<p>We study the existence of positive solutions for the Berger plate equation with Navier boundary condition <Equation ID="Equ1"> <EquationNumber>P</EquationNumber> <EquationSource Format="TEX">$\begin{aligned} \left\{ \begin{array}{ll} \Delta ^2 u-m\Big (x, \int _\Omega |\nabla u|^2{\text {d}}x\Big )\Delta u =f(x,u,|\nabla u|,\Delta u), & x\in \Omega ,\\ u=\Delta u=0, & x\in \partial \Omega ,\\ \end{array} \right. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mrow> <msup> <mi mathvariant="normal">Δ</mi> <mn>2</mn> </msup> <mi>u</mi> <mo>-</mo> <mi>m</mi> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">(</mo> </mrow> <mi>x</mi> <mo>,</mo> <msub> <mo>∫</mo> <mi mathvariant="normal">Ω</mi> </msub> <msup> <mrow> <mo stretchy="false">|</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> <mtext>d</mtext> <mi>x</mi> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">)</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>u</mi> <mo>,</mo> <mo stretchy="false">|</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">|</mo> <mo>,</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <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> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mi>x</mi> <mo>∈</mo> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mtd> </mtr> </mtable> </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$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$N\in \mathbb {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation>, with a smooth boundary <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\partial \Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$m:\Omega \times \mathbb {R}^+ \rightarrow \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>:</mo> <mi mathvariant="normal">Ω</mi> <mo>×</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mo>+</mo> </msup> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> is a continuous function, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$f:\Omega \times \mathbb {R}\times \mathbb {R}^+\times \mathbb {R}\rightarrow \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <mi mathvariant="normal">Ω</mi> <mo>×</mo> <mi mathvariant="double-struck">R</mi> <mo>×</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mo>+</mo> </msup> <mo>×</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> is a Carathéodory function. The proofs of the main results are based on the topological degree and continuum theory.</p>

Positive Solutions for a Class of Berger Plate Equations

期刊论文
发表日期：

2025年12月22日

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Ruyun Ma,
Tingting Zhang,
Meng Yan

摘要

We study the existence of positive solutions for the Berger plate equation with Navier boundary condition P $\begin{aligned} \left\{ \begin{array}{ll} \Delta ^2 u-m\Big (x, \int _\Omega |\nabla u|^2{\text {d}}x\Big )\Delta u =f(x,u,|\nabla u|,\Delta u), & x\in \Omega ,\\ u=\Delta u=0, & x\in \partial \Omega ,\\ \end{array} \right. \end{aligned}$ $\begin{matrix} \{\begin{matrix} Δ^{2} u - m (x, \int_{Ω} {| \nabla u |}^{2} d x) Δ u = f (x, u, | \nabla u |, Δ u), & x \in Ω, \\ u = Δ u = 0, & x \in \partial Ω, \end{matrix}) \end{matrix}$ where $\Omega $ $Ω$ is a bounded domain in $\mathbb {R}^N$ $R^{N}$ , $N\in \mathbb {N}$ $N \in N$ , with a smooth boundary $\partial \Omega $ $\partial Ω$ , $m:\Omega \times \mathbb {R}^+ \rightarrow \mathbb {R}$ $m : Ω \times R^{+} \to R$ is a continuous function, $f:\Omega \times \mathbb {R}\times \mathbb {R}^+\times \mathbb {R}\rightarrow \mathbb {R}$ $f : Ω \times R \times R^{+} \times R \to R$ is a Carathéodory function. The proofs of the main results are based on the topological degree and continuum theory.