Positive solutions for a nonlinear Schrödinger-Poisson system

In this paper, we study the following nonlinear Schrodinger-Poisson system \begin{document}$\left\{\begin{array}{ll} -\Delta u+u+\epsilon K(x)\Phi(x)u = f(u),& x\in \mathbb{R}^{3} , \\ -\Delta \Phi = K(x)u^{2},\,\,& x\in \mathbb{R}^{3}, \\\end{array}\right.$ \end{document} where \begin{document} $K(x)$ \end{document} is a positive and continuous potential and \begin{document} $f(u)$ \end{document} is a nonlinearity satisfying some decay condition and some non-degeneracy condition, respectively. Under some suitable conditions, which are given in section 1, we prove that there exists some \begin{document} $\epsilon_{0}>0$ \end{document} such that for \begin{document} $0 , the above problem has infinitely many positive solutions by applying localized energy method. Our main result can be viewed as an extension to a recent result Theorem 1.1 of Ao and Wei in [ 3 ] and a result of Li, Peng and Wang in [ 26 ].

1. Introduction and the main result. In this paper, we are mainly motivated by the following problem where is the Planck constant, i is the imaginary unit, m is a positive constant, E is a real number, > 0, ψ : R 3 × [0, T ] → C. It is well known that this type of equations has a strong physical meaning because it appears in quantum mechanics models (see e.g. [10,12,28]) and in semiconductor theory (see [8,29,30]). (1.1) Since in (1.1), the first equation is a nonlinear stationary equation (where the nonlinear term simulates the interaction between many particles) that is coupled with a Poisson equation to be satisfied by Φ, meaning that the potential is determined by the charge of the wave function, we refer (1.1) as a nonlinear Schrödinger-Poisson system. When → 0 in (1.1), the existence of the so-called semi-classical states in nonlinear Schrödinger-Poisson system, we refer the readers to [22][23][24] and the references therein. When 2 2m = 1, (1.1) becomes which was first introduced in [9] as a model describing solitary waves for the nonlinear stationary Schrödinger equations integrating with the electrostatic field. In recent years, when = 1 and K(x) ≡ 1, problem (1.2) with V (x) ≡ 1 or being radially symmetric, has been widely studied under various conditions on f (see [2,17,35]). The case of positive and bounded nonradial potential V has been considered in [37], when f is asymptotically linear, in [5,6], when f (x, u) = |u| p−1 u, with 3 < p < 5. Moreover, in [5], the existence of ground state solutions for problem (1.2) has been proved in several situations, including the positive constant potential case. In [6], Azzollini and Pomponio considered problem (1.2) with a class of more general potential which may be unbounded from below, and the existence of ground state solutions was proved. When 3 < p < 5 and V was not a constant, the existence of ground state solutions was established in [5,6,14]. For the autonomous case, namely K(x) ≡ 1 and when f (u) = |u| p−2 u, it has been investigated in [9,29]. In [2,35], it was proved that (1.2) has infinitely many pairs radial solutions for 3 < p < 6, and has multiple solutions (but not infinitely many) for small and 2 < p < 3. In [36], when = 1, Ruiz proved that (1.2) admits no nontrivial solutions for 2 < p ≤ 3 and possesses a positive radial solution for 3 < p < 6. In [11,17], they used the mountain pass theorem and obtained a radial positive solution for 4 ≤ p < 6. In [14], when = 1 and V (x) ≡ 1, Cerami and Vaira studied (1.2) with f (x, u) = Q(x)|u| p−1 u, 3 < p < 5. They have proved the existence of positive solutions to (1.2) when Q and K were nonsymmetric and nonnegative functions satisfying lim |x|→∞ Q(x) = Q ∞ and lim |x|→∞ K(x) = 0. When V (x) ≡ 1 and f (x, u) = Q(x)|u| p−1 u, 1 < p < 5, in [25], Li, Peng and Yan proved that (1.2) has infinitely many non-radial positive solutions under the assumptions that K(x), Q(x) were positive bounded radial functions in R 3 satisfying some decaying conditions. In [26], applying the finite reduction method, Li, Peng and Wang proved that there exists (m) such that, for 0 < < (m), (1.2) with f (x, u) = |u| p−2 u, 2 < p < 6, has an m-bump solution under the assumption that (K) K(x) ∈ C(R 3 , R + ), lim |x|→∞ K(x) = 0 and lim |x|→∞ ln(K(x)) |x| = 0.
Recently in [13], Cerami, Passasseo and Solimini developed a localized Nahari's manifold argument and localized variational method to prove the existence of infinitely many positive solutions of the following equation where the potential K satisfies suitable decay assumptions(see below (K 1 ) − (K 2 )). Unlike [13], Ao and Wei in [3] used localized energy method to prove that there existed some such that 0 < < 0 , (1.3) with more general nonlinearity f (u) had infinitely many solutions. This generalized and gave a new proof of the results by Cerami, Passasseo and Solimini in [13]. They also used the new techniques to establish the existence of infinitely many positive bound states for elliptic systems. Motivated by [3,26], in this paper we study the case 2 2m = V (x) = 1 and f (x, u) = f (u), i.e., the following system of Schrödinger-Poisson equations In order to state our main result, we give the conditions imposed on K(x)(similar to [3,13]) and f : (K 1 ) K(x) ∈ C(R 3 , R + ), lim |x|→∞ K(x) = 0; (K 2 ) ∃ 0 < α < 1, lim |x|→∞ K(x)e α|x| = +∞; (f 1 ) f : R → R is of class C 1+δ for some 0 < δ ≤ 1 and f (u) = 0 for u ≤ 0, f (0) = 0; (f 2 ) The equation has a nondegenerate solution w, i.e., From the well-known results of [21], we know that w is radially symmetric with exponential decay. Moreover, we have the following asymptotic behavior of w for r large, where A N > 0 is a generic constant.
Note that the function f (w) = w p − aw q , for t ≥ 0 with a constant a ≥ 0 satisfies the above conditions (f 1 ) − (f 2 ) if 1 < q < p < 5 (see Appendix C of [33], also [3].) Nondegeneracy is a generic condition. We want to point out that there do exist nonlinearities with degenerate ground states; the first example seems to be given by Dancer [16]. See also Polacik [34].

