Ground State for the Schrödinger Operator with the Weighted Hardy Potential

Abstract

We establish the existence of ground states on ℝ^𝑁 for the Laplace operator involving the Hardy-type potential. This gives rise to the existence of the principal eigenfunctions for the Laplace operator involving weighted Hardy potentials. We also obtain a higher integrability property for the principal eigenfunction. This is used to examine the behaviour of the principal eigenfunction around 0.

1. Introduction

In this paper, we investigate the existence of ground states of the Schrödinger operator associated with the quadratic form 𝑄_𝑉(𝑢)=_ℝ𝑁||∇𝑢||²−Λ_𝑉𝑉(𝑥)𝑢²𝑑𝑥,𝑢∈𝐶^∞∘ℝ^𝑁,𝑁≥3,(1.1) where 𝑉 belongs to the Lorentz space 𝐿^𝑁/2,∞(ℝ^𝑁) and Λ_𝑉 is the largest constant (whenever exists) for which the form 𝑄_𝑉 is nonnegative. This assumption implies, when 𝑉≥0, that the potential term ∫_ℝ𝑁𝑉(𝑥)𝑢²𝑑𝑥 is continuous in 𝐷^1,2(ℝ^𝑁), where 𝐷^1,2(ℝ^𝑁) is the Sobolev space obtained as the completion of 𝐶^∞∘(ℝ^𝑁) with respect to the norm ‖𝑢‖^2𝐷_1,2=_ℝ𝑁||∇𝑢||²𝑑𝑥.(1.2) We are mainly interested in the case of the Hardy-type potential 𝑉(𝑥)=𝑚(𝑥)/|𝑥|² with 𝑚∈𝐿^∞(ℝ^𝑁). Assuming that 𝑉 is positive on a set of positive measure, the constant Λ_𝑉 is given by the variational problem Λ_𝑉=inf_{𝑢∈𝐷1,2ℝ𝑁,∫ℝ𝑁𝑉𝑢2𝑑𝑥=1}_ℝ𝑁||∇𝑢||²𝑑𝑥,(1.3) and the continuity of ∫_ℝ𝑁𝑉(𝑥)𝑢²𝑑𝑥 implies that Λ_𝑉>0. If problem (1.3) has a minimizer 𝑢, then it satisfies −Δ𝑢−Λ_𝑉𝑉(𝑥)𝑢=0.(1.4) A solution of (1.4) is understood in the weak sense _ℝ𝑁∇𝑢∇𝜙𝑑𝑥=Λ_𝑉_ℝ𝑁𝑉(𝑥)𝑢𝜙𝑑𝑥,(1.5) for every 𝜙∈𝐷^1,2(ℝ^𝑁).

Since |𝑢| is also a minimizer for Λ_𝑉, we may assume that 𝑢≥0 a.e. on ℝ^𝑁. In particular, when 𝑉(𝑥)=𝑚(𝑥)/|𝑥|² with 𝑚∈𝐿^∞(ℝ^𝑁), then 𝑢>0 on ℝ^𝑁 by the Harnack inequality [1]. If the potential term is weakly continuous in 𝐷^1,2(ℝ^𝑁), for example, when 𝑉(𝑥)=𝑚(𝑥)/|𝑥|² with 𝑚∈𝐿^∞(ℝ^𝑁) and lim_|𝑥|→∞𝑚(𝑥)=lim_𝑥→0𝑚(𝑥)=0, then there exists a minimizer for Λ_𝑉. We will call the minimizer of (1.3) a ground state of finite energy. In general, (1.3) may not have a minimizer. This is the case for the Hardy potential 𝑉(𝑥)=1/|𝑥|² with the corresponding optimal constant Λ_𝑉=Λ_𝑁=((𝑁−2)/2)². In fact, the ground state of finite energy is a particular case of the generalized ground state, defined as follows (see [2–4]).

Definition 1.1. Let Ω⊂ℝ^𝑁 be an open set, and let 𝑄_𝑉 be as in (1.1). A sequence of nonnegative functions 𝑣_𝑘∈𝐶^∞∘(Ω) is said to be a null sequence for the functional 𝑄_𝑉 if 𝑄_𝑉(𝑣_𝑘)→0, as 𝑘→∞, and there exists a nonnegative function 𝜓∈𝐶^∞∘(Ω) such that ∫_Ω𝜓𝑣_𝑘𝑑𝑥=1 for each 𝑘.

Let us recall that the capacity of a compact set 𝐸 relative to an open set Ω⊂ℝ^𝑁, with 𝐸⊂Ω, is given by cap(𝐸,Ω)=inf_Ω||∇𝑢||²𝑑𝑥;𝑢∈𝐶^∞∘(Ω),with𝑢(𝑥)≥1on𝐸.(1.6) In the case Ω=ℝ^𝑁, we use notation cap(𝐸) (see [5]).

We can now formulate the following “ground state alternative” (see [3, 4]).

Theorem 1.2. Let 𝑉 be a measurable function bounded on every compact subset of Ω=ℝ^𝑁−𝑍, where 𝑍 is a closed set of capacity zero, and assume that 𝑄_𝑉(𝑢)≥0 for all 𝑢∈𝐶^∞∘(Ω). Then, if 𝑄_𝑉 admits a null sequence 𝑣_𝑘, then the sequence 𝑣_𝑘 converges weakly in 𝐻^1loc(ℝ^𝑁) to a unique (up to a multiplicative constant) positive solution of (1.4).

This theorem gives rise to the definition of the generalized ground state.

Definition 1.3. A unique positive solution 𝑣 of (1.4) is called a generalized ground state of the functional 𝑄_𝑉, if the functional admits a null sequence weakly convergent to 𝑣.

If 𝑉(𝑥)=1/|𝑥|², then the functional 𝑄_𝑉 has a ground state 𝑣(𝑥)=|𝑥|^(2−𝑁)/2 of infinite 𝐷^1,2 norm, while (1.3) has no minimizer in 𝐷^1,2(ℝ^𝑁).

It is important to note that the functional 𝑄_𝑉 with the optimal constant Λ_𝑉 does not necessarily have a ground state. We quote the following statement from [4].

Theorem 1.4. Let 𝑉 be a measurable function bounded on every compact subset of Ω=ℝ^𝑁−𝑍, where 𝑍 is a closed set of capacity zero, and assume that 𝑄_𝑉(𝑢)≥0 for all 𝑢∈𝐶^∞∘(Ω). Then either Q_𝑉 admits a null sequence, or there exists a function 𝑊, positive and continuous on Ω, such that 𝑄_𝑉(𝑢)≥_ℝ𝑁𝑊(𝑥)𝑢²𝑑𝑥.(1.7)

For example, let 𝑚 be a continuous function on ℝ^𝑁−{0} such that 𝑚(𝑥)=1/|𝑥|² for 0<|𝑥|≤1, 𝑚(𝑥)∈[1/2,1] for |𝑥|∈(1,2) and 𝑚(𝑥)=1/2|𝑥|² for |𝑥|≥2. Then, Λ_𝑉=((𝑁−2)/2)² and the functional 𝑄_𝑉 does not admit a null sequence. From Theorem 1.4 follows that 𝑄_𝑉 satisfies (1.7) with some function 𝑊 positive on ℝ^𝑁−{0}.

Obviously, ground states of finite 𝐷^1,2 norm are principal eigenfunctions of (1.4). There is a quite extensive literature on principal eigenfunctions with indefinite weight functions for elliptic operators on ℝ^𝑁 or on unbounded domains of ℝ^𝑁, with the Dirichlet boundary conditions. We mention papers [2, 6–13], where the existence of principal eigenfunctions has been established under various assumptions on weight functions. These conditions require that a potential belongs to some Lebesgue space, for example 𝐿^𝑝(ℝ^𝑁) with 𝑝>𝑁/2. These results have been recently greatly improved in papers [14, 15], where potentials from the Lorentz spaces have been considered. To describe the results from [14, 15] we recall the definition of the Lorentz space [16–18].

Let 𝑓∶ℝ^𝑁→ℝ be a measurable function. We define the distribution function 𝛼_𝑓 and a nonincreasing rearrangement 𝑓^∗ of 𝑓 in the following way 𝛼_𝑓(𝑠)=||𝑥∈ℝ^𝑁;||𝑓(𝑥)||>𝑠||,𝑓^∗(𝑡)=inf𝑠>0;𝛼_𝑓(𝑠)≤𝑡.(1.8) We now set ‖𝑓‖^{∗(𝑝,𝑞)}=⎧⎪⎨⎪⎩^∞0𝑡^1/𝑝𝑓^∗(𝑡)^𝑞𝑑𝑡𝑡^1/𝑞sup_𝑡>0,if1≤𝑝,𝑞<∞,𝑡^1/𝑝𝑓^∗(𝑡),if1≤𝑝≤∞,𝑞=∞.(1.9) The Lorentz space 𝐿^𝑝,𝑞(ℝ^𝑁) is defined by 𝐿^𝑝,𝑞ℝ^𝑁=𝑓∈𝐿^1locℝ^𝑁;‖𝑓‖^{∗(𝑝,𝑞)}<∞.(1.10) The functional ‖𝑓‖^{∗(𝑝,𝑞)} is only a quasinorm. To obtain a norm we replace 𝑓 by 𝑓^∗∗(𝑡)=(1/𝑡)∫^𝑡0𝑓^∗(𝑠)𝑑𝑥 in the definition of ‖𝑓‖^{∗(𝑝,𝑞)}, that is, the norm is given by ‖𝑓‖_(𝑝,𝑞)=⎧⎪⎨⎪⎩^∞0𝑡^1/𝑝𝑓^∗∗(𝑡)^𝑞𝑑𝑡𝑡^1/𝑞sup_𝑡>0,if1≤𝑝,𝑞<∞,𝑡^1/𝑝𝑓^∗∗(𝑡),if1≤𝑝≤∞,𝑞=∞.(1.11)𝐿^𝑝,𝑞(ℝ^𝑁)equipped with the norm ‖𝑓‖_(𝑝,𝑞) is a Banach space.

In paper [15] the existence of principal eigenfunctions has been established for weights belonging to ⋃_1≤𝑞<∞𝐿^𝑁/2,𝑞(ℝ^𝑁). This was extended in [14] to a larger class of weights ℱ_𝑁/2 obtained as the completion of 𝐶^∞∘(ℝ^𝑁) in norm ‖⋅‖_𝑁/2,∞.

However, these conditions do not cover the singular weight functions considered in this paper. By contrast, in our approach, we give an exact upper bound for the principal eigenvalue which allows us to prove the existence of the principal eigenfunction. We point out that if 𝑉∈𝐿^𝑁/2,∞(ℝ^𝑁), then the functional ∫_ℝ𝑁𝑉(𝑥)𝑢²𝑑𝑥 is continuous on 𝐷^1,2(ℝ^𝑁), but not necessarily weakly continuous.

The paper is organized as follows. In Section 2, we prove the existence of minimizers with finite norm 𝐷^1,2(ℝ^𝑁) and also with infinite norm 𝐷^1,2(ℝ^𝑁). In Section 3 we discuss perturbation of a given quadratic form 𝑄_𝑉∘ with 𝑉_∘∈𝐿^𝑁/2,∞(ℝ^𝑁). We show that if 𝑄_𝑉∘ has ground state, then this property is stable under small perturbations of 𝑉_∘. This is not true if 𝑄_𝑉∘ does not have a ground state; rather it is stable under larger perturbation of 𝑉_∘. The final Section is devoted to a higher integrability property of minimizers of 𝑄_𝑉∘ in the case where 𝑉_∘(𝑥)=𝑚(𝑥)/|𝑥|² with 𝑚∈𝐿^∞(ℝ^𝑁). We also examine the behaviour of the principal eigenfunction around 0.

Throughout this paper, in a given Banach space, we denote strong convergence by “→” and weak convergence by “⇀”. The norms in the Lebesgue space 𝐿^𝑝(Ω), 1≤𝑝≤∞, are denoted by ‖𝑢‖_𝑝.

2. Existence of Minimizers

We consider the Hardy-type potential 𝑉(𝑥)=𝑚(𝑥)/|𝑥|² with 𝑚∈𝐿^∞(ℝ^𝑁). In Theorem 2.2 we formulate conditions on 𝑚 guaranteeing the existence of a principal eigenfunction. Let 𝛾₊>1 and 𝛾₋>1. In our approach to problem (1.3), the following two limits play an important role: it is assumed that the following limits exist a.e. 𝑚𝑚₊(𝑥)=lim_{𝑗∈ℕ,𝑗→∞}𝑚𝛾^𝑗+𝑥,(2.1)₋(𝑥)=lim_{𝑗∈ℕ,𝑗→∞}𝑚𝛾^−−𝑗𝑥.(2.2) Both functions 𝑚_± satisfy 𝑚_±(𝛾_±𝑥)=𝑚_±(𝑥). We now define the following infima: Λ_𝑚=inf_{𝑢∈𝐷1,2ℝ𝑁−{0}}∫_ℝ𝑁||∇𝑢||²𝑑𝑥∫_ℝ𝑁𝑚(𝑥)/|𝑥|²𝑢²𝑑𝑥,(2.3) (we use the notation Λ_𝑚 instead of Λ_𝑉) and Λ_±=inf_{𝑢∈𝐷1,2ℝ𝑁−{0}}∫_ℝ𝑁||∇𝑢||²𝑑𝑥∫_ℝ𝑁𝑚_±(𝑥)/|𝑥|²𝑢²𝑑𝑥.(2.4)

Lemma 2.1. The following holds true Λ_𝑚≤minΛ₊,Λ₋.(2.5)

Proof. Let 𝑢∈𝐷^1,2(ℝ^𝑁)−{0}. Testing Λ_𝑚 with 𝛾^{+−(𝑁−2)/2}𝑢(𝛾^+−𝑗𝑥) gives Λ_𝑚≤∫^ℝ𝑁||∇𝑢||²𝑑𝑥∫_ℝ𝑁𝑚𝛾^𝑗+𝑥/|𝑥|²𝑢²𝑑𝑥.(2.6) Letting 𝑗→∞ and using the Lebesgue dominated convergence theorem, we obtain Λ_𝑚≤∫^ℝ𝑁||∇𝑢||²𝑑𝑥∫_ℝ𝑁𝑚₊(𝑥)/|𝑥|²𝑢²𝑑𝑥.(2.7) The inequality Λ_𝑚≤Λ₊ follows. The proof of the inequality Λ_𝑚≤Λ₋ is similar.

In the case when the inequality (2.5) is strict problem, (2.2) has a minimizer.

Theorem 2.2. Assume that the convergence in (2.1) is uniform on sets {𝑥∈ℝ^𝑁;|𝑥|≥𝑅} for every 𝑅>0 and that the convergence in (2.2) is uniform on sets {𝑥∈ℝ^𝑁;|𝑥|≤𝜌} for every 𝜌>0. If Λ_𝑚<𝑚𝑖𝑛(Λ₊,Λ₊), then problem (2.3) has a minimizer.

Proof. Let {𝑢_𝑘}⊂𝐷^1,2(ℝ^𝑁) be a minimizing sequence for Λ_𝑚, that is, _ℝ𝑁||∇𝑢_𝑘||²𝑑𝑥⟶Λ_𝑚,_ℝ𝑁𝑚(𝑥)||𝑋||²𝑢^2𝑘𝑑𝑥=1.(2.8) We can assume, up to a subsequence, that 𝑢_𝑘⇀𝑤 in 𝐷^1,2(ℝ^𝑁), 𝐿²(ℝ^𝑁,𝑑𝑥/|𝑥|²), and 𝑢_𝑘→𝑤 in 𝐿^2loc(ℝ^𝑁) for some 𝑤∈𝐷^1,2(ℝ^𝑁). Let 𝑣_𝑘=𝑢_𝑘−𝑤. We then have 1=_ℝ𝑁𝑚(𝑥)|𝑥|²𝑢^2𝑘𝑑𝑥=_ℝ𝑁𝑚(𝑥)|𝑥|²𝑤²𝑑𝑥+_ℝ𝑁𝑚(𝑥)|𝑥|²Λ𝑣^2𝑘𝑑𝑥+𝑜(1),(2.9)_𝑚=_ℝ𝑁||∇𝑢_𝑘||²𝑑𝑥+𝑜(1)=_ℝ𝑁||∇𝑤||²𝑑𝑥+_ℝ𝑁||∇𝑣_𝑘||²𝑑𝑥+𝑜(1).(2.10) We define a radial function 𝜒^𝑗+∈𝐶¹(ℝ^𝑁) such that 0≤𝜒^𝑗+(𝑥)≤1, 𝜒^𝑗+(𝑥)=0 for |𝑥|≤𝛾^−−2𝑗 and 𝜒^𝑗+(𝑥)=1 for |𝑥|>𝛾^+2𝑗. Let 𝜒^𝑗−(𝑥)=1−𝜒^𝑗+(𝑥). In what follows, we use 𝑜^{(𝑗)𝑘→∞}(1) to denote a quantity such that for each 𝑗∈ℕ, 𝑜^{(𝑗)𝑘→∞}(1)→0 as 𝑘→∞. Thus, _ℝ𝑁𝑚(𝑥)|𝑥|²𝑣^2𝑘𝑑𝑥=_ℝ𝑁𝑚(𝑥)|𝑥|²𝑣_𝑘𝜒^𝑗−²𝑑𝑥+_ℝ𝑁𝑚(𝑥)|𝑥|²=𝑣_𝑘𝜒^𝑗+²𝑑𝑥+𝑜^{(𝑗)𝑘→∞}(1)_ℝ𝑁𝑚𝛾^−−𝑗𝑥|𝑥|²𝑣^−𝑘²𝑑𝑥+_ℝ𝑁𝑚𝛾^𝑗+𝑥|𝑥|²𝑣^+𝑘²𝑑𝑥+𝑜^{(𝑗)𝑘→∞}(1),(2.11) where 𝑣𝑣^−𝑘_+𝑘(𝑥)=𝛾^{−(−(𝑁−2)/2)𝑗}𝑣_𝑘𝛾^−−𝑗𝑥𝜒₋𝛾^−−𝑗𝑥,(𝑥)=𝛾^{+((𝑁−2)/2)𝑗}𝑣_𝑘𝛾^𝑗+𝑥𝜒₊𝛾^𝑗+𝑥.(2.12) We now estimate the integrals involving 𝑣^−𝑘 and 𝑣^+𝑘. We have ||||_ℝ𝑁𝑚𝛾^−−𝑗𝑥|𝑥|²𝑣^−𝑘²𝑑𝑥−_ℝ𝑁𝑚₋(𝑥)|𝑥|²≤||||𝑣^−𝑘²𝑑𝑥_{|𝑥|<𝛾−−𝑗}||||𝑚𝛾^−−𝑗𝑥−𝑚₋(𝑥)|𝑥|²𝑣^−𝑘²𝑑𝑥||||+||||_{𝛾−−2𝑗<|𝑥|<𝛾+2𝑗}𝑚𝛾^−−𝑗𝑥−𝑚₋(𝑥)|𝑥|²=𝐽₁𝑣^−𝑘²𝑑𝑥||||+𝐽₂.(2.13) By the uniform convergence of 𝑚(𝛾^−−𝑗𝑥) to 𝑚₋(𝑥), we see that 𝐽₁≤𝜖 for 𝑗 sufficiently large uniformly in 𝑘. For 𝐽₂ we have 𝐽₂≤2‖𝑚‖_∞_𝛾^−−2𝑗_<|𝑥|<𝛾^+2𝑗𝑣^2𝑘|𝑥|²𝑑𝑥.(2.14) It is clear that 𝐽₂ is a quantity of type 𝑜^{(𝑗)𝑘→∞}(1). Therefore, we have ||||_ℝ𝑁𝑚𝛾^−−𝑗𝑥|𝑥|²𝑣^−𝑘²𝑑𝑥−_ℝ𝑁𝑚₋(𝑥)|𝑥|²𝑣^−𝑘²𝑑𝑥||||≤𝜖+𝑜^{(𝑗)𝑘→∞}(1).(2.15) In a similar way, we obtain |||||_ℝ𝑁𝑚𝛾^𝑗+𝑥|𝑥|²𝑣^+𝑘²𝑑𝑥−_ℝ𝑁𝑚₊(𝑥)|𝑥|²𝑣^+𝑘²𝑑𝑥|||||≤𝜖+𝑜^{(𝑗)𝑘→∞}(1)(2.16) for 𝑗 sufficiently large. We now fix 𝑗∈ℕ so that (2.15) and (2.16) hold. Consequently, we have 1≤_ℝ𝑁𝑚(𝑥)|𝑥|²𝑤²𝑑𝑥+_ℝ𝑁𝑚₋(𝑥)|𝑥|²𝑣^−𝑘²𝑑𝑥+_ℝ𝑁𝑚₊(𝑥)|𝑥|²𝑣^+𝑘²𝑑𝑥+2𝜖+𝑜^{(𝑗)𝑘→∞}(1).(2.17) We now estimate ∫_ℝ𝑁|∇𝑣_𝑘|²𝑑𝑥 in the following way _ℝ𝑁||∇𝑣_𝑘||²𝑑𝑥==_ℝ𝑁||∇𝑣_𝑘𝜒^𝑗−+𝑣_𝑘𝜒^𝑗+||²𝑑𝑥_ℝ𝑁+2||∇_ℝ𝑁𝑣_𝑘𝜒^𝑗−||²𝑑𝑥+_ℝ𝑁||∇𝑣_𝑘𝜒^𝑗+||²𝑑𝑥=∇𝑣_𝑘𝜒^𝑗−∇𝑣_𝑘𝜒^𝑗+𝑑𝑥_ℝ𝑁+2||∇_ℝ𝑁𝑣^−𝑘||²𝑑𝑥+_ℝ𝑁||∇𝑣^+𝑘||²𝑑𝑥+2_ℝ𝑁||∇𝑣_𝑘||²𝜒^𝑗−𝜒^𝑗+𝑑𝑥≥𝑣_𝑘∇𝑣_𝑘∇𝜒^𝑗−𝜒^𝑗+𝑑𝑥+2_ℝ𝑁𝑣_𝑘∇𝑣_𝑘𝜒^𝑗−∇𝜒^𝑗+𝑑𝑥+2_ℝ𝑁𝑣^2𝑘∇𝜒^𝑗−∇𝜒^𝑗+𝑑𝑥_ℝ𝑁+2||∇_ℝ𝑁𝑣^−𝑘||²𝑑𝑥+_ℝ𝑁||∇𝑣^+𝑘||²𝑑𝑥+2_ℝ𝑁𝑣_𝑘∇𝑣_𝑘∇𝜒^𝑗−𝜒^𝑗+𝑑𝑥𝑣_𝑘∇𝑣_𝑘𝜒^𝑗−∇𝜒^𝑗+𝑑𝑥+2_ℝ𝑁𝑣^2𝑘∇𝜒^𝑗−∇𝜒^𝑗+𝑑𝑥.(2.18) Since 𝑣_𝑘→0 in 𝐿^2loc(ℝ^𝑁), we obtain the following estimate _ℝ𝑁||∇𝑣_𝑘||²𝑑𝑥≥_ℝ𝑁||∇𝑣^−𝑘||²𝑑𝑥+_ℝ𝑁||∇𝑣^+𝑘||²𝑑𝑥+𝑜^{(𝑗)𝑘→∞}(1).(2.19) This, combined with (2.9), gives the following estimate Λ_𝑚≥≥Λ_ℝ𝑁𝑚||∇𝑤||²𝑑𝑥+_ℝ𝑁||∇𝑣^−𝑘||²𝑑𝑥+_ℝ𝑁||∇𝑣^+𝑘||²𝑑𝑥+𝑜^{(𝑗)𝑘→∞}(1)_ℝ𝑁𝑚(𝑥)|𝑥|²𝑤²𝑑𝑥+Λ₋_ℝ𝑁𝑚₋(𝑥)|𝑥|²+Λ₊𝑣^−𝑘²𝑑𝑥_ℝ𝑁𝑚₊(𝑥)|𝑥|²𝑣^+𝑘²𝑑𝑥+𝑜^{(𝑗)𝑘→∞}(1).(2.20) Let Λ_∗=min(Λ₋,Λ₊). We deduce from (2.17) and (2.20) that Λ_∗−Λ_𝑚_ℝ𝑁𝑚₋(𝑥)|𝑥|²𝑣^−𝑘²𝑑𝑥+𝑚⁺(𝑥)|𝑥|²𝑣^+𝑘²𝑑𝑥≤2𝜖Λ_𝑚+𝑜^{(𝑗)𝑘→∞}(1).(2.21) Letting 𝑘→∞, we obtain limsup_𝑘→∞_ℝ𝑁𝑚₋(𝑥)|𝑥|²𝑣^−𝑘²𝑑𝑥+𝑚⁺(𝑥)|𝑥|²𝑣^+𝑘²𝑑𝑥≤2𝜖Λ^𝑚Λ_∗−Λ_𝑚.(2.22) It then follows from (2.17) that 1≤_ℝ𝑁𝑚(𝑥)|𝑥|²𝑤²𝑑𝑥+2𝜖Λ^𝑚Λ_∗−Λ_𝑚.(2.23) Since 𝜖>0 is arbitrary, we get ∫_ℝ𝑁(𝑚(𝑥)/|𝑥|²)𝑤²𝑑𝑥=1, and the result follows.

In what follows, we denote 𝑚(∞)=lim_|𝑥|→∞𝑚(𝑥), assuming that this limit exists. As a direct consequence of Theorem 2.2, we obtain the following result.

Theorem 2.3. Let 𝑚∈𝐿^∞(ℝ^𝑁), and assume that 𝑚 is continuous at 0. Further, suppose that 𝑚(∞)>0 and 𝑚(0)>0. If Λ_𝑚<Λ_𝑁min(1/𝑚(∞),1/𝑚(0)), then there exists a minimizer for Λ_𝑚.

Remark 2.4. Λ_𝑚 has a minimizer also in the following cases, corresponding formally to Λ₊ or Λ₋ taking the value +∞.(i)Let 𝑚(0)=0 and 𝑚(∞)>0. If Λ_𝑚<Λ_𝑁/𝑚(∞), then a minimizer for Λ₁(𝑚) exists.(ii)Let 𝑚(0)>0 and 𝑚(∞)=0. If Λ_𝑚<Λ_𝑁/𝑚(0), then a minimizer for Λ₁(𝑚) exists.(iii)If 𝑚(0)=𝑚(∞)=0, 𝑚(𝑥)≥0 and ≢ 0 on ℝ^𝑁, then Λ_𝑚 has a minimizer.

We point out that Theorem 2.3 and the results described in Remark 2.4 can be deduced from [19, Theorem  1.2]. Unlike in paper [19], to obtain Theorem 2.3 we avoided the use of the concentration-compactness principle.

We now give examples of weight functions 𝑚 satisfying conditions of Theorems 2.2 and 2.3. In general, functions satisfying this condition have large local maxima.

Example 2.5. Let 𝑚_𝐴(𝑥)=⎧⎪⎨⎪⎩𝑚¹𝐴𝑚(₂𝑥),for0<|𝑥|<1,𝑚(𝑥),for1≤|𝑥|≤2,₃(𝑥),for2<|𝑥|,(2.24) where 𝐴>0 is a constant to be chosen later and 𝑚₁∶𝐵(0,1)−{0}→[0,∞), 𝑚₂∶(1≤|𝑥|≤2)→[0,∞), and 𝑚₃∶ℝ^N⧵𝐵(0,2)→[0,∞) are continuous bounded functions satisfying the following conditions: 𝑚₁(𝑥)=0 for |𝑥|=1, 𝑚₂(𝑥)=0 for |𝑥|=1, 𝑚₂(𝑥)=0 for |𝑥|=2, and 𝑚₂(𝑥)>0 for 1<|𝑥|<2,𝑚₃(𝑥)=0 for |𝑥|=2. Further we assume that 𝑚₃(𝑥)=𝑎+||𝑥¹||||𝑥₂||+⋯+||𝑥_𝑁−1||||𝑥_𝑁||𝑏+|𝑥|²,(2.25) for |𝑥|≥𝑅>2, where 𝑎>0, 𝑏>0 and 𝑅 constants. A function 𝑚₁(𝑥) for small 𝛿>0 is given by 𝑚₁(𝑥)=||𝑥¹||+⋯+||𝑥_𝑁|||𝑥|,(2.26) for 0<|𝑥|≤𝛿<1. We have lim_𝑗→∞𝑚_𝐴𝛾^𝑗+𝑥=lim_𝑗→∞𝛾^+−2𝑗𝑎+||𝑥₁||||𝑥₂||+⋯+||𝑥_𝑁−1||||𝑥_𝑁||𝛾^+−2𝑗𝑏+|𝑥|²=||𝑥¹||||𝑥₂||+⋯+||𝑥_𝑁−1||||𝑥_𝑁|||𝑥|²lim_𝑗→∞=𝑚₊(𝑥),𝑚_𝐴𝛾^−−𝑗𝑥=||𝑥¹||+⋯+||𝑥_𝑁|||𝑥|=𝑚⁻(𝑥).(2.27) Both limits are uniform. Since 𝑚₋ and 𝑚₊ are bounded, Λ₋ and Λ₊ are positive and finite. We have Λ_𝑚=inf_{𝐷1,2ℝ𝑁−{0}}∫_ℝ𝑁||∇𝑢||²𝑑𝑥≤1∫_ℝ𝑁𝑚_𝐴(𝑥)/|𝑥|²𝑢²𝑑𝑥𝐴inf_𝐷^1,2_ℝ^𝑁_−{0}∫_ℝ𝑁||∇𝑢||²𝑑𝑥∫_{1≤|𝑥|≤2}𝑚₂(𝑥)/|𝑥|²𝑢²𝑑𝑥<𝑚𝑖𝑛Λ⁻,Λ₊,(2.28) for 𝐴 large. By Theorem 2.2, Λ_𝑚 with 𝑚=𝑚_𝐴 has a minimizer.

Example 2.6. Consider a sequence of functions of the form 𝑚_𝑘(𝑥)=𝐵𝑀_𝑘(𝑥)+𝐴𝑓(𝑥),𝑘=1,2,…, where 𝐴>0, 𝐵>0 are constants and 𝑀_𝑘 and 𝑓 are continuous functions satisfying the following conditions: (a)𝑀_𝑘(0)=1, 𝑀_𝑘(𝑥)>0 on ℝ^𝑁, 𝑀_𝑘(∞)=0, for 𝑘=1,2,…, (b)𝑀_𝑘(𝑥)=𝑘 on 1<|𝑥|<2 for 𝑘=1,2,…, (c)𝑓(𝑥)≥0 on ℝ^𝑁, 𝑓(0)=0 and 𝑓(∞)=1. Then 𝑚_𝑘(0)=𝐵 and 𝑚_𝑘(∞)=𝐴 for 𝑘=1,2,…. We show that for 𝑘 sufficiently large 𝑚_𝑘 satisfies the conditions of Theorem 2.3. Let 𝑢(𝑥)=exp(−|𝑥|) (one can take any other function from 𝐷^1,2(ℝ^𝑁) which is ≢0 on (1<|𝑥|<2)). Thus Λ_𝑚𝑘≤∫^ℝ𝑁||∇(exp(−|𝑥|))||²𝑑𝑥≤∫_ℝ𝑁∫𝐵𝑀_𝑘(𝑥)+𝐴𝑓(𝑥)/|𝑥|²exp(−2|𝑥|)𝑑𝑥_ℝ𝑁||∇(exp(−|𝑥|))||²𝑑𝑥𝐵∫_ℝ𝑁𝑀_𝑘(𝑥)/|𝑥|²exp(−2|𝑥|)𝑑𝑥⟶0,(2.29) as 𝑘→∞. So we can find 𝑘_∘≥1 so that Λ_𝑚𝑘<Λ_𝑁min1𝐴,𝐵1for𝑘≥𝑘^∘.(2.30)

In Proposition 2.7, we described a class of weight functions 𝑚 satisfying conditions of Theorem 2.3.

Proposition 2.7. Let 𝑚∈𝐶(ℝ^𝑁). Suppose that 𝑚(𝑥)≥0, 𝑚(0)>0, and 𝑚(∞)>0. Assume that there exists a ball 𝐵(𝑥_𝑀,𝑟) such that 𝑚(𝑥)≥𝑚(𝑥_𝑀)>0 for 𝑥∈𝐵(𝑥_𝑀,𝑟) and 0∉𝐵(𝑥_𝑀,𝑟). If 𝑚(0)𝑚𝑥_𝑀,𝑚𝑚(𝑥∞_𝑀)<𝑟²(𝑁−2)²2𝑟+||𝑥_𝑀||²(𝑁+1)(𝑁+2).(2.31) Then Λ_𝑚<Λ_𝑁min(1/𝑚(0),1/𝑚(∞)). (Hence, there exists a minimizer for Λ_𝑚.)

Proof. Let 𝑢∈𝐻^1∘(𝐵(𝑥_𝑀,𝑟))−{0}. Then _{𝐵(𝑥𝑀,𝑟)}𝑚(𝑥)|𝑥|²𝑢²𝑑𝑥≥𝑚𝑥_𝑀_{𝐵(𝑥𝑀)}𝑢²|𝑥|²𝑑𝑥≥𝑚𝑥^𝑀𝑟+||𝑥_𝑀||²_{𝐵(𝑥𝑀,𝑟)}𝑢²𝑑𝑥.(2.32) Hence, ∫_{𝐵(𝑥𝑀,𝑟)}||∇𝑢||²𝑑𝑥∫_{𝐵(𝑥𝑀,𝑟)}𝑚(𝑥)/|𝑥|²𝑑𝑥≤𝑟+||𝑥^𝑀||²∫_{𝐵(𝑥𝑀,𝑟)}||∇𝑢||²𝑑𝑥𝑚𝑥_𝑀∫_{𝐵(𝑥𝑀,𝑟)}𝑢²𝑑𝑥.(2.33) Since 𝐻^1∘(𝐵(𝑥_𝑀,𝑟))−{0}⊂{u∈𝐷^1,2(ℝ^𝑁);∫_ℝ𝑁(𝑚(𝑥)/|𝑥|²)𝑢²𝑑𝑥>0}, we deduce from the above inequality that Λ_𝑚≤𝑟+||𝑥^𝑀||²𝑚𝑥_𝑀𝜆^𝐷1𝐵𝑥_𝑀,𝑟,(2.34) where 𝜆^𝐷1(𝐵(𝑥_𝑀,𝑟)) denotes the first eigenvalue for “−Δ” in 𝐵(𝑥_𝑀,𝑟) with the Dirichlet boundary conditions. We now estimate 𝜆^𝐷1=𝜆^𝐷1(𝐵(𝑥_𝑀,𝑟)). We test 𝜆^𝐷1 with 𝑣(𝑥)=𝑟−|𝑥−𝑥_𝑀| for 𝑥∈𝐵(𝑥_𝑀,𝑟). We have _{𝐵(𝑥𝑀,𝑟)}𝑣²𝑑𝑥=_𝐵(0,𝑟)(𝑟−|𝑥|)²𝑑𝑥=𝜔_𝑁^𝑟0(𝑟−𝑠)²𝑠^𝑁−1𝑑𝑠=2𝜔^𝑁𝑟^𝑁+2𝑁(𝑁+1)(𝑁+2)_{𝐵(𝑥𝑀,𝑟)},||∇𝑣||²𝑑𝑥=𝜔^𝑁𝑟^𝑁𝑁.(2.35) Hence 𝜆^𝐷1≤∫^{𝐵(𝑥𝑀,𝑟)}||∇𝑣||²𝑑𝑥∫_{𝐵(𝑥𝑀,𝑟)}𝑣²𝑑𝑥=(𝑁+12)𝑟(²𝑁+2).(2.36) Combining this with (2.34), we derive Λ_𝑚≤(𝑁+1)(𝑁+2)𝑟+||𝑥^𝑀||²2𝑟²𝑚𝑥_𝑀.(2.37) Therefore Λ_𝑚<Λ_𝑁min(1/(𝑚(0)),1/(𝑚(∞))) if (2.31) holds.

The estimate (2.31) has terms that are easy to compute but are of course not optimal. In particular, the factor ((𝑁+1)(𝑁+2))/2 can be replaced by the first eigenvalue of the Laplacian on a unit ball with Dirichlet boundary conditions.

If 𝑚(𝑥) is a continuous bounded and nonnegative function such that 𝑚(𝑥)≤𝑚(0) on ℝ^𝑁 and 𝑚(0)>0 (or 𝑚(𝑥)≤𝑚(∞) on ℝ^𝑁, 𝑚(∞)>0), then Λ_𝑚 does not have a minimizer. Indeed, suppose that 𝑚(𝑥)≤𝑚(0) on ℝ^𝑁 and that Λ_𝑚 has a minimizer 𝑢. Then, by the Hardy inequality, we obtain Λ_𝑁𝑚(0)≥∫^ℝ𝑁||∇𝑢||²𝑑𝑥∫_ℝ𝑁𝑚(𝑥)/|𝑥|²𝑢²𝑑𝑥≥∫^ℝ𝑁||∇𝑢||²𝑑𝑥𝑚(0)∫_ℝ𝑁𝑢²/|𝑥|²𝑑𝑥≥𝑚Λ(^𝑁0).(2.38) So 𝑢 is a minimizer for Λ_𝑁, which is impossible.

We now construct a ground state with infinite 𝐷^1,2 norm.

Theorem 2.8. Let 𝛾>1, and assume that the function 𝑚∈𝐿^∞(ℝ^𝑁) satisfies 𝑚(𝛾𝑥)=𝑚(𝑥)for𝑥∈ℝ^𝑁.(2.39) Then the form 𝑄_𝑉 with 𝑉(𝑥)=𝑚(𝑥)/|𝑥|² and Λ_𝑉=Λ_∘ (see (2.41) below) admits a ground state 𝑣 satisfying 𝑣(𝛾𝑥)=𝛾^(2−𝑁)/2𝑣(𝑥)for𝑥∈ℝ^𝑁.(2.40) The function 𝑣 is uniquely defined by its values on 𝐴_𝛾={𝑥∈ℝ^𝑁;1<|𝑥|<𝛾}, and, moreover, the function 𝑣_|𝐴𝛾 is a minimizer for the problem Λ_∘=inf⎧⎪⎨⎪⎩∫^𝐴𝛾||∇𝑣||²𝑑𝑥∫_𝐴𝛾𝑚(𝑥)/|𝑥|²𝑢²𝑑𝑥;𝑢∈𝐻¹𝐴_𝛾−{0},𝑢(𝛾𝑥)=𝛾^(2−𝑁)/2𝑢(𝑥)for|𝑥|=1⎫⎪⎬⎪⎭.(2.41)

Proof. The problem (2.41) is a compact variational problem that has a minimizer 𝑣 which satisfies −Δ𝑣=Λ_∘𝑚(𝑥)|𝑥|²𝑣,𝑥∈𝐴_𝛾,(2.42) with the Neumann boundary conditions. Since the test functions satisfy 𝑢(𝛾𝑥)=𝛾^(2−𝑁)/2𝑢(𝑥) for |𝑥|=1, one has 𝜕𝑣𝜕𝑟(𝛾𝑥)=𝛾^−𝑁/2𝜕𝑣𝜕𝑟(𝑥)for|𝑥|=1.(2.43) Note that |𝑣| is also a minimizer, so we may assume that 𝑣 is nonnegative. We now extend the function 𝑣 from 𝐴_𝛾 to ℝ^𝑁−{0} by using (2.40) and denote the extended function again by 𝑣. Since 𝑣 satisfies (2.41), the extended function 𝑣 is of class 𝐶¹(ℝ^𝑁−{0}) and satisfies −Δ𝑣=Λ_∘𝑚(𝑥)|𝑥|²𝑣,(2.44) in a weak sense. From this and the Harnack inequality on bounded subsets of ℝ^𝑁−{0} it follows that 𝑣 is positive on ℝ^𝑁−{0} and subsequently there exists a constant 𝐶>0 such that 𝐶⁻¹|𝑥|^(2−𝑁)/2≤𝑣(𝑥)≤𝐶|𝑥|^(2−𝑁)/2.(2.45) We can now explain the choice of the exponent (2−𝑁)/2 in the constraint 𝑢(𝛾𝑥)=𝛾^(2−𝑁)/2𝑢(𝑥) from (2.41): with any other choice, the resulting Neumann condition would not yield the continuity of the derivatives of the extended function 𝑣 on the spheres |𝑥|=𝛾^𝑗,𝑗∈ℕ. Finally, we show that 𝑣 is a ground state for the corresponding quadratic form 𝑄 with 𝑉(𝑥)=Λ_∘𝑚(𝑥)/|𝑥|². Using the ground state formula (2.10), from [20] and (2.45), we have 𝑤_𝑘(𝑥)=|𝑥|^1/𝑘 for |𝑥|≤1 and 𝑤_𝑘(𝑥)=|𝑥|^−1/𝑘 for |𝑥|≥1, 𝑄𝑣𝑤_𝑘=≤𝐶_ℝ𝑁𝑣²||∇𝑤_𝑘||²𝑑𝑥≤𝐶_ℝ𝑁|𝑥|^2−𝑁||∇𝑤_𝑘||²𝑑𝑥𝑘²¹⁰𝑟^{−1+(2/𝑘)}𝑑𝑟+𝐶𝑘²^∞1𝑟^{−1−(2/𝑘)}𝑑𝑟≤𝐶𝑘⟶0,(2.46) as 𝑘→∞. Since 𝑣𝑤_𝑘→𝑣 uniformly on compact sets, this implies that 𝑣 is a ground state for 𝑄. By (2.45) and the Sobolev inequality, 𝑣∉𝐷^1,2(ℝ^𝑁).

3. Perturbations from Virtual Ground States

In this section, we show that if a potential term admits a (generalized or large or virtual) ground state, then its weakly continuous perturbations in the suitable direction will admit a ground state with the finite 𝐷^1,2 norm. Then, we investigate potentials that do not give rise to a ground state with finite 𝐷^1,2 norm.

We need the following existence result.

Proposition 3.1. Let 𝑉_∘∈𝐿^𝑁/2,∞(ℝ^𝑁) be positive on a set of positive measure, and let Λ_∘=inf_{𝑢∈𝐷1,2ℝ𝑁,∫ℝ𝑁𝑉∘𝑢2𝑑𝑥=1}_ℝ𝑁||∇𝑢||²𝑑𝑥.(3.1) Assume that 𝑉₁∈𝐿^𝑁/2,∞(ℝ^𝑁) is positive on a set of positive measure and that the functional ∫_ℝ𝑁(𝑉₁(𝑥)−𝑉_∘(𝑥))𝑢²𝑑𝑥 is weakly continuous in 𝐷^1,2(ℝ^𝑁), and let Λ₁=inf_{𝑢∈𝐷1,2ℝ𝑁,∫ℝ𝑁𝑉1𝑢2𝑑𝑥=1}_ℝ𝑁||∇𝑢||²𝑑𝑥.(3.2) If Λ₁<Λ_∘, then there exists a minimizer for Λ₁.

