PDE Seminar
Large solutions to elliptic equations involving fractional Laplacian
Dr. Huyuan Chen (NYU Shanghai)
2014年10月22日（周三）下午2:00-3:00 闵行校区行政楼12楼PDE中心1202报告厅
Abstract. In this talk, we will discuss boundary blow up solutions for semilinear fractional elliptic equations
\begin{equation}
\begin{cases}
\;(-\Delta)^{\alpha} u(x)+|u|^{p-1}u(x)=0,\quad & x\in\Omega,\\[2mm]
\phantom{ (-\Delta)^{\alpha} u(x)+|u|^{p-1}}\\
\;u(x)=0,\quad & x\in\bar\Omega^c,\\[2mm]
\phantom{ (-\Delta)^{\alpha} }\\
\;\lim_{x\in\Omega, x\to\partial\Omega}u(x)=+\infty,\\
\end{cases}
\end{equation}
where $p>1$, $\Omega$ is an open bounded $C^2$ domain of $\mathbb{R}^N(N\geq2)$ and the operator $(-\Delta)^{\alpha}$ with $\alpha\in(0,1)$ is the fractional Laplacian. We show that the above fractional problem admits a solution with behavior $d(x)^{-\frac{2\alpha}{p-1}}$ when $1$ $+2\alpha$ $<$ $p$ $<$ $1-$ $\frac{2\alpha}{\tau_0(\alpha)}$ for some $\tau_0(\alpha)\in(-1,0)$ and has infinitely many solutions with behavior $d(x)^{\tau_0(\alpha)}$ when max $\{1-$ $\frac{2\alpha}{\tau_0(\alpha)}$ + $\frac{\tau_0(\alpha)+1}{\tau_0(\alpha)}$,$1\}$ $<$ $p$ $<1$ - $\frac{2\alpha}{\tau_0(\alpha)}$.