Polyharmonic inequalities with nonlocal terms

We study the existence and non-existence of classical solutions for inequalities of type $$ \pm \Delta^m u \geq \big(\Psi(|x|)*u^p\big)u^q \quad\mbox{ in } {\mathbb R}^N (N\geq 1). $$ Here, $\Delta^m$ $(m\geq 1)$ is the polyharmonic operator, $p, q>0$ and $*$ denotes the convolution operator, where $\Psi>0$ is a continuous non-increasing function. We devise new methods to deduce that solutions of the above inequalities satisfy the poly-superharmonic property. This further allows us to obtain various Liouville type results. Our study is also extended to the case of systems of simultaneous inequalities.


Introduction and the main results
In this paper we are concerned with the following elliptic inequality Typical examples of functions Ψ include: The operator * in (1.1) is the standard convolution operator, that is, By a non-negative solution u of (1.1) we understand a function u ∈ C 2m (R N ), u ≥ 0, such thatˆ| y|>1 u p (y)Ψ |y| 2 dy < ∞ (1. 3) and u satisfies (1.1) pointwise. Note that condition (1.3) is weaker than the condition u ∈ L p (R N ) and that (1.3) is (almost) necessary and sufficient in order to ensure that the convolution term Ψ(|x|) * u p is finite for all x ∈ R N . Indeed, for any x ∈ R N we have Ψ(|x|) * u p =ˆ| y|≤2|x| u p (y)Ψ(|y − x|)dy +ˆ| y|>2|x| u p (y)Ψ(|y − x|)dy ≤ max |z|≤2|x| u(z) pˆB 3|x| Ψ(|y|)dy +ˆ| y|>2|x| u p (y)Ψ |y| 2 dy < ∞, by the fact that Ψ(|x|) ∈ L 1 loc (R) is non-increasing and (1.3). We prefer to separate the analysis of (1.1) into two distinct inequalities as follows: and We study first the inequality (1.4). Our main result in this case reads as follows.
The above result is new even in the semilinear case m = 1.
We are next concerned with the inequality (1.5). The related semilinear problem − ∆u ≥ |x| −α * u p u q in R N (1. 8) was completely investigated in [20]. The equality case in (1.8) bears the name Choquard-Pekar (or simply Choquard) equation and was introduced in [25] as a model in quantum theory. Since then, the prototype model (1.8) has been used to describe many phenomena arising in mathematical physics (see, e.g., [15,19,21] for further details). Quasilinear versions of (1.8) (including the case of p-Laplace or mean curvature operator) are discussed in [2,9,10]. Also, singular solutions of (1.8) are considered in [3,4,8,11,12]. It is obtained in [20,Theorem 1] (see also [9,Corollary 2.5] for an extension to quasilinear inequalities) that if p ≥ 1 and q > 1 then (1.8) has non-negative solutions if and only if An important matter in the study of polyharmonic problems is whether non-negative solutions of (−∆) m u ≥ f (u) in R N enjoy the so-called poly-superharmonic property, that is, whether (−∆) j u ≥ 0 in R N for all 1 ≤ j ≤ m. This has been shown to be true under some general conditions for nonlinearities f (u) without non-local terms (see, e.g. [5,14,16,17,22]). In [14,17] the authors use a contradiction argument and construct a suitable sequence of power functions acting as lower barriers for the spherical average of the solution u. In turn, the approach in [5,16] relies on a re-centering argument which at each step in the proof one brings forward a new center of the spherical average of the functions (−∆) j u.
In this paper we show that the poly-superharmonic property is still preserved in case of polyharmonic inequalities of type (1.5) in the presence of convolution terms. More precisely we have: Theorem 1.2. Assume N, m ≥ 1 and either p + q ≥ 2 or p ≥ 1.
If u ∈ C 2m (R N ) is a non-negative solution of (1.5), then, for all 1 ≤ j ≤ m we have Our approach is different from the methods already devised in [5,16,17,22]. More precisely, if p + q ≥ 2 we use a general integral estimate which we establish in Lemma 3.1 below. On the other hand, if p ≥ 1, we exploit the integral condition (1.3) to construct our argument.
Particularly useful are the integral representation formulae obtained in [1] (see also [6,7]) for solutions of (−∆) m = µ in D ′ (R N ), where µ is a Radon measure on R N , N > 2m. Such an approach was recently adopted in [24] for the polyharmonic Hardy-Hénon equation (−∆) m u = |x| σ u p in R N , p > 1. We believe that this is still an underexploited direction of research in the study of higher order elliptic equations and inequalities. We shall make use of these facts which we recall in Proposition 2.3.
We next focus on Liouville-type results for the inequality (1.5).
Theorem 1.3. Assume N, m ≥ 1 and let u ∈ C 2m (R N ) be a non-negative solution of (1.5).
