论文标题
非本地liouville型方程的爆破解决方案
Blowing-up solutions for a nonlocal Liouville type equation
论文作者
论文摘要
我们考虑非局部liouville类型方程$$（-Δ）^{\ frac {1} {2}} u = \varepsilonκ（x） i，$$，$ i $是$ d \ geq 2 $脱节的间隔的结合，$κ$是一个平稳的界限功能，具有正值，$ \ varepsilon> 0 $是一个小参数。对于任何整数$ 1 \ leq m \ leq d $，我们构建了一个解决方案$（u_ \ varepsilon）_ {\ varepsilon} $，它以$ m $ $ i $的$ m $内部不同点爆炸，$ i $ us $ \ \ \ \ \ varepsilon \ varepsilon \ int_i -int_i-int_iκe^$^u _ \ vareps aurr preaps aurps aurr preeps aurp， $ \ varepsilon \至0 $。此外，我们表明，当$ d = 2 $和$ m $适当大时，就不可能进行这种结构。
We consider the nonlocal Liouville type equation $$ (-Δ)^{\frac{1}{2}} u = \varepsilon κ(x) e^u, \quad u > 0, \quad \mbox{in } I, \qquad u = 0, \quad \mbox{in } \mathbb{R} \setminus I, $$ where $I$ is a union of $d \geq 2$ disjoint bounded intervals, $κ$ is a smooth bounded function with positive infimum and $\varepsilon > 0$ is a small parameter. For any integer $1 \leq m \leq d$, we construct a family of solutions $(u_\varepsilon)_{\varepsilon}$ which blow up at $m$ interior distinct points of $I$ and for which $\varepsilon \int_I κe^{u_\varepsilon} \, \rightarrow 2 m π$, as $\varepsilon \to 0$. Moreover, we show that, when $d = 2$ and $m$ is suitably large, no such construction is possible.
