Singular solutions for coercive quasilinear elliptic inequalities with nonlocal terms

We study the inequality $$ {\rm div}\big(|x|^{-\alpha}|\nabla u|^{m-2}\nabla u\big)\geq (I_\beta\ast u^p)u^q \quad\mbox{ in } B_1\setminus\{0\}\subset {\mathbb R}^N, $$ where $\alpha>0$, $N\geq 1$, $m>1$, $p, q>m-1$ and $I_\beta$ denotes the Riesz potential of order $\beta\in(0, N)$. We obtain sharp conditions in terms of these parameters for which positive singular solutions exist. We further establish the asymptotic profile of singular solutions to the double inequality $$ a(I_\beta\ast u^p)u^q\geq {\rm div}\big(|x|^{-\alpha}|\nabla u|^{m-2}\nabla u\big)\geq b(I_\beta\ast u^p)u^q \quad\mbox{ in } B_1\setminus\{0\}\subset {\mathbb R}^N, $$ where $a\geq b>0$ are constants.


Introduction and the main results
In this paper we are concerned with the following quasilinear elliptic inequality div |x| −α |∇u| m−2 ∇u ≥ (I β * u p )u q in B 1 \ {0} ⊂ R N , (1.1) and with the double inequality where α > 0, β ∈ (0, N ), m > 1, N ≥ 1, p > 0, q > m − 1 and a ≥ b > 0. Throughout this paper, B R (z) denotes the open ball in R N , N ≥ 1, with center at z ∈ R N and having radius R > 0. When z = 0, we simply use B R instead of B R (0).
The quantity I β * u p represents the convolution operation (I β * u p )(x) = B 1 I β (x − y)u p (y)dy, where I β : R N → R is the Riesz potential of order β ∈ (0, N ) given by By a positive solution of (1.1) we understand a function u ∈ W 1,m loc (B 1 \ {0}) ∩ C(B 1 \ {0}) which satisfies: • u > 0, u ∈ L p (B 1 ), div(|x| −α |∇u| m−2 ∇u), (I β * u p )u q ∈ L 1 loc (B 1 \ {0}); • for any φ ∈ C ∞ c (Ω), φ ≥ 0 we have Remark. Let us point out that the condition u ∈ L p (B 1 ) is needed to ensure I β * u p is finite almost everywhere. In fact, these two conditions are equivalent since for x ∈ B 1 \ {0} we have so u ∈ L p (B 1 ). Conversely, if u ∈ L p (B 1 ) then, by standard properties of convolution (see, e.g., [20,Chapter 2]) one has I β * u p ∈ L 1 (B 1 ). The study of quasilinear elliptic inequalities has received constant attention in the last decades, one general example is the inequality − div[A(x, u, ∇u)] ≥ f (x, u) in Ω, (1.4) which has appeared in many research papers under various structural hypotheses on A. The work by Mitidieri and Pohozaev [21] contains many results in this direction and provides the reader with a range of methods to investigate the nonexistence of a solution. The equality case in (1.4) naturally leads to a proper differential equation and has even a longer history. We only mention here the seminal work of Gidas and Spruck [19] for the semilinear case with power type nonlinearity but also some more recent results [10], [15], [23] dealing with other different situations. A systematic study of the inequality L A u = −div[A(x, u, ∇u)] ≥ |x| σ u q in Ω, along with the corresponding system is carried out in [2] for various domains Ω ⊂ R N , such as open balls and their complements, half balls and half spaces (see also [6] for the case of general nonlinearities). More recently, quasilinear elliptic inequalities and systems integrate the gradient term in the nonlinearity: the authors in [9] and [11] discuss coercive quasilinear inequalities in the form div(g(x)|∇u| p−2 ∇u) ≥ h(x)f (u)ℓ(|∇u|) in R N , and respectively div(h(x)g(u)A(|∇u|)∇u) ≥ f (x, u, ∇u) in R N .

