Existence of solutions for some quasilinear p → ( x ) -elliptic problem with Hardy potential

Elhoussine Azroul; Mohammed Bouziani; Hassane Hjiaj; Ahmed Youssfi

Mathematica Bohemica (2019)

  • Volume: 144, Issue: 3, page 299-324
  • ISSN: 0862-7959

Abstract

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We consider the anisotropic quasilinear elliptic Dirichlet problem - ∑ i = 1 N D i a i ( x , u , ∇ u ) + | u | s ( x ) - 1 u = f + λ | u | p 0 ( x ) - 2 u | x | p 0 ( x ) in Ω , u = 0 on ∂ Ω , where Ω is an open bounded subset of ℝ N containing the origin. We show the existence of entropy solution for this equation where the data f is assumed to be in L 1 ( Ω ) and λ is a positive constant.

How to cite

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Azroul, Elhoussine, et al. "Existence of solutions for some quasilinear $\vec{p}(x)$-elliptic problem with Hardy potential." Mathematica Bohemica 144.3 (2019): 299-324. <http://eudml.org/doc/294314>.

@article{Azroul2019,
abstract = {We consider the anisotropic quasilinear elliptic Dirichlet problem \[ \{\left\lbrace \begin\{array\}\{ll\} \displaystyle -\sum \_\{i=1\}^\{N\} D^\{i\} a\_\{i\}(x,u,\nabla u) + |u|^\{s(x)-1\}u= f +\lambda \frac\{|u|^\{p\_\{0\}(x)-2\}u\}\{|x|^\{p\_\{0\}(x)\}\}&\text\{in\}\ \Omega ,\\ u = 0 & \text\{on\}\ \partial \Omega , \end\{array\}\right.\} \] where $\Omega $ is an open bounded subset of $\mathbb \{R\}^N$ containing the origin. We show the existence of entropy solution for this equation where the data $f$ is assumed to be in $L^\{1\}(\Omega )$ and $\lambda $ is a positive constant.},
author = {Azroul, Elhoussine, Bouziani, Mohammed, Hjiaj, Hassane, Youssfi, Ahmed},
journal = {Mathematica Bohemica},
keywords = {anisotropic variable exponent Sobolev space; quasilinear elliptic equation; Hardy potential; entropy solution; $L^\{1\}$-data},
language = {eng},
number = {3},
pages = {299-324},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Existence of solutions for some quasilinear $\vec\{p\}(x)$-elliptic problem with Hardy potential},
url = {http://eudml.org/doc/294314},
volume = {144},
year = {2019},
}

TY - JOUR
AU - Azroul, Elhoussine
AU - Bouziani, Mohammed
AU - Hjiaj, Hassane
AU - Youssfi, Ahmed
TI - Existence of solutions for some quasilinear $\vec{p}(x)$-elliptic problem with Hardy potential
JO - Mathematica Bohemica
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 144
IS - 3
SP - 299
EP - 324
AB - We consider the anisotropic quasilinear elliptic Dirichlet problem \[ {\left\lbrace \begin{array}{ll} \displaystyle -\sum _{i=1}^{N} D^{i} a_{i}(x,u,\nabla u) + |u|^{s(x)-1}u= f +\lambda \frac{|u|^{p_{0}(x)-2}u}{|x|^{p_{0}(x)}}&\text{in}\ \Omega ,\\ u = 0 & \text{on}\ \partial \Omega , \end{array}\right.} \] where $\Omega $ is an open bounded subset of $\mathbb {R}^N$ containing the origin. We show the existence of entropy solution for this equation where the data $f$ is assumed to be in $L^{1}(\Omega )$ and $\lambda $ is a positive constant.
LA - eng
KW - anisotropic variable exponent Sobolev space; quasilinear elliptic equation; Hardy potential; entropy solution; $L^{1}$-data
UR - http://eudml.org/doc/294314
ER -

References

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