A Liouville theorem for indefinite fractional diffusion equations and its application to existence of solutions

Abstract

In this work we obtain a Liouville theorem for positive, bounded solutions of the equationwhere (−∆) s stands for the fractional Laplacian with s ∈ (0, 1), and the functions h and f are nondecreasing. The main feature is that the function h changes sign in R, therefore the problem is sometimes termed as indefinite. As an application we obtain a priori bounds for positive solutions of some boundary value problems, which give existence of such solutions by means of bifurcation methods.

Fractional equations with indefinite nonlinearities

“…(ii) In the subcritical case 1 < p < n+α n−α , u ≡ 0. This greatly improves the result in Corollary 1 by extending the range of α from [1,2) to (0, 2).…”

“…Remark 1. (i) Upon the completion of the work, we noticed that a similar theorem was obtained in [1]. However, our method is different than their's and our conditions are weaker.…”

“…In order the moving of plane has a stating point, in [1], the authors simple assumed that a(t) ≤ 0, for t ≤ 0 (8) under which they were able to apply a maximum principle to show that…”

Nonexistence of solutions for indefinite fractional parabolic equations

“…Subsequently, Barrios etc. [3] generalized this result to 0 < s < 1 by the method of moving planes in integral forms, where a and f are nondecreasing functions satisfying some additional conditions. At almost the same time, Chen, Li, and Zhu [15] proved the Liouville theorem for (1.3) by a direct method of moving planes under much weaker conditions than that in [3].…”

Monotonicity of solutions for fractional parabolic equations

“…For more results concerning the parabolic equations, please refer to e.g. [2][3][4]6,11,16,17] and the references therein.…”