Large solutions to elliptic equations involving fractional Laplacian

Chen, Huyuan; Felmer, Patricio; Quaas, Alexander

Chen, Huyuan ; Felmer, Patricio ; Quaas, Alexander

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015), p. 1199-1228 / Harvested from Numdam

Résumé

The purpose of this paper is to study boundary blow up solutions for semi-linear fractional elliptic equations of the form $\begin{matrix} {\begin{matrix} {(- Δ)}^{α} u (x) + {| u |}^{p - 1} u (x) = f (x), & x \in Ω, \\ u (x) = 0, & x \in {\bar{Ω}}^{c}, \\ \lim_{x \in Ω, x \to \partial Ω} u (x) = + \infty, \end{matrix} & (0.1) \end{matrix}$ where $p > 1$ , Ω is an open bounded $C^{2}$ domain of $ℝ^{N}$ , $N \geq 2$ , the operator ${(- Δ)}^{α}$ with $α \in (0, 1)$ is the fractional Laplacian and $f : Ω \to ℝ$ is a continuous function which satisfies some appropriate conditions. We obtain that problem (0.1) admits a solution with boundary behavior like $d {(x)}^{- \frac{2 α}{p - 1}}$ , when $1 + 2 α < p < 1 - \frac{2 α}{τ_{0} (α)}$ , for some $τ_{0} (α) \in (- 1, 0)$ , and has infinitely many solutions with boundary behavior like $d {(x)}^{τ_{0} (α)}$ , when $\max {1 - \frac{2 α}{τ_{0}} + \frac{τ_{0} (α) + 1}{τ_{0}}, 1} < p < 1 - \frac{2 α}{τ_{0}}$ . Moreover, we also obtained some uniqueness and non-existence results for problem (0.1).

Publié le : 2015-01-01

MR 3425260 | Zbl 06520570

DOI : https://doi.org/10.1016/j.anihpc.2014.08.001

@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},
     volume = {32},
     year = {2015},
     pages = {1199-1228},
     doi = {10.1016/j.anihpc.2014.08.001},
     mrnumber = {3425260},
     zbl = {06520570},
     language = {en},
     url = {http://dml.mathdoc.fr/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, Tome 32 (2015) pp. 1199-1228. doi : 10.1016/j.anihpc.2014.08.001. http://gdmltest.u-ga.fr/item/AIHPC_2015__32_6_1199_0/

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