ESTIMATES FOR THE GREEN FUNCTION AND SINGULAR SOLUTIONS FOR POLYHARMONIC NONLINEAR EQUATION

We establish a new form of the 3G theorem for polyharmonic Green function on the unit ball of Rn (n≥ 2) corresponding to zero Dirichlet boundary conditions. This enables us to introduce a new class of functions Km,n containing properly the classical Kato class Kn. We exploit properties of functions belonging to Km,n to prove an infinite existence result of singular positive solutions for nonlinear elliptic equation of order 2m.


Introduction
In [2], Boggio gave an explicit expression for the Green function G m,n of (− ) m on the unit ball B of R n (n ≥ 2) with Dirichlet boundary conditions where ∂/∂ν is the outward normal derivate and m is a positive integer.
In fact, he proved that for each x, y in B, we have obtained the following inequality called 3G theorem: there exists a constant a m,n > 0 such that for each x, y,z ∈ B, (1. 3) The Green function for the Laplacian (m = 1) satisfies the above inequality in an arbitrary bounded C 1,1 domain Ω in R n .In fact, for the case n ≥ 3, Zhao proved in [19] the existence of a positive constant C n such that for each x, y,z in Ω, (1.4) Moreover, for the case n = 2, Chung and Zhao showed in [3] the existence of a positive constant C 2 such that for each x, y,z in Ω, G 1,2 (x,z)G 1,2 (z, y) G 1,2 (x, y) ≤ C 2 max 1,log 1 |x − z| + max 1,log 1 |y − z| . (1.5) The 3G theorem related to G 1,n has been exploited in the study of functions belonging to the Kato class K n (Ω) (see Definition 1.1), which was widely used in the study of some nonlinear differential equations (see [15,18]).More properties pertaining to this class can be found in [1,3].
Definition 1.1 (see [1,3]).A Borel measurable function ϕ in Ω belongs to the Kato class K n (Ω) if ϕ satisfies the following conditions: (1.6) The purpose of this paper is two-folded.One is to give a new form of the 3G theorem to the Green function G m,n in B 2 which improves (1.3) and enables us to introduce a new Kato class K m,n := K m,n (B) in the sense of Definition 1.2.The Imed Bachar et al. 717 second purpose is to investigate the existence of infinitely many singular positive solutions for the following nonlinear elliptic problem: ∆ m u = (−1) m f (•,u) in B \ {0} (in the sense of distributions), ∂ν m−1 u = 0 on ∂B, u(x) ∼ cρ(x), near x = 0, for any sufficiently small c > 0, (1.7) where and f is required to satisfy suitable assumptions related to the class K m,n which will be specified later.
The existence of infinitely many singular positive solutions for problem (1.7) in the case m = 1, for an arbitrary bounded C 1,1 domain Ω in R n (n ≥ 3), has been established by Zhang and Zhao in [18] for the special nonlinearity f (x,t) = p(x)t µ , µ>1, (1.9) where the function p satisfies (1.10) This result has been recently extended by Mâagli and Zribi in [14], where f satisfies some appropriate conditions related to the class K 1,n (Ω).
Here we extend these results to the high order.The outline of the paper is as follows.In Section 2, we find again by a simpler argument some estimates on the Green function G m,n given by Grunau and Sweers in [7] and we give further ones, including the following: (1.11) Next, we establish the 3G theorem in this form: there exists C m,n > 0 such that for each x, y,z ∈ B, which improves (1.3).We note that, for m = 1, (1.12) holds for an arbitrary bounded domain Ω in R n .This was proved by Kalton and Verbitsky in [10] for n ≥ 3 and by Selmi in [16] for the case n = 2.
In Section 3, we define and study some properties of functions belonging to the class K m,n .Definition 1.2.A Borel measurable function ϕ in B belongs to the class K m,n if ϕ satisfies the following condition: In particular, we show that K m,n contains properly K j,n , for 1 ≤ j ≤ m − 1, which contains properly K n (B).We close this section by giving a characterization of the radial functions belonging to the class K m,n .
For the case m = 1, this class has been extensively studied for an arbitrary bounded C 1,1 domain in R n , in [14], for n ≥ 3, and in [12,17] for n = 2.To study problem (1.7) in Section 4, we assume that f satisfies the following hypotheses: (H 1 ) f is a Borel measurable function on B × (0,∞), continuous with respect to the second variable; (H 2 ) | f (x,t)| ≤ tq(x,t), where q is a nonnegative Borel measurable function in B × (0,∞), such that the function t → q(x,t) is nondecreasing on (0,∞) and lim t→0 q(x,t) = 0; (H 3 ) the function g, defined on B by g(x) = q(x,G m,n (x,0)), belongs to the class K m,n .We point out that in the case m = 1 and f (x,t) = p(x)t µ , the assumption (1.10) implies (H 3 ).
In order to simplify our statements, we define some convenient notation.
(ii) We denote s ∧ t = min(s,t) and s ∨ t = max(s,t) for s,t ∈ R.
(iii) For x, y ∈ B, (1. (iv) Let f and g be positive functions on a set S.
We call f ∼ g if there is c > 0 such that We call f g if there is c > 0 such that (1.17) The following properties will be used several times: (i) for s,t ≥ 0, we have (ii) let λ, µ > 0 and 0 < γ ≤ 1, then we have

