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Article

Azroul, Elhoussine ; Bouziani, Mohammed ; Hjiaj, Hassane ; Youssfi, Ahmed
Existence of solutions for some quasilinear $\vec {p}(x)$-elliptic problem with Hardy potential. (English). Mathematica Bohemica, vol. 144 (2019), issue 3, pp. 299-324
MSC: 35J15, 35J62 | MR 3985859 | Zbl 07088853 | DOI: 10.21136/MB.2018.0093-17
Keywords:
anisotropic variable exponent Sobolev space; quasilinear elliptic equation; Hardy potential; entropy solution; $L^{1}$-data
Summary:
We consider the anisotropic quasilinear elliptic Dirichlet problem $$ \begin {cases} \displaystyle -\sum _{i=1}^{N} D^{i} a_{i}(x,u,\nabla u) + |u|^{s(x)-1}u= f +\lambda \frac {|u|^{p_{0}(x)-2}u}{|x|^{p_{0}(x)}}&\text {in}\ \Omega ,\\ u = 0 & \text {on}\ \partial \Omega , \end {cases} $$ where $\Omega $ is an open bounded subset of $\Bbb R^N$ 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.
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