<p>We study the existence of positive solutions for the semipositone biharmonic equation with Navier boundary conditions <Equation ID="Equ1"> <EquationNumber>P</EquationNumber> <EquationSource Format="TEX">$\begin{aligned} \left\{ \begin{array}{ll} \Delta ^2 u=\lambda f(t,u),\ \ \ & t\in \Omega ,\\[2ex] u=\Delta u=0,\ \ \ & t\in \partial \Omega , \end{array} \right. \end{aligned}$</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>λ</mi> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="3.33333pt" /> <mspace width="3.33333pt" /> <mspace width="4pt" /> <mspace width="4pt" /> <mspace width="4pt" /> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mi>t</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mo stretchy="false">[</mo> <mn>2</mn> <mi>e</mi> <mi>x</mi> <mo stretchy="false">]</mo> <mi>u</mi> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="3.33333pt" /> <mspace width="3.33333pt" /> <mspace width="4pt" /> <mspace width="4pt" /> <mspace width="4pt" /> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mi>t</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">$\Omega \subset \mathbb {R}^n (n\ge 1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <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> is a smooth bounded domain, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\lambda >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$f:\Omega \times \mathbb {R^+}\rightarrow \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <mi mathvariant="normal">Ω</mi> <mo>×</mo> <msup> <mi mathvariant="double-struck">R</mi> <mo>+</mo> </msup> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> is a continuous function with <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$f(t,0)<0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo><</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathbb {R^+}=[0,\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi mathvariant="double-struck">R</mi> <mo>+</mo> </msup> <mo>=</mo> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We obtain existence results for positive solutions of problem (<i>P</i>) under different growth conditions. The proofs of the main results are based on bifurcation theory and degree theory combined with a rescaling argument.</p>

Existence of positive solutions for the semipositone biharmonic equation with Navier boundary conditions

期刊论文
发表日期：

2025年9月30日

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Meng Yan,
Tingting Zhang,
Ruyun Ma

摘要

We study the existence of positive solutions for the semipositone biharmonic equation with Navier boundary conditions P $\begin{aligned} \left\{ \begin{array}{ll} \Delta ^2 u=\lambda f(t,u),~~\ \ \ & t\in \Omega ,\\[2ex] u=\Delta u=0,~~\ \ \ & t\in \partial \Omega , \end{array} \right. \end{aligned}$ $\begin{matrix} \{\begin{matrix} Δ^{2} u = λ f (t, u), & t \in Ω, \\ [2 e x] u = Δ u = 0, & t \in \partial Ω, \end{matrix}) \end{matrix}$ where $\Omega \subset \mathbb {R}^n (n\ge 1)$ $Ω \subset R^{n} (n \geq 1)$ is a smooth bounded domain, $\lambda >0$ $λ > 0$ and $f:\Omega \times \mathbb {R^+}\rightarrow \mathbb {R}$ $f : Ω \times R^{+} \to R$ is a continuous function with $f(t,0)<0$ $f (t, 0) < 0$ in $\Omega $ $Ω$ , $\mathbb {R^+}=[0,\infty )$ $R^{+} = [0, \infty)$ . We obtain existence results for positive solutions of problem (P) under different growth conditions. The proofs of the main results are based on bifurcation theory and degree theory combined with a rescaling argument.