<p>We consider the non-uniformly elliptic problems with <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1132_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="TEX">$(L^{p(\cdot )}, L^{q(\cdot )})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msup> <mi>L</mi> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> <msup> <mi>L</mi> <mrow> <mi>q</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-logarithmic growth, where the variable exponents <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1132_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="115" /> </InlineMediaObject> <EquationSource Format="TEX">$1<p(x)\le q(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> satisfy the strong log-Hölder continuity and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1132_Article_IEq5.gif" Format="GIF" Height="28" Rendition="HTML" Resolution="72" Type="Linedraw" Width="87" /> </InlineMediaObject> <EquationSource Format="TEX">$\frac{q(x)}{p(x)}\le 1+\frac{\alpha }{n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mrow> <mi>q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> <mo>≤</mo> <mn>1</mn> <mo>+</mo> <mfrac> <mi>α</mi> <mi>n</mi> </mfrac> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1132_Article_IEq6.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="214" /> </InlineMediaObject> <EquationSource Format="TEX">$0\le a(x) \in C^{0,\,\alpha }(\Omega ), \alpha \in (0,1]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>≤</mo> <mi>a</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msup> <mi>C</mi> <mrow> <mn>0</mn> <mo>,</mo> <mspace width="0.166667em" /> <mi>α</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mi>α</mi> <mo>∈</mo> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Under some sharp conditions on the nonlinear operators, we prove its local Calderón-Zygmund type estimate in the setting of Lorentz spaces.</p>

Regularity for non-uniformly elliptic problems with $(L^{p(\cdot )}, L^{q(\cdot )})$-logarithmic growth

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发表日期：

2025年9月24日

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Shuang Liang,
Shenzhou Zheng

摘要

We consider the non-uniformly elliptic problems with $(L^{p(\cdot )}, L^{q(\cdot )})$ $(L^{p (\cdot)}, L^{q (\cdot)})$ -logarithmic growth, where the variable exponents $1<p(x)\le q(x)$ $1 < p (x) \leq q (x)$ satisfy the strong log-Hölder continuity and $\frac{q(x)}{p(x)}\le 1+\frac{\alpha }{n}$ $\frac{q (x)}{p (x)} \leq 1 + \frac{α}{n}$ for $0\le a(x) \in C^{0,\,\alpha }(\Omega ), \alpha \in (0,1]$ $0 \leq a (x) \in C^{0, α} (Ω), α \in (0, 1]$ . Under some sharp conditions on the nonlinear operators, we prove its local Calderón-Zygmund type estimate in the setting of Lorentz spaces.

Regularity for non-uniformly elliptic problems with \((L^{p(\cdot )}, L^{q(\cdot )})\)-logarithmic growth

摘要