We study the existence and optimal summability of solutions to the nonlinear elliptic equation <Equation ID="Equ33"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1025_Article_Equ33.gif" Format="GIF" Height="42" Rendition="HTML" Resolution="72" Type="Linedraw" Width="463" /> </MediaObject> <EquationSource Format="TEX">$ -\hbox {div}\,(a(x, u, \nabla u))=\frac{\lambda }{|x|^p}|u|^{p-2}u+f\quad \hbox { in }\;\Omega ,\qquad u=0\;\hbox { on }\;\partial \Omega , $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mo>-</mo> <mtext>div</mtext> <mspace width="0.166667em" /> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>u</mi> <mo>,</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mfrac> <mi>λ</mi> <msup> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> <mi>p</mi> </msup> </mfrac> <msup> <mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>p</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mi>u</mi> <mo>+</mo> <mi>f</mi> <mspace width="1em" /> <mspace width="0.333333em" /> <mtext>in</mtext> <mspace width="0.333333em" /> <mspace width="0.277778em" /> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mspace width="2em" /> <mi>u</mi> <mo>=</mo> <mn>0</mn> <mspace width="0.277778em" /> <mspace width="0.333333em" /> <mtext>on</mtext> <mspace width="0.333333em" /> <mspace width="0.277778em" /> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mrow> </math></EquationSource> </Equation>influenced by the value of the parameter <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1025_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation> and the summability of f. Here <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1025_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> is an open bounded domain in <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1025_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {R}^N$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation> containing the origin and a can be a nonlinear function.

Hardy potential in existence and optimal summability in nonlinear elliptic equations

期刊论文
发表日期：

2025年2月6日

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Juan C. Ortiz Chata

摘要

We study the existence and optimal summability of solutions to the nonlinear elliptic equation $ -\hbox {div}\,(a(x, u, \nabla u))=\frac{\lambda }{|x|^p}|u|^{p-2}u+f\quad \hbox { in }\;\Omega ,\qquad u=0\;\hbox { on }\;\partial \Omega , $ $- div (a (x, u, \nabla u)) = \frac{λ}{{| x |}^{p}} {| u |}^{p - 2} u + f in Ω, u = 0 on \partial Ω,$ influenced by the value of the parameter $\lambda $ $λ$ and the summability of f. Here $\Omega $ $Ω$ is an open bounded domain in $\mathbb {R}^N$ $R^{N}$ containing the origin and a can be a nonlinear function.