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

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 𝑢_𝑛 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 {𝑢_𝑛}_{𝑛=0,1,2,…}, where for each 𝑛 the 𝑛th eigenfunction 𝑢_𝑛 possesses precisely 𝑛 zeros in [0,1), is a basis in 𝐿^𝑟₂(𝐵) (𝐿^𝑟₂(𝐵) is the subspace of 𝐿₂(𝐵) that consists of radial functions from 𝐿₂(𝐵). In addition, in the latter case, the sequence {𝑢_𝑛/‖𝑢_𝑛‖_𝐿2(𝐵)}_{𝑛=0,1,2,…} is a Bari basis in the same space.

1. Introduction, Notation and Definitions, and Results

In the present paper, we consider the problem −Δ𝑢+𝑓(𝑢)=𝜆𝑢,𝑢=𝑢(|𝑥|),𝑥∈𝐵={𝑥∈ℝ³∶|𝑥|<1},𝑢(0)=𝑝>0,𝑢|_|𝑥|=1=0, (1.1)(1.2)(1.3) 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 {𝑢_𝑛}_{𝑛=0,1,2,…} such that for each integer 𝑛 the 𝑛th eigenfunction 𝑢_𝑛 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 𝐿^𝑟₂(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 𝑢_𝑛, where for each integer 𝑛≥0 the 𝑛th eigenfunction 𝑢_𝑛, 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 𝐿^𝑟₂(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 {𝜑_𝑛}_{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 {𝜑_𝑛/‖𝜑_𝑛‖_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 some notation 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)=(⋅,⋅)_𝐿^1/2_2(𝐵). By 𝐿^𝑟₂=𝐿^𝑟₂(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 𝐿_2,𝑟2(0,1) denote the usual weighted Lebesgue space of functions 𝑔 measurable in (0,1) for which ‖𝑔‖_{𝐿22,𝑟(0,1)}∶={¹₀𝑟²𝑔²(𝑟)𝑑𝑟}^1/2<∞. (1.4) The space 𝐿_2,𝑟2(0,1) is equipped with the corresponding scalar product. In fact, 𝐿₂(𝐵), 𝐿^𝑟₂, and 𝐿_2,𝑟2(0,1) are Hilbert spaces.

Let 𝐻 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 {ℎ_𝑛}_{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 {𝑎_𝑛}_{𝑛=0,1,2,…} such that ℎ=^∞_𝑛=0𝑎_𝑛ℎ_𝑛in𝐻. (1.5) Two sequences {ℎ_𝑛}_{𝑛=0,1,2,…} and {𝑒_𝑛}_{𝑛=0,1,2,…} from 𝐻 are called quadratically close to each other (or the sequence {ℎ_𝑛}_{𝑛=0,1,2,…} is called quadratically close to the sequence {𝑒_𝑛}_{𝑛=0,1,2,…}) if ^∞_𝑛=0‖ℎ_𝑛−𝑒_𝑛‖²_𝐻<∞. (1.6) A basis {ℎ_𝑛}_{𝑛=0,1,2,…} in 𝐻 quadratically close to an orthonormal basis {𝑒_𝑛}_{𝑛=0,1,2,…} in 𝐻 is called a Bari basis in 𝐻. According to Corollary 2.5 in [1], if {ℎ_𝑛}_{𝑛=0,1,2,…} is an arbitrary sequence of elements of 𝐻 and if ^∞_𝑛=0‖ℎ_𝑛−𝑒_𝑛‖²_𝐻<1, (1.7) where {𝑒_𝑛}_{𝑛=0,1,2,…} is an orthonormal basis in 𝐻, then {ℎ_𝑛}_{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 {𝑢_𝑛}_{𝑛=0,1,2,…} of problem (1.1)–(1.3) standard if for each integer 𝑛≥0 the 𝑛th eigenfunction 𝑢_𝑛 regarded as a function of 𝑟 possesses precisely 𝑛 zeros in the interval [0,1). Everywhere we assume the following.

(f) Let 𝑓 be a continuously differentiable odd function in ℝ and let 𝑓^′(0)=0.

