Nonexistence of positive solutions for an indefinite fractional p-Laplacian equation

In this paper, we consider the following fractional p-Laplacian equation with indefinite nonlinearities (-Delta)(p)(s)u(x) = x(1)u(q)(x), x is an element of R-n, where 0 < s < 1, p = q+1 > 2, we use the direct method of moving planes to prove the solution of above fractional p-Laplacian equation is monotone increasing along x(1) direction, then obtain that the above equation possess no positive bounded solution on the whole space. There is no decay condition at infinity on solution for the fractional p-Laplacian, and the traditional approaches of translation and taking limit there no longer work for the nonlocal nonlinear fractional p-Laplacian here. We also cannot find a suitable auxiliary function for this fractional p-Laplacian. To circumvent these difficulties, we use the following two estimate to study the property of solution. (I) We estimate the singular integrals defining the fractional p-Laplacian along a sequence of approximate maxima; (II) We estimate the lower and upper bound of the fractional p-Laplacian equation. (C) 2020 Elsevier Ltd. All rights reserved.