Regularity of extremal solutions of nonlocal elliptic systems

Abstract

We examine regularity of the extremal solution of nonlinear nonlocal eigenvalue problem    Lu = λF (u, v) in Ω, Lv = γG (u, v) in Ω,with an integro-differential operator, including the fractional Laplacian, of the formwhen J is a nonnegative measurable even jump kernel. In particular, we consider jump kernels of the form of J(y) = a(y/|y|) |y| n+2s where s ∈ (0, 1) and a is any nonnegative even measurable function in L 1 (S n−1 ) that satisfies ellipticity assumptions. We first establish stability inequalities for minimal solutions of the above system for a general nonlinearity and a general kernel. Then, we prove regularity of the extremal solution in dimensions n < 10s and n < 2s + 4s p∓1 [p + p(p ∓ 1)] for the Gelfand and Lane-Emden systems when p > 1 (with positive and negative exponents), respectively. When s → 1, these dimensions are optimal. However, for the case of s ∈ (0, 1) getting the optimal dimension remains as an open problem. Moreover, for general nonlinearities, we consider gradient systems and we establish regularity of the extremal solution in dimensions n < 4s. As far as we know, this is the first regularity result on the extremal solution of nonlocal system of equations.

A universal Hölder estimate up to dimension 4 for stable solutions to half-Laplacian semilinear equations

“…The estimates for stable solutions in [21] depend on f , in contrast with ours, and do not provide any estimate for n = 1 if s is small. The results from [21] have been recently extended to the case of systems of two equations by Fazly [12].…”

On finite Morse index solutions of higher order fractional elliptic equations

“…Also, the finite Morse index solutions to (1.1) with 0 < s < 1 and λ > 0 was examined in our previous work [25]. On the other hand, Fazly [15] examined regularity of the extremal solution of nonlinear nonlocal eigenvalue system with an integro-differential operator, including the fractional Laplacian. They established various stability inequalities for minimal solutions of systems with a general nonlocal operator with a nonnegative measurable even jump kernel.…”