International Journal of Differential Equations

International Journal of Differential Equations / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 358087 | 26 pages | https://doi.org/10.1155/2011/358087

Ground State for the SchrΓΆdinger Operator with the Weighted Hardy Potential

Academic Editor: Kanishka Perera
Received06 May 2011
Accepted17 Jul 2011
Published15 Sep 2011

Abstract

We establish the existence of ground states on ℝ𝑁 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 SchrΓΆdinger operator associated with the quadratic form 𝑄𝑉(ξ€œπ‘’)=ℝ𝑁||||βˆ‡π‘’2βˆ’Ξ›π‘‰π‘‰(π‘₯)𝑒2𝑑π‘₯,π‘’βˆˆπΆβˆžβˆ˜ξ€·β„π‘ξ€Έ,𝑁β‰₯3,(1.1) where 𝑉 belongs to the Lorentz space 𝐿𝑁/2,∞(ℝ𝑁) and Λ𝑉 is the largest constant (whenever exists) for which the form 𝑄𝑉 is nonnegative. This assumption implies, when 𝑉β‰₯0, that the potential term βˆ«β„π‘π‘‰(π‘₯)𝑒2𝑑π‘₯ is continuous in 𝐷1,2(ℝ𝑁), where 𝐷1,2(ℝ𝑁) is the Sobolev space obtained as the completion of 𝐢∞∘(ℝ𝑁) with respect to the norm ‖𝑒‖2𝐷1,2=ξ€œβ„π‘||||βˆ‡π‘’2𝑑π‘₯.(1.2) We are mainly interested in the case of the Hardy-type potential 𝑉(π‘₯)=π‘š(π‘₯)/|π‘₯|2 with π‘šβˆˆπΏβˆž(ℝ𝑁). Assuming that 𝑉 is positive on a set of positive measure, the constant Λ𝑉 is given by the variational problem Λ𝑉=infπ‘’βˆˆπ·1,2ℝ𝑁,βˆ«β„π‘π‘‰π‘’2𝑑π‘₯=1ξ€œβ„π‘||||βˆ‡π‘’2𝑑π‘₯,(1.3) and the continuity of βˆ«β„π‘π‘‰(π‘₯)𝑒2𝑑π‘₯ implies that Λ𝑉>0. If problem (1.3) has a minimizer 𝑒, then it satisfies βˆ’Ξ”π‘’βˆ’Ξ›π‘‰π‘‰(π‘₯)𝑒=0.(1.4) A solution of (1.4) is understood in the weak sense ξ€œβ„π‘βˆ‡π‘’βˆ‡πœ™π‘‘π‘₯=Ξ›π‘‰ξ€œβ„π‘π‘‰(π‘₯)π‘’πœ™π‘‘π‘₯,(1.5) for every πœ™βˆˆπ·1,2(ℝ𝑁).

Since |𝑒| is also a minimizer for Λ𝑉, we may assume that 𝑒β‰₯0 a.e. on ℝ𝑁. In particular, when 𝑉(π‘₯)=π‘š(π‘₯)/|π‘₯|2 with π‘šβˆˆπΏβˆž(ℝ𝑁), then 𝑒>0 on ℝ𝑁 by the Harnack inequality [1]. If the potential term is weakly continuous in 𝐷1,2(ℝ𝑁), for example, when 𝑉(π‘₯)=π‘š(π‘₯)/|π‘₯|2 with π‘šβˆˆπΏβˆž(ℝ𝑁) and lim|π‘₯|β†’βˆžπ‘š(π‘₯)=limπ‘₯β†’0π‘š(π‘₯)=0, then there exists a minimizer for Λ𝑉. 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(π‘₯)=1/|π‘₯|2 with the corresponding optimal constant Λ𝑉=Λ𝑁=((π‘βˆ’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 Ξ©βŠ‚β„π‘ be an open set, and let 𝑄𝑉 be as in (1.1). A sequence of nonnegative functions π‘£π‘˜βˆˆπΆβˆžβˆ˜(Ξ©) is said to be a null sequence for the functional 𝑄𝑉 if 𝑄𝑉(π‘£π‘˜)β†’0, as π‘˜β†’βˆž, and there exists a nonnegative function πœ“βˆˆπΆβˆžβˆ˜(Ξ©) such that βˆ«Ξ©πœ“π‘£π‘˜π‘‘π‘₯=1 for each π‘˜.

Let us recall that the capacity of a compact set 𝐸 relative to an open set Ξ©βŠ‚β„π‘, with πΈβŠ‚Ξ©, is given by ξ‚»ξ€œcap(𝐸,Ξ©)=infΞ©||||βˆ‡π‘’2𝑑π‘₯;π‘’βˆˆπΆβˆžβˆ˜ξ‚Ό(Ξ©),with𝑒(π‘₯)β‰₯1on𝐸.(1.6) In the case Ξ©=ℝ𝑁, we use notation cap(𝐸) (see [5]).

We can now formulate the following β€œground state alternative” (see [3, 4]).

Theorem 1.2. Let 𝑉 be a measurable function bounded on every compact subset of Ξ©=β„π‘βˆ’π‘, where 𝑍 is a closed set of capacity zero, and assume that 𝑄𝑉(𝑒)β‰₯0 for all π‘’βˆˆπΆβˆžβˆ˜(Ξ©). Then, if 𝑄𝑉 admits a null sequence π‘£π‘˜, then the sequence π‘£π‘˜ converges weakly in 𝐻1loc(ℝ𝑁) 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 𝑄𝑉, if the functional admits a null sequence weakly convergent to 𝑣.

