Fractional KPZ equations with fractional gradient term and Hardy potential

Abstract

<abstract><p>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</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left\{ \begin{array}{rcll} (-\Delta )^s u &amp; = &amp;\lambda \dfrac{u}{|x|^{2s}}+ (\mathfrak{F}(u)(x))^p+ \rho f &amp; \text{ in } \Omega,\\ u&amp;&gt;&amp;0 &amp; \text{ in }\Omega,\\ u&amp; = &amp;0 &amp; \text{ in }(\mathbb{R}^N\setminus\Omega), \end{array}\right. $\end{document} </tex-math></disp-formula></p> <p>where $ \Omega\subset \mathbb{R}^N $ is a $ C^{1, 1} $ bounded domain, $ N &gt; 2s, \rho &gt; 0 $, $ 0 &lt; s &lt; 1 $, $ 1 &lt; p &lt; \infty $ and $ 0 &lt; \lambda &lt; \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 &gt; p_{+}(\lambda, s) $, there is no positive solution, 2) if $ p &lt; 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 $.</p></abstract>