Nonlinear Inequalities with Double Riesz Potentials

We investigate the nonnegative solutions to the nonlinear integral inequality u ≥ Iα ∗((Iβ ∗ up)uq) a.e. in ℝN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathbb R}^{N}$\end{document}, where α, β ∈ (0, N), p, q > 0 and Iα, Iβ denote the Riesz potentials of order α and β respectively. Our approach relies on a nonlocal positivity principle which allows us to derive optimal ranges for the parameters α, β, p and q to describe the existence and the nonexistence of a solution. The optimal decay at infinity for such solutions is also discussed.

By a nonnegative solution of (1) we understand a function u ∈ L 1 loc (R N ), u ≥ 0, such that the right hand side of (1) is well-defined, i.e.
Integral inequalities and equations featuring a single Riesz potential have been extensively investigated in the past two decades.The prototype model has been largely investigated starting with the seminal works in [3,4].More recently, various techniques have been devised to deal with systems of equations or inequalities that incorporate anisotropic or more general potentials [2,6,9,14,15].Our aim in this paper is to provide an optimal description for the existence and nonexistence of positive solutions to the integral inequality (1).
For any α > 0, the fractional Laplacian (−∆) α/2 is defined by means of the Fourier transform for all u ∈ S ′ such that |ξ| α u ∈ S ′ , here S ′ stands for the space of tempered distributions on R N which is the dual of the Schwartz space S.
Since for α ∈ (0, N ) the Riesz potential I α can be interpreted as the inverse of (−∆) α/2 (cf.[13,Sect.5.1] or [2, Section 2.1]), under some extra integrability conditions on u ≥ 0, inequality (1) is equivalent to the elliptic inequality provided that both (1) and ( 4) are well-defined.This is the case, for instance, if (3) holds and u belongs to the homogeneous Sobolev space Ḣα/2 (R N ), so that ( 4) is understood in the weak sense.Pointwise interpretations of the inequality (4) for non-integer α/2 are also possible, cf.[2, Theorem 2.13].For a comparison of different definitions of the higher order fractional Laplacian (−∆) α/2 see [1].Inequality (4) is a Choquard type inequality.Equations and inequalities of such structure originate from mathematical physics and have attracted considerable interest of mathematicians in the past decade, see [12] for a survey.In the 2nd order elliptic case α = 2 optimal regimes for the existence and nonexistence of positive solutions to inequality (4) were fully investigated in [11].The higher-order polyharmonic case α/2 = m ∈ N was recently studied in [5], where (amongst other results) optimal existence and nonexistence regimes for the equation (4) were obtained for the exponents p ≥ 1 and q > 1, see [5,Theorem 1.4].
In this work we extend earlier results in [11] and [5] to the full admissible range α ∈ (0, N ) and exponents p, q > 0. Our approach is different from the techniques in [5], which were based on the poly-superharmonic properties of (−∆) m in the elliptic framework of equation (4).Instead, we work entirely with the double-nonlocal inequality (1).Our analysis of (1) employs only direct Riesz kernel estimates, and a new version of the nonlocal positivity principle in Lemma 3.1, inspired by [11,Proposition 3.2].This has the advantage of incorporating the fractional case of noninteger α/2 in a seemingly effortless way, and does not rely on comparison principles or Harnack type inequalities, which are commonly used for similar Liouville type results in the elliptic framework, but which are generally speaking not available in the case of the higher-order fractional Laplacians (−∆) α/2 with α > 2.
The main result of this work related to the existence of positive solutions to (1) reads as follows.
Theorem 1.1.Let p, q > 0.Then, inequality (1) has a nontrivial nonnegative solution The necessary part of the proof follows directly from Propositions 2.3, 4.1-4.5 below.The sufficiency follows from Propositions 5.4-5.7,where we construct explicitly smooth positive radial solutions to (1).In the case α = 2 our results are fully consistent with the results established in [11,Theorem 1] for the 2nd order elliptic inequality (4).The nonexistence of positive solutions to double-nonlocal inequality (1) with p > 1, q > 0 and p + q ≤ N +β N −α was established by different methods in [8, Theorem 1].

