# On the Existence, Uniqueness, and Basis  Properties of Radial Eigenfunctions of a Semilinear Second-Order Elliptic Equation in a Ball

Authors: Peter Zhidkov
Journal: International Journal of Mathematics and Mathematical Sciences
 (2009)

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## Abstract

We consider the following eigenvalue problem:−Δu+f(u)=λu, u=u(x), x∈B={x∈ℝ3:|x|<1},  u(0)=p>0,  u||x|=1=0, where p is an arbitrary fixed parameter and f is an odd smooth function. First, we prove that for each integer n≥0 there exists a radially symmetric eigenfunction un which possesses precisely n zeros being regarded as a function of r=|x|∈[0,1). For p>0 sufficiently small, such an eigenfunction is unique for each n. Then, we prove that if p>0 is sufficiently small, then an arbitrary sequence of radial eigenfunctions {un}n=0,1,2,…, where for each n the nth eigenfunction un possesses precisely n zeros in [0,1), is a basis in  L2r(B) (L2r(B) is the subspace of L2(B) that
consists of radial functions from L2(B). In addition, in the latter case, the sequence {un/∥un∥L2(B)}n=0,1,2,… is a Bari basis in the same space.


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## Body

## 1. Introduction, Notation and Definitions, and Results

In the present paper, we consider the problem(1.1)-Δu+f(u)=λu,u=u(|x|),x∈B={x∈ℝ3:|x|<1},(1.2)u(0)=p>0,(1.3)u||x|=1=0,
where f is an odd continuously differentiable function, λ∈ℝ is a spectral parameter, and p is an arbitrary positive fixed parameter. Problems of this type may arise in particular in the solid state physics, heat and diffusion theory, in the theory of nonlinear waves, and so forth. Hereafter in the paper, all the quantities we deal with are real.We restrict our attention to the radial eigenfunctions of problem (1.1)–(1.3), that is, to the eigenfunctions u that depend only on r=|x|. Under our assumptions, the problem has an infinite sequence of radial eigenfunctions {un}n=0,1,2,… such that for each integer n the nth eigenfunction un regarded as a function of r has precisely n zeros in the interval [0,1). The main question we are interested in the present  paper is whether such a sequence of eigenfunctions is a basis in a commonly used space, such as the subspace L2r(B) of L2(B) that consists of all radial functions from L2(B). According to our result below, this is true if p>0 is sufficiently small.For a discussion of the pertinence of our formulation of the problem (note that problem (1.1)–(1.3) includes an unusual normalization condition (1.2)) and for a longer list of references, we refer the reader to our quite recent review paper [1]. Here we note only that our formulation of the problem is “good’’ in the sense that, as in the linear case, our problem has an infinite sequence of radial eigenfunctions un, where for each integer n≥0 the nth eigenfunction un, regarded as a function of the argument r∈[0,1), possesses precisely n zeros and, if p>0 is sufficiently small, such a sequence of eigenfunctions is a basis in the space L2r(B). But if one excludes the normalization condition (1.2) from the statement of the problem, then the set of all eigenfunctions becomes too wide; it would contain “a lot of’’ bases. It is a separate question what normalization condition should be imposed. The author believes that this question may be answered only in the future if/when the field becomes developed sufficiently; in particular, an applied problem may give such an answer. In this context, the reader may consider our system (1.1)–(1.3) as a model problem.We mention especially our paper [2] (see also [1]) in which a problem analogous to (1.1)–(1.3) was studied in the spatial dimension 1. It is proved in these two articles that if assumption (f) is valid (see below) and if in addition f(u)/u is a nondecreasing function of u>0, then this one-dimensional problem has a unique sequence of eigenfunctions {φn}n=0,1,2,… such that for each n the nth eigenfunction has precisely n zeros in [0,1) and this sequence of eigenfunctions is a basis (in addition, a Riesz basis) in L2(0,1) while the sequence of normalized eigenfunctions {φn/∥φn∥L2(0,1)}n=0,1,2,… is a Bari basis in the same space (we will establish precise definitions in what follows).Now, we will introduce somenotation and definitions. Let L2=L2(B) be the standard Lebesgue space of functions g,h,… square integrable over B, equipped with the scalar product (g,h)L2(B)=∫Bg(x)h(x)dx and the corresponding norm ∥·∥L2(B)=(·,·)L2(B)1/2. By L2r=L2r(B) we denote the subspace of the space L2(B) that consists of all radial functions from L2(B) and is equipped with the same scalar product and the norm. Let L2,r2(0,1) denote the usual weighted Lebesgue space of functions g measurable in (0,1) for which
