Liouville theorem for poly-harmonic functions on $${{\mathbb {R}}}^{n}_{+}$$

Super poly-harmonic properties, Liouville theorems and classification of nonnegative solutions to equations involving higher-order fractional Laplacians

“…In this section, we will prove Theorem 1.8 by way of contradiction and the method of scaling spheres developed by Dai and Qin [21] (see also [22,23,25]). For more related literature on the method of moving planes (spheres), we refer to [1,2,3,5,6,7,8,9,11,13,16,17,18,20,24,26,28,29,30,32,36,37] and the references therein. Now suppose, on the contrary, that u ≥ 0 satisfies integral equations (1.7) but u is not identically zero, then there exists a ponit x ∈ R n such that u(x) > 0.…”

“…First, we will investigate the super poly-harmonic properties for nonnegative solutions to (1.1). It is well known that the super poly-harmonic properties of nonnegative solutions play a crucial role in establishing the integral representation formulae, Liouville type theorems and classification of solutions to higher order PDEs in R n or R n + (see [1,2,3,4,10,17,18,19,20,22,24,25,29,31,36] and the references therein).…”

“…For integer higher-order equations (i.e., α = 0 in (1.1)), the super poly-harmonic properties for nonnegative solutions usually can be derived via the "spherical average, re-centers and iteration" arguments in conjunction with careful ODE analysis (we refer to [4,29,31,36], see also [2,3,10,19,22,24] and the references therein). However, for the fractional higher-order equation (1.1), so far there is no result on the super poly-harmonic properties.…”

“…For PDEs (1.1) and IEs (1.7), we say the Hardy-Hénon type nonlinearities f (x, u) = |x| a u p is subcritical if 0 < p < p c (a) := n+2m+α+2a n−2m−α , critical if p = p c (a) and super-critical if p > p c (a). There are also lots of literature on Liouville type theorems for fractional order or higher order Hardy-Hénon type equations in the subcritical cases, and we refer to [3,4,5,6,11,17,18,20,21,23,24,25,29,33,36,38] and the references therein. It should be noted that, all the known results focused on the cases m = 0 or α = 0, hence Liouville type theorems for general fractional higher-order cases m ≥ 1 and 0 < α < 2 are still open.…”

Maximum principles and the method of moving planes for the uniformly elliptic nonlocal Bellman operator and applications

“…Later, it was further developed by Serrin [61], Gidas, Ni and Nirenberg [52], Caffarelli, Gidas and Spruck [18], Chen and Li [21], Li [57], Lin [58], Chen, Li and Ou [25] and many others. For more literatures on the methods of moving planes, see [3,4,11,15,16,19,20,22,24,29,34,38,39,41,42,43,44,45,55,56,59,64] and the references therein.…”