Opuscula Math. 41, no. 1 (2021), 5-23
https://doi.org/10.7494/OpMath.2021.41.1.5

Opuscula Mathematica

# Some existence results for a nonlocal non-isotropic problem

Rachid Bentifour

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.

Mathematics Subject Classification: 35A15, 35B09, 35E15, 35J20.

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• Rachid Bentifour
• https://orcid.org/0000-0003-3177-663X
• Université de Tlemcen, Dépatement GEE, Laboratoire d'Analyse Non Linéaire et Mathématiques Appliquées, BP 119 Tlemcen, 13000, Algeria
• Sofiane El-Hadi Miri (corresponding author)
• https://orcid.org/0000-0001-6572-4366
• Université de Tlemcen, Dépatement GEE, Laboratoire d'Analyse Non Linéaire et Mathématiques Appliquées, BP 119 Tlemcen, 13000, Algeria
• Communicated by Vicentiu D. Radulescu.
• Revised: 2020-11-23.
• Accepted: 2020-11-24.
• Published online: 2021-02-08.