Note that the assumption that 𝑓^′(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: −Δ𝑣=𝜇𝑣,𝑣=𝑣(|𝑥|),𝑥∈𝐵,𝑣(0)=𝑝,𝑣|_|𝑥|=1=0, (1.8) where 𝜇∈ℝ is a spectral parameter. Denote by {𝑣_𝑛}_{𝑛=0,1,2,…} the sequence of the radial eigenfunctions of problem (1.8) where, for each integer 𝑛≥0, the 𝑛th eigenfunction 𝑣_𝑛 regarded as a function of 𝑟∈[0,1) possesses precisely 𝑛 zeros. By 𝜇₀<𝜇₁<⋯<𝜇_𝑛<⋯ (1.9) we denote the corresponding sequence of eigenvalues. Note that {𝑣_𝑛}_{𝑛=0,1,2,…} is an orthogonal basis in 𝐿^𝑟₂. Our main results here are as follows.

Theorem 1.1. Under assumption (f)(a)for any integer 𝑛≥0 problem (1.1)–(1.3) has a radial eigenfunction 𝑢_𝑛 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 𝑢_𝑛 is unique;(d)under assumption (f) there exists 𝑝₀>0 such that for any 𝑝∈(0,𝑝₀] and any integer 𝑛≥0 the radial eigenfunction 𝑢_𝑛 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 exists 𝜅=𝜅(𝑝)>0 defined for all 𝑝>0 and going to 0 as 𝑝→+0 such that for any 𝑝>0^∞_𝑛=0‖𝑢_𝑛‖𝑢_𝑛‖_𝐿2−𝑣_𝑛‖𝑣_𝑛‖_𝐿2‖²_𝐿2<𝜅(𝑝) (1.10) for an arbitrary standard sequence {𝑢_𝑛}_{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 𝐿^𝑟₂ and, in addition, the sequence {𝑢_𝑛/‖𝑢_𝑛‖_𝐿2}_{𝑛=0,1,2,…} is a Bari basis in the same space.

Remark 1.3. In view of Theorem 1.2 and the Bari's theorem (see [1, 3]), if one proves the linear independence, in the sense of the space 𝐿^𝑟₂, 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 𝐿^𝑟₂ 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 Theorem 1.1, and in Section 3, Theorem 1.2.

2. Proof of Theorem 1.1

Proofs of results of the type of Theorem 1.1(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: −(𝑟²𝑢^′)^′+𝑟²𝑓(𝑢)=𝜆𝑟²𝑢𝑢,𝑢=𝑢(𝑟),𝑟∈(0,1),𝑢(0)=𝑝,_′(0)=𝑢(1)=0, (2.1)(2.2)(2.3) where the prime means the derivative in 𝑟. Equation (2.1) can also be rewritten in the following equivalent form: −𝑢^′′−2𝑟𝑢^′+𝑓(𝑢)=𝜆𝑢,𝑢=𝑢(𝑟),𝑟∈(0,1). (2.4) We supply (2.4) with the following initial data: 𝑢(0)=𝑝,𝑢^′(0)=0. (2.5) 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 𝑢_𝜆=𝑑𝑢/𝑑𝜆 exists, is continuous and it satisfies the equations −(𝑟²𝑢^′_𝜆)^′+𝑟²𝑓^′(𝑢(𝑟))𝑢_𝜆=𝜆𝑟²𝑢_𝜆+𝑟²𝑢,𝑢_𝜆=𝑢_𝜆𝑢(𝑟),𝑟∈(0,1),_𝜆(0)=𝑢^′_𝜆(0)=0. (2.6)(2.7) 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 𝑢_𝜆, the solution 𝑢_𝜆 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 Theorem 1.1. Multiply (2.4) by 𝑢^′(𝑟) and integrate the result from 0 to 𝑟. Then, we obtain the identity {12𝑢^′2𝜆(𝑟)+2[𝑢²(𝑟)−𝑝²]+𝐹(𝑝)−𝐹(𝑢(𝑟))}^′2=−𝑟𝑢^′2(𝑟), (2.8) where 𝐹(𝑢)=∫^𝑢₀𝑓(𝑠)𝑑𝑠. Denote 𝐸(𝑢(𝑟))=(1/2)𝑢^′2(𝑟)+(𝜆/2)[𝑢²(𝑟)−𝑝²]+𝐹(𝑝)−𝐹(𝑢(𝑟)).

Lemma 2.1. If 𝜆∈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 that 𝑢^′′(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. Let 𝑢^′′(0)≤0. Then, |𝑢(𝑟)|≤𝑝 for all 𝑟∈(0,1].

Proof. It can be made by analogy (if 𝑢^′′(0)=0, then 𝑢(𝑟)≡𝑝 by the uniqueness theorem).

