We prove the existence of weak solutions of the homogeneous Dirichlet problem associated with a degenerate elliptic equation whose model is <Equation ID="Equ37"> <EquationSource Format="TEX">$ -\textrm{div} \bigg (\frac{\nabla u}{(1+|u|)\log ^q (e+|u|)}\bigg )+u=f, \quad \text {in }\Omega $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mo>-</mo> <mtext>div</mtext> <mrow> <mo maxsize="2.047em" minsize="2.047em" stretchy="true">(</mo> </mrow> <mfrac> <mrow> <mi mathvariant="normal">∇</mi> <mi>u</mi> </mrow> <mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> <mo stretchy="false">)</mo> </mrow> <msup> <mo>log</mo> <mi>q</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>e</mi> <mo>+</mo> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> <mo stretchy="false">)</mo> </mrow> </mrow> </mfrac> <mrow> <mo maxsize="2.047em" minsize="2.047em" stretchy="true">)</mo> </mrow> <mo>+</mo> <mi>u</mi> <mo>=</mo> <mi>f</mi> <mo>,</mo> <mspace width="1em" /> <mtext>in</mtext> <mspace width="0.333333em" /> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Omega \subset \mathbb {R}^N$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$N \ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, is an open, bounded set, q is a positive number and f satisfies a suitable summability assumption. The existence result will be extended also to a nonlinear (with respect to <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\nabla u$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">∇</mi> <mi>u</mi> </mrow> </math></EquationSource> </InlineEquation>) degenerate elliptic equation.

A Dirichlet Problem with Degenerate Coercivity

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发表日期：

2026年4月13日

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Lucio Boccardo,
Giuseppa Rita Cirmi

摘要

We prove the existence of weak solutions of the homogeneous Dirichlet problem associated with a degenerate elliptic equation whose model is $ -\textrm{div} \bigg (\frac{\nabla u}{(1+|u|)\log ^q (e+|u|)}\bigg )+u=f, \quad \text {in }\Omega $ $- div (\frac{\nabla u}{(1 + | u |) {log}^{q} (e + | u |)}) + u = f, in Ω$ where $\Omega \subset \mathbb {R}^N$ $Ω \subset R^{N}$ , $N \ge 3$ $N \geq 3$ , is an open, bounded set, q is a positive number and f satisfies a suitable summability assumption. The existence result will be extended also to a nonlinear (with respect to $\nabla u$ $\nabla u$ ) degenerate elliptic equation.