Systems of quasilinear elliptic inequalities of type
are studied in [7] and [8] respectively. To the best of our knowledge, the first results dealing with quasilinear elliptic inequalities in the presence of nonlocal terms appear in [3]. The authors in [3] obtain local estimates and Liouville type results for where K ∈ L 1 loc (R N ), K ≥ 0 and q > 0. Extensions to these results were recently obtained in [14] in the case K(x) = |x| −β , β ∈ (0, N ). The related equation is known in the literature under the name of Choquard (or Choquard-Pekar) equation and arises in various fields ranging from quantum physics to one-component plasma and Newtonian relativity. A survey on the mathematical results on the Choquard equation is presented in [22]. Solutions to the Choquard equation featuring isolated singularities are studied in [4] and [5]. In [17] and [13] it is investigated the behaviour around the origin of singular solutions to respectively. Returning to inequality (1.1), we are now ready to state our first main result.
We next proceed to the study of the double inequality (1.2). To formulate our main result on (1.2) we introduce the exponent be the fundamental solution of the weighted m-Laplace operator for m > 1. Note that Φ m,α satisfies the distributional equality for some positive constant c.
Given two positive functions f, g defined on B 1 \ {0}, by f ≍ g we understand that the quotient f /g is bounded on B 1 \ {0} between two positive constants.
In case σp < N we have the following result on (1.2).
Our asymptotic behaviour (1.9)-(1.10) is in line with [24, Theorem 1.1] (see also [12,Theorem 2.1]) where the authors considered the equation It is obtained in [24, Theorem 1.1] that any singular solution u of (1.11) satisfies the following behaviour at the origin: where δ 0 denotes the Dirac delta mass concentrated at the origin.
In the case of (1.2) such exact behaviour seems difficult to capture due to the presence of the nonlocal term I β * u p . Our approach relies on establishing several a priori estimates for the behavior of the singular solutions to (1.1). These combine the Keller-Osserman type estimates (Proposition 2.5), the Harnack inequality (Propositions 2.2 and 2.3) and various estimates for the convolution term I β * u p . We collect all these results in the next section. Sections 3 and 4 contain the proofs of our main results.
Throughout this paper by c, C, C 1 , C 2 , ... we denote positive generic constants whose values may vary on each occasion. Also, all integrals are computed in the Riemann sense even if we omit the dx or dy symbol.

Preliminary Results
A key tool in our approach is the use of a priori estimates for solutions u ∈ W 1,m where Ω ⊂ R N is an open set and f ∈ L 1 loc (Ω), f ≥ 0. Solutions u of (2.1) are understood in the weak sense, that is, div |x| −α |∇u| m−2 ∇u ∈ L 1 loc (Ω) and Let u ∈ C(Ω) ∩ W 1,1 loc (Ω) be a positive solution of (2.1).
Then, for any ℓ > m − 1 there exists Λ = Λ(m, ℓ) such that for any λ > Λ there exists In the next results we recall the strong and the weak Harnack inequality for the weighted m-Laplace operator.

Proposition 2.3. (Weak Harnack inequality)
Let R > 0 and a, b, c be real numbers such that Then, for any ℓ > m − 1, there exists a constant C > 0 independent of R such that which satisfies the structural assumptions in [25]. Let Thus, Then, either u is bounded near the origin, or there exist C > 0 and r 0 ∈ (0, 1/2) such that where Φ m,α is defined in (1.7).
Proof. Assume that (2.7) does not hold. Hence, Then, for any k ≥ 1 there exists r k ∈ (0, 1/2), with r k → 0 as k → ∞, such that A comparison principle in the annular region B 1/2 \ B r k shows that for all k ≥ 1 we have Letting k → ∞ in the above estimate we deduce that u is bounded in the ball B 1/2 .
The result below provides a first important estimate for solutions to (1.1).
Proof. We use Proposition 2.
Similar to Proposition 2.5 we have: for some constant C > 0.
Since supp φ ⊂ B 4R \ B R/2 , from the above estimate and the weak Harnack inequality (2.6) with a = 7/4, b = 5/4 and c = 1/8 it follows that From here, we easily deduce (2.12). Then, Then, any solution of (1.1) is bounded around the origin.
