$$p(\cdot )$$-Laplacian problem with nonlinear singular terms

Elharrar, N.; Igbida, Jaouad; Talibi, H.

doi:10.1007/s12215-021-00604-y

Cited by 3 publications

(3 citation statements)

References 15 publications

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“…The problem (P ω ) or special cases of it has been extensively studied by many authors in elliptic or parabolic case, we refer the reader to [1,3,4,5,9,11,14,15].…”

Section: Introductionmentioning

confidence: 99%

Rothe Time-Discretization Method for Nolinear Parabolic Problems in Weighted Sobolev Space with Variable Exponents

Elharrar¹,

Igbida²

2023

Az. J. Math.

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“…The problem (P ω ) or special cases of it has been extensively studied by many authors in elliptic or parabolic case, we refer the reader to [1,3,4,5,9,11,14,15].…”

Section: Introductionmentioning

confidence: 99%

Rothe Time-Discretization Method for Nolinear Parabolic Problems in Weighted Sobolev Space with Variable Exponents

Elharrar¹,

Igbida²

2023

Az. J. Math.

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“…In the last years, the problem (P ) or special cases of it have been extensively treated by many authors in elliptic or parabolic case, we invite the reader to see for example the works [2,3,4,5,10,13,16,18].…”

Section: Introductionmentioning

confidence: 99%

Rothe time-discretization method for Nolinear Parabolic Problems

Elharrar,

Igbida,

Talibi

2022

Methods of Functional Analysis and Topology

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In this paper we consider a class of nonlinear parabolic problems whose model is(\ast ) Using time discretization technique and Rothe method we prove existence and uniqueness results for bounded weak solutions.Використовуючи технiку дискретизацiї за часом та метод Рота, доведено iснування та єдинiсть слабкого обмеженого розв'язку для нелiнiйних параболiчних задач вигляду (\ast ).

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“…Recently, in the variable exponents cases, the authors in [24] improved the results of [6,30] and studied the problem (1.2) in general form depending on γ(•) > 0, µ = 0, m(•) = m ≥ 1, 2 < p(•) ≤ N and p(•)-Laplace operator. They proved the existence and regularity results of problem (1.2) by adapting some techniques used in [14], nevertheless it was a new spirit of idea different from that used in [12], ( when γ(•) = γ ≥ 1, see [25] for a similar paper ). In [28], the authors generalized the work [2] by considering nonlinear elliptic equations with a singular nonlinearity, lower order terms and L 1 datum in the setting of Sobolev spaces with variable exponents.…”

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confidence: 99%

Existence and Uniqueness Results for a Singular Nonlinear Elliptic Equations with Variable Exponents

Abdelaziz

2024

Preprint

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This paper deals with study of the nonlinear singular elliptic equations in a bounded domain $\Omega\subset\mathbb{R}^N$, $(N\geq2)$ with Lipschitz boundary $\partial\Omega$, $$ Au=\frac{f}{u^{\gamma(\cdot)}}+\mu, $$ Where $A:=-\mathrm{div}\left(\widehat{a}(\cdot,Du)\right)$ is a Leray-Lions type operator which maps continuously $W^{1,p(\cdot)}_0(\Omega)$ into its dual $W^{-1,p’(\cdot)}(\Omega)$, whose simplest model is the $p(\cdot)$-laplacian type operator ( i.e. $\widehat{a}(\cdot,\xi)=|\xi|^{p(\cdot)-2}\xi$ ), such taht $f$ is a nonnegative function belonging to the Lebesgue space with variable exponents $L^{m(\cdot)}(\Omega)$, with $m(\cdot)$ being small ( or $L^{1}(\Omega)$ ) and $\mu$ is a nonnegative bounded Radon measure, while $m:\overline{\Omega}\to (1,+\infty)$, $\gamma:\overline{\Omega}\to (0,1)$ are continuous functions satisfying certain conditions depend on $p(\cdot)$. We prove the existence, uniqueness and regularity of nonnegative weak solutions or this class of problems with $p(\cdot)$-growth conditions. More precisely, we will discuss that the nonlinear singular term has some regularizing effects on the solutions of our problem which depends on the summability of $f$, $m(\cdot)$ and the value of $\gamma(\cdot)$. The functional framework involves Sobolev spaces with variable exponents as well as Lebesgue spaces with variable exponents. Our results can be seen as a generalization of some results given in the constant exponents case.

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$$p(\cdot )$$-Laplacian problem with nonlinear singular terms

Cited by 3 publications

References 15 publications

Rothe Time-Discretization Method for Nolinear Parabolic Problems in Weighted Sobolev Space with Variable Exponents

Rothe Time-Discretization Method for Nolinear Parabolic Problems in Weighted Sobolev Space with Variable Exponents

Rothe time-discretization method for Nolinear Parabolic Problems

Existence and Uniqueness Results for a Singular Nonlinear Elliptic Equations with Variable Exponents

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