<p>We study the existence and uniqueness of solutions for a fractional Laplacian problem with nonlinear singular terms having variable exponent <Equation ID="Equ10"> <EquationSource Format="TEX">$ (-\Delta )^s u + \gamma (x) = \frac{f(x)}{u^{\alpha (x)}}, \quad \text {in } \Omega . $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">)</mo> </mrow> <mi>s</mi> </msup> <mi>u</mi> <mo>+</mo> <mi>γ</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>u</mi> <mrow> <mi>α</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mfrac> <mo>,</mo> <mspace width="1em" /> <mtext>in</mtext> <mspace width="0.333333em" /> <mi mathvariant="normal">Ω</mi> <mo>.</mo> </mrow> </math></EquationSource> </Equation>We use the regularization method combined with Schauder’s fixed point theorem, as well as the notion of 2-fractional capacity to deal with aspects related to singular measures. Our work builds on and extends the previous works of [<CitationRef CitationID="CR2">2</CitationRef>, <CitationRef CitationID="CR11">11</CitationRef>, <CitationRef CitationID="CR18">18</CitationRef>], and [<CitationRef CitationID="CR20">20</CitationRef>] to the fractional framework with variable exponents. Here <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$H^s_0(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>H</mi> <mn>0</mn> <mi>s</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denotes the standard fractional Sobolev space obtained as the closure of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$C_0^\infty (\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>C</mi> <mn>0</mn> <mi>∞</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> under the Gagliardo seminorm.</p>

Fractional Laplacian problem with nonlinear singular terms having variable exponent

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发表日期：

2026年7月22日

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J. Igbida,
Y. El Ghali,
N. Elharrar,
A. Kaddar,
M. Moukhliss

摘要

We study the existence and uniqueness of solutions for a fractional Laplacian problem with nonlinear singular terms having variable exponent $ (-\Delta )^s u + \gamma (x) = \frac{f(x)}{u^{\alpha (x)}}, \quad \text {in } \Omega . $ ${(- Δ)}^{s} u + γ (x) = \frac{f (x)}{u^{α (x)}}, in Ω .$ We use the regularization method combined with Schauder’s fixed point theorem, as well as the notion of 2-fractional capacity to deal with aspects related to singular measures. Our work builds on and extends the previous works of [2, 11, 18], and [20] to the fractional framework with variable exponents. Here $H^s_0(\Omega )$ $H_{0}^{s} (Ω)$ denotes the standard fractional Sobolev space obtained as the closure of $C_0^\infty (\Omega )$ $C_{0}^{\infty} (Ω)$ under the Gagliardo seminorm.