CHUNHUA WANG AND JING YANG
Under the nondegeneracy assumption (f 2 ), the spectrum of the linearized oper- where each of the eigenfunction corresponding to the positive eigenvalue λ j decays expoentially. These eigenfunctions will play important role in our secondray Liapunov-Schmidt reduction(see Section 3 below).
In the sequel, the Sobolev space H 1 (R 3 ) is endowed with the standard norm which is induced by the inner product ∇u, ∇v = (∇u∇v + uv).
Let | · | p be the usual norm of L p (R 3 ). Define D 1,2 (R 3 ) to be the completion of 2 . We will look for the solutions (u, Φ) ∈ H 1 (R 3 )×D 1,2 (R 3 ). Now we reduce (1.4) to a single equation with a non-local term. By the assumption of (K 1 ) and u ∈ L q (R 3 ) for all q ∈ [2, 6], then Ku 2 ∈ L 6 5 (R 3 ) for all u ∈ H 1 (R 3 ), and there holds By Riesz theorem, there exists a unique Φ u ∈ D 1,2 (R 3 ) such that It follows that Φ u satisfies the Poisson equation Moreover, taking v = Φ u in (1.10), we have Now we consider the functional I : H 1 (R 3 ) → R given by Noting that so I is a well defined C 1 functional. If u ∈ H 1 (R 3 ) is a critical point of it, then the pair (u, Φ u ) is a classical solution of (1.4).
Our main result of this paper is as follows: Then there exists 0 > 0 such that 0 < < 0 , problem (1.4) has infinitely many positive solutions. Before we close this introduction, let us outline the main idea in the proof of Theorem 1.1.
First we introduce some notations. Let µ > 0 be a real number such that w(x) ≤ ce −|x| for |x| > µ and some constant c independent of µ large. Now we define the configuration space (1.14) Let w be the nondegenerate solution of (1.5) and m ≥ 1 be an integer. Define the sum of m spikes as Let the operator be Fixing P m = (P 1 , · · · , P m ) ∈ Ω m , we define the following functions as the approximate kernels: where η j (x) = η( 2|x−Pj | µ−1 ) and η(t) is a cut off function, such that η(t) = 1 for |t| ≤ 1 and η(t) = 0 for |t| ≥ µ 2 µ 2 −1 . Note that the support of Q j,k belongs to B µ 2 2(µ+1) (P j ).
Applying w Pm as the approximate solution and performing the Liapunov-Schmidt reduction, we can show that there exists a constant µ 0 , such that for µ ≥ µ 0 , and < c µ , for some constant c µ depending on µ but independent of m and P m , we can find a ϕ Pm such that and we can show that ϕ Pm is C 1 in P m . This is done in Section 2. After that, for any m, we define a new function and we maximize M(P m ) overΩ m . At the maximum point of M(P m ), we show that c jk = 0 for all j, k. Therefore we prove that the corresponding w Pm + ϕ Pm is a solution of (1.12). And (w Pm + ϕ Pm , Φ w Pm +ϕ Pm ) is a pair solution of (1.4). By the arguments before, we know that there exists µ 0 large such that µ ≥ µ 0 and < c µ and for any m, there exists a spike solution to (1.12) with m spikes in Ω m . Considering that m is arbitrary, then there exists infinitely many spikes solutions for < c µ0 independent of m.
There are three main difficulties in the maximization process. Firstly, we need to show that the maximum points will not go to infinity. This is guaranteed by the slow decay assumption on the potential K(x). Secondly, we have to detect the difference in the energy when the spikes move to the boundary of the configuration space. Here we use the induction method and detect the difference of the m-spikes energy and the m + 1-spikes energy. A crucial estimate is Lemma 3.2, where we prove that the accumulated error can be controlled from step m to step m + 1. To this end, we make a secondary Liapunov-Schmidt reduction. This is done in Section 3. Thirdly, the Poisson potential brings some new difficulties which involves many complex and technical estimates.
Motivated by [3,4,27,32], our main idea is to use the Liapunov-Schmidt reduction method. We want to point out that the only assumption we need is the nondegeneracy of the bump. We have no requirements on the structure of the nonlinearity. Unlike in [3], in order to deal with Poisson potential, we have to introduce two different norms (see (2.1) and (2.2) in section 2 below). Moreover, the Poisson potential makes our problem is more difficult than the single Schrodinger equation in [3].
Our paper is organized as follows. In section 2, we carry out Lyapunov-Schmidt reduction. Then we perform a second Liapunov-Schmidt reduction in section 3. Finally, we prove our main result in section 4. We put some technical estimates in Appendix A.
2. Finite-dimensional reduction. In this section, we perform a finite-dimensional reduction.
Lemma 2.1. ( [20], Lemma 3.4) There exists a constant C 3 = 6 3 such that for any m ∈ N + and any P m = (P 1 , P 2 , · · · , P m ) ∈ R 3m , for all x ∈ R 3 and all l ∈ N. Particularly, we have In the following, β will denote a positive constant depending on and γ but independent of µ, m, P m and may vary from line to line.