Preliminaries
In this section we collect some useful facts for our approach.
Lemma 2.1.Let f : R N → R be a nonnegative measurable function.Then, the Riesz potential I α * f of order α ∈ (0, N ) is well defined, in the sense that Moreover, if (7) fails then I α * f = +∞ everywhere in R N , see [7, p.61-62].We present the proof of the lemma for completeness.
Proof.Assume first that (7) holds.Then, for any x, y ∈ R N , x = 0 we have Thus, for any x ∈ R N \ {0} such that (7) holds, we have which yields (8).
One of the elementary yet important for our approach consequences of ( 9) is the following estimate, which we will be using frequently, and which to some extent is the counterpart of the Harnack inequalities on the annuli in the study of (4).Lemma 2.2.Let α ∈ (0, N ), θ > 0 and 0 ≤ f ∈ L 1 (1 + |x|) −(N −α) dx, R N .Then for all R > 0 we have (10) Proof.Follows from (9) by integration.
An obvious implication of ( 9) is that I α * f can not decay faster than I α at infinity, even if the function f is compactly supported.Recall also that if f ≥ 0 then an elementary estimate shows that for every x ∈ R N , ( 11) As a consequence, if u ≥ 0 is a solution of (1) that is positive on a set of positive measure, then u is everywhere strictly positive on R N and the following lower bounds must hold: On the other hand, (2) requires Combining the competing upper and lower bounds immediately leads to the following nonexistence result.
Remark 2.4.We will see in Proposition 4.3 below that q ≤ β N −α − 1 is suboptimal for the nonexistence and could be refined.

Nonlocal positivity principle
The nonexistence result in Proposition 2.3 "decouples" the values of p and q.In order to deduce an estimate which involves the quantity p + q which appears in Theorem 1.1, we need the following lemma, inspired by [10, Proposition 2.1] and [11,Section 3].
Then for every R > 0 and 0 Proof.Take ψ := ϕ 2 u as a test function in (16).Then which completes the proof.
Remark 3.2.Nonlocal inequality (16) can be interpreted as the "inversion" of the fractional Schrödinger inequality In this context Lemma 3.1 can be seen as a higher-order version of the fractional Agmon-Allegretto-Piepenbrink's positivity principle: if (16) has a positive supersolution then a certain variational inequality which involves the potential V must hold.We will see that Lemma 3.1 alongside with the standard integral estimate (10) of the Riesz potentials are sufficient for the complete analysis of the existence and nonexistence of positive solutions of the nonlinear inequality (1).
Using Lemma 3.1 we establish the following estimate.
Proposition 3.3.Let p, q > 0 and u > 0 be a solution of (1).Then, for every R > 0 and every Proof.For every ϕ ∈ C ∞ c (B R ), by Lemma 3.1 with V = (I β * u p )u q−1 , and using a similar inequality to (18) for I β * u p , we obtain One of the principal tools in the subsequent analysis is the following decay estimate on the solutions of (1), which is an adaptation of (19).Note that for q < 1 our estimate contains a lower bound on the solution, since the 2nd integral involves a negative power of u.Corollary 3.4.Let p, q > 0 and u > 0 be a solution of (1).Then for every R > 0, (20)

Nonexistence
In this section we derive several nonexistence result for (1).Our approach is inspired by [11] which studied the inequality (4) in the semilinear the case α = 2, yet with substantial modifications.In particular, in this work we completely avoid the use of the comparison principle and Harnack's inequalities, which are not applicable in the framework of the doublenonlocal inequality (1).It turns out that Harnack inequality estimates in the context of (1) can be replaced by the estimate (10).Proposition 4.1.Let p, q > 0 and assume that p + q < 1.If u ≥ 0 is a solution of (1) then u ≡ 0.