Nonlinear fractional Laplacian problems with nonlocal "gradient terms" (1812.00414v1)
Abstract: Let $\Omega \subset \mathbb{R}N$, $N \geq 2$, be a smooth bounded domain. For $s \in (1/2,1)$, we consider a problem of the form [ \left{\begin{aligned} (-\Delta)s u & = \mu(x)\, \mathbb{D}s{2}(u) + \lambda f(x)\,, & \quad \mbox{in} \Omega,\ u & = 0\,, & \quad \mbox{in} \mathbb{R}N \setminus \Omega, \end{aligned} \right. ] where $\lambda > 0$ is a real parameter, $f$ belongs to a suitable Lebesgue space, $\mu \in L{\infty}(\Omega)$ and $\mathbb{D}_s2$ is a nonlocal "gradient square" term given by [ \mathbb{D}_s2 (u) = \frac{a{N,s}}{2}\mbox{p.v.} \int_{\mathbb{R}N} \frac{|u(x)-u(y)|2}{|x-y|{N+2s}} dy \,. ] Depending on the real parameter $\lambda > 0$, we derive existence and non-existence results. The proof of our existence result relies on sharp Calder\'on-Zygmund type regularity results for the fractional Poisson equation with low integrability data. We also obtain existence results for related problems involving different nonlocal diffusion terms.