Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian
Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian
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分数拉普拉斯半线性椭圆方程径向稳定解的正则性
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发表时间：
2017
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通讯作者：
Tomás Sanz
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作者：
Tomás Sanz
我们研究了问题$$ \left\{ \begin{array}{rcll} (-\Delta)^s u &=& f(u) & \text{in} \quad B_1\,, u &\equiv&0 & \text{in} \quad \mathbb R^n\setminus B_1\,, \end{array} \right. $$的稳定解的正则性，其中$s\in(0,1)$。我们的主要结果为这个问题在$2 \leq n < 2(s+2+\sqrt{2(s+1)})$维上的稳定和径向递减的$H^s$解建立了一个$L^\infty$界。特别地，这个估计适用于维度$2 \leq n\leq 6$中的所有$s\in(0,1)$。它适用于所有非线性方程$f\in C^2$。
对于这些参数$s$和$n$，我们的结果导致当$f$被$\lambda > 0$代替$\lambda f$时极值解的规律性。这是一个广泛研究$s=1$的问题，在$s=1$和$s<1$的非径向情况下，这个问题仍然很大程度上是开放的。
We study the regularity of stable solutions to the problem $$ \left\{ \begin{array}{rcll} (-\Delta)^s u &=& f(u) & \text{in} \quad B_1\,, u &\equiv&0 & \text{in} \quad \mathbb R^n\setminus B_1\,, \end{array} \right. $$ where $s\in(0,1)$. Our main result establishes an $L^\infty$ bound for stable and radially decreasing $H^s$ solutions to this problem in dimensions $2 \leq n < 2(s+2+\sqrt{2(s+1)})$. In particular, this estimate holds for all $s\in(0,1)$ in dimensions $2 \leq n\leq 6$. It applies to all nonlinearities $f\in C^2$.
For such parameters $s$ and $n$, our result leads to the regularity of the extremal solution when $f$ is replaced by $\lambda f$ with $\lambda > 0$. This is a widely studied question for $s=1$, which is still largely open in the nonradial case both for $s=1$ and $s<1$.