Proof. Let {𝑢_𝑘}⊂𝐷^1,2(ℝ^𝑁) be a minimizing sequence for (3.2), that is, ∫_ℝ𝑁𝑉₁(𝑥)𝑢^2𝑘𝑑𝑥=1 and ∫_ℝ𝑁|∇𝑢_𝑘|²𝑑𝑥→Λ₁. We may assume that, up to a subsequence, 𝑢_𝑘⇀𝑤 in 𝐷^1,2(ℝ^𝑁) and 𝐿²(ℝ^𝑁,𝑉₁(𝑥)𝑑𝑥). Let 𝑣_𝑘=𝑢_𝑘−𝑤. Then, 1=+_ℝ𝑁𝑉₁(𝑥)𝑢^2𝑘𝑑𝑥=_ℝ𝑁𝑉₁(𝑥)𝑣^2𝑘𝑑𝑥+_ℝ𝑁𝑉₁(𝑥)𝑤²𝑑𝑥+𝑜(1)=_ℝ𝑁𝑉₁(𝑥)𝑤²𝑑𝑥_ℝ𝑁=𝑉₁(𝑥)−𝑉_∘(𝑥)𝑣^2𝑘𝑑𝑥+_ℝ𝑁𝑉_∘(𝑥)𝑣^2𝑘𝑑𝑥+𝑜(1)_ℝ𝑁𝑉_∘(𝑥)𝑣^2𝑘𝑑𝑥+_ℝ𝑁𝑉₁(𝑥)𝑤²𝑑𝑥+𝑜(1).(3.3) Let 𝑡=∫_ℝ𝑁𝑉₁(𝑥)𝑤²𝑑𝑥. Then ∫_ℝ𝑁𝑉_∘(𝑥)𝑣^2𝑘𝑑𝑥→1−𝑡. Assuming that 𝑡<1 we get Λ₁=_ℝ𝑁||∇𝑣_𝑘||²𝑑𝑥+_ℝ𝑁||∇𝑤||²𝑑𝑥+𝑜(1)≥Λ_∘(1−𝑡)+Λ₁𝑡+𝑜(1).(3.4) From this, we deduce that Λ₁≥Λ_∘ which is impossible. Hence, ∫_ℝ𝑁𝑉₁(𝑥)𝑤²𝑑𝑥=1. From this and the lower semicontinuity of the norm with respect to weak convergence, we derive that 𝑤 is a minimizer and 𝑢_𝑘→𝑤 in 𝐷^1,2(ℝ^𝑁).
Proposition 3.1 is related to [19, Theorem  1.7] which asserts that a potential of the form 𝑉(𝑥)=(1/|𝑥|²)+𝑔(𝑥), with a subcritical potential 𝑔 (for the definition of a subcritical potential see [19]), has a principal eigenfunction. This follows from the fact that 𝑔 is weakly continuous in 𝐷^1,2(ℝ^𝑁) (see [12]) and the potential 𝑔 admits a principal eigenfunction.

Remark 3.2. (i) If 𝑉₁>𝑉_∘, then Λ₁≤Λ_∘, but not necessarily Λ₁<Λ_∘.
(ii) If, in Proposition 3.1, assumption Λ₁<Λ_∘ is replaced by Λ_∘<Λ₁, then Λ_∘ is attained.

Example 3.3. Let 𝑀 be a continuous function ℝ^𝑁 such that 𝑀≥0, ≢0 on ℝ^𝑁 and 𝑀(0)=𝑀(∞)=0. Define 𝑚_𝐴,𝐵(𝑥)=𝐵𝑀(𝑥)+𝐴, where 𝐴>0 and 𝐵>0 are constants. Let 𝑉₁(𝑥)=𝑚_𝐴,𝐵(𝑥)/|𝑥|² and 𝑉_∘(𝑥)=𝐴/|𝑥|². The functional ∫_ℝ𝑁(𝑉₁(𝑥)−𝑉_∘(𝑥))𝑢²𝑑𝑥=∫_ℝ𝑁(𝐵𝑀(𝑥)/|𝑥|²)𝑢²𝑑𝑥 is weakly continuous in 𝐷^1,2(ℝ^𝑁). It is easy to show that for every 𝐴>0 there exists 𝐵_∘>0 such that Λ₁<Λ_∘ for 𝐵>𝐵_∘. By Proposition 3.1  Λ₁ has a minimizer for 𝐵>𝐵_∘.

We now give a sufficient condition for the inequality Λ₁<Λ_∘.

Theorem 3.4. Suppose that 𝑉₁ and 𝑉_∘ satisfy assumptions of Proposition 3.1. Moreover, assume that the quadratic form 𝑄_𝑉∘ has a positive ground state 𝑣, possibly with infinite 𝐷^1,2 norm, and that if {𝑣_𝑘}⊂𝐶^∞∘(ℝ^𝑁) is a null sequence corresponding to Λ_∘, then limsup_𝑘→∞_ℝ𝑁𝑉₁(𝑥)−𝑉_∘(𝑥)𝑣^2𝑘𝑑𝑥>0.(3.5) Then Λ₁<Λ_∘ and Λ₁ has a minimizer.

Proof. It suffices to show that the inequality _ℝ𝑁||∇𝑢||²𝑑𝑥−Λ_∘_ℝ𝑁𝑉₁(𝑥)𝑢²𝑑𝑥≥0(3.6) fails for some 𝑢∈𝐷^1,2(ℝ^𝑁). We have _ℝ𝑁||∇𝑣_𝑘||²𝑑𝑥−Λ_∘_ℝ𝑁𝑉₁(𝑥)𝑣^2𝑘𝑑𝑥=𝑄_𝑉∘𝑣_𝑘−Λ_∘_ℝ𝑁𝑉₁=𝑜(1)−Λ_∘(𝑥)−𝑉_∘(𝑥)𝑣^2𝑘𝑑𝑥_ℝ𝑁𝑉₁(𝑥)−𝑉_∘(𝑥)𝑣^2𝑘𝑑𝑥<0,(3.7) for sufficiently large 𝑘, which completes the proof of the theorem.

Note that the conditions of Theorem 3.4 are satisfied if, in particular, 𝑉₁≥𝑉_∘ on ℝ^𝑁, with the strict inequality on a set of positive measure. Indeed, the sequence {𝑣_𝑘} converges weakly in 𝐻^1loc(ℝ^𝑁) to 𝑣>0, and the condition limsup_𝑘→∞∫_ℝ𝑁(𝑉₁(𝑥)−𝑉_∘(𝑥))𝑣^2𝑘𝑑𝑥>0 follows from the Fatou lemma.

The situation becomes different if 𝑄_𝑉∘ does not have a ground state. The absence of the ground state is stable property under small (in some sense) compact perturbation, but not under compact perturbations that are not small.

Theorem 3.5. Assume that 𝑉_∘ satisfies the conditions of Proposition 3.1 and that (1.7) holds. (This occurs under conditions of Theorem 1.4 if 𝑄_𝑉∘ has no ground state.) Let 𝑊 be as in (1.7). Then, for every 𝑡∈(0,1/Λ_∘), the functional 𝑄_{𝑉∘+𝑡𝑊} has no ground state and Λ_{𝑉∘+𝑡𝑊}=Λ_𝑉∘. Furthermore, if the functional ∫_ℝ𝑁𝑊(𝑥)𝑢²𝑑𝑥 is weakly continuous in 𝐷^1,2(ℝ^𝑁), the same conclusion holds for −∞<𝑡<0.

Proof. First, we observe that the constants Λ_∘ and Λ₁ corresponding to 𝑉_∘ and 𝑉₁=𝑉_∘+𝑡𝑊, respectively, are equal. Indeed, since 𝑉₁>𝑉_∘, one has Λ₁≤Λ_∘ by monotonicity. On the other hand, it follows from (1.7) that _ℝ𝑁||∇𝑢||²𝑑𝑥−Λ_∘_ℝ𝑁𝑉_∘(𝑥)+𝑡𝑊(𝑥)𝑢²𝑑𝑥≥0,(3.8) for 𝑡∈(0,1/Λ_∘) which implies Λ₁≥Λ_∘. Let 𝑣_𝑘∈𝐶^∞∘(ℝ^𝑁−𝑍) satisfy 𝑄_𝑉1(𝑣_𝑘)→0. Then 1−Λ_∘𝑡_ℝ𝑁𝑊𝑣^2𝑘𝑑𝑥≤𝑄_𝑉1𝑣_𝑘⟶0,(3.9) which implies that, up to subsequence, 𝑣_𝑘→0 a.e. If 𝑣_𝑘 were a null sequence, it would converge in 𝐻^1loc(ℝ^𝑁) and it would have a limit zero. Therefore, 𝑄_𝑉1 admits no null sequence and consequently no ground state. Assume now that the functional ∫_ℝ𝑁𝑊(𝑥)𝑢²𝑑𝑥 is weakly continuous in 𝐷^1,2(ℝ^𝑁). Let {𝑤_𝑘}⊂𝐷^1,2(ℝ^𝑁) be a minimizing sequence for Λ_∘. If {𝑤_𝑘} has a subsequence weakly convergent in 𝐷^1,2(ℝ^𝑁) to some 𝑤≠0, then it is easy to see that |𝑤| would be a minimizer for Λ_∘ and thus a ground state for 𝑄_Λ∘. Therefore, 𝑤_𝑘⇀0. By the weak continuity of ∫_ℝ𝑁𝑊(𝑥)𝑢²𝑑𝑥, we get _ℝ𝑁𝑉₁(𝑥)𝑤^2𝑘𝑑𝑥=_ℝ𝑁𝑉_∘(𝑥)𝑤^2𝑘𝑑𝑥+𝑜(1)=1+𝑜(1),(3.10) and thus Λ₁≤_ℝ𝑁||∇𝑤_𝑘||²𝑑𝑥=Λ_∘+𝑜(1).(3.11) This yields Λ₁≤Λ_∘. Then, _ℝ𝑁||∇𝑢||²𝑑𝑥−Λ₁_ℝ𝑁≥Λ𝑉₁(𝑥)𝑢²𝑑𝑥₁Λ_∘_ℝ𝑁||∇𝑢||²𝑑𝑥−Λ_∘_ℝ𝑁=Λ𝑉₁(𝑥)𝑢²𝑑𝑥₁Λ_∘𝑄^𝑉∘(𝑢)−𝑡Λ_∘_ℝ𝑁𝑊(𝑥)𝑢²𝑑𝑥≥Λ₁_ℝ𝑁Λ^∘−1−𝑡𝑊(𝑥)𝑢²𝑑𝑥.(3.12) Since 𝑡<0, this implies that 𝑄_𝑉1 has no ground state.

Theorem 3.5 concerns with small perturbations of a potential that does not change the constant Λ or the absence of a ground state. The next theorem shows that a large compact perturbation of the potential term yields a ground state of finite 𝐷^1,2(ℝ^𝑁) norm.

Theorem 3.6. Assume that 𝑉_∘ satisfies conditions of Proposition 3.1 and that 𝑊∈𝐿^2,∞(ℝ^𝑁) is such that the functional ∫_ℝ𝑁𝑊(𝑥)𝑢²𝑑𝑥 is weakly continuous in 𝐷^1,2(ℝ^𝑁). Then, for every 𝜆∈(0,Λ_∘) there exists 𝜎∈ℝ such that 𝑄_{𝑉∘+𝜎𝑊} has a ground state of finite 𝐷^1,2(ℝ^𝑁) norm corresponding to the energy constant (3.2).

