Harnack‐type inequality for fractional elliptic equations with critical exponent

Abstract

In this paper, we use the method of moving planes to derive the Harnack‐type inequality maxBRu·minB2Ru≤CRN−2α, for nonnegative solutions to fractional semilinear elliptic equations false(−normalΔfalse)αu=ffalse(ufalse),1em1emx∈B3Rfalse(0false),ufalse(xfalse)>0,1em1emx∈B3Rfalse(0false),ufalse(xfalse)=0,1em1emx∈double-struckRN∖B3Rfalse(0false), with 0<R<1, 0<α<1, and 2α<N. We obtain an asymptotic blow‐up rate estimate, as an application, for solutions to the above equation with a singular set.

Monotonicity and Symmetry of Solutions to Fractional Laplacian in Strips

“…Due to applications in physics, chemistry, biology, probability, and finance, differential equations involving the fractional Laplacian ð−ΔÞ α/2 have received growing attention from the mathematical communicity in recent years (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14]). There are many papers devoted to the study of qualitative properties of fractional Laplacian equations in bounded or unbounded domains, but seldom are concerned with slab-like domains.…”

“…It is not assumed that Ω is bounded. The function f appearing in (5) will always be assumed to be (globally) Lipschitz continuous. We firmly believe that the result introduced here is of great importance, and the ideals and methods can be applied to study a variety of nonlocal problems with more general operators and nonlinearities.…”

Existence of Solutions to Elliptic Problem with Convection Term and Lower-Order Term

“…The constant C E lies on E, α, and Ω. Some other results about noncoercivity elliptic problems see [4][5][6][7][8][9][10][11][12][13][14][15][16] and reference therein.…”

Non-existence of solutions to some degenerate coercivity elliptic equations involving measures data

“…Note that, in problem (12), they required that γ > 1. It is worth pointing out that different ranges of γ have an important impact on the behavior of solutions to problem (12), more details see [25,18,1,17,13,4].…”