Proof. We use some tools from [26, Proposition 1.2]. Let and let u be a positive solution of (1.1). We note that since u ∈ L p (B 1 ), u satisfies . Using Proposition 2.6 (with θ = 0 and being q ≥ ν > m − 1) we deduce In particular, again by q ≥ ν, it follows that for some C > 0.
In order to proceed to the proof of Proposition 2.8 we need the following result. . (2.18) Letting k → ∞, by Fatou's lemma we find By Hölder's inequality and since (u − M ) + |∇u| = (u − M ) + |∇(u − M ) + |, we estimate the right hand-side of (2.19) as We are now ready to proceed to the proof of Proposition 2.8 whose arguments will be divided into two steps.

(2.22)
By the definition of ζ k and the fact that |∇ζ k | ≤ ck we have Using this fact in (2.22) together with ζ 2k = 1 in A k and ζ k ≥ 0 in A 2k , we further estimate , (2.23) where in the last inequality we have used (2.18) with φ = ζ 2k and the fact that ∇ζ 2k = 0 in Hence, from (2.23) we deduce We now replace η in (2.21) by a sequence {η n } such that η n (t) → sign + (t) as n → ∞.
Letting n → ∞ and then k → ∞ in (2.21), since supp ζ k = B 2/3 and ζ k → 1 in B 1/2 , we find . We return to the estimate (2.23) and split our analysis into two cases.
• Case 2.2: ν < m. From (2.16) we have We now return to (2.21) and let k → ∞ to deduce Since η ≥ 0, it follows that u ≤ M in B 1/2 , so u ∈ L ∞ loc (B 1 ) which completes our proof.
Lemma 2.10. Let a, b ∈ (0, N ) and θ ≥ 0. Then, there exists C > c > 0 such that: The proof of the above lemma will be given in the Appendix.
(ii) Let u be a positive singular solution of (1.1). Using Propositions 2.4 and 2.5, there exists C > 0 such that for small R > 0 we find We claim that both inequalities are strict. Assume by contradiction that σ = N −m−α ≥ C|x| β−σp (by estimate (3.4)).
For any k ≥ 3 let v k ∈ C 1 (B 1/2 \ B 1/k ) be a radial function such that Observe that u is a subsolution while cΦ m,α is a supersolution of the above problem for suitable c > 0. By the maximum principle we find that k −→ v k is increasing and Also v is radial (since v k is radial) and from (3.5) we find Using this inequality it is easy to see that v satisfies the conditions of Proposition 2.2 with
To derive the first inequality in (1.5) we combine the weak Harnack inequality and Proposition 2.4 with the regularity condition u ∈ L p (B 1 ). We find N −m−α , by Proposition 2.8 we deduce u ∈ L ∞ (B 1 ), which is not possible since u is singular. Hence, q < N (m−1) N −m−α . Conversely, assume that (1.5) holds. We construct a singular radially symmetric solution u of (1.1) in the form u(x) = κ|x| −γ , with κ, γ > 0 to be determined.
Note that this choice of γ is possible thanks to (1.5) 1 and to our assumption σ > N −m−α m−1 . Also, u(x) = κ|x| −γ satisfies (3.2), where now the positivity of A follows from the lower bound of γ.
Let us observe first that this condition is equivalent to Indeed, by replacing in σp ≤ β the value of σ given in (1.6), we get (m + α)p ≤ β(q − m + 1).
Adding (m + α)(q − m + 1) on both sides of the above inequality we find thus the required lower bound for σ follows, as the upper bound trivially holds since we are in Case 1b.
Conversely, assume now that either N ≤ m + α or N > m + α and (1.5) holds. Let σ be defined by (1.6) and τ = 1 p+q−m+1 > 0. We claim that is a solution of (1.2). A straightforward calculation using Remark 2.11 yields To see this we first note that (1.5) 2 implies Thus, the coefficient A defined in (2.27) (in which γ = σ) satisfies A > 0. Also, by Lemma 2.10(i)-(iii) (we use θ = τ p ∈ (0, 1) if σp = β) we have where in the latter case σp < β, from Case 2 in the proof of Theorem 1.1(ii), we have that (3.10) holds with the strict sign so that we fall in Case(iii) of Lemma 2.10. From the above estimates we have div |x| −α |∇u| m−2 ∇u ≍ (I α * u p )u q and thus, for suitable constants a ≥ b > 0 we have that u satisfies (1.2).
