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<p>We establish a sufficient condition for the removability of isolated singularities for solutions to the <i>p</i>(<i>x</i>)-Laplacian equation <Equation ID="Equ19"> <EquationSource Format="TEX">\( \textrm{div}\big (|\nabla u|^{p(x)-2}\,\nabla u \big )=0 \)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msup> <mrow> <mtext>div</mtext> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mo stretchy="false">|</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mspace width="0.166667em" /> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </Equation>under the following conditions on the variable exponent function: <Equation ID="Equ20"> <EquationSource Format="TEX">\( \underset{B_{r}(x_0)}{\text {osc}} p(x)\leqslant \frac{L}{\log \frac{1}{r}},\qquad 0&lt;r&lt;\frac{1}{2}, \qquad 0&lt;L&lt;\infty , \)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <munder> <mtext>osc</mtext> <mrow> <msub> <mi>B</mi> <mi>r</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </munder> <mi>p</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>⩽</mo> <mfrac> <mi>L</mi> <mrow> <mo>log</mo> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> </mfrac> <mo>,</mo> <mspace width="2em" /> <mn>0</mn> <mo>&lt;</mo> <mi>r</mi> <mo>&lt;</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>,</mo> <mspace width="2em" /> <mn>0</mn> <mo>&lt;</mo> <mi>L</mi> <mo>&lt;</mo> <mi>∞</mi> <mo>,</mo> </mrow> </math></EquationSource> </Equation><Equation ID="Equ21"> <EquationSource Format="TEX">\( 1&lt; p^{-} \le p(x) \le p^{+} &lt; n. \)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mn>1</mn> <mo>&lt;</mo> <msup> <mi>p</mi> <mo>-</mo> </msup> <mo>≤</mo> <mi>p</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <msup> <mi>p</mi> <mo>+</mo> </msup> <mo>&lt;</mo> <mi>n</mi> <mo>.</mo> </mrow> </math></EquationSource> </Equation></p>

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ON THE REMOVABILITY OF ISOLATED SINGULARITIES FOR THE p(x)-LAPLACIAN EQUATION

  • 期刊论文

  • 发表日期：

    2025年11月1日

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  • Mariia Savchenko

摘要

We establish a sufficient condition for the removability of isolated singularities for solutions to the p(x)-Laplacian equation \( \textrm{div}\big (|\nabla u|^{p(x)-2}\,\nabla u \big )=0 \) div ( | ∇ u | p ( x ) - 2 ∇ u ) = 0 under the following conditions on the variable exponent function: \( \underset{B_{r}(x_0)}{\text {osc}} p(x)\leqslant \frac{L}{\log \frac{1}{r}},\qquad 0<r<\frac{1}{2}, \qquad 0<L<\infty , \) osc B r ( x 0 ) p ( x ) ⩽ L log 1 r , 0 < r < 1 2 , 0 < L < ∞ , \( 1< p^{-} \le p(x) \le p^{+} < n. \) 1 < p - ≤ p ( x ) ≤ p + < n .