<p>This paper investigates a class of strongly nonlinear elliptic problems with singular terms within the framework of variable exponent Sobolev spaces. The problem is formulated as a differential inclusion <Equation ID="Equ23"> <EquationSource Format="TEX">$ \zeta (u) + A(u) + H(x,u,\nabla u) \ni \dfrac{f}{u^{\gamma }}, $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi>ζ</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>A</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>H</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>u</mi> <mo>,</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>∋</mo> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>f</mi> <msup> <mi>u</mi> <mi>γ</mi> </msup> </mfrac> </mstyle> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where the operator <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ A $</EquationSource> <EquationSource Format="MATHML"><math> <mi>A</mi> </math></EquationSource> </InlineEquation> is of Leray–Lions type acting between the variable exponent Sobolev space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ W_0^{1,p(\cdot )}(\Omega ) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>W</mi> <mn>0</mn> <mrow> <mn>1</mn> <mo>,</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and its dual <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ W^{-1,p'(\cdot )}(\Omega ) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>W</mi> <mrow> <mo>-</mo> <mn>1</mn> <mo>,</mo> <msup> <mi>p</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. The multivalued mapping <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ \zeta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ζ</mi> </math></EquationSource> </InlineEquation> is maximal monotone with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$ 0 \in \zeta (0) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>∈</mo> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, and the nonlinearity <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$ H(x, s, \xi ) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>H</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>s</mi> <mo>,</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> exhibits natural growth with respect to <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$ \xi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ξ</mi> </math></EquationSource> </InlineEquation> of order <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$ p(x) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. The singular term involves <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$ f \in L^{\infty }(\Omega ) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <msup> <mi>L</mi> <mi>∞</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$ \gamma > 0 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. Under appropriate assumptions on the variable exponent <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$ p(\cdot ) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, we establish the existence of weak solutions using a comprehensive approximation scheme. The proof employs truncation methods, Yosida approximations of the maximal monotone operator, careful energy estimates, and sophisticated convergence analysis in variable exponent spaces. Our results extend and generalize previous work in fixed exponent Sobolev spaces to the more flexible and physically relevant setting of variable exponents.</p>

On strongly nonlinear elliptic equations with singularities in variable exponent spaces

期刊论文
发表日期：

2026年3月4日

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Mohamed Bahadi,
Morad Ouboufettal,
Youssef Akdim

摘要

This paper investigates a class of strongly nonlinear elliptic problems with singular terms within the framework of variable exponent Sobolev spaces. The problem is formulated as a differential inclusion $ \zeta (u) + A(u) + H(x,u,\nabla u) \ni \dfrac{f}{u^{\gamma }}, $ $ζ (u) + A (u) + H (x, u, \nabla u) ∋ \frac{f}{u^{γ}},$ where the operator $ A $ $A$ is of Leray–Lions type acting between the variable exponent Sobolev space $ W_0^{1,p(\cdot )}(\Omega ) $ $W_{0}^{1, p (\cdot)} (Ω)$ and its dual $ W^{-1,p'(\cdot )}(\Omega ) $ $W^{- 1, p^{'} (\cdot)} (Ω)$ . The multivalued mapping $ \zeta $ $ζ$ is maximal monotone with $ 0 \in \zeta (0) $ $0 \in ζ (0)$ , and the nonlinearity $ H(x, s, \xi ) $ $H (x, s, ξ)$ exhibits natural growth with respect to $ \xi $ $ξ$ of order $ p(x) $ $p (x)$ . The singular term involves $ f \in L^{\infty }(\Omega ) $ $f \in L^{\infty} (Ω)$ and $ \gamma > 0 $ $γ > 0$ . Under appropriate assumptions on the variable exponent $ p(\cdot ) $ $p (\cdot)$ , we establish the existence of weak solutions using a comprehensive approximation scheme. The proof employs truncation methods, Yosida approximations of the maximal monotone operator, careful energy estimates, and sophisticated convergence analysis in variable exponent spaces. Our results extend and generalize previous work in fixed exponent Sobolev spaces to the more flexible and physically relevant setting of variable exponents.