If 𝑉(π‘₯)=1/|π‘₯|2, then the functional 𝑄𝑉 has a ground state 𝑣(π‘₯)=|π‘₯|(2βˆ’π‘)/2 of infinite 𝐷1,2 norm, while (1.3) has no minimizer in 𝐷1,2(ℝ𝑁).

It is important to note that the functional 𝑄𝑉 with the optimal constant Λ𝑉 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 Ξ©=β„π‘βˆ’π‘, where 𝑍 is a closed set of capacity zero, and assume that 𝑄𝑉(𝑒)β‰₯0 for all π‘’βˆˆπΆβˆžβˆ˜(Ξ©). Then either Q𝑉 admits a null sequence, or there exists a function π‘Š, positive and continuous on Ξ©, such that 𝑄𝑉(ξ€œπ‘’)β‰₯β„π‘π‘Š(π‘₯)𝑒2𝑑π‘₯.(1.7)

For example, let π‘š be a continuous function on β„π‘βˆ’{0} such that π‘š(π‘₯)=1/|π‘₯|2 for 0<|π‘₯|≀1, π‘š(π‘₯)∈[1/2,1] for |π‘₯|∈(1,2) and π‘š(π‘₯)=1/2|π‘₯|2 for |π‘₯|β‰₯2. Then, Λ𝑉=((π‘βˆ’2)/2)2 and the functional 𝑄𝑉 does not admit a null sequence. From Theorem 1.4 follows that 𝑄𝑉 satisfies (1.7) with some function π‘Š positive on β„π‘βˆ’{0}.

Obviously, ground states of finite 𝐷1,2 norm are principal eigenfunctions of (1.4). There is a quite extensive literature on principal eigenfunctions with indefinite weight functions for elliptic operators on ℝ𝑁 or on unbounded domains of ℝ𝑁, 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 𝐿𝑝(ℝ𝑁) 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 π‘“βˆΆβ„π‘β†’β„ be a measurable function. We define the distribution function 𝛼𝑓 and a nonincreasing rearrangement π‘“βˆ— of 𝑓 in the following way 𝛼𝑓||ξ€½(𝑠)=π‘₯βˆˆβ„π‘;||𝑓||ξ€Ύ||(π‘₯)>𝑠,π‘“βˆ—ξ€½(𝑑)=inf𝑠>0;𝛼𝑓.(𝑠)≀𝑑(1.8) We now set β€–π‘“β€–βˆ—(𝑝,π‘ž)=⎧βŽͺ⎨βŽͺβŽ©ξ‚΅ξ€œβˆž0𝑑1/π‘π‘“βˆ—ξ€»(𝑑)π‘žπ‘‘π‘‘π‘‘ξ‚Ά1/π‘ž,if1≀𝑝,π‘ž<∞,sup𝑑>0𝑑1/π‘π‘“βˆ—(𝑑),if1β‰€π‘β‰€βˆž,π‘ž=∞.(1.9) The Lorentz space 𝐿𝑝,π‘ž(ℝ𝑁) is defined by 𝐿𝑝,π‘žξ€·β„π‘ξ€Έ=ξ‚†π‘“βˆˆπΏ1locℝ𝑁;β€–π‘“β€–βˆ—(𝑝,π‘ž)<∞.(1.10) The functional β€–π‘“β€–βˆ—(𝑝,π‘ž) is only a quasinorm. To obtain a norm we replace 𝑓 by π‘“βˆ—βˆ—βˆ«(𝑑)=(1/𝑑)𝑑0π‘“βˆ—(𝑠)𝑑π‘₯ in the definition of β€–π‘“β€–βˆ—(𝑝,π‘ž), that is, the norm is given by ‖𝑓‖(𝑝,π‘ž)=⎧βŽͺ⎨βŽͺβŽ©ξ‚΅ξ€œβˆž0𝑑1/π‘π‘“βˆ—βˆ—ξ€»(𝑑)π‘žπ‘‘π‘‘π‘‘ξ‚Ά1/π‘ž,if1≀𝑝,π‘ž<∞,sup𝑑>0𝑑1/π‘π‘“βˆ—βˆ—(𝑑),if1β‰€π‘β‰€βˆž,π‘ž=∞.(1.11)𝐿𝑝,π‘ž(ℝ𝑁)equipped with the norm ‖𝑓‖(𝑝,π‘ž) is a Banach space.

In paper [15] the existence of principal eigenfunctions has been established for weights belonging to ⋃1β‰€π‘ž<βˆžπΏπ‘/2,π‘ž(ℝ𝑁). This was extended in [14] to a larger class of weights ℱ𝑁/2 obtained as the completion of 𝐢∞∘(ℝ𝑁) 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 π‘‰βˆˆπΏπ‘/2,∞(ℝ𝑁), then the functional βˆ«β„π‘π‘‰(π‘₯)𝑒2𝑑π‘₯ is continuous on 𝐷1,2(ℝ𝑁), but not necessarily weakly continuous.

The paper is organized as follows. In Section 2, we prove the existence of minimizers with finite norm 𝐷1,2(ℝ𝑁) and also with infinite norm 𝐷1,2(ℝ𝑁). In Section 3 we discuss perturbation of a given quadratic form π‘„π‘‰βˆ˜ with π‘‰βˆ˜βˆˆπΏπ‘/2,∞(ℝ𝑁). We show that if π‘„π‘‰βˆ˜ has ground state, then this property is stable under small perturbