Some remarks about the summability of nonlocal nonlinear problems

Abstract

In this note, we will study the problemwhere < s < , (−∆) s p is the nonlocal p-Laplacian de ned below, Ω is a smooth bounded domain. The main point studied in this work is to prove, adapting the techniques used in [ ] for the case p = to the general case p ∈ ( , +∞), the summability of the nite energy solutions in terms of the summability of a source term f(x). The aim of this note is to present the results in a way as elementary as possible.

Caffarelli–Kohn–Nirenberg type inequalities of fractional order with applications

“…Proof. The case β = 0 is obtained in [21] if p = 2 and in [6] if p = 2. For the reader convenience we include some details for the general case β = 0.…”

Fractional superharmonic functions and the Perron method for nonlinear integro-differential equations

“…Much has been written about this topic and the connection with the theory of degenerate elliptic equations; we refer the reader to the exhaustive book [15] by Heinonen, Kilpeläinen and Martio, and to the useful lecture notes [31] by Lindqvist. However -though many important physical contexts can be surely modeled using potentials satisfying the Laplace equation or via partial differential equations as in (1) with the leading term given by a nonlinear operator as for instance the p-Laplacian with coefficients -other contexts, as e. g. from Biology and Financial Mathematics, are naturally described by the fractional counterpart of (1), that is, the fractional Laplacian operator (−∆) s . Recently, a great attention has been focused on the study of problems involving fractional Sobolev spaces and corresponding nonlocal equations, both from a pure mathematical point of view and for concrete applications, since they naturally arise in many contexts when the interactions coming from far are determinant 1 .…”

Some integrability estimates for solutions of the fractional p-Laplace equation

“…Following [1], we can also give the proofs with a stampacchia's type result. We omit here for their similarity.…”

“…The Dirichlet boundary problem of L α,2 has been intensively investigated and many fundamental results have been proved, we refer the reader to [2,4,8,12,14] and the references therein for a fuller treatment of this topic. As a nonlinear generalization of L α,2 , L α,p has been extensively explored in recent years ( [1,5,10]). …”

“…In [1], Barrios et al studied the summability of the finite energy solutions to (1.1) in terms of the summability of g for αp < n by adapting the ideas used in [12] for p = 2. In this paper, highly inspired by their methods and some known estimates for bounded Lipschitz domain, we obtain the following results for αp n.…”