Note that Lemmas 2.1 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 that 𝑢^′′(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 integer 𝑛≥0 and denote by Λ_𝑛 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 𝜆_𝑛=infΛ_𝑛. Denote by 𝑢_𝑛(𝑟) the solution of problem (2.4)-(2.5) taken with 𝜆=𝜆_𝑛. Then, |𝑢_𝑛(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 𝑢_𝑛(𝑟) are isolated and hence, 𝑢_𝑛 has a finite number 𝑚 of zeros in (0,1). If 𝑚>𝑛, then there exists 𝜆<𝜆_𝑛 sufficiently close to 𝜆_𝑛 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 Λ_𝑛. By analogy, if 𝑚<𝑛 or if 𝑚=𝑛 and 𝑢_𝑛(1)≠0, then any solution of (2.4) and (2.5) taken for 𝜆>𝜆_𝑛 sufficiently close to 𝜆_𝑛 has no more than 𝑛 zeros in (0,1) which contradicts our definition of the set Λ_𝑛. Thus, 𝑢_𝑛(𝑟) has precisely 𝑛 zeros in the interval (0,1) and 𝑢_𝑛(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 eigenfunctions 𝑢¹ and 𝑢² of problem (2.1)–(2.3) corresponding to the eigenvalues 𝜆¹ and 𝜆², respectively, where 𝜆¹<𝜆². By (2.8), 𝑢^𝑖′(𝑟)<0 for any 𝑟∈(0,1]. Indeed, if it would be 𝑢^𝑖(𝑟)=𝑢^𝑖′(𝑟)=0, then 𝑢^𝑖≡0, while if 𝑢^𝑖(𝑟)>0 and 𝑢^𝑖′(𝑟)<0 for 𝑟∈(0,𝑟₀) and if 𝑢^𝑖′(𝑟₀)=0, then 𝑢^𝑖(𝑟)≥𝑢^𝑖(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 has 𝑢^2′(𝑟)/𝑢²(𝑟)<𝑢^1′(𝑟)/𝑢¹(𝑟) for any 𝑟∈(0,1).

Proof. We have 𝑢^2′′(0)<𝑢^1′′(0)<0, therefore 𝑢^1′(𝑟)𝑢¹>𝑢(𝑟)^2′(𝑟)𝑢²(𝑟),𝑢²(𝑟)<𝑢¹(𝑟)<𝑝 (2.9) 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 ln𝑢¹(𝑟)−ln𝑝>ln𝑢²(𝑟)−ln𝑝, (2.10) 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), (𝑢^1′(𝑟)𝑢¹)(𝑟)^′|_𝑟=𝑎𝑢>(^2′(𝑟)𝑢²)(𝑟)^′|_𝑟=𝑎, (2.11) hence, by continuity (2.11) is still valid in a left half-neighborhood of the point 𝑟=𝑎 so that it must be 𝑢^1′(𝑎)/𝑢¹(𝑎)>𝑢^2′(𝑎)/𝑢²(𝑎), which is a contradiction.

Now, suppose that (2.9) holds everywhere in (0,1) and that 𝑢^𝑖(1)=0, 𝑖=1,2. Denote 𝑡₁=𝑢^1′(1), 𝑡₂=𝑢^1′′(1), 𝑠₁=𝑢^2′(1), and 𝑠₂=𝑢^2′′(1). From (2.4), 𝑡₂=−2𝑡₁,𝑠₂=−2𝑠₁. (2.12) In a neighborhood of the point 𝑟=1, one has 𝑢^1′(𝑟)𝑢¹=𝑡(𝑟)¹−2𝑡₁(𝑟−1)+𝑂((𝑟−1)²)𝑡₁(𝑟−1)−𝑡₁(𝑟−1)²+𝑂((𝑟−1)³)=1𝑟−1−1+𝑂(𝑟−1), (2.13) and by analogy 𝑢^2′(𝑟)𝑢²=1(𝑟)𝑟−1−1+𝑂(𝑟−1). (2.14) So, we see that the difference 𝑟⁻¹(𝑢^1′(𝑟)𝑢¹−𝑢(𝑟)^2′(𝑟)𝑢²)(𝑟) (2.15) goes to 0 as 𝑟→1−0. But then (𝑢^1′(𝑟)/𝑢¹(𝑟))^′−(𝑢^2′(𝑟)/𝑢²(𝑟))^′>0 in a left half-neighborhood of the point 𝑟=1 (because by (2.9) (𝑢^1′(𝑟)/𝑢¹(𝑟))²−(𝑢^2′(𝑟)/𝑢²(𝑟))²≤0 in (0,1)), and since in addition 𝑢^1′(𝑟)/𝑢¹(𝑟)>𝑢^2′(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 Theorem 1.1. Let us prove the following.

Lemma 2.4. There exist 𝑝>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 ‖𝑢_𝜆‖²_{𝐿22,𝑟(0,𝑎)}≤𝐶‖𝑢‖²_{𝐿22,𝑟(0,𝑎)}. (2.16)

Proof. First, multiply (2.6) by 𝑢^′_𝜆 and integrate the result from 0 to 𝑟∈(0,𝑎]. Then, after integration by parts, we obtain 12𝑟²𝑢_𝜆^′2𝜆(𝑟)+2𝑟²𝑢²_𝜆(𝑟)+^𝑟₀𝑠𝑢_𝜆^′2(𝑠)𝑑𝑠=𝜆^𝑟₀𝑠𝑢²_𝜆(𝑠)𝑑𝑠+^𝑟₀𝑠²𝑓^′(𝑢(𝑠))𝑢_𝜆(𝑠)𝑢^′_𝜆(𝑠)𝑑𝑠−^𝑟₀𝑠²𝑢(𝑠)𝑢^′_𝜆(𝑠)𝑑𝑠. (2.17) Second, multiply (2.6) by 𝑟⁻¹𝑢_𝜆(𝑟) and integrate the result from 0 to 𝑟. Then, after integration by parts, 12𝑢²_𝜆(𝑟)+𝜆^𝑟₀𝑠𝑢²_𝜆(𝑠)𝑑𝑠=^𝑟₀𝑠𝑢_𝜆^′2(𝑠)𝑑𝑠−𝑟𝑢_𝜆(𝑟)𝑢^′_𝜆+(𝑟)_𝑟0𝑠𝑓^′(𝑢(𝑠))𝑢²_𝜆(𝑠)𝑑𝑠−^𝑟₀𝑠𝑢(𝑠)𝑢_𝜆(𝑠)𝑑𝑠. (2.18) Now, add (2.18) to (2.17). Then, we obtain the following identity: 12𝑟²𝑢_𝜆^′2𝜆(𝑟)+2𝑟²𝑢²_𝜆1(𝑟)+2𝑢²_𝜆(𝑟)=−𝑟𝑢_𝜆(𝑟)𝑢^′_𝜆(𝑟)+^𝑟₀𝑠𝑓^′(𝑢(𝑠))𝑢²_𝜆(𝑠)𝑑𝑠−^𝑟₀𝑠²𝑢(𝑠)𝑢^′_𝜆+(𝑠)𝑑𝑠_𝑟0𝑠²𝑓^′(𝑢(𝑠))𝑢_𝜆(𝑠)𝑢^′_𝜆(𝑠)𝑑𝑠−^𝑟₀𝑠𝑢(𝑠)𝑢_𝜆(𝑠)𝑑𝑠. (2.19)
Now, take an arbitrary 𝑝>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, 12^𝑟₀𝑠²[𝑢_𝜆^′2(𝑠)+𝜆𝑢²_𝜆1(𝑠)]𝑑𝑠+2𝑟𝑢²_𝜆(𝑟)≤𝐶₁(𝑝)^𝑟₀𝑑𝑡(^𝑡₀(𝑠+𝑠²)𝑢²_𝜆1(𝑠)𝑑𝑠)+2^𝑟₀𝑑𝑡(^𝑡₀𝑠²𝑢_𝜆^′2+1(𝑠)𝑑𝑠)2^𝑟₀𝑑𝑡(^𝑡₀𝑢²_𝜆3(𝑠)𝑑𝑠)+2^𝑟₀𝑑𝑡(^𝑡₀𝑠²𝑢²(𝑠)𝑑𝑠)≤𝐶₂(𝑝)𝑟^𝑟₀𝑠𝑢²_𝜆𝑟(𝑠)𝑑𝑠+2^𝑟₀𝑠²𝑢_𝜆^′2+𝑟(𝑠)𝑑𝑠2^𝑟₀𝑢²_𝜆3(𝑠)𝑑𝑠+2𝑟^𝑟₀𝑠²𝑢²(𝑠)𝑑𝑠 (2.20) 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 𝑢²_𝜆(𝑟)≤𝐶₃(𝑝)^𝑟₀𝑢²_𝜆(𝑠)𝑑𝑠+𝐶₄^𝑟₀𝑠²𝑢²(𝑠)𝑑𝑠, (2.21) 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 exists 𝑝>0 such that for any 𝑝∈(0,𝑝] there is no 𝑎∈(0,1], and 𝜆∈ℝ for which 𝑢(𝑎)=𝑢_𝜆(𝑎)=0.

