Large solutions to elliptic equations involving fractional Laplacian
Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 6, p. 1199-1228
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The purpose of this paper is to study boundary blow up solutions for semi-linear fractional elliptic equations of the form $\begin{array}{cc}\left\{\begin{array}{cc}{\left(-\Delta \right)}^{\alpha }u\left(x\right)+{|u|}^{p-1}u\left(x\right)=f\left(x\right),\hfill & x\in \Omega ,\hfill \\ u\left(x\right)=0,\hfill & x\in {\overline{\Omega }}^{c},\hfill \\ \underset{x\in \Omega ,x\to \partial \Omega }{\mathrm{lim}}u\left(x\right)=+\infty ,\hfill \end{array}& \text{(0.1)}\end{array}$ where $p>1$, Ω is an open bounded ${C}^{2}$ domain of ${ℝ}^{N}$, $N\ge 2$, the operator ${\left(-\Delta \right)}^{\alpha }$ with $\alpha \in \left(0,1\right)$ is the fractional Laplacian and $f:\Omega \to ℝ$ is a continuous function which satisfies some appropriate conditions. We obtain that problem (0.1) admits a solution with boundary behavior like $d{\left(x\right)}^{-\frac{2\alpha }{p-1}}$, when $1+2\alpha , for some ${\tau }_{0}\left(\alpha \right)\in \left(-1,0\right)$, and has infinitely many solutions with boundary behavior like $d{\left(x\right)}^{{\tau }_{0}\left(\alpha \right)}$, when $\mathrm{max}\left\{1-\frac{2\alpha }{{\tau }_{0}}+\frac{{\tau }_{0}\left(\alpha \right)+1}{{\tau }_{0}},1\right\}. Moreover, we also obtained some uniqueness and non-existence results for problem (0.1).

DOI : https://doi.org/10.1016/j.anihpc.2014.08.001
Keywords: Large solutions, Fractional Laplacian, Existence, Uniqueness, Non-existence infinite existence
@article{AIHPC_2015__32_6_1199_0,
author = {Chen, Huyuan and Felmer, Patricio and Quaas, Alexander},
title = {Large solutions to elliptic equations involving fractional Laplacian},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Elsevier},
volume = {32},
number = {6},
year = {2015},
pages = {1199-1228},
doi = {10.1016/j.anihpc.2014.08.001},
zbl = {06520570},
mrnumber = {3425260},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2015__32_6_1199_0}
}

Chen, Huyuan; Felmer, Patricio; Quaas, Alexander. Large solutions to elliptic equations involving fractional Laplacian. Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 6, pp. 1199-1228. doi : 10.1016/j.anihpc.2014.08.001. http://www.numdam.org/item/AIHPC_2015__32_6_1199_0/

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