Regularity of radial extremal solutions for some non local semilinear equations

Details Regularity of radial extremal solutions for some non local semilinear equations Regularity of radial extremal solutions for some non local semilinear equations

2010-04-12

arxiv.org/abs/1004.1906v1

Mathematics

Analysis of PDEs

Scientific paper

We investigate stable solutions of elliptic equations of the type \begin{equation*} \left \{ \begin{aligned} (-\Delta)^s u&=\lambda f(u) \qquad {\mbox{ in $B_1 \subset \R^{n}$}} \\ u&= 0 \qquad{\mbox{ on $\partial B_1$,}}\end{aligned}\right . \end{equation*} where $n\ge2$, $s \in (0,1)$, $\lambda \geq 0$ and $f$ is any smooth positive superlinear function. The operator $(-\Delta)^s$ stands for the fractional Laplacian, a pseudo-differential operator of order $2s$. According to the value of $\lambda$, we study the existence and regularity of weak solutions $u$.

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