(1.4)∥g∥L2,r2(0,1):={∫01r2g2(r)dr}1/2<∞.
The space L2,r2(0,1) is equipped with the corresponding scalar product. In fact, L2(B), L2r, and L2,r2(0,1) are Hilbert spaces.LetH be a separable Hilbert space over the field of real numbers in which the scalar product and the norm are denoted (·,·)H and ∥·∥H, respectively. We recall that a sequence {hn}n=0,1,2,…⊂H is called a (Schauder) basis in H if for any h∈H there exists a unique sequence of real numbers {an}n=0,1,2,… such that
(1.5)h=∑n=0∞anhninH.
Two sequences {hn}n=0,1,2,… and {en}n=0,1,2,… from H are called quadratically close to each other (or the sequence {hn}n=0,1,2,… is called quadratically close to the sequence {en}n=0,1,2,…) if
(1.6)∑n=0∞∥hn-en∥H2<∞.
A basis {hn}n=0,1,2,… in H quadratically close to an orthonormal basis {en}n=0,1,2,… in H is called a Bari basis in H. According to Corollary 2.5 in [1], if {hn}n=0,1,2,… is an arbitrary sequence of elements of H and if
(1.7)∑n=0∞∥hn-en∥H2<1,
where {en}n=0,1,2,… is an orthonormal basis in H, then {hn}n=0,1,2,… is a Bari basis in H. Bari bases being compared with bases have additional nice properties that we do not discuss in the present paper (on this subject, see, e.g., [3]). Some general aspects of the theory of nonorthogonal expansions in a Hilbert space are considered in [1, 3].We call a sequence of radial eigenfunctions{un}n=0,1,2,… of problem (1.1)–(1.3) standard if for each integer n≥0 the nth eigenfunction un regarded as a function of r possesses precisely n zeros in the interval [0,1). Everywhere we assume the following.(f)
Letfbe a continuously differentiable odd function inℝand letf′(0)=0.Note that the assumption thatf′(0)=0 is not restrictive. One can achieve this for an arbitrary odd continuously differentiable function by a shift of the spectrum.Consider the following linear eigenvalue problem:(1.8)-Δv=μv,v=v(|x|),x∈B,v(0)=p,v||x|=1=0,
where μ∈ℝ is a spectral parameter. Denote by {vn}n=0,1,2,… the sequence of the radial eigenfunctions of problem (1.8) where, for each integer n≥0, the nth eigenfunction vn regarded as a function of r∈[0,1) possesses precisely n zeros. By
(1.9)μ0<μ1<⋯<μn<⋯
we denote the corresponding sequence of eigenvalues. Note that {vn}n=0,1,2,… is an orthogonal basis in L2r. Our main results here are as follows.Theorem 1.1.
Under assumption (f)(a)
for any integern≥0 problem (1.1)–(1.3) has a radial eigenfunction un which, being regarded as a function of r, possesses precisely n zeros in the interval [0,1);(b)
|u(r)|≤p for any r∈[0,1] and for an arbitrary radial eigenfunction u of problem (1.1)–(1.3);(c)
let in addition to assumption (f)f(u)/u be a nondecreasing function of u>0. Then, the positive radial eigenfunction un is unique;(d)
under assumption (f) there existsp0>0 such that for any p∈(0,p0] and any integer n≥0 the radial eigenfunction un of problem (1.1)–(1.3) that, being regarded as a function of r, has precisely n zeros in [0,1) is unique.Theorem 1.2.
Let assumption (f) be valid. Then, there existsκ=κ(p)>0 defined for all p>0 and going to 0 as p→+0 such that for any p>0(1.10)∑n=0∞∥un∥un∥L2-vn∥vn∥L2∥L22<κ(p)
for an arbitrary standard sequence {un}n=0,1,2,… of eigenfunctions of problem (1.1)–(1.3). Consequently, if p>0 is sufficiently small, an arbitrary standard sequence of eigenfunctions, which is unique for p>0 sufficiently small by Theorem 1.1, is a basis in L2r and, in addition, the sequence {un/∥un∥L2}n=0,1,2,… is a Bari basis in the same space.Remark 1.3.
In view of Theorem1.2 and the Bari's theorem (see [1, 3]), if one proves the linear independence, in the sense of the space L2r, of a standard sequence of eigenfunctions of problem (1.1)–(1.3) when p>0 is not necessarily sufficiently small, then this sequence of eigenfunctions will be proved to be a basis in L2r and the sequence of the normalizations of these eigenfunctions to 1 is a Bari basis in the same space, too. However, in the present paper, we leave open the question about the linear independence of such a system when p>0 is not small.In the next section, we will prove Theorem1.1, and in Section 3, Theorem 1.2.


