Local Analysis of Solutions of Fractional Semi-Linear Elliptic Equations with Isolated Singularities

Abstract

In this paper, we study the local behaviors of nonnegative local solutions of fractional order semi-linear equations (−∆) σ u = u n+2σ n−2σ with an isolated singularity, where σ ∈ (0, 1). We prove that all the solutions are asymptotically radially symmetric. When σ = 1, these have been proved in [3] by Caffarelli, Gidas and Spruck .

Liouville theorem and isolated singularity of fractional Laplacian system with critical exponents

“…Together with blow up analysis and the Pohozaev integral, we get the upper and lower bound of the local solutions in B 1 \{0}. Our results is an extension of the classical work by Caffarelli et al [6,7], Chen et al [16] .…”

“…For more details about the Liouville Theorem, please see [13,15] and the references therein. Caffarelli-Jin-Sire-Xiong [7] studied the global behaviors of positive solutions of the fractional Yamabe equations…”

Harnack‐type inequality for fractional elliptic equations with critical exponent

“…[3][4][5][6][7][8][9][10][11] There are some results about the Harnack-type inequality present in the literature for nonlocal fractional elliptic problem (1.1). In Caffarelli et al, 12 via the extension formulations for fractional Laplacian, 13 moving spheres method, and blow-up analysis, they established the Harnack-type inequality of positive solutions to { −div(t 1−2 ∇U) = 0,…”

Classification of isolated singularities of nonnegative solutions to fractional semi-linear elliptic equations and the existence results

“…Involving the fractional Laplacian defined by harmonic extension, Caffarelli, Jin, Sire and Xiong in classified the isolated singularities of solutions to by considering the related harmonic extended problem. This harmonic extension, also called Dirichlet to Neuman map, provides a monotonicity formula for Lane–Emden equation in the fractional Laplacian setting, see (see also for bi‐Laplacian problem).…”

“…Furthermore, u k is a very weak solution of −Δu + u p = kδ 0 in Ω, u = 0 on ∂Ω (1.5) and u ∞ is the limit of {u k } k as k → +∞. Involving the fractional Laplacian defined by harmonic extension, Caffarelli, Jin, Sire and Xiong in [7] classified the isolated singularities of solutions to…”