Lemma 2.2.
Let h with h * * bounded and assume that (ϕ Pm , {c jk }) is a solution to (2.7). Then there exist positive numbers µ 0 and C, such that for all 0 < < e −2µ , µ > µ 0 and P m ∈ Ω m , one has where C is a positive constant independent of µ, m and P m ∈ Ω m .
Proof. We prove it by contradiction. Assume that there exists a solution ϕ Pm to (2.7) and h * * → 0, ϕ Pm * = 1.
Multiplying the equation in (2.7) by Q j,k and integrating in R 3 , we get Considering the exponential decay at infinity of ∂ x k w and the definition of Q j,k , we have , as µ → ∞. (2.12)

CHUNHUA WANG AND JING YANG
On the other hand, applying (2.6), we can check that (2.13) Here and in what follows, C stands for a positive constant independent of and µ, as → 0. Now if we writeQ j,k = ∂w P j ∂x k , then we have Moreover, by Lemma A.1 we have for some β > 0.

POSITIVE SOLUTIONS FOR A NONLINEAR SCHRÖDINGER-POISSON SYSTEM 5469
by Lemma A.1 we have for some β > 0. Finally, by the assumption of we have 18) for some β > 0, where we use the following estimates whose proofs we put in Appendix A It follows from (2.11) to (2.18) that Now let ϑ ∈ (0, 1). It is easy to check that the function E 2 (defined in (2.2)) satisfies providedμ is large enough but independent of µ. Indeed, by Lemma 2.1 we have Then Noting that here x ∈ R 3 \ ∪ m j=1 Bμ(P j ), then Moreover, it follows from Lemmas A.1 to A.5 that From (2.23) and (2.24), by simple computation we have which yields that (2.22) is true. Hence the function E 2 can be used as a barrier to prove ϕ Pm (x) ∈ C(R 3 \ ∪ m j=1 Bμ(P j )). Then applying the maximum principle for the linear operator

POSITIVE SOLUTIONS FOR A NONLINEAR SCHRÖDINGER-POISSON SYSTEM 5471
which implies that for all x ∈ R 3 \ ∪ m j=1 Bμ(P j ). Now we prove it by contradiction. We assume that there exist a sequence of tending to 0, µ tending to ∞ and a sequence of solutions of (2.7) for which (2.10) is not true. The problem being linear, we can reduce to the case where we have a sequence (n) tending to 0, µ (n) tending to ∞ and sequences h (n) , ϕ (n) , {c for some fixed constant C > 0. Applying elliptic estimates together with Ascoli-Arzela's theorem, we can find a sequence P (n) j and we can extract, from the sequence which is bounded by a constant times e −γ|x| , with γ > 0. Moreover, recall that ϕ (n) Pm satisfies the orthogonality conditions in (2.7). Hence the limit function ϕ ∞ also satisfies Since w is non-degenerate, we have that ϕ ∞ ≡ 0 which contradicts to (2.26).
From Lemma 2.2, we can obtain the following result.
Proposition 2.3. Then there exist positive numbers γ ∈ (0, 1), µ 0 > 0 and C > 0, such that for all 0 < < e −2µ , µ > µ 0 and for any given h with h * * norm bounded, there is a unique solution (ϕ Pm , {c jk }) to problem (2.7). Moreover, Proof. Here we consider the space Observe that problem (2.7) in ϕ Pm is rewritten as In the sequel, if ϕ Pm is the unique solution given by Proposition 2.3, we denote We come to the main result in this section.
Proposition 2.4. Given 0 < γ < 1. There exist positive numbers µ 0 , C and β > 0(independent of µ, m and P m ∈ Ω m ) such that for all µ ≥ µ 0 , and for any P m ∈ Ω m , < e −2µ , there is a unique solution (ϕ Pm , {c jk }) to problem (2.31). Furthermore, ϕ Pm is C 1 in P m and we have (2.32) Note that the first equation in (2.31) can be rewritten as where S(·) is defined as (1.19), (2.35) In order to use the contraction mapping theorem to prove that (2.33) is uniquely solvable in the set that ϕ Pm * is small, we need to estimate S(w Pm ) * * and N (ϕ Pm ) * * respectively. Lemma 2.5. Given 0 < γ < 1. For µ large enough, and any P m ∈ Ω m , < e −2µ , we have S(w Pm ) * * ≤ Ce −βµ (2.36) for some constants β > 0 and C independent of µ, m and P m . (2.37) Similar to (2.5) and (2.6) of section 2.1 in [3], we can prove for a proper choice of β > 0.

POSITIVE SOLUTIONS FOR A NONLINEAR SCHRÖDINGER-POISSON SYSTEM 5473
Moreover, by the assumption of , Lemmas A.1 to A.4, we can prove that for some β > 0. In fact, on one hand, fix j ∈ {1, 2, · · ·, m} and consider the region Λ j . In this region, if x ∈ R 3 \{P 1 , P 2 , · · · , P m }, we have where we use the fact that And if x = P j0 , j 0 ∈ {1, 2, · · · , m}, then we also have On the other hand, considering the region If x = P j0 , j 0 ∈ {1, 2, · · · , m}, then we also have It follows from (2.38) and (2.39) that for some β > 0 independent of µ, m and P m . and Proof. By direct computation and applying the mean-value theorem, we have Since f is Hölder continuous with the exponent δ, we deduce Also, similar to (2.39), x = P j0 , j 0 ∈ {1, 2, · · · , m}, applying Lemmas A.1 to A.4 we have which implies that By direct computation, we have  Define where τ > 0 small enough. We will prove that T is a contraction mapping from B to itself. On one hand, for any ϕ Pm ∈ B, it follows from Lemmas 2.5 and 2.6 that On the other hand, taking ϕ 1 Pm and ϕ 2 Pm in B, by (2.42) in Lemma 2.6 we have

POSITIVE SOLUTIONS FOR A NONLINEAR SCHRÖDINGER-POISSON SYSTEM 5479
Hence by the contraction theorem there exists a unique ϕ Pm ∈ B such that (2.48) holds. So Combining (2.21), (2.36) and (2.41) we have Now we leave to prove that ϕ Pm is C 1 in P m . Consider the following mapping H : Then problem (2.31) is equivalent to H(P m , ϕ Pm , c) = 0. We know that, given P m ∈ Ω m , there is a unique solution (ϕ Pm , {c jk }). We prove that the operator is invertible for µ large. Then ϕ Pm ∈ C 1 follows from the implicit function theorem. Indeed, observe that Since ϕ Pm * is small, the same proof as in that of Lemma 2.2 shows that ∂H(P m , ϕ Pm , c) is invertible for µ large. This completes the proof of Proposition 2.4.

