Fractional KPZ equations with critical growth in the gradient respect to Hardy potential

In this work we study the existence of positive solution to the fractional quasilinear problem, $$ \left\{ \begin{array}{rcll} (-\Delta )^s u &=&\lambda \dfrac{u}{|x|^{2s}}+ |\nabla u|^{p}+ \mu f &\inn \Omega,\\ u&>&0 & \inn\Omega,\\ u&=&0 & \inn(\mathbb{R}^N\setminus\Omega), \end{array}\right. $$ where $\Omega$ is a $C^{1,1}$ bounded domain in $\mathbb{R}^N$, $N> 2s, \mu>0$, $\frac{1}{2}0$, there exists a critical exponent $p_{+}(\lambda, s)$ such that for $p> p_{+}(\lambda,s)$ there is no positive solution. Moreover, $p_{+}(\lambda,s)$ is optimal in the sense that, if $p

Keywords:

• Lambda
• Combinatorics
• Bounded function
• Nabla symbol
• Critical exponent
• Mathematical analysis
• Omega
• Mathematics

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