Inequalities for the Green function
We first find another expression of G m,n given by Hayman and Korenblum in [8], which will be used later.
where α m,n is some fixed positive constant.
Moreover, from formula (1.2), we may prove, by simpler argument, the following estimates on G m,n given in [7].
In the sequel, for a nonnegative measurable function f on B, we put (2.32) Remark 2.6.Let m ≥ n.Then there exists a positive constant C 1 such that, for each f ∈ L 1 + (B) and x ∈ B, we have (2.33) In particular, we have V m,n 1(x) ∼ (δ(x)) m .Moreover, let 1 ≤ m < n.Then there exists a positive constant C 2 such that for each f ∈ L p + (B) with p > n/m and x ∈ B, we have Next, we aim to prove inequality (1.12).So, we need the following key lemma.

A(x, y) A(x,z) + A(y,z). (2.46)
To show the claim, we separate the proof into three cases.Case 1.For 2m < n, using Proposition 2.2, we have (2.47) We distinguish the following subcases:  Then the result holds by arguments similar to that of Case 2(i).

The Kato class K m,n
In this section, we will study properties of functions belonging to the class K m,n .We first compare the classes K j,n for j ≥ 1.

Imed Bachar et al. 729
Remark 3.2.Let j,m ∈ N such that 1 ≤ j < m, then we have Indeed, by a similar argument as above, we prove that, on B 2 , Proof.Let ϕ ∈ K m,n , then by (1.13), there exists α > 0 such that for each Let x 1 ,...,x p be in B such that B ⊂ ∪ 1≤i≤p B(x i ,α).Then by (2.25), there exists C > 0 such that for all i ∈ {1, ..., p} and y ∈ B(x i ,α) ∩ B, we have Hence, we have This completes the proof.
In the sequel, we use the notation Imed Bachar et al. 731 Thus by Fatou's lemma and (1.12), we deduce that which completes the proof.Remark 3.7.We recall (see [1]) that for m = 1 and n ≥ 3, a radial function ϕ is in the classical Kato class K n (B) if and only if Similarly, we will give in the sequel a characterization of the radial functions belonging to K m,n , which asserts, in particular, that inclusions (3.5) are proper.More precisely, we will prove in the next proposition that a radial function ϕ is in K m,n if and only if (3.20) is satisfied.Proposition 3.8.Let ϕ be a radial function in B, then the following assertions are equivalent: (1) (3) for 2m < n, (3.23) where σ is the normalized measure on the unit sphere S n−1 of R n .Now, using Corollary 2.3 and the fact that for each y ∈ B, [0, y] = 1, we deduce that (3.29) Using [4, Theorem 2.4], we have (3.30) On the other hand, by (3.1), we have Now, by elementary calculus, we obtain that Example 3.9.let q be the function defined in B by (3.36)By Proposition 3.8, q ∈ K m,n if and only if λ < 2m and V m,n q is bounded if and only if λ < m + 1.In fact, we give in the next proposition more precise estimates on the m-potential V m,n q.
Proposition 3.10.On B, the following estimates hold: Proof.Let λ < m + 1.Then from (2.25), we have which implies the lower estimates.
Imed Bachar et al. 735 This implies that (3.39) So, by elementary calculus, we obtain that This completes the proof.

Positive singular solutions of the equation
In this section, we are interested in the existence of positive singular solutions for problem (1.7).We present in the next theorem the main result of this section.

. 6 ) 4 . 3 . 3 .
which implies that K j,n ⊂ K m,n .The first inclusion in (3.5) holds by putting m = 1 in Corollary 2.Lemma Let ϕ be a function in K m,n .Then the function