Symmetry of solutions to singular fractional elliptic equations and applications

Abstract

In this article, we study the symmetry of positive solutions to a class of singular semilinear elliptic equations whose prototype is \begin{align*} (P) \quad \left\lbrace \begin{array}{ll} (-\Delta )^{s}u = \frac{1}{u^\delta } + f(u), \; u>0\quad & \text{ in }\Omega ; \\ u=0 & \text{ in } \mathbb{R}^n\setminus \Omega ,\\ \end{array} \right. \end{align*} where $0, $n\ge 2s$, $\Omega = B_r(0) \subset \mathbb{R}^n, \; \delta >0$, $f(u)$ is a locally Lipschitz function. We prove that classical solutions are radial and radially decreasing (see Theorem 1). The proof uses the moving plane method adapted to the non local setting. We then give two applications of this main result: Theorem 2 establishes the uniform apriori bound for classical solutions in case of polynomial growth nonlinearities whereas Theorem 3 ensures in case of exponential growth nonlinearities the convergence of large solutions with unbounded energy to a singular solution.