Fractional KPZ equations with fractional gradient term and Hardy potential

Boumediene Abdellaoui; Kheireddine Biroud; Ana Primo; Fernando Soria; Abdelbadie Younes; Boumediene Abdellaoui; Kheireddine Biroud; Ana Primo; Fernando Soria; Abdelbadie Younes

doi:10.3934/mine.2023042

Mathematics in Engineering

2023, Volume 5, Issue 2: 1-36. doi: 10.3934/mine.2023042

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Fractional KPZ equations with fractional gradient term and Hardy potential

1.
Laboratoire d'Analyse Nonlinéaire et Mathématiques Appliquées, Département de Mathématiques, Université Abou Bakr Belkaïd, Tlemcen, Tlemcen 13000, Algeria
2.
Ecole superieure de management, Tlemcen. Tlemcen 13000, Algeria
3.
Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049, Madrid, Spain

Received: 24 December 2021 Revised: 05 June 2022 Accepted: 14 June 2022 Published: 27 June 2022

In this work we address the question of existence and non existence of positive solutions to a class of fractional problems with non local gradient term. More precisely, we consider the problem

$ \left\{ \begin{array}{rcll} (-\Delta )^s u & = &\lambda \dfrac{u}{|x|^{2s}}+ (\mathfrak{F}(u)(x))^p+ \rho f & \text{ in } \Omega,\\ u&>&0 & \text{ in }\Omega,\\ u& = &0 & \text{ in }(\mathbb{R}^N\setminus\Omega), \end{array}\right. $

where $ \Omega\subset \mathbb{R}^N $ is a $ C^{1, 1} $ bounded domain, $ N > 2s, \rho > 0 $, $ 0 < s < 1 $, $ 1 < p < \infty $ and $ 0 < \lambda < \Lambda_{N, s} $, the Hardy constant defined below. We assume that $ f $ is a non-negative function with additional hypotheses. Here $ \mathfrak{F}(u) $ is a nonlocal "gradient" term. In particular, if $ \mathfrak{F}(u)(x) = |(-\Delta)^{\frac s2}u(x)| $, then we are able to show the existence of a critical exponents $ p_{+}(\lambda, s) $ such that: 1) if $ p > p_{+}(\lambda, s) $, there is no positive solution, 2) if $ p < p_{+}(\lambda, s) $, there exists, at least, a positive supersolution solution for suitable data and $ \rho $ small. Moreover, under additional restriction on $ p $, there exists a solution for general datum $ f $.
- fractional elliptic equations,
- nonlocal gradient term,
- Hardy potential,
- stationary Kardar-Parisi-Zhang equations,
- existence and nonexistence results
Citation: Boumediene Abdellaoui, Kheireddine Biroud, Ana Primo, Fernando Soria, Abdelbadie Younes. Fractional KPZ equations with fractional gradient term and Hardy potential[J]. Mathematics in Engineering, 2023, 5(2): 1-36. doi: 10.3934/mine.2023042

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Abstract

In this work we address the question of existence and non existence of positive solutions to a class of fractional problems with non local gradient term. More precisely, we consider the problem

$ \left\{ \begin{array}{rcll} (-\Delta )^s u & = &\lambda \dfrac{u}{|x|^{2s}}+ (\mathfrak{F}(u)(x))^p+ \rho f & \text{ in } \Omega,\\ u&>&0 & \text{ in }\Omega,\\ u& = &0 & \text{ in }(\mathbb{R}^N\setminus\Omega), \end{array}\right. $

where $ \Omega\subset \mathbb{R}^N $ is a $ C^{1, 1} $ bounded domain, $ N > 2s, \rho > 0 $, $ 0 < s < 1 $, $ 1 < p < \infty $ and $ 0 < \lambda < \Lambda_{N, s} $, the Hardy constant defined below. We assume that $ f $ is a non-negative function with additional hypotheses. Here $ \mathfrak{F}(u) $ is a nonlocal "gradient" term. In particular, if $ \mathfrak{F}(u)(x) = |(-\Delta)^{\frac s2}u(x)| $, then we are able to show the existence of a critical exponents $ p_{+}(\lambda, s) $ such that: 1) if $ p > p_{+}(\lambda, s) $, there is no positive solution, 2) if $ p < p_{+}(\lambda, s) $, there exists, at least, a positive supersolution solution for suitable data and $ \rho $ small. Moreover, under additional restriction on $ p $, there exists a solution for general datum $ f $.

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