Existence of positive weak solutions for fractional Lane–Emden equations with prescribed singular sets

Abstract

In this paper, we consider the problem of the existence of positive weak solutions of $$\begin{aligned}{\left\{ \begin{array}{ll} (-\Delta )^s{u}=u^p&{}\text {in}~\Omega \\ u=0&{}\text {on}~{\mathbb R}^n \backslash \Omega \end{array}\right. }\\\end{aligned}$$ having prescribed isolated interior singularities. We prove that if $$\frac{n}{n-2s}<p<p_1$$ for some critical exponent $$p_1$$ defined in the introduction which is related to the stability of the singular solution $$u_s$$ , and if S is a closed subset of $$\Omega $$ , then there are infinitely many positive weak solutions with S as its singular set. We also show the existence of solutions to the fractional Yamabe problem with singular set to be the whole space $${\mathbb R}^n$$ . These results are the extension of Chen and Lin’s result (Chen and Lin in J Geom Anal 9(2):221–246, 1999) to the fractional case.