3.
A secondary Liapunov-Schmidt reduction. In this section, as in [3,4,27], applying secondary Lyapunov-Schmidt reduction we give a key estimate on the difference between the solutions in the m-th step and (m + 1)-th step. For P m ∈ Ω m , we denote u Pm as w Pm + ϕ Pm , where ϕ Pm is the unique solution given by Proposition 2.4. The main estimate below shows that the difference between u Pm+1 and u Pm + w Pm+1 is small globally in H 1 (R 3 ) norm. For this purpose, we now write (3.5) But the estimate (3.5) is not sufficient. We need a crucial estimate for φ m+1 which will be given later. (In the following we will always assume that γ > 1 2 .) In order to obtain the crucial estimate, we will need the following lemma.
Proof. In order to prove (3.8), we have to make a further decomposition. As Lemma 2.2 in [3], the basic idea is as follows: around each spike, we project φ m+1 into the orthogonal space of the unstable eigenfunctions and kernels. By this way, we get a linear operator which is possibly definite. Therefore we have to estimate three components of φ m+1 : the coefficients of projections to the unstable eigenfunctions and kernels, and the orthogonal part. Now we perform this procedure in details.
By the non-degeneracy (f 2 ), the following eigenvalue problem admits the following set of eigenvalues (3.10) We denote the eigenfunctions corresponding to the positive eigenvalues λ j as ϕ j 0 (x), j = 1, · · · , n. By (f 2 ), we conclude that there is a positive generic constant c 0 such that

20) and
where we use the orthogonality conditions satisfied by ϕ Pm and ϕ Pm+1 . Hence by Proposition 2.4, we have
We claim that for some constant c 0 > 0 (independent of m and P m+1 ).
Since the approximate solution is exponentially decay away from the points P j , we have Now we only need to prove the above estimate in the domain ∪ j B µ 2 (P j ). We prove it by contradiction. Otherwise, there exists a sequence µ n → ∞, and P Then we can extract from the sequence ψ n (· − P (n) j ) a subsequence which will converge weakly in H 1 (R 3 ) to ψ ∞ such that and ψ ∞ ϕ l 0 = ψ ∞ ∂w ∂x j = 0, for j = 1, 2, 3, l = 1, 2, · · · , n.  (3.37)

POSITIVE SOLUTIONS FOR A NONLINEAR SCHRÖDINGER-POISSON SYSTEM 5485
From (3.15) and (3.37), recalling that γ > 1 2 , we get (3.38) Since we choose γ > 1 2 , by the definition of the configuration space, we have It follows from (3.38) and (3.39) that (3.40) Hence (3.8) holds. Moreover, from the estimates (3.22) and (3.28), and taking into consideration that η j is supposed in B µ 2 (P j ), using Hölder inequality, we can get a more accurate estimate on φ m+1 , Observe that M(P m ) is continuous in P m . We will prove below that the maximization problem has a solution. Denote M(P m ) as the maximum whereP m = (P 1 , · · ·,P m ) ∈Ω m , that is and we denote the solution by uP m . First we show that the maximum can be attained at finite points for each C m . (ii) There holds C m+1 > C m + I(w), (4.5) where I(w) is the energy of w, Proof. Here we follows the proofs in [3,13] and we need to use the estimate (3.8).
We divide the proof into the following two steps.
Step 1. C 1 > I(w), and C 1 can be attained at a finite point. First applying standard Liapnunov-Schmidt reduction, we have Supposing that |P | → ∞, then by (4.7) we have where we use the following estimates (4.8) and whose proof we put in the Appendix A.