Proof. Assume without loss of generality that 𝑊 is positive on a set of positive measure. Let 0<𝜆<Λ_∘ and consider 𝜎=inf_{𝑢∈𝐷1,2ℝ𝑁,∫ℝ𝑁𝑊(𝑥)𝑢2𝑑𝑥=1}𝜆⁻¹_ℝ𝑁||∇𝑢||²𝑑𝑥−𝜆_ℝ𝑁𝑉_∘(𝑥)𝑢²𝑑𝑥.(3.13) Since (∫_ℝ𝑁|∇𝑢|²𝑑𝑥−𝜆∫_ℝ𝑁𝑉_∘(𝑥)𝑢²𝑑𝑥)^1/2 defines an equivalent norm on 𝐷^1,2(ℝ^𝑁), it is easy to show that there exists a minimizer for 𝜎. It is clear that this minimizer is also a ground state of 𝑄_{𝑉∘+𝜎𝑊} corresponding to the optimal constant 𝜆.

If we assume additionally that 𝑊 is positive on a set of positive measure, then it is easy to show that 𝜎 is a continuous decreasing function of 𝜆 with lim_𝜆→0𝜎(𝜆)=+∞ and 𝜎_∘=lim_𝜆→Λ∘𝜎(𝜆)≥0. In particular, if (1.7) holds with a weight 𝑊_∘ satisfying 𝑊_∘≥𝛼𝑊, then 𝜎_∘≥𝛼. In other words, given 𝑉_∘ and 𝑊 as in Theorem 3.6, the potential 𝑉_∘+𝜎𝑊 admits a ground state whenever 𝜎≥𝜎_∘.

For further results of that nature, we refer to paper [19].

4. Behaviour of a Ground State Around 0

In what follows we consider the potential of the Hardy-type 𝑉(𝑥)=𝑚(𝑥)/|𝑥|², where 𝑚(𝑥) is continuous and 𝑚(0)>0 and 𝑚(∞)>0. The corresponding ground state, if it exists, is denoted by 𝜙₁, which is chosen to be positive on ℝ^𝑁. Obviously the ground state 𝜙₁ satisfies Δ𝑢=Λ_𝑚𝑚(𝑥)|𝑥|²𝑢inℝ^𝑁(4.1) in a weak sense.

We need the following extension of the Hardy inequality: let Ω⊂ℝ^𝑁 be a bounded domain and 0∈Ω, then for every 𝛿>0, there exists a constant 𝐴(𝛿,Ω)>0 such that _Ω𝑢²|𝑥|²𝑑𝑥≤1Λ_𝑁+𝛿_Ω||∇𝑢||²𝑑𝑥+𝐴(𝛿,Ω)_Ω𝑢²𝑑𝑥,(4.2) for every 𝑢∈𝐻¹(Ω) (see [21]).

Proposition 4.1. Let Λ_𝑚<Λ_𝑁min1𝑚(0),𝑚(1∞).(4.3) Then 𝜙₁∈𝐿^2∗(1+𝛿)(𝐵(0,𝑟)) for some 𝛿>0 and 𝑟>0.

Proof. Let Φ∈𝐶¹(ℝ^𝑁) be such that Φ(𝑥)=1 on 𝐵(0,𝑟), Φ(𝑥)=0onℝ^𝑁−𝐵(0,2𝑟), 0≤Φ(𝑥)≤1 on ℝ^𝑁 and |∇Φ(𝑥)|≤2/𝑟. For simplicity, we set 𝜆=Λ_𝑚, 𝑢=𝜙₁. We define 𝑣=Φ²𝑢min(𝑢,𝐿)^𝑝−2=Φ²𝑢𝑢^𝐿𝑝−2, where 𝐿>0 and 𝑝>2. Testing (4.1) with 𝑣, we get _ℝ𝑁||∇𝑢||²𝑢^𝐿𝑝−2Φ²+(𝑝−2)∇𝑢∇𝑢_𝐿𝑢^𝐿𝑝−2Φ²+2∇𝑢∇Φ𝑢𝑢^𝐿𝑝−2Φ𝑑𝑥=𝜆_ℝ𝑁𝑚(𝑥)|𝑥|²𝑢²𝑢^𝐿𝑝−2Φ²𝑑𝑥.(4.4) Applying the Young inequality to the third term on the left side, we get (1−≤𝜂)𝜆_{ℝ𝑁ℝ𝑁}||∇𝑢||²𝑢^𝐿𝑝−2Φ²𝑑𝑥+(𝑝−2)_ℝ𝑁∇𝑢∇𝑢_𝐿𝑢^𝐿𝑝−2Φ²𝑑𝑥𝑚(𝑥)|𝑥|²𝑢²𝑢^𝐿𝑝−2Φ²𝑑𝑥+𝐶(𝜂)_ℝ𝑁𝑢²𝑢^𝐿𝑝−2||∇Φ||²𝑑𝑥,(4.5) where 𝜂>0 is a small number to be suitably chosen. Since the second integral on the left side is nonnegative, this inequality can be rewritten in the following form: (1−≤𝜂)𝜆_{ℝ𝑁ℝ𝑁}||∇𝑢||²𝑢^𝐿𝑝−2Φ²𝑑𝑥+(1−𝜂)(𝑝−2)_ℝ𝑁∇𝑢∇𝑢_𝐿𝑢^𝐿𝑝−2Φ²𝑑𝑥𝑚(𝑥)|𝑥|²𝑢²𝑢^𝐿𝑝−2Φ²𝑑𝑥+𝐶(𝜂)_ℝ𝑁𝑢²𝑢^𝐿𝑝−2||∇Φ||²𝑑𝑥.(4.6) Multiplying this inequality by (𝑝+2)/4 and noting that (𝑝+2)/4>1, we get (1−𝜂)_ℝ𝑁||∇𝑢||²𝑢^𝐿𝑝−2Φ²𝑑𝑥+𝑝²−44_ℝ^𝑁≤𝜆(𝑝+2)∇𝑢∇𝑢_𝐿𝑢^𝐿𝑝−2Φ²𝑑𝑥4_ℝ^𝑁𝑚(𝑥)|𝑥|²+𝐶(𝜂)(𝑝+2)𝑢²𝑢^𝐿𝑝−2Φ²𝑑𝑥4_ℝ^𝑁𝑢²𝑢^𝐿𝑝−2||∇Φ||²𝑑𝑥.(4.7) We now observe that _ℝ𝑁|||∇𝑢𝑢_{𝐿(𝑝/2)−1}|||²Φ²𝑑𝑥=_ℝ𝑁||∇𝑢||²𝑢^𝐿𝑝−2Φ²𝑑𝑥+𝑝²−44_ℝ^𝑁||∇𝑢_𝐿||²𝑢^𝐿𝑝−2Φ²𝑑𝑥.(4.8) Hence, (4.7) takes the form (1−𝜂)_ℝ𝑁|||∇𝑢𝑢_{𝐿(𝑝/2)−1}|||²Φ²𝑑𝑥≤𝜆(𝑝4+2)_ℝ^𝑁𝑚(𝑥)|𝑥|²+𝐶(𝜂)(𝑝+2)𝑢²𝑢^𝐿𝑝−2Φ²𝑑𝑥4_ℝ^𝑁𝑢²𝑢^𝐿𝑝−2||∇Φ||²𝑑𝑥.(4.9) Since 𝜆𝑚(0)/Λ_𝑁<1, we can choose 𝜖₁>0 so that (𝜆/Λ_𝑁)(𝑚(0)+𝜖₁)<1. By the continuity of 𝑚, there exists 0<𝑟₁<𝑟 such that 𝑚(𝑥)≤𝑚(0)+𝜖₁ for 𝑥∈𝐵(0,𝑟₁). This is now used to estimate the first integral on the right side of (4.9): 𝜆(𝑝+2)4_ℝ^𝑁𝑚(𝑥)|𝑥|²𝑢²𝑢^𝐿𝑝−2Φ²𝑑𝑥≤𝜆(𝑝+2)4_{𝐵(0,𝑟1)}𝑚(0)+𝜖₁|𝑥|²+𝜆(𝑝+2)‖𝑚‖_∞𝑢²𝑢^𝐿𝑝−2𝑑𝑥4𝑟²¹_{𝐵(0,2𝑟)}𝑢²𝑢^𝐿𝑝−2𝑑𝑥.(4.10) Applying the Hardy inequality (4.2), we get 𝜆(𝑝+2)4_𝐵(0,𝑟)𝑚(𝑥)|𝑥|²𝑢²𝑢^𝐿𝑝−2𝑑𝑥≤𝜆(𝑝+2)4𝑚(0)+𝜖¹Λ1_𝑁+𝜖_{𝐵(0,𝑟1)}|||∇𝑢𝑢_{𝐿(𝑝/2)−1}+𝜆(𝑝+2)|||²𝑑𝑥4𝐴𝐵0,𝑟¹,𝜖+𝜆(𝑝+42𝑟)₂₁‖𝑚‖^∞_{𝐵(0,2𝑟)}𝑢𝑢^{𝐿𝑝/2−1}²𝑑𝑥,(4.11) for every 𝜖>0. Inserting this estimate into (4.9), we obtain 1−𝜂−𝜆(𝑝4+2)𝑚(0)+𝜖¹Λ1_𝑁+𝜖×_𝐵(0,𝑟)|||∇𝑢𝑢_{𝐿(𝑝/2)−1}|||²𝑑𝑥≤𝐶₁_{𝐵(0,2𝑟)}𝑢𝑢^{𝐿(𝑝/2)−1}²𝑑𝑥,(4.12) where 𝐶₁=(𝜆(𝑝+2)/4)𝐴(𝐵(0,𝑟₁),𝜖)+((𝜆(𝑝+2)‖𝑚‖_∞)/(4𝑟²¹))+(𝑝+2)𝐶(𝜂)/𝑟². We put 𝑝=2+𝛿, 𝛿>0. We now observe that we can choose 𝛿 and 𝜖 so small that 𝜆1+𝛿4𝑚(0)+𝜖¹Λ1_𝑁+𝜖=𝜆Λ_𝑁1+𝛿4𝑚(0)+𝜖¹+𝜆𝜖1+𝛿4𝑚(0)+𝜖¹<1.(4.13) We point out that we have used here the inequality (𝜆/Λ_𝑁)(𝑚(0)+𝜖₁)<1. With this choice of 𝜖 and 𝛿, we now choose 𝜂>0 so small that 𝐶₂∶=1−𝜂−𝜆1+𝛿4𝑚(0)+𝜖¹Λ1_𝑁+𝜖>0.(4.14) Finally, we apply the Sobolev inequality in 𝐻¹(𝐵(0,𝑟)) and deduce 𝑆𝐶₂_𝐵(0,𝑟)||𝑢𝑢^{𝐿(𝑝/2)−1}||^2∗𝑑𝑥^2/2∗≤𝐶₁+𝐶₂_{𝐵(0,2𝑟)}𝑢𝑢^{𝐿(𝑝/2)−1}²𝑑𝑥,(4.15) where 𝑆 denotes the best Sobolev constant of the embedding of 𝐻¹(𝐵(0,𝑟)) into 𝐿^2∗(𝐵(0,𝑟)). Letting 𝐿→∞ we deduce that 𝑢∈𝐿^{2∗(1+(𝛿/2))}(𝐵(0,𝑟)). So the assertion holds with 𝛿_∘=𝛿/2.