(ii) Let u be a singular solution of (1.2). We divide our argument into two steps.
Note first that u satisfies the inequality where c = 2 α−N B 1 u p dx > 0. Applying Proposition 2.6 with θ = 0 we find Using the above estimate (if σp < β) and (2.8) (if σp > β), from Lemma 2.10(i),(iii) we obtain where We are exactly in the frame of Proposition 2.2 which yields (2.4).
Let c > 0 be such that u(x) ≤ cΦ m,α (x) in B 1 \ {0}. By Lemma 2.10 we have and τ > 0 is chosen small enough such that 1 Also, by the definition (4.7) of θ and (1.5) we have 0 ≤ θ < m + α, this latter condition is required in the statement of Lemma 2.7. Indeed, this is easy to check > β then we observe that from (1.5) 2 and q > m − 1 we find Since u is a singular solution of (1.2), there exists a decreasing sequence {r k } ⊂ (0, 1), r k → 0 (as k → ∞) such that sup |x|=r k u(x) → ∞ as k → ∞.
Using the strong Harnack inequality (2.4) we also have inf |x|=r k u(x) → ∞ as k → ∞. (4.8) For any k ≥ 1 let w k ∈ C 1 (B 1 \ B r k ) be a radial function such that Since u satisfies (4.6), by the maximum principle we find that k −→ w k is increasing and Also w is singular since by (4.8) we have Thus, by (4.5) and Lemma 2.7 (which can be applied since in the first and in the second case of (4.7), condition (1.8) implies (2.13) being σ > (N − m − α)/(m − 1) by virtue of (1.5) 2 , the third case of (4.7) condition (1.8) is exactly (2.13)) we deduce u ≍ Φ m,α .
Hence, one may find a decreasing sequence {r k } ⊂ (0, 1), r k → 0 (as k → ∞) such that Using the strong Harnack inequality (2.4) for u and the fact that Φ m,α (x) = Φ m,α (|x|), one has Recall that u satisfies (4.3)-(4.4). For any k ≥ 1 let w k ∈ C 1 (B 1 \ B r k ) be a radial function such that where ϕ is defined in (4.3). By the maximum principle we find that k −→ w k is increasing and u ≥ w k in B 1 \ B r k . Thus, there exists w(x) = lim k→∞ w k (x) for all x ∈ B 1 \ {0} and div |x| −α |∇w| m−2 ∇w = Cϕ(x)w q in B 1 \ {0}.
In particular, w satisfies (2.14) with θ = (σp − β) + < m + α since q > m − 1, and (4.10) Using the above estimates and (4.9) it follows that By Lemma 2.7 it follows that It is enough to establish the above inequality for all x ∈ B 1/2 \ {0}. Then, since all involved functions are continuous on B 1 \ B 1/2 we may take a smaller constant c > 0 such that the above estimate still holds for all x ∈ B 1 \ {0}. Indeed, if we denote by φ(x) the function on RHS of the above estimate, then, for where This shows that the inequality holds true on B 1 \ B 1/2 so we need only to prove it on Observe that where ω N is the surface area of the unit ball in R N . From here we estimate as follows: (i2) If a + b = N then, for any 0 < |x| < 1/2 we have Indeed, if θ = 1 then In order to establish the upper bounds in the estimates (2.24)-(2.26) we proceed as in [16,Lemma 3.6] (see also [18,Lemma 10.4]). Let r = |x| ∈ (0, 1) and use the the change of variables x = rζ, y = rη. In particular, we have |ζ| = 1. Thus Next, a straightforward calculation leads to the desired estimates in the upper bounds of (i)-(iii). Indeed, we proceed as follows.
(ii3) When a + b < N , the upper bound in (2.26) follows immediately since for all 0 < r < 1.