POSITIVE SOLUTIONS FOR A NONLINEAR SCHRÖDINGER-POISSON SYSTEM 5487
By the assumption (K 2 ), we have So we have C 1 ≥ J(u P ) > I(w). (4.10) Now we will prove that C 1 can be attained at a finite point. Let {P j } be a sequence such that lim j→∞ M(P j ) = C 1 , and assume that |P j | → +∞, By the assumption (K 1 ), we have Then we obtain which contradicts to (4.10). Thus C 1 can be attained at a finite point.
Step 2. Assume that there existsP m = (P 1 , ···,P m ) ∈ Ω m such that C m = M(P m ) and we denote the solution by uP m . Next we prove that there exists P m+1 ∈ Ω m+1 such that C m+1 can be attained. Let P (n) m+1 be a sequence such that We claim that P (n) m+1 is bounded. We prove it by contradiction. Without loss of generality, we assume that |P (n) m+1 | → ∞ as n → ∞. In the following we omit the index n for simplicity. By direct computation, we have CHUNHUA WANG AND JING YANG (4.12) since similar to (4.9), we have Moreover, we have

POSITIVE SOLUTIONS FOR A NONLINEAR SCHRÖDINGER-POISSON SYSTEM 5491
(4.20) Hence by Lemma 3.1, we have (4.21) Combing (4.12), (4.13), (4.14) and (4.21), we obtain (4.22) By the assumption that |P On the other hand, since by the assumption, C m can be attained at (P 1 , · · ·,P m ), so there exists other point P m+1 which is far away from the m points which be determined later. Next let's consider the solution concentrated at the points (P 1 , · · ·,P m , P m+1 ), and we denote the solution by uP m,Pm+1 , then similar with the above argument, applying the estimate (3.41) of φ m+1 instead of (3.20), we have the following estimate:

POSITIVE SOLUTIONS FOR A NONLINEAR SCHRÖDINGER-POISSON SYSTEM 5493
(4.26) By the asymptotic behavior of K at infinity, i.e. lim |x|→∞ K(x)e α|x| = 0 as |x| → ∞, for some α < 1, we choose γ > α, then we can choose P m+1 such that Then we can get It follows from (4.25) and (4.28) that which is impossible. Hence we prove that C m+1 can be attained at finite points in Ω m+1 . has a solution P m ∈ Ω • m , i.e., the interior of Ω m .
Proof. We prove it by an indirect method. Assume thatP m = (P 1 ,P 2 , · · · ,P m ) ∈ ∂Ω m . Then there exists (j, k) such that |P j −P k | = µ. Without loss of generality, we assume (j, k) = (j, m). Then following the estimates (4.12), (4.13), (4.14) and 5494 CHUNHUA WANG AND JING YANG (4.21), we have (4.30) By the definition of the configuration set, we observe that given a ball of size µ, there are at most C 3 := 6 3 number of non-overlapping ball of size µ surrounding this ball. Since |P j −P k | = µ, we have if C 3 < µ 2 , which is true for µ large enough. Hence, we have which contradicts to (4.5) in Lemma 4.1.
Now we are in position to prove our main result.
For s = j, by the definition of Q j,k , we have For each (s, l), the off-diagonal term yields (4.36) It follows from (4.35) and (4.36) that equation (4.34) becomes a system of homogeneous equations for c sl , and the matrix of the system is nonsingular. Hence c sl = 0 for s = 1, · · · , m, l = 1, 2, 3. Therefore u P 0 = w P 0 + ϕ P 0 is a solution of (1.12). Similar to the argument in Section 6 of [27], we can get that u P 0 > 0 and it has exactly m local maximum points for µ large enough. Therefore (u P 0 , Φ u P 0 ) is a positive pair solution of (1.4).
Appendix A. Some technical estimates. In this section, we give some technical estimates which are used before. Recall that Lemma A.1. For any x ∈ Λ j (j = 1, · · · , m) and any ϑ > 0, we have (A.1) For any x ∈ Λ c , we have Proof. Note that given a ball of size µ, there are at most C 3 := 6 3 number of nonoverlapping ball of size µ surrounding this ball. For any x ∈ Λ j , i.e. |x − P j | ≤ µ 2 , we have Then we have if ln C 3 < ϑµ 2 , which is true for µ large enough. The proof of (A.4) is similar.
By the same arguments as above, we also have Lemma A.2. For any x ∈ Λ j (j = 1, · · · , m), we have For any x ∈ Λ c , we have For any x ∈ Λ j (j = 1, · · · , m) and any ϑ 1 , ϑ 2 > 0, we have For any x ∈ Λ c , we have The following lemma is very crucial.
Since We will estimate I 21 , I 22 , I 23 and I 24 respectively. Similar to I 1 , we have Finally, we estimate I 24 . Then we have When x = P j0 , j 0 ∈ {1, 2, · · · , m}, just by the same argument as above, we can show that this case is also true.
Just by the same argument as Lemma A.4, we also have the following estimate.