<p>We discuss the existence and regularity of solutions to an elliptic system, whose basic example is <Equation ID="Equ16"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2025_840_Article_Equ16.gif" Format="GIF" Height="95" Rendition="HTML" Resolution="72" Type="Linedraw" Width="244" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} {\left\{ \begin{array}{ll} \,-\Delta u =\frac{f(x)\hspace{0.55542pt}v^{\theta } }{u^{\theta }} & \text{ in }\;\; \Omega ,\\ \,-\Delta v = g(x)(1+u)^{\gamma } & \text{ in }\;\; \Omega ,\\ \,u>0,\;v>0 & \text{ in } \;\;\Omega ,\\ \,u = 0,\; v=0 & \text{ on } \;\; \partial \Omega , \end{array}\right. } \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mrow> <mspace width="0.166667em" /> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>=</mo> <mfrac> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="0.55542pt" /> <msup> <mi>v</mi> <mi>θ</mi> </msup> </mrow> <msup> <mi>u</mi> <mi>θ</mi> </msup> </mfrac> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mspace width="0.333333em" /> <mtext>in</mtext> <mspace width="0.333333em" /> <mspace width="0.277778em" /> <mspace width="0.277778em" /> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mspace width="0.166667em" /> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>v</mi> <mo>=</mo> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mi>γ</mi> </msup> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mspace width="0.333333em" /> <mtext>in</mtext> <mspace width="0.333333em" /> <mspace width="0.277778em" /> <mspace width="0.277778em" /> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mspace width="0.166667em" /> <mi>u</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mspace width="0.277778em" /> <mi>v</mi> <mo>></mo> <mn>0</mn> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mspace width="0.333333em" /> <mtext>in</mtext> <mspace width="0.333333em" /> <mspace width="0.277778em" /> <mspace width="0.277778em" /> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mspace width="0.166667em" /> <mi>u</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="0.277778em" /> <mi>v</mi> <mo>=</mo> <mn>0</mn> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mspace width="0.333333em" /> <mtext>on</mtext> <mspace width="0.333333em" /> <mspace width="0.277778em" /> <mspace width="0.277778em" /> <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"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2025_840_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="141" /> </InlineMediaObject> <EquationSource Format="TEX">$0\leqslant f\leqslant g\in L^{m}(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>⩽</mo> <mi>f</mi> <mo>⩽</mo> <mi>g</mi> <mo>∈</mo> <msup> <mi>L</mi> <mi>m</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> are not identically zero, and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2025_840_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">$\theta ,\gamma >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mo>,</mo> <mi>γ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> are fixed constants.</p>

On an elliptic system with singular nonlinearity

期刊论文
发表日期：

2025年6月20日

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Genival da Silva

摘要

We discuss the existence and regularity of solutions to an elliptic system, whose basic example is $\begin{aligned} {\left\{ \begin{array}{ll} \,-\Delta u =\frac{f(x)\hspace{0.55542pt}v^{\theta } }{u^{\theta }} & \text{ in }\;\; \Omega ,\\ \,-\Delta v = g(x)(1+u)^{\gamma } & \text{ in }\;\; \Omega ,\\ \,u>0,\;v>0 & \text{ in } \;\;\Omega ,\\ \,u = 0,\; v=0 & \text{ on } \;\; \partial \Omega , \end{array}\right. } \end{aligned}$ $\begin{matrix} \{\begin{matrix} - Δ u = \frac{f (x) v^{θ}}{u^{θ}} & in Ω, \\ - Δ v = g (x) {(1 + u)}^{γ} & in Ω, \\ u > 0, v > 0 & in Ω, \\ u = 0, v = 0 & on \partial Ω, \end{matrix}) \end{matrix}$ where $0\leqslant f\leqslant g\in L^{m}(\Omega )$ $0 ⩽ f ⩽ g \in L^{m} (Ω)$ are not identically zero, and $\theta ,\gamma >0$ $θ, γ > 0$ are fixed constants.