On some singular nonlinear problems for monotone elliptic operators

Chipot, Michel

doi:10.4171/rlm/848

Cited by 3 publications

(1 citation statement)

References 12 publications

(14 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…the existence and nonexistence of solutions for problem (1.2) has been proved in [6]. Similar results, with different proofs, were obtained in [5,12,13,20,21,33,37,44]. In [11] the authors studied problem (1.2) when p(•) = p = 2, µ = 0, m(•) = m > 1.…”

supporting

confidence: 65%

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

Abdelaziz

2024

Preprint

View full text Add to dashboard Cite

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.

show abstract