One solution to a fourth-order Leray–Lions problem involving <i>p</i> ( <i>x</i> )-Hardy potential
Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">D</m:mi> <m:mo>⊂</m:mo> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> </m:math> $\mathcal{D}\subset {\mathbb{R}}^{N}$ ( N ≥ 2) be a smooth bounded domain. Consider a class of fourth-order problems involving a p ( x )-Hardy potential of the form <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <m:mfenced close="" open="{"> <m:mrow> <m:mtable class="cases"> <m:mtr> <m:mtd columnalign="left"> <m:mi mathvariant="normal">Δ</m:mi> <m:mfenced close=")" open="("> <m:mrow> <m:mi>a</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mfenced> <m:mo>=</m:mo> <m:mi>μ</m:mi> <m:mi>θ</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mfrac> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>u</m:mi> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>δ</m:mi> <m:msup> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mi>p</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:msup> </m:mrow> </m:mfrac> <m:mo>+</m:mo> <m:mi>λ</m:mi> <m:mi>f</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>,</m:mo> </m:mtd> <m:mtd columnalign="left"> <m:mtext> </m:mtext> <m:mtext>in</m:mtext> <m:mspace width="0.3333em"/> <m:mi mathvariant="script">D</m:mi> <m:mo>,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="left"> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> </m:mtd> <m:mtd columnalign="left"> <m:mtext> </m:mtext> <m:mtext>on</m:mtext> <m:mspace width="0.3333em"/> <m:mi>∂</m:mi> <m:mi mathvariant="script">D</m:mi> <m:mo>,</m:mo> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:math> $$\begin{cases}\left(a\left(x,{\Delta}u\right)\right)=\mu \theta \left(x\right)\frac{\vert u{\vert }^{p\left(x\right)-2}u}{\delta {\left(x\right)}^{2p\left(x\right)}}+\lambda f\left(x,u\right),\hfill & \quad \text{in} \mathcal{D},\hfill \\ u={\Delta}u=0,\hfill & \quad \text{on} \partial \mathcal{D},\hfill \end{cases}$$ here <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="normal">Δ</m:mi> <m:mfenced close=")" open="("> <m:mrow> <m:mi>a</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>u</m:mi> </m:mrow>
Authors
- Anouar Bahrouni (ORCID: https://orcid.org/0000-0002-5465-4064)
- Nguyen Thanh Chung (ORCID: https://orcid.org/0000-0001-7345-620X)
- Khaled Kefi (ORCID: https://orcid.org/0000-0001-9277-5820)
Institutions
- Northern Border University (SA)
- University of Monastir (TN)
- University of Da Nang (VN)
Publication Details
- Journal
- Mathematica Slovaca
- Published
- 2026-07-24
- DOI
- https://doi.org/10.1515/ms-2026-0468
- Primary Topic
- Holomorphic and Operator Theory
- Type
- article
- Field-Weighted Citation Impact
- 0.00