# Ngô Quốc Anh

## August 29, 2010

### Achieving regularity results via bootstrap argument, 4

Filed under: Giải Tích 6 (MA5205), PDEs — Tags: — Ngô Quốc Anh @ 2:36

Let us consider the following equation

$\displaystyle u(x) = \int_{{\mathbb{R}^n}} {\frac{{u{{(y)}^{\frac{{n + \alpha }}{{n - \alpha }}}}}}{{{{\left| {x - y} \right|}^{n - \alpha }}}}dy} , \quad x \in {\mathbb{R}^n}$

for $n\geqslant 1$ and $0<\alpha. In this entry, by using boothstrap argument, we show that

Theorem. If positive function $u \in L_{loc}^\frac{2n}{n-\alpha}(\mathbb R^n)$ solves the equation, then $u \in C^\infty(\mathbb R^n)$.

In the process of proving the result, we need the following result

Proposition. Let $V \in L^\frac{n}{\alpha}(B_3)$ be a non-negative function and set

$\displaystyle \delta(V)=\|V\|_{L^\frac{n}{\alpha}(B_3)}$.

For $\nu >r>\frac{n}{n-\alpha}$, there exist positive constants $\overline \delta<1$ and $C \geqslant 1$ depending only on $n, \alpha, r$ and $\nu$ such that for any $0 \leqslant V \in L^\frac{n}{\alpha}(B_3)$ with $\delta(V) \leqslant \overline \delta$, $h \in L^\nu(B_2)$ and $0 \leqslant u \in L^r(B_3)$ satisfying

$\displaystyle u(x) \leqslant \int_{{B_3}} {\frac{{V(y)u(y)}}{{{{\left| {x - y} \right|}^{n - \alpha }}}}dy} + h(x), \quad x \in {B_2}$

we have

$\displaystyle {\left\| u \right\|_{{L^\nu }({B_{1/2}})}} \leqslant C\left( {{{\left\| u \right\|}_{{L^r}({B_3})}} + {{\left\| h \right\|}_{{L^\nu }({B_2})}}} \right)$.