## 2. Proof of Theorem1.1

Proofs of results of the type of Theorem1.1(a) are known now (on this subject, see, e.g., [4]), so we will establish only a sketch of the proof of this claim. In the class of radial solutions, problem (1.1)–(1.3) reduces to the following one:
(2.1)-(r2u′)′+r2f(u)=λr2u,u=u(r),r∈(0,1),(2.2)u(0)=p,(2.3)u′(0)=u(1)=0,
where the prime means the derivative in r. Equation (2.1) can also be rewritten in the following equivalent form:
(2.4)-u′′-2ru′+f(u)=λu,u=u(r),r∈(0,1).
We supply (2.4) with the following initial data:
(2.5)u(0)=p,u′(0)=0.
A solution of (2.4) and (2.5) that satisfies the condition u(1)=0 is a solution of problem (2.1)–(2.3). In (2.1) and (2.4), r=0 is a singular point. However, for problem (2.4)-(2.5), local existence, uniqueness, and continuous dependence theorems in their usual form are valid (for proofs of these claims, see, e.g., [4]). Let u(r) be a solution of (2.4)-(2.5). Then, u′′(0)=(1/3)[f(p)-λp], the derivative uλ=du/dλ exists, is continuous and it satisfies the equations
(2.6)-(r2uλ′)′+r2f′(u(r))uλ=λr2uλ+r2u,uλ=uλ(r),r∈(0,1),(2.7)uλ(0)=uλ′(0)=0.
As above, for problem (2.6)-(2.7) local existence and uniqueness theorems in their usual form, as far as the theorem of the continuous dependence on λ, take place. In addition, since (2.6) is linear with respect to uλ, the solution uλ of problem (2.6)-(2.7) exists for all those values of r>0, for which the solution u(r) of problem (2.4)-(2.5) exists.Let us prove statement (a) of Theorem1.1. Multiply (2.4) by u′(r) and integrate the result from 0 to r. Then, we obtain the identity
(2.8){12u′2(r)+λ2[u2(r)-p2]+F(p)-F(u(r))}′=-2ru′2(r),
where F(u)=∫0uf(s)ds. Denote E(u(r))=(1/2)u′2(r)+(λ/2)[u2(r)-p2]+F(p)-F(u(r)).Lemma 2.1.
Ifλ∈ℝ is such that u′′(0)>0, where u(r) is the corresponding solution of problem (2.4)-(2.5), then there is no point r>0 such that u(r)=p. In particular, u(r)>0 for any r>0 so that u is not an eigenfunction of problem (2.1)–(2.3).Proof.
On the contrary, suppose thatu′′(0)>0 and there exists r>0 such that u(r)=p. But then, u′2(r)≥0, therefore E(u(r))≥0 which contradicts (2.8).Lemma 2.2.
Letu′′(0)≤0. Then, |u(r)|≤p for all r∈(0,1].Proof.
It can be made by analogy (ifu′′(0)=0, then u(r)≡p by the uniqueness theorem).Note that Lemmas2.1 and 2.2 yield statement (b) of Theorem 1.1. Note in addition that if u′′(0)≤0, then the solution of problem (2.4)-(2.5) is global (i.e., it can be continued on the entire half-line r>0).Observe now thatu′′(0)>0 for all sufficiently large |λ|, λ<0, and u′′(0)<0 for all sufficiently large λ>0. By Lemma 2.2, |u(r)|≤p for all λ>0 sufficiently large. Therefore, comparing (2.4)-(2.5) and (1.8) (one should rewrite system (1.8) in the form analogous to (2.4)-(2.5)), we see that, according to the standard comparison theorem, the number of zeros in (0,1) of u(r) increases unboundedly when λ>0 unboundedly increases.Take an arbitrary integern≥0 and denote by Λn the set of all values of λ for each of which the