(ii) If N > 2m and one of the following conditions holds: then, u ≡ 0.
has non-negative non-trivial solutions if and only if (1.11) In the particular case m = 1, Theorem 1.4 retrieves the result in [20, Theorem 1] which yields the optimal condition (1.9) for the existence of non-negative solutions of the semilinear inequality (1.8).
Assume next that the adjacency matrix (e ij ) is given by

Preliminaries
In this section we collect some auxiliary results which will be useful in our proofs.
A crucial result in our approach is the following representation formula for distributional solutions of the polyharmonic operator.
and for a.e. x ∈ R N we have is a distributional solution of (2.1), essinf u = ℓ and u is weakly polysuperharmonic in the sense that Using Proposition 2.3 we deduce: where p ≥ 1 and Ψ is a function which satisfies (1.2). Then, essinf v = 0 and for some constant (2.6) Take R > |x|. By the property (1.2) on Ψ we have as R → ∞. Thus, we may further estimate in (2.6) to deduce If p = 1 we simply use the property (1.2) and (1.3) to estimatê 3 Proof of Theorem 1.1 We first establish an estimate which holds for the general inequality (1.1).
Then, there exists C > 0 such that any non-negative solution u of (1.1) satisfieŝ (3.1) Proof. It is easy to check that where C > 0 is a positive constant. This yields We multiply by ϕ 2 in (1.1) and integrate. Using the above estimate we find We next estimate the left-hand side of (3.2). By inter-changing the variables and Hölder's inequality one gets where c > 0 is a constant. Now, splitting the integrals according to x and y variables we deduceˆR Using this last inequality in (3.2) we deduce (3.1).
Using this last estimate in (3.1) we find In particular, since ϕ = 1 on B R we deducê By Lemma 2.2 it now follows that u ≡ 0 which concludes our proof in this case.
Case 2: N > 2m and p ≥ 1.We apply Lemma 2.4 for v = −u. It follows in particular that v = −u ≥ 0 which yields u ≡ 0.
4 Proof of Theorem 1.2 The proof of Theorem 1.2 follows from the result below which will also be useful in the study of the system (1.12).
Step 1: , v i (r)) the spherical average of u and v (resp u i and v i ) on the sphere ∂B r (x 0 ), that is,

By integration one gets
for all r ≥ 0. We can rewrite the last estimate as Case 1: m is odd. From (4.5) one has ∆ m−1ū (r) ≤ ∆ m−1ū (0) < 0 for all r ≥ 0.
Integrating twice the above inequality we obtain for all r ≥ 0 and proceeding further we deducē which contradicts the fact that u ≥ 0. Case 2: m is even. Hence m ≥ 2. From (4.5) we find ∆ m−1ū (r) ≥ ∆ m−1ū (0) > 0 for all r ≥ 0.
In the same manner as we derived (4.6) it follows that for any 1 ≤ i ≤ m one has In particular, for i = m we find Since ∆ m−1ū (0) > 0 and m ≥ 2 it follows from the above estimate that u(r) ≥ C 1 r 2(m−1) − C 2 for all r ≥ 0, (4.7) for some constants C 1 , C 2 > 0. Case 2a: Assume that (4.2) holds. Let ϕ be as in Lemma 3.1. In the same way as in the proof of (3.1) we have where τ = p 1 + q 1 ≥ 2 and θ = p 2 + q 2 ≥ 2.
By Hölder's inequality we find (4.9) From (4.8) and (4.9) we deduce (4.10) We use the second estimate of (4.10) in the first one to obtain which we arrange as Using (4.7) we find where C 3 , C 4 > 0 and σ N denotes the surface area of the unit sphere in R N . Comparing the exponents of R in the above inequality we raise a contradiction, since C 3 > 0. This finishes the proof of Step 1 in this case. Case 2b: Assume that (4.3) holds. From (4.7) one can find r 0 > 0 and a constant c > 0 such thatū (r) ≥ cr 2(m−1) for all r ≥ r 0 . (4.11) Using the fact that r N Ψ(r) → ∞ as r → ∞, by taking r 0 > 1 large enough we may also assume that To raise a contradiction, we next return to condition (1.3) for u (in which we replace p with p 2 ). From (4.11)-(4.12), co-area formula and Jensen's inequality we obtain: which is a contradiction and concludes the proof in Step 1.
Step 2: Suppose to the contrary that there exists x 0 ∈ R N so that u m−2 (x 0 ) < 0. We next take the spherical average with respect to spheres centred at x 0 and proceed as in Step 1 by discussing separately the cases m is odd and m is even in order to raise a contradiction. Thus, (−∆) m−2 u ≥ 0, (−∆) m−2 v ≥ 0 in R N . We proceed further until we get −∆u ≥ 0, −∆v ≥ 0 in R N .
