positive solutions, multiple negative solutions, and multiple sign-changing solutions, depending on $\lambda$ $λ$ .