solution u(r) of (2.4)-(2.5) has at least (n+1) zeros in (0,1). Let λn=infΛn. Denote by un(r) the solution of problem (2.4)-(2.5) taken with λ=λn. Then, |un(r)|≤p for all r so that this solution is global. Observe that, if u(r)≢0 is a solution of (2.4) and if u(r0)=0, then u′(r0)≠0 by the uniqueness theorem. Therefore, zeros of un(r) are isolated and hence, un has a finite number m of zeros in (0,1). If m>n, then there exists λ<λn sufficiently close to λn such that the corresponding solution u(r) of (2.4)-(2.5) has no less than m>n zeros in this interval, which contradicts our definition of the set Λn. By analogy, if m<n or if m=n and un(1)≠0, then any solution of (2.4) and (2.5) taken for λ>λn sufficiently close to λn has no more than n zeros in (0,1) which contradicts our definition of the set Λn. Thus, un(r) has precisely n zeros in the interval (0,1) and un(1)=0. So, claim (a) of Theorem 1.1 is proved.Let us prove claim (c). On the contrary, suppose that there exist two positive eigenfunctionsu1 and u2 of problem (2.1)–(2.3) corresponding to the eigenvalues λ1 and λ2, respectively, where λ1<λ2. By (2.8), ui′(r)<0 for any r∈(0,1]. Indeed, if it would be ui(r)=ui′(r)=0, then ui≡0, while if ui(r)>0 and ui′(r)<0 for r∈(0,r0) and if ui′(r0)=0, then ui(r)≥ui(r0) for all r>r0 in view of (2.4) and by the same arguments as in the proof of Lemma 2.1.Now, we apply a variant of the result from [5].Lemma 2.3.
One hasu2′(r)/u2(r)<u1′(r)/u1(r) for any r∈(0,1).Proof.
We haveu2′′(0)<u1′′(0)<0, therefore
(2.9)u1′(r)u1(r)>u2′(r)u2(r),u2(r)<u1(r)<p
in a right half-neighborhood of the point r=0. Let us prove that (2.9) holds everywhere in (0,1). Suppose that the first inequality (2.9) holds in some interval (0,a), where a∈(0,1). Integrate it from 0 to r∈(0,a]. Then
(2.10)lnu1(r)-lnp>lnu2(r)-lnp,
therefore u1(r)>u2(r) until the first inequality (2.9) holds.
Suppose that the first inequality (2.9) is valid in an interval (0,a), a∈(0,1), and that it is violated at the point r=a. Note that, as is proved above, u1(a)>u2(a). But then, by (2.4),
(2.11)(u1′(r)u1(r))′|r=a>(u2′(r)u2(r))′|r=a,
hence, by continuity (2.11) is still valid in a left half-neighborhood of the point r=a so that it must be u1′(a)/u1(a)>u2′(a)/u2(a), which is a contradiction.Now, suppose that (2.9) holds everywhere in (0,1) and that ui(1)=0, i=1,2. Denote t1=u1′(1), t2=u1′′(1), s1=u2′(1), and s2=u2′′(1). From (2.4),
(2.12)t2=-2t1,s2=-2s1.
In a neighborhood of the point r=1, one has
(2.13)u1′(r)u1(r)=t1-2t1(r-1)+O((r-1)2)t1(r-1)-t1(r-1)2+O((r-1)3)=1r-1-1+O(r-1),
and by analogy
(2.14)u2′(r)u2(r)=1r-1-1+O(r-1).
So, we see that the difference
(2.15)r-1(u1′(r)u1(r)-u2′(r)u2(r))
goes to 0 as r→1-0. But then (u1′(r)/u1(r))′-(u2′(r)/u2(r))′>0 in a left half-neighborhood of the point r=1 (because by (2.9) (u1′(r)/u1(r))2-(u2′(r)/u2(r))2≤0 in (0,1)), and since in addition u1′(r)/u1(r)>u2′(r)/u2(r) in (0,1), and we arrive at a contradiction as earlier. Thus, claim (c) of Theorem 1.1 is proved.Now, we turn to proving claim (d) of Theorem1.1. Let us prove the following.Lemma 2.4.
There existp¯>0 and C>0 such that for any p∈(0,p¯],  a∈(0,1], λ∈ℝ, and the corresponding solution u(r) of (2.4)-(2.5) which satisfies u(a)=0, one has
(2.16)∥uλ∥L2,r2(0,a)2≤C∥u∥L2,r2(0,a)2.Proof.
First, multiply (2.6) by uλ′ and integrate the result from 0 to r∈(0,a]. Then, after integration by parts, we obtain
(2.17)12r2uλ′2(r)+λ2r2uλ2(r)+∫0rsuλ′2(s)ds=λ∫0rsuλ2(s)ds+∫0rs2f′(u(s))uλ(s)uλ′(s)ds-∫0rs2u(s)uλ′(s)ds.
Second, multiply (2.6) by r-1uλ(r) and integrate the result from 0 to r. Then, after integration by parts,
(2.18)12uλ2(r)+λ∫0rsuλ2(s)ds=∫0rsuλ′2(s)ds-ruλ(r)uλ′(r)+∫0rsf′(u(s))uλ2(s)ds-∫0rsu(s)uλ(s)ds.
Now, add (2.18) to (2.17). Then, we obtain the following identity:
(2.19)12r2uλ′2(r)+λ2r2uλ2(r)+12uλ2(r)=-ruλ(r)uλ′(r)+∫0rsf′(u(s))uλ2(s)ds-∫0rs2u(s)uλ′(s)ds+∫0rs2f′(u(s))uλ(s)uλ′(s)ds-∫0rsu(s)uλ(s)ds.
Now, take an arbitraryp¯>0 and integrate (2.19) for one more time from 0 to r. Then, applying the inequality ab≤(1/2t)a2+(t/2)b2, t>0,
(2.20)12∫0rs2[uλ′2(s)+λuλ2(s)]ds+12ruλ2(r)≤C1(p¯)∫0rdt(∫0t(s+s2)uλ2(s)ds)+12∫0rdt(∫0ts2uλ′2(s)ds)+12∫0rdt(∫0tuλ2(s)ds)+32∫0rdt(∫0ts2u2(s)ds)≤C2(p¯)r∫0rsuλ2(s)ds+r2∫0rs2uλ′2(s)ds+r2∫0ruλ2(s)ds+32r∫0rs2u2(s)ds
for constants C1=C1(p¯)>0 and C2=C2(p¯)>0 independent of p∈(0,p¯], a, r, and λ.  Consequently, since λ>0 for p¯>0 sufficiently small by the comparison theorem, for p¯>0 sufficiently small
(2.21)uλ2(r)≤C3(p¯)∫0ruλ2(s)ds+C4∫0rs2u2(s)ds,
where the constants C3(p¯)>0 and C4>0 do not depend on p∈(0,p¯], a, r, and λ. Now, the statement of Lemma 2.4 follows by the Gronwell's lemma.Lemma 2.5.
There existsp̂>0 such that for any p∈(0,p̂] there is no a∈(0,1], and λ∈ℝ for which u(a)=uλ(a)=0.Proof.
On the contrary, suppose that there exist arbitrary smallp>0, a∈(0,1], and λ∈ℝ for which u(a)=uλ(a)=0. Multiply (2.4) by uλ(r), (2.6) by u(r), subtract the results from each other and integrate the obtained identity from 0 to a. Then, after integration by parts,
(2.22)∫0as2u2(s)ds=∫0as2[f′(u(s))-f(u(s))u(s)]u(s)uλ(s)ds≤C5(p)[∥u∥L2,r2(0,a)2+∥uλ∥L2,r2(0,a)2],
where C5(p)→+0 as p→+0 because f(0)=f′(0)=0. Therefore
(2.23)∥u∥L2,r2(0,a)2≤2C5(p)∥uλ∥L2,r2(0,a)
for p>0 sufficiently small, which contradicts Lemma 2.4.Let us prove Theorem1.1(d). Let now p∈(0,p̂]. Take an arbitrary integer n≥0 and let λn=infΛn, as earlier. Then, a simple corollary of Lemma 2.5 is that there exists a right half-neighborhood In of λn belonging to Λn in which the (n+1)st zero rn+1 of the solution u(r) of problem (2.4)-(2.5) is a strictly decreasing function of λ∈In (so that in particular rn+1∈(0,1)). Letting λ increase further, by Lemma 2.5, again, one sees that the (n+1)st zero of the corresponding solution u(r) of (2.4)-(2.5) continue to decrease strictly so that in the half-line (λn,+∞) there is no value λ for which u(r) has precisely n zeros in the interval (0,1) and u(1)=0. Claim (d) of Theorem 1.1 is proved. Our proof of Theorem 1.1 is complete.