We now establish the higher integrability property of the principal eigenfunction on ℝ^𝑁⧵𝐵(0,𝑅). Although this will not be used in the sequel, we add it for the sake of completeness. We denote by 𝐷^1,2(ℝ^𝑁⧵𝐵(0,𝑅)) the Sobolev space defined by𝐷^1,2ℝ^𝑁⧵𝐵(0,𝑅)=𝑢∶∇𝑢∈𝐿²ℝ^𝑁⧵𝐵(0,𝑅)and𝑢∈𝐿^2∗ℝ^𝑁⧵𝐵(0,𝑅).(4.16)

Lemma 4.2. For every 𝛿>0, there exists a constant 𝐴=𝐴(𝛿,𝑅)>0 such that _{|𝑥|≥𝑅}𝑢²|𝑥|²𝑑𝑥≤1Λ_𝑁+𝛿_{|𝑥|≥𝑅}||∇𝑢||²𝑑𝑥+𝐴_{𝑅≤|𝑥|≤𝑅+1}𝑢²𝑑𝑥,(4.17) for every 𝑢∈𝐷^1,2(ℝ^𝑁⧵𝐵(0,𝑅)).

Proof. Let Φ∈𝐶¹(ℝ^𝑁) be such that Φ(𝑥)=0 on𝐵(0,𝑅), Φ(𝑥)=1 on ℝ^𝑁⧵𝐵(0,𝑅+1), 0≤Φ(𝑥)≤1 on ℝ^𝑁⧵𝐵(0,𝑅)and |∇Φ(𝑥)|≤2/𝑅 on ℝ^𝑁. Then, 𝑢Φ∈𝐷^1,2(ℝ^𝑁), and, by the Hardy and Young inequalities, we have _{|𝑥|≥𝑅}𝑢²|𝑥|²𝑑𝑥=_{|𝑥|≥𝑅}(𝑢Φ)²|𝑥|²𝑑𝑥+_{|𝑥|≥𝑅}1−Φ²𝑢²|𝑥|²≤Λ_𝑁−1𝑑𝑥_{|𝑥|≥𝑅}||∇(𝑢Φ)||²𝑑𝑥+1𝑅²_{𝑅≤|𝑥|≤𝑅+1}≤Λ_𝑁−1𝑢²𝑑𝑥_{|𝑥|≥𝑅}||∇𝑢||²Φ²𝑑𝑥+Λ^𝑁−1_{|𝑥|≥𝑅}+2Λ_𝑁−1𝑢²||∇Φ||²𝑑𝑥_{|𝑥|≥𝑅}𝑢Φ∇𝑢∇Φ𝑑𝑥+1𝑅²_{𝑅≤|𝑥|≤𝑅+1}≤Λ𝑢²𝑑𝑥_𝑁−1+𝛿_{|𝑥|≥𝑅}||∇𝑢||²𝑑𝑥+Λ^𝑁−1+𝐶(𝛿)_{|𝑥|≥𝑅}𝑢²||∇Φ||²𝑑𝑥+1𝑅²_{𝑅≤|𝑥|≤𝑅+1}𝑢²𝑑𝑥,(4.18) and the result follows with 𝐴(𝛿,𝑅)=(4/𝑅²)(Λ^𝑁−1+𝐶(𝛿))+1/𝑅².

Proposition 4.3. Suppose that 𝑚(∞)>0 and Λ_𝑚<Λ_𝑁min(1/𝑚(0),1/𝑚(∞)). Let 𝜙₁ be the principal eigenfunction of problem (4.1). Then there exist 𝛿>0 and 𝑅>0 such that 𝜙∈𝐿^2∗(1+𝛿)(ℝ^𝑁⧵𝐵(0,𝑅)).

Proof. We modify the argument used in the proof of Proposition 4.1. Since Λ_𝑚<(Λ_𝑁/𝑚(∞)), there exist 𝜖>0 and 𝑅>0 such that (Λ_𝑚/Λ_𝑁)(𝑚(∞)+𝜖)<1 and 𝑚(𝑥)<(𝑚(∞)+𝜖) for |𝑥|≥𝑅. Let Ψ∈𝐶¹(ℝ^𝑁) be such that Ψ(𝑥)=0 on 𝐵(0,𝑅), Ψ(𝑥)=1 on ℝ^𝑁−𝐵(0,𝑅+1), 0≤Ψ(𝑥)≤1 on ℝ^𝑁, and |∇Ψ(𝑥)|≤(2/𝑅) on ℝ^𝑁. Let 𝜆=Λ_𝑚, 𝑢=𝜙₁, and 𝑣=𝑢𝑢^𝐿𝑝−2Ψ², where 𝐿>1, 𝑝>2, and 𝑢_𝐿=min(𝑢,𝐿). It is clear that 𝑣∈𝐷^1,2(ℝ^𝑁). Testing (4.1) with 𝑣 and applying the Young inequality, we obtain (1−≤𝜂)𝜆_{ℝ𝑁ℝ𝑁}||∇𝑢||²𝑢^𝐿𝑝−2Ψ²𝑑𝑥+(𝑝−2)_ℝ𝑁∇𝑢∇𝑢_𝐿𝑢^𝐿𝑃−2Ψ²𝑑𝑥𝑚(𝑥)|𝑥|²𝑢²𝑢^𝐿𝑝−2Ψ²𝑑𝑥+𝐶(𝜂)_ℝ𝑁𝑢²𝑢^𝐿𝑝−2||∇Ψ||²𝑑𝑥.(4.19) From this, as in the proof of Proposition 4.1, we derive that (1−𝜂)_ℝ𝑁|||∇𝑢𝑢_{𝐿(𝑝/2)−1}|||²Ψ²𝑑𝑥≤𝜆(𝑝4+2)_ℝ^𝑁𝑚(𝑥)|𝑥|²+C(𝜂)(𝑝+2)𝑢²𝑢^𝐿𝑝−2Ψ²𝑑𝑥4_ℝ^𝑁𝑢²𝑢^𝐿𝑝−2||∇Ψ||²𝑑𝑥.(4.20) We now estimate the first integral on the right side of (4.20). Using Lemma 4.2, we have, for every 𝜖₁>0, _ℝ𝑁𝑚(𝑥)|𝑥|²𝑢²𝑢^𝐿𝑝−2Ψ²𝑑𝑥≤(𝑚(∞)+𝜖)_{|𝑥|≥𝑅+1}𝑢𝑢^{𝐿(𝑝/2)−1}²|𝑥|²×𝑑𝑥+(𝑚(∞)+𝜖)_{𝑅≤|𝑥|≤𝑅+1}𝑢𝑢^{𝐿(𝑝/2)−1}²|𝑥|²≤Λ𝑑𝑥_𝑁−1×+𝜖₁(𝑚(∞)+𝜖)_{|𝑥|≥𝑅+1}|||∇𝑢𝑢_{𝐿(𝑝/2)−1}+(𝑚(∞)+𝜖)|||²𝑑𝑥+𝐴𝜖₁,𝑅(𝑚(∞)+𝜖)_{𝑅+1≤|𝑥|≤𝑅+2}𝑢𝑢^{𝐿(𝑝/2)−1}²𝑑𝑥𝑅²×_{𝑅≤|𝑥|≤𝑅+1}𝑢𝑢^{𝐿(𝑝/2)−1}²𝑑𝑥(4.21) Inserting this into (4.20), we obtain 1−𝜂−𝜆(𝑝4+2)Λ^𝑁−1+𝜖₁(𝑚(∞)+𝜖)_{|𝑥|≥𝑅+1}|||∇𝑢𝑢_{𝐿(𝑝/2)−1}≤𝐶₁|||²𝑑𝑥𝛿,𝜖₁,𝑅_{𝑅≤|𝑥|≤𝑅+2}𝑢𝑢^{𝐿(𝑝/2)−1}²𝑑𝑥,(4.22) where 𝐶₁𝛿,𝜖₁,𝑅∶=𝜆(𝑝+2)4(𝑚(∞)+𝜖)𝐴𝜖¹,𝑅+𝜆(𝑝+2)4𝑅²(𝑚(∞)+𝜖)+𝐶(𝜂)(𝑝+2)𝑅².(4.23) We now set 𝑝=2+𝛿. We choose 𝛿>0 and 𝜖₁>0 such that 𝜆1+𝛿4Λ^𝑁−1+𝜖₁(𝑚(∞)+𝜖)<1.(4.24) Then we choose 𝜂>0 small enough to guarantee the inequality 𝐶₂∶=1−𝜂−𝜆1+𝛿4Λ^𝑁−1+𝜖₁(𝑚(∞)+𝜖)>0.(4.25) Having chosen 𝜖₁ and 𝛿, we apply the Sobolev inequality to deduce from (4.22) SC₂_{|𝑥|≥𝑅+1}||𝑢𝑢^{𝐿(𝑝/2)−1}||^2∗𝑑𝑥^2/2∗≤𝐶₁_{𝑅≤|𝑥|≤𝑅+1}𝑢𝑢^{𝐿(𝑝/2)−1}²𝑑𝑥,(4.26) where 𝑆 is the best Sobolev constant for the embedding of 𝐷^1,2(ℝ^𝑁−𝐵(0,𝑅+1)) into 𝐿^2∗(ℝ^𝑁−𝐵(0,𝑅+1)). Letting 𝐿→∞, the result follows.

Continuing with the above notations 𝜆=Λ_𝑚, 𝑢=𝜙₁, we put 𝑢=|𝑥|^−𝑠𝑣, with 𝑠>0 to be chosen later. We have div|𝑥|^−2𝑠∇𝑣=−𝜆|𝑥|^−2−𝑠𝑚(𝑥)𝑢+𝑢−𝑠²|𝑥|^−𝑠−2+𝑠𝑁|𝑥|^−𝑠−2−2𝑠|𝑥|^−𝑠−2.(4.27) We now consider the above equation in a small ball 𝐵(0,𝑟). Since 𝜆=Λ_𝑚<Λ_𝑁min1𝑚(0),𝑚(1∞)≤𝑚Λ(^𝑁0),(4.28) there exists 𝑟>0 (small enough) such that 𝜆max_{𝑥∈𝐵(0,𝑟)}𝑚(𝑥)<Λ_𝑁. Let 𝑠=√Λ_𝑁−√Λ_𝑁−𝜆𝑚_𝑟 with 𝑚_𝑟=max_{𝑥∈𝐵(0,𝑟)}𝑚(𝑥), then −div|𝑥|^−2𝑠∇𝑣≤0in𝐵(0,𝑟).(4.29) Let 𝑚_𝑟=min_{𝑥∈𝐵(0,𝑟)}𝑚(𝑥), and set 𝑠=√Λ_𝑁−√Λ_𝑁−𝜆𝑚_𝑟. Then −div|𝑥|^−2𝑠∇𝑣≥0in𝐵(0,𝑟).(4.30)

Proposition 4.4. Let 𝑚(0)>0 and Λ_𝑚<Λ_𝑁min1𝑚(0),𝑚(1∞).(4.31) Then, there exists 𝑟>0 such that 𝑀₁|𝑥|^{√−(Λ𝑁−√Λ𝑁−𝜆𝑚𝑟)}≤𝜙₁(𝑥)≤𝑀₂|𝑥|^{√−(Λ𝑁−√Λ𝑁−𝜆𝑚𝑟)},(4.32) for 𝑥∈𝐵(0,𝑟) and some constants 𝑀₁>0, 𝑀₂>0.

