Fractional KPZ equations with fractional gradient term and Hardy potential

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 $.

Keywords

• fractional elliptic equations
• nonlocal gradient term
• hardy potential
• stationary kardar-parisi-zhang equations
• existence and nonexistence results