## 3. Proof of Theorem1.2

Denotewn(r)=un(r)-vn(r). Then
(3.1)(-r2wn′)′=μnr2wn+r2gn(r),r∈(0,1),(3.2)wn(0)=wn′(0)=wn(1)=0,
where gn(r)=(λn-μn)r2un(r)-r2f(un(r)). By the comparison theorem and Theorem 1.1(b)
(3.3)|λn-μn|≤C1(p),
where C1(p)>0 goes to 0 as p→+0 and C1 does not depend on n. In addition, by the standard comparison theorem, again, there exists C2>0 such that
(3.4)C2-1(n+1)2≤μn≤C2(n+1)2
for all n. By (3.3) and Theorem 1.1(b),
(3.5)|gn(r)|≤C3(p)|un(r)|,
where C3(p)>0 goes to 0 as p→+0 uniformly in n.Now, we proceed as when proving Lemma2.4. First, multiply (3.1) by wn′(r) and integrate the result from 0 to r. Then, after integration by parts,
(3.6)12r2wn′2(r)+μn2r2wn2(r)+∫0rswn′2(s)ds=μn∫0rswn2(s)ds-∫0rs2gn(s)wn′(s)ds.
Second, multiply (3.1) by r-1wn(r) and integrate the result from 0 to r. Then, in view of the boundary conditions (3.2), after integration by parts,
(3.7)12wn2(r)+μn∫0rswn2(s)ds=∫0rswn′2(s)ds-rwn(r)wn′(r)-∫0rsgn(s)wn(s)ds.
Add (3.7) to (3.6). Then, we obtain
(3.8)12r2wn′2(r)+μn2r2wn2(r)+12wn2(r)=-rwn(r)wn′(r)-∫0rs2gn(s)wn′(s)ds-∫0rsgn(s)wn(s)ds.
Integrate (3.8) for one more time from 0 to r. Then, as when deriving (2.20),
(3.9)12∫0rs2[wn′2(s)+μnwn2(s)]ds+12rwn2(r)≤r2∫0rwn2(s)ds+r2∫0rs2wn′2(s)ds+r∫0rs2gn2(s)ds.
Hence, in view of (3.3) and (3.5),
(3.10)wn2(r)≤∫0rwn2(s)ds+C4(p)∫0rs2un2(s)ds,
where C4(p)>0 is a constant independent of n and r and C4(p)→0 as p→+0. Now, we obtain from (3.10) by the Gronwell's lemma the following:
(3.11)wn2(r)≤C5(p)∥un∥L2,r2(0,r)2,r∈[0,1],
where C5(p)>0 does not depend on n and r and goes to 0 as p→+0. Thus, finally from (3.4) and (3.9), we have
(3.12)∥wn∥L22≤C6(p)(n+1)-2∥un∥L22
for a constant C6>0 independent of n and going to 0 as p→+0.Now, it follows from (3.12) that
(3.13)∑n=0∞∥un∥un∥L2-vn∥un∥L2∥L22<C¯(p),
where C¯(p)>0 is defined for all p>0 and goes to 0 as p→+0. So, we have
(3.14)∑n=0∞∥un∥un∥L2-vn∥vn∥L2∥L22≤2∑n=0∞{∥un∥un∥L2-vn∥un∥L2∥L22+∥vn∥un∥L2-vn∥vn∥L2∥L22}=2{C¯(p)+∑n=0∞(∥vn∥L2∥un∥L2-1)2}.
In view of this estimate, to prove Theorem 1.2, it suffices to show that
(3.15)∑n=0∞(∥vn∥L2∥un∥L2-1)2≤Ĉ(p),
where Ĉ(p)>0 goes to 0 as p→+0. But we have
(3.16)∑n=0∞(∥vn∥L2∥un∥L2-1)2≤∑n=0∞∥un∥un∥L2-vn∥un∥L2∥L22≤C¯(p)
by the proved part of Theorem 1.2, where C¯(p)→+0 as p→+0. Thus, (3.15) follows. Our proof of Theorem 1.2 is complete.



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