Large solutions to elliptic equations involving fractional Laplacian

Chen, Huyuan; Felmer, Patricio; Quaas, Alexander

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 6, pp. 1199-1228

The purpose of this paper is to study boundary blow up solutions for semi-linear fractional elliptic equations of the form

$$\begin{array}{cc}\{\begin{array}{cc}{(-\Delta )}^{\alpha }u\left(x\right)+{\left|u\right|}^{p-1}u\left(x\right)=f\left(x\right),\hfill & x\in \Omega ,\hfill \\ u\left(x\right)=0,\hfill & x\in {\overline{\Omega }}^c,\hfill \\ \underset{x\in \Omega ,x\to \partial \Omega }{\lim }u\left(x\right)=+\infty ,\hfill \end{array}& \text{(0.1)}\end{array}$$

where $p>1$, Ω is an open bounded $C^2$ domain of ${ℝ}^N$, $N\ge 2$, the operator ${(-\Delta )}^{\alpha }$ with $\alpha \in (0,1)$ 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 $1+2\alpha <p<1-\frac{2\alpha }{{\tau }_0\left(\alpha \right)}$, for some ${\tau }_0\left(\alpha \right)\in (-1,0)$, and has infinitely many solutions with boundary behavior like $d{\left(x\right)}^{{\tau }_0\left(\alpha \right)}$, when $\max \{1-\frac{2\alpha }{{\tau }_0}+\frac{{\tau }_0\left(\alpha \right)+1}{{\tau }_0},1\}<p<1-\frac{2\alpha }{{\tau }_0}$. Moreover, we also obtained some uniqueness and non-existence results for problem (0.1).

Keywords: Large solutions, Fractional Laplacian, Existence, Uniqueness, Non-existence infinite existence

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