<p>This paper is devoted to study the following strongly nonlinear elliptic problem <Equation ID="Equ69"> <EquationSource Format="TEX">$\begin{aligned} \left\{ \begin{array}{ll} - \sum _{i=1}^{N} D^{i}a_{i}(x, u, \nabla u)+ g(x,u,\nabla u)=f & \quad \text{ in } \Omega , \\ u=0 & \quad \text{ on } \partial \Omega , \end{array}\right. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mrow> <mo>-</mo> <msubsup> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </msubsup> <msup> <mi>D</mi> <mi>i</mi> </msup> <msub> <mi>a</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>u</mi> <mo>,</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>u</mi> <mo>,</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>f</mi> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mspace width="1em" /> <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>=</mo> <mn>0</mn> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mspace width="1em" /> <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>in a suitable anisotropic Sobolev space, where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$f \in L^{1}(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <msup> <mi>L</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and the lower order term <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$g(x,s,\xi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>s</mi> <mo>,</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> satisfies some singular growth condition. We show the existence of renormalized solutions for this elliptic equation. Moreover, we will provide some regularity results.</p>

Renormalized Solutions for Singular Elliptic Problems with Degenerate Coercivity

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发表日期：

2025年11月28日

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Aymane El Janathi,
Hassane Hjiaj

摘要

This paper is devoted to study the following strongly nonlinear elliptic problem $\begin{aligned} \left\{ \begin{array}{ll} - \sum _{i=1}^{N} D^{i}a_{i}(x, u, \nabla u)+ g(x,u,\nabla u)=f & \quad \text{ in } \Omega , \\ u=0 & \quad \text{ on } \partial \Omega , \end{array}\right. \end{aligned}$ $\begin{matrix} \{\begin{matrix} - \sum_{i = 1}^{N} D^{i} a_{i} (x, u, \nabla u) + g (x, u, \nabla u) = f & in Ω, \\ u = 0 & on \partial Ω, \end{matrix}) \end{matrix}$ in a suitable anisotropic Sobolev space, where $f \in L^{1}(\Omega )$ $f \in L^{1} (Ω)$ and the lower order term $g(x,s,\xi )$ $g (x, s, ξ)$ satisfies some singular growth condition. We show the existence of renormalized solutions for this elliptic equation. Moreover, we will provide some regularity results.