@Article{Bentifour2021,
author = "Bentifour, Rachid and El-Hadi Miri, Sofiane",
title = "Some existence results for a nonlocal non-isotropic problem",
journal = "Opuscula Math.",
year = "2021",
volume = "41",
number = "1",
pages = "5--23",
doi = "https://doi.org/10.7494/OpMath.2021.41.1.5",
url = "https://doi.org/10.7494/OpMath.2021.41.1.5",
abstract = "In this paper we deal with the following problem \[\begin{cases}-\sum\limits_{i=1}^{N}\left[ \left( a+b\int\limits_{\, \Omega }\left\vert \partial _{i}u\right\vert ^{p_{i}}dx\right) \partial _{i}\left( \left\vert \partial _{i}u\right\vert ^{p_{i}-2}\partial _{i}u\right) \right]=\frac{f(x)}{u^{\gamma }}\pm g(x)u^{q-1} & in\ \Omega, \\ u\geq 0 & in\ \Omega, \\ u=0 & on\ \partial \Omega, \end{cases}\] where \(\Omega\) is a bounded regular domain in \(\mathbb{R}^{N}\). We will assume without loss of generality that \(1\leq p_{1}\leq p_{2}\leq \ldots\leq p_{N}\) and that \(f\) and \(g\) are non-negative functions belonging to a suitable Lebesgue space \(L^{m}(\Omega)\), \(1\lt q\lt \overline{p}^{\ast}\), \(a\gt 0\), \(b\gt 0\) and \(0\lt\gamma \lt 1.\)",
keywords = "anisotropic operator, integro-differential problem, variational methods",
issn = "1232-9274"
}