This paper investigates a class of nonlinear elliptic boundary value problems combining Hardy-type singular potentials, Leray–Lions operators with Orlicz growth, and logarithmic source terms. We study the Dirichlet problem <Equation ID="Equa"> <EquationSource Format="MATHML"><math> <mrow> <mo>{</mo> <mtable columnalign="right left" columnspacing="0.2em"> <mtr> <mtd> <mo>−</mo> <mi mathvariant="normal">div</mi> <mo maxsize="2.4ex" minsize="2.4ex" stretchy="true">(</mo> <mi mathvariant="script">A</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">)</mo> <mo maxsize="2.4ex" minsize="2.4ex" stretchy="true">)</mo> </mtd> <mtd> <mo>+</mo> <mi>α</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mfrac> <mrow> <mo stretchy="false">|</mo> <mi>u</mi> <msup> <mo stretchy="false">|</mo> <mrow> <mi>p</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mi>u</mi> </mrow> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <msup> <mo stretchy="false">|</mo> <mi>p</mi> </msup> </mrow> </mfrac> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> <mi>λ</mi> <mspace width="0.2em" /> <mi>a</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">|</mo> <mi>u</mi> <msup> <mo stretchy="false">|</mo> <mrow> <mi>p</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mi>u</mi> <mo>log</mo> <mo maxsize="2.4ex" minsize="2.4ex" stretchy="true">(</mo> <mn>1</mn> <mo>+</mo> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> <mo maxsize="2.4ex" minsize="2.4ex" stretchy="true">)</mo> <mo>+</mo> <mi>λ</mi> <mspace width="0.2em" /> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">|</mo> <mi>u</mi> <msup> <mo stretchy="false">|</mo> <mrow> <mi>r</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mi>u</mi> <mo>,</mo> <mspace width="1em" /> <mtext>in </mtext> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>u</mi> </mtd> <mtd> <mo>=</mo> <mn>0</mn> <mspace width="1em" /> <mtext>on </mtext> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> <EquationSource Format="TEX">$ \left \{ \begin{aligned} -\mathrm{div}\big(\mathcal{A}(x,\nabla u)\big) &+\alpha (x) \frac{|u|^{p-2}u}{|x|^{p}} \\ &=\lambda \, a(x)|u|^{p-2}u\log \big(1+|u|\big) +\lambda \, h(x)|u|^{r-2}u, \quad \text{in }\Omega , \\ u&=0 \quad \text{on }\partial \Omega , \end{aligned} \right . $</EquationSource> </Equation> where <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mi mathvariant="double-struck">R</mi> <mi>N</mi> </msup> </math></EquationSource> <EquationSource Format="TEX">$\Omega \subset \mathbb{R}^{N}$</EquationSource> </InlineEquation> is bounded, contains the origin, and <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> <EquationSource Format="TEX">$\mathcal{A}$</EquationSource> </InlineEquation> exhibits Orlicz growth.The interaction between singular coercivity, nonpolynomial diffusion, and logarithmic reaction yields a nonstandard variational structure. Within an Orlicz–Sobolev framework, we prove the existence of nontrivial weak solutions using Hardy-type inequalities and compact embedding results. A Nehari-manifold approach adapted to the logarithmic nonlinearity is then developed to establish multiplicity of solutions for small values of λ. Finally, a conforming finite element discretization is proposed, and numerical experiments are presented to illustrate the convergence and stability of the scheme in the presence of strong singularities and nonstandard growth.

Existence and multiplicity results for singular elliptic problems with logarithmic source terms and Orlicz growth

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2026年6月25日

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Salah Boulaaras

摘要

This paper investigates a class of nonlinear elliptic boundary value problems combining Hardy-type singular potentials, Leray–Lions operators with Orlicz growth, and logarithmic source terms. We study the Dirichlet problem ${\begin{aligned} - div (A (x, \nabla u)) & + α (x) \frac{| u |^{p - 2} u}{| x |^{p}} \\ = λ a (x) | u |^{p - 2} u log (1 + | u |) + λ h (x) | u |^{r - 2} u, in Ω, \\ u & = 0 on \partial Ω, \end{aligned}$ $ \left \{ \begin{aligned} -\mathrm{div}\big(\mathcal{A}(x,\nabla u)\big) &+\alpha (x) \frac{|u|^{p-2}u}{|x|^{p}} \\ &=\lambda \, a(x)|u|^{p-2}u\log \big(1+|u|\big) +\lambda \, h(x)|u|^{r-2}u, \quad \text{in }\Omega , \\ u&=0 \quad \text{on }\partial \Omega , \end{aligned} \right . $ where $Ω \subset R^{N}$ $\Omega \subset \mathbb{R}^{N}$ is bounded, contains the origin, and $A$ $\mathcal{A}$ exhibits Orlicz growth.

The interaction between singular coercivity, nonpolynomial diffusion, and logarithmic reaction yields a nonstandard variational structure. Within an Orlicz–Sobolev framework, we prove the existence of nontrivial weak solutions using Hardy-type inequalities and compact embedding results. A Nehari-manifold approach adapted to the logarithmic nonlinearity is then developed to establish multiplicity of solutions for small values of λ. Finally, a conforming finite element discretization is proposed, and numerical experiments are presented to illustrate the convergence and stability of the scheme in the presence of strong singularities and nonstandard growth.