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<n. 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).

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