Stable solution and extremal solution for the fractional p-Laplacian equation

To our knowledge, this paper is the first attempt to consider the stable solution and extremal solution for the fractional p-Laplacian equation: (-) ₚˢ u= f (u), \; u> 0 ~in~;\; u=0\;in~ RN, where p>1, s (0, 1), N>sp, >0 and is a bounded domain with continuous boundary. We first construct the notion of stable solution, and then we prove that when f is of class C¹, nondecreasing and such that f (0) >0 and t f (t) t^{p-1}=, there exists an extremal parameter ^* (0, ) such that a bounded minimal solution u_ exists if (0, ^*), and no bounded solution exists if >^*, no W₀^s, p () solution exists if in addition f (t) ^1{p-1} is convex. Moreover, this family of minimal solutions are stable, and nondecreasing in, therefore the extremal function u^*: =^*u_ exists. For the regularity of the extremal function, we first show the Lʳ-estimates for the equation (-) ₚˢu=g with g W₀^s, p () ^* Lq (), q 1. When f is a power-like nonlinearity, we derive the W₀^s, p () regularity of u^* in all dimension and L^ () regularity of u^* in some low dimensions. For more general nonlinearities, when f is class of C² and such that some convexity assumptions, then u^* W₀^s, p () if Np-2p-1, the results above can be improved as: u^* W₀^s, p () for all dimensions, and u^* L^ () if N<sp+4spp-1.