<p>This manuscript is dedicated to the study of the nonlinear and singular anisotropic elliptic problem defined by <Equation ID="Equ107"> <EquationSource Format="TEX">$ {\left\{ \begin{array}{ll} \displaystyle Au+H(x, \nabla u)+g(x,u,\nabla u)=f& \text{ in } \Omega , \\ u=0 & \text{ on } \partial \Omega , \end{array}\right. } $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mi>A</mi> <mi>u</mi> <mo>+</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>u</mi> <mo>,</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> </mrow> </mstyle> </mtd> <mtd columnalign="left"> <mrow> <mspace width="0.333333em" /> <mtext>in</mtext> <mspace width="0.333333em" /> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mi>u</mi> <mo>=</mo> <mn>0</mn> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mspace width="0.333333em" /> <mtext>on</mtext> <mspace width="0.333333em" /> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\displaystyle Au=- \sum _{i=1}^{N} D^{i}a_{i}(x, u, \nabla u)+|u|^{p_{0}-2}u $</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mi>A</mi> <mi>u</mi> <mo>=</mo> <mo>-</mo> <munderover> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <msup> <mi>D</mi> <mi>i</mi> </msup> <msub> <mi>a</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>u</mi> <mo>,</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mi>u</mi> </mrow> </mstyle> </math></EquationSource> </InlineEquation>, the lower-order terms <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$g(x,s,\xi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>s</mi> <mo>,</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$H(x, \xi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>H</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> satisfy some growth conditions, and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$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>. We prove the existence of entropy solutions for this nonlinear and singular elliptic problem.</p>

STUDY OF SOME NONLINEAR ELLIPLIC PROBLEMS WITH SINGULAR TERMS

期刊论文
发表日期：

2026年4月16日

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Aymane El Janathi,
Hassane Hjiaj

摘要

This manuscript is dedicated to the study of the nonlinear and singular anisotropic elliptic problem defined by $ {\left\{ \begin{array}{ll} \displaystyle Au+H(x, \nabla u)+g(x,u,\nabla u)=f& \text{ in } \Omega , \\ u=0 & \text{ on } \partial \Omega , \end{array}\right. } $ $\{\begin{matrix} A u + H (x, \nabla u) + g (x, u, \nabla u) = f & in Ω, \\ u = 0 & on \partial Ω, \end{matrix})$ where $\displaystyle Au=- \sum _{i=1}^{N} D^{i}a_{i}(x, u, \nabla u)+|u|^{p_{0}-2}u $ $A u = - \sum_{i = 1}^{N} D^{i} a_{i} (x, u, \nabla u) + {| u |}^{p_{0} - 2} u$ , the lower-order terms $g(x,s,\xi )$ $g (x, s, ξ)$ and $H(x, \xi )$ $H (x, ξ)$ satisfy some growth conditions, and $f\in L^{1}(\Omega )$ $f \in L^{1} (Ω)$ . We prove the existence of entropy solutions for this nonlinear and singular elliptic problem.