Proof of Theorem 1.2. Let Ψ = Φ and (p 1 , q 1 ) = (p 2 , q 2 ) = (p, q). Suppose u is a nonnegative solution of (1.5). If u ≡ 0, then the conclusion clearly holds. If u ≡ 0, then (u, u) is a non-negative solution of (4.1). By Theorem 4.1 we see that, for all 1 ≤ i ≤ m, 5 Proof of Theorem 1.3 (i) By Theorem 1.2 we see that, for 1 ≤ j ≤ m, (−∆) j u ≥ 0 in R N . In particular −∆u ≥ 0 in R N . Since N = 1, 2, it is well known that a nonnegative superharmonic function is constant. Thus, u = c in R N . By (1.5) it follows that (Ψ(|x|) * u p )u q = 0 in R N . This clearly yields u = 0 in R N , otherwise there would exist x 0 ∈ R N such that u(x 0 ) > 0 and hence (Ψ(|x|) * u p )u q > 0 at x 0 .
Assume now that (1.11) holds and let us construct a positive solution to (1.10). First, we write (1.11) in the form Thus, we can choose κ ∈ (0, N − 2m) such that Then, where b j (a) ∈ R. In particular, for a = 0 we find On the other hand, by direct computation one has since 0 < κ < N − 2m. Comparing (6.2) and (6.3) we find b 2m (0) > 0. By the continuous dependence on the data, we can find now a > 0 such that b 2m (a) > 0. Also, Thus, there exist c > 0 and R > 1 such that Let now v(x) = (a + |x| 2 ) −κ/2 , where a > 0 satisfies b 2m (a) > 0. By the above estimates we have where γ 0 > 0 is a normalizing constant such that and δ 0 denotes the Dirac mass concentrated at the origin.
Thus, V ∈ C 2m (R N ), V > 0 in R N and from (6.4) we have Also, by taking M > 1 large enough we have Observe that for x ∈ R N \ B 4R and y ∈ B 2R we have |x − y| ≥ |x| − |y| ≥ |x|/2. Thus, Using this estimate in the definition of V together with 0 < κ < N − 2m it follows that for some constant C 0 > 0. We next evaluate the convolution term (|x| −α * V p )V q and indicate how to construct a positive solution to (1.10). Using (6.7) we can apply Lemma 2.1 for f = V p , β = κp > N − α and ρ = R/2. It follows that for any x ∈ R N \ B R we have Using this last estimate together with (6.5) and (6.1) 1 , (6.1) 3 we deduce for some C 1 > 0. Since (−∆) m V and (|x| −α * V p )V q are continuous and positive functions on the compact B R (see (6.6)), one can find C 2 > 0 such that Thus, letting C = min{C 1 , C 2 } > 0 and U = C 1/(p+q−1) V , it follows that U ∈ C 2m (R N ) is positive and that from (6.8)-(6.9) one has which concludes our proof.
7 Proof of Theorem 1.5 and Theorem 1.7 Proof of Theorem 1.5. Let L > 0 denote the positive limit in (1.16). Thus, one can find an increasing sequence {R i } ⊂ (0, ∞) that tends to infinity and such that for all i ≥ 1 one has Let φ be the positive eigenfunction of −∆ in the unit ball B 1 corresponding to the eigenvalue λ 1 > 0. We normalize φ such that 0 ≤ φ ≤ 1 in B 1 and max B 1 φ(x) = 1. Let ϕ i (x) = φ(x/R i ). Multiplying by ϕ i in the inequality of (1.12) that corresponds to u k we find where we used (−∆) m−1 u k ≥ 0 by Theorem 4.1 and that, by Hopf lemma, ∂ϕ i /∂n < 0 on ∂B R i . Proceeding further one findŝ Let us next estimate the integral in the left-hand side of (7.3). If x ∈ B R i , then one has Thus, by the fact that 0 ≤ ϕ i ≤ 1 one haŝ Let τ = p kℓ + q kℓ ≥ 2. By Hölder's inequality we have Again by Hölder's inequality we derivê (7.7) By (7.5), (7.6) and (7.7) we have Using this last estimate in (7.3) we deduce Ψ kℓ (|x|) * u p kℓ ℓ u q kℓ ℓ ∈ L 1 (R N ), and sô B R i \B R i /2 Ψ kℓ (|x|) * u p kℓ ℓ u q kℓ ℓ ϕ i → 0 as i → ∞. (7.10) We may estimate the above integral as we did in (7.4) to obtain Finally, we use (7.2) in the above inequality to deducê for large i, which contradicts (7.10) and concludes our proof.
The proofs of Corollaries 1.6 and 1.8 (i) follow immediately.
Proof of Theorem 1.7. The proof of Theorem 1.7 can be carried out in the same way as above. The only difference is that we cannot apply Theorem 4.1 to derive that u k and u ℓ satisfy (7.2). Instead, we apply Lemma 2.4 to deduce that u k and u ℓ are poly-superharmonic. Further, by the estimate (2.5) one has that (7.2) holds. From now on, we follow the above proof line by line.