The lower bound follows from [22, Proposition  2.2]. To apply it, we need inequality (4.30). To establish the upper bound, we modify the argument used in paper [23]. Let 𝜂 be a 𝐶¹ function such that 𝜂(𝑥)=1 on 𝐵(0,𝑟), 𝜂(𝑥)=0 on ℝ^𝑁⧵𝐵(0,𝜌), and |∇𝜂(𝑥)|≤2/(𝜌−𝑟) on ℝ^𝑁, where 0<𝑟<𝜌. We use as a test function in (4.29) 𝑤=𝜂²𝑣𝑣^{𝑙2(𝑡−1)}=𝜂²𝑣min(𝑣,𝑙)^2(𝑡−1), where 𝑙,𝑡>1. Substituting into (4.29), we obtain _ℝ𝑁|𝑥|^−2𝑠2𝜂𝑣𝑣^{𝑙2(𝑡−1)}∇𝑣∇𝜂+𝜂²𝑣^{𝑙2(𝑡−1)}||∇𝑣||²+2(𝑡−1)𝜂²𝑣^{𝑙2(𝑡−1)}||∇𝑣_𝑙||²𝑑𝑥≤0,(4.33) where 𝑠=√Λ_𝑁−√Λ_𝑁−𝜆𝑚_𝑟. By the Young inequality, for every 𝜖>0, there exists 𝐶(𝜖)>0 such that 2_ℝ𝑁|𝑥|^−2𝑠𝜂𝑣𝑣^{𝑙2(𝑡−1)}∇𝜂∇𝑣𝑑𝑥≤+𝜖𝐶(_ℝ𝑁𝜖|)𝑥|^−2𝑠_ℝ𝑁𝜂²𝑣^{𝑙2(𝑡−1)}||∇𝑣||²𝑑𝑥|𝑥|^−2𝑠||∇𝜂||²𝑣²𝑣^{𝑙2(𝑡−1)}𝑑𝑥.(4.34) Taking 𝜖=1/2, we derive from (4.33) that _ℝ𝑁|≤𝑥|𝐶^−2𝑠_ℝ𝑁𝜂²𝑣^{𝑙2(𝑡−1)}||∇𝑣||²+2(𝑡−1)𝜂²𝑣^{𝑙2(𝑡−1)}||∇𝑣_𝑙||²𝑑𝑥|𝑥|^−2𝑠||∇𝜂||²𝑣²𝑣^{𝑙2(𝑡−1)}𝑑𝑥,(4.35) where 𝐶>0 is a constant independent of 𝑙. To proceed further we use the Caffarelli-Kohn-Nirenberg inequality [24]: _𝐵(0,𝜌)|𝑥|^−𝑏𝑝|𝑤|^𝑝𝑑𝑥^2/𝑝≤𝐶_𝑎,𝑏_𝐵(0,𝜌)|𝑥|^−2𝑎||∇𝑤||²𝑑𝑥,(4.36) for every 𝑤∈𝐻^1∘(𝐵(0,𝜌),|𝑥|^−2𝑎𝑑𝑥), where −∞<𝑎<(𝑁−2)/2, 𝑎≤𝑏≤(𝑎+1), 𝑝=2𝑁/((𝑁−2)+2(𝑏−𝑎)), and 𝐶_𝑎,𝑏>0 is a constant depending on 𝑎 and b. We choose 𝑎=𝑏=√Λ_𝑁−Λ_𝑁−𝜆𝑚_𝑟<𝑁−22.(4.37) In this case we have 𝑝=2^∗. We then deduce from (4.35) and (4.36) with 𝑤=𝜂𝑣𝑣^𝑙𝑡−1 that_ℝ𝑁|𝑥|^−2∗𝑠||𝜂𝑣𝑣^𝑙𝑡−1||^2∗𝑑𝑥^2/2∗≤𝐶_𝑎,𝑏_ℝ𝑁≤2𝐶_𝑎,𝑏|𝑥|^−2𝑠||∇𝜂𝑣𝑣^𝑙𝑡−1||²𝑑𝑥_ℝ𝑁≤𝐶𝑡_ℝ𝑁|𝑥|^−2𝑠||∇𝜂||²𝑣²𝑣^{𝑙2(𝑡−1)}+𝜂²𝑣^{𝑙2(𝑡−1)}||∇𝑣||²+(𝑡−1)²𝜂²𝑣^{𝑙2(𝑡−1)}||∇𝑣_𝑙||²𝑑𝑥|𝑥|^−2∗𝑠||∇𝜂||²𝑣²𝑣^{𝑙2(𝑡−1)}𝑑𝑥.(4.38) We now observe that_ℝ𝑁|𝑥|^−2∗𝑠||𝜂||^2∗𝑣²𝑣^{2∗𝑙𝑡−2}𝑑𝑥≤_ℝ𝑁|𝑥|^−2∗𝑠||𝜂𝑣𝑣^𝑙𝑡−1||^2∗𝑑𝑥.(4.39) Indeed, to show this, we need to check that 𝑣²𝑣^{2∗𝑙𝑡−2}≤𝑣^{2∗𝑙(𝑡−1)}𝑣^2∗ on supp𝜂. This can be verified by considering the cases 𝑣_𝑙=𝑙 and 𝑣_𝑙=𝑣. The above inequality allows us to rewrite (4.38) as _ℝ𝑁|𝑥|^−2∗𝑠||𝜂||^2∗𝑣²𝑣^{2∗𝑙𝑡−2}𝑑𝑥^2/2∗≤𝐶𝑡_ℝ𝑁|𝑥|^−2∗𝑠||∇𝜂||²𝑣²𝑣^{𝑙2(𝑡−1)}𝑑𝑥.(4.40) Due to the properties of the function 𝜂, the above inequality becomes _𝐵(0,𝑟)|𝑥|^−2∗𝑠𝑣²𝑣^{2∗𝑙𝑡−2}𝑑𝑥^2/2∗≤𝐶𝑡(𝜌−𝑟)²_𝐵(0,𝜌)|𝑥|^−2∗𝑠𝑣²𝑣^{𝑙2(𝑡−1)}𝑑𝑥.(4.41) One can easily check that the resulting integral on the right side is of (4.41) is finite. We now choose 𝑁/(𝑁−2)<𝑡^∗<(1+𝛿_∘)(𝑁/(𝑁−2)), where 𝛿_∘ is a constant from Proposition 4.1. We define the sequence 𝑡_𝑗=𝑡^∗(2^∗/2)^𝑗, 𝑗=0,1,…. Setting 𝑡=𝑡_𝑗 in (4.41), we obtain _𝐵(0,𝑟)|𝑥|^−2∗𝑠𝑣²𝑣^{2𝑡𝑗+1𝑙−2}𝑑𝑥^{1/2𝑡𝑗+1}≤𝐶𝑡^𝑗(𝜌−𝑟)²^1/2𝑡𝑗_𝐵(0,𝜌)|𝑥|^−2∗𝑠𝑣²𝑣^{2𝑡𝑗𝑙−2}𝑑𝑥^1/2𝑡𝑗.(4.42) We put 𝑟_𝑗=𝜌_∘(1+𝜌^𝑗∘), 𝑗=0,1,… with 𝜌_∘ small. Substituting in the last inequality 𝜌=𝑟_𝑗, 𝑟=𝑟_𝑗+1, we obtain_{𝐵(0,𝑟𝑗+1)}|𝑥|^−2∗𝑠𝑣²𝑣^{2𝑡𝑗+1𝑙−2}𝑑𝑥^{1/2𝑡𝑗+1}≤𝐶𝑡^𝑗𝜌_∘−𝜌^2∘²𝜌^∘2𝑗^1/2𝑡𝑗_{𝐵(0,𝑟𝑗)}|𝑥|^−2∗𝑠𝑣²𝑣^{2𝑡𝑗𝑙−2}𝑑𝑥^1/2𝑡𝑗.(4.43) Iterating gives _{𝐵(0,𝑟𝑗+1)}|𝑥|^−2∗𝑠𝑣²𝑣^{2𝑡𝑗+1𝑙−2}𝑑𝑥^{1/2𝑡𝑗+1}≤𝐶𝜌_∘−𝜌^2∘^{∑∞𝑗=01/𝑡𝑗}𝜌^{−∑∞𝑗=0𝑗/𝑡𝑗}_∘∞_𝑗=0𝑡^{𝑗/2𝑡𝑗}_𝑗_{𝐵(0,𝑟∘)}|𝑥|^−2∗𝑠𝑣²𝑣^{2𝑡∗𝑙−2}𝑑𝑥^1/2𝑡∗.(4.44) We now notice that infinite sums and the infinite product in the above inequality are finite. Since 2^∗<2𝑡^∗<(1+𝛿_∘)2^∗, we have _{𝐵(0,𝑟∘)}|𝑥|^−2∗𝑠𝑣²𝑣^{2𝑡∗𝑙−2}𝑑𝑥≤_{𝐵(0,𝑟∘)}|𝑥|^{(2𝑡∗−2∗)𝑠}|𝑢|^2𝑡∗𝑑𝑥≤𝑟^{(2𝑡∗−2∗∘)𝑠}_{𝐵(0,𝑟∘)}|𝑢|^2∗𝑡∗𝑑𝑥<∞.(4.45) We now deduce from (4.44) and (4.45) that‖‖𝑣_𝑙‖‖_{𝐿2𝑡𝑗+1(𝐵(0,𝜌∘))}≤‖‖𝑣_𝑙‖‖_{𝐿2𝑡𝑗+1(𝐵(0,𝑟𝑗+1))}≤𝑟^{(2∗𝑠)/2𝑡𝑗+1}_∘_{𝐵(0,𝑟𝑗+1)}|𝑥|^−2∗𝑠𝑣²𝑣^{𝑡𝑗+1𝑙−2}𝑑𝑥^{1/2𝑡𝑗+1}≤𝐶,(4.46)

where 𝐶>0 is a constant independent of 𝑙 and 𝑗. Letting 𝑡_𝑗→∞, we get ‖𝑣_𝑙‖_{𝐿∞(𝐵(0,𝜌∘))}≤𝐶. Finally, if 𝑙→∞, we obtain ‖𝑣‖_𝐿^∞_{(𝐵(0,𝜌∘))}≤𝐶, and this completes the proof of Proposition 4.4.