<p>We discuss the existence and regularity of solutions to the following Dirichlet problem:<Equation ID="Equ1"> <EquationNumber>1</EquationNumber> <EquationSource Format="TEX">$\begin{aligned} {\left\{ \begin{array}{ll} -\textrm{div}\left( \frac{Du}{(1+|u|)^{\theta }}\right) = -\textrm{div}\left( |u|^{\gamma }E(x)\right) +f(x) \qquad & \text{ in } \Omega ,\\ u (x) = 0 & \text{ on } \partial \Omega , \end{array}\right. } \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mrow> <mo>-</mo> <mtext>div</mtext> <mfenced close=")" open="("> <mfrac> <mrow> <mi mathvariant="italic">Du</mi> </mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> <mo stretchy="false">)</mo> </mrow> <mi>θ</mi> </msup> </mfrac> </mfenced> <mo>=</mo> <mo>-</mo> <mtext>div</mtext> <mfenced close=")" open="("> <msup> <mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mi>γ</mi> </msup> <mi>E</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfenced> <mo>+</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="2em" /> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mspace width="0.333333em" /> <mtext>in</mtext> <mspace width="0.333333em" /> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mspace width="0.333333em" /> <mtext>on</mtext> <mspace width="0.333333em" /> <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">$\theta ,\gamma >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mo>,</mo> <mi>γ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. An interesting feature of this problem is the interplay between the two nonlinearities, the degeneracy and the power nonlinearity.</p>

An elliptic equation with power nonlinearity and degenerate coercivity

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发表日期：

2025年5月10日

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Genival da Silva

摘要

We discuss the existence and regularity of solutions to the following Dirichlet problem: 1 $\begin{aligned} {\left\{ \begin{array}{ll} -\textrm{div}\left( \frac{Du}{(1+|u|)^{\theta }}\right) = -\textrm{div}\left( |u|^{\gamma }E(x)\right) +f(x) \qquad & \text{ in } \Omega ,\\ u (x) = 0 & \text{ on } \partial \Omega , \end{array}\right. } \end{aligned}$ $\begin{matrix} \{\begin{matrix} - div (\frac{Du}{{(1 + | u |)}^{θ}}) = - div ({| u |}^{γ}, E, (x)) + f (x) & in Ω, \\ u (x) = 0 & on \partial Ω, \end{matrix}) \end{matrix}$ where $\theta ,\gamma >0$ $θ, γ > 0$ . An interesting feature of this problem is the interplay between the two nonlinearities, the degeneracy and the power nonlinearity.