TY  - JOUR
ID  - Bentifour2021
LB  - Bentifour2021
AU  - Bentifour, Rachid
AU  - El-Hadi Miri, Sofiane
TI  - Some existence results for a nonlocal non-isotropic problem
ST  - Some existence results for a nonlocal non-isotropic problem
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2021
VL  - 41
IS  - 1
SP  - 5
EP  - 23
DO  - https://doi.org/10.7494/OpMath.2021.41.1.5
UR  - https://doi.org/10.7494/OpMath.2021.41.1.5
AB  - 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 \(0\lt\gamma \lt 1.\)
KW  - anisotropic operator
KW  - integro-differential problem
KW  - variational methods
SN  - 1232-9274
ER  -