Proof. On the contrary, suppose that there exist arbitrary small 𝑝>0, 𝑎∈(0,1], and 𝜆∈ℝ for which 𝑢(𝑎)=𝑢_𝜆(𝑎)=0. Multiply (2.4) by 𝑢_𝜆(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, ^𝑎₀𝑠²𝑢²(𝑠)𝑑𝑠=^𝑎₀𝑠²[𝑓^′(𝑢(𝑠))−𝑓(𝑢(𝑠))𝑢(𝑠)]𝑢(𝑠)𝑢_𝜆(𝑠)𝑑𝑠≤𝐶₅(𝑝)[‖𝑢‖²_{𝐿22,𝑟(0,𝑎)}+‖𝑢_𝜆‖²_{𝐿22,𝑟(0,𝑎)}], (2.22) where 𝐶₅(𝑝)→+0 as 𝑝→+0 because 𝑓(0)=𝑓^′(0)=0. Therefore ‖𝑢‖²_{𝐿22,𝑟(0,𝑎)}≤2𝐶₅(𝑝)‖𝑢_𝜆‖_{𝐿22,𝑟(0,𝑎)} (2.23) for 𝑝>0 sufficiently small, which contradicts Lemma 2.4.

Let us prove Theorem 1.1(d). Let now 𝑝∈(0,𝑝]. Take an arbitrary integer 𝑛≥0 and let 𝜆_𝑛=infΛ_𝑛, as earlier. Then, a simple corollary of Lemma 2.5 is that there exists a right half-neighborhood 𝐼_𝑛 of 𝜆_𝑛 belonging to Λ_𝑛 in which the (𝑛+1)st zero 𝑟_𝑛+1 of the solution 𝑢(𝑟) of problem (2.4)-(2.5) is a strictly decreasing function of 𝜆∈𝐼_𝑛 (so that in particular 𝑟_𝑛+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 (𝜆_𝑛,+∞) 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 Theorem 1.2

Denote 𝑤_𝑛(𝑟)=𝑢_𝑛(𝑟)−𝑣_𝑛(𝑟). Then (−𝑟²𝑤^′_𝑛)^′=𝜇_𝑛𝑟²𝑤_𝑛+𝑟²𝑔_𝑛𝑤(𝑟),𝑟∈(0,1),_𝑛(0)=𝑤^′_𝑛(0)=𝑤_𝑛(1)=0, (3.1)(3.2) where 𝑔_𝑛(𝑟)=(𝜆_𝑛−𝜇_𝑛)𝑟²𝑢_𝑛(𝑟)−𝑟²𝑓(𝑢_𝑛(𝑟)). By the comparison theorem and Theorem 1.1(b) |𝜆_𝑛−𝜇_𝑛|≤𝐶₁(𝑝), (3.3) 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 𝐶₂⁻¹(𝑛+1)²≤𝜇_𝑛≤𝐶₂(𝑛+1)² (3.4) for all 𝑛. By (3.3) and Theorem 1.1(b), |𝑔_𝑛(𝑟)|≤𝐶₃(𝑝)|𝑢_𝑛(𝑟)|, (3.5) where 𝐶₃(𝑝)>0 goes to 0 as 𝑝→+0 uniformly in 𝑛.

