On the Existence of a Non-trivial Non-negative Global Radial Weak Solution to a Fractional Laplacian Problem with a Singular Potential

Masoud Bayrami¹ ; Mahmoud Hesaaraki¹

¹ Department of Mathematical Sciences Sharif University of Technology P.O. Box 11365-9415, Tehran, Iran

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 64 (2016) no. 2-3, pp. 175-183

Résumé

We prove the existence of a non-trivial non-negative radial weak solution to the problem \begin{equation*} \begin{cases} (-\Delta) ^{\alpha} u+bu=\lambda \dfrac{u}{|x|^{2\alpha}} +|u|^{p-1}u+\mu |u|^{r-1}u \mathrm{in} \ \mathbb{R}^N,\\ \lim\limits_{|x| \to \infty} u(x) = 0. \end{cases} \end{equation*} Here $N \gt 2\alpha $, $ \alpha \in ({1}/{2},1)$, $1 \lt r \lt p \lt \frac{N+2\alpha}{N-2\alpha}$ and $\mu \in \mathbb{R}$. We also assume that $b \gt 0$ and $ 0 \lt \lambda \lt 4^\alpha \frac{\Gamma^2(\frac{N+2\alpha}4)}{\Gamma^2(\frac{N-2\alpha}4)}$.

Détail

DOI : 10.4064/ba8070-9-2016

Keywords: prove existence non trivial non negative radial weak solution problem begin equation* begin cases delta alpha lambda dfrac alpha p r mathrm mathbb lim limits infty end cases end equation* here alpha alpha frac alpha n alpha mathbb assume lambda alpha frac gamma frac alpha gamma frac n alpha