Regularity for mixed-order nonlinear fractional equations with degenerate coefficients^†

Abstract

We consider a class of nonlinear integro-differential equations whose leading operator is modeled on a superposition of (−∆_p)^s and (−∆_p)^t, where 0 < s < t < 1 < p < ∞, weighted via two possibly degenerate coefficients a(·, ·) ≥ 0 and b(·, ·) ≥ 0, respectively. We prove local boundedness and Hölder regularity of its weak solutions under natural assumptions on the coefficients a(·, ·), b(·, ·) and the powers s, t, p. Moreover, when a(·, ·) ≡ 1, we also prove a Harnack inequality for weak solutions.

Keywords

nonlinear nonlocal operator / mixed order / degenerate coefficient / regularity / Harnack inequality

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