Now, we proceed as when proving Lemma 2.4. First, multiply (3.1) by 𝑤^′_𝑛(𝑟) and integrate the result from 0 to 𝑟. Then, after integration by parts, 12𝑟²𝑤_𝑛^′2𝜇(𝑟)+^𝑛2𝑟²𝑤²_𝑛(𝑟)+^𝑟₀𝑠𝑤_𝑛^′2(𝑠)𝑑𝑠=𝜇_𝑛^𝑟₀𝑠𝑤²_𝑛(𝑠)𝑑𝑠−^𝑟₀𝑠²𝑔_𝑛(𝑠)𝑤^′_𝑛(𝑠)𝑑𝑠. (3.6) Second, multiply (3.1) by 𝑟⁻¹𝑤_𝑛(𝑟) and integrate the result from 0 to 𝑟. Then, in view of the boundary conditions (3.2), after integration by parts, 12𝑤²_𝑛(𝑟)+𝜇_𝑛^𝑟₀𝑠𝑤²_𝑛(𝑠)𝑑𝑠=^𝑟₀𝑠𝑤_𝑛^′2(𝑠)𝑑𝑠−𝑟𝑤_𝑛(𝑟)𝑤^′_𝑛(𝑟)−^𝑟₀𝑠𝑔_𝑛(𝑠)𝑤_𝑛(𝑠)𝑑𝑠. (3.7) Add (3.7) to (3.6). Then, we obtain 12𝑟²𝑤_𝑛^′2𝜇(𝑟)+^𝑛2𝑟²𝑤²_𝑛1(𝑟)+2𝑤²_𝑛(𝑟)=−𝑟𝑤_𝑛(𝑟)𝑤^′_𝑛(𝑟)−^𝑟₀𝑠²𝑔_𝑛(𝑠)𝑤^′_𝑛(𝑠)𝑑𝑠−^𝑟₀𝑠𝑔_𝑛(𝑠)𝑤_𝑛(𝑠)𝑑𝑠. (3.8) Integrate (3.8) for one more time from 0 to 𝑟. Then, as when deriving (2.20), 12^𝑟₀𝑠²[𝑤_𝑛^′2(𝑠)+𝜇_𝑛𝑤²_𝑛1(𝑠)]𝑑𝑠+2𝑟𝑤²_𝑛≤𝑟(𝑟)2^𝑟₀𝑤²_𝑛𝑟(𝑠)𝑑𝑠+2^𝑟₀𝑠²𝑤_𝑛^′2(𝑠)𝑑𝑠+𝑟^𝑟₀𝑠²𝑔²_𝑛(𝑠)𝑑𝑠. (3.9) Hence, in view of (3.3) and (3.5), 𝑤²_𝑛(𝑟)≤^𝑟₀𝑤²_𝑛(𝑠)𝑑𝑠+𝐶₄(𝑝)^𝑟₀𝑠²𝑢²_𝑛(𝑠)𝑑𝑠, (3.10) 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: 𝑤²_𝑛(𝑟)≤𝐶₅(𝑝)‖𝑢_𝑛‖²_{𝐿22,𝑟(0,𝑟)},𝑟∈[0,1], (3.11) 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 ‖𝑤_𝑛‖²_𝐿2≤𝐶₆(𝑝)(𝑛+1)⁻²‖𝑢_𝑛‖²_𝐿2 (3.12) for a constant 𝐶₆>0 independent of 𝑛 and going to 0 as 𝑝→+0.

Now, it follows from (3.12) that ^∞_𝑛=0‖𝑢_𝑛‖𝑢_𝑛‖_𝐿2−𝑣_𝑛‖𝑢_𝑛‖_𝐿2‖²_𝐿2<𝐶(𝑝), (3.13) where 𝐶(𝑝)>0 is defined for all 𝑝>0 and goes to 0 as 𝑝→+0. So, we have ^∞_𝑛=0‖𝑢_𝑛‖𝑢_𝑛‖_𝐿2−𝑣_𝑛‖𝑣_𝑛‖_𝐿2‖²_𝐿2≤2^∞_𝑛=0𝑢{‖^𝑛‖𝑢_𝑛‖_𝐿2−𝑣_𝑛‖𝑢_𝑛‖_𝐿2‖²_𝐿2𝑣+‖^𝑛‖𝑢_𝑛‖_𝐿2−𝑣_𝑛‖𝑣_𝑛‖_𝐿2‖²_𝐿2}=2{𝐶(𝑝)+^∞_𝑛=0(‖𝑣_𝑛‖_𝐿2‖𝑢_𝑛‖_𝐿2−1)²}. (3.14) In view of this estimate, to prove Theorem 1.2, it suffices to show that ^∞_𝑛=0(‖𝑣_𝑛‖_𝐿2‖𝑢_𝑛‖_𝐿2−1)²≤𝐶(𝑝), (3.15) where 𝐶(𝑝)>0 goes to 0 as 𝑝→+0. But we have ^∞_𝑛=0(‖𝑣_𝑛‖_𝐿2‖𝑢_𝑛‖_𝐿2−1)²≤^∞_𝑛=0‖𝑢_𝑛‖𝑢_𝑛‖_𝐿2−𝑣_𝑛‖𝑢_𝑛‖_𝐿2‖²_𝐿2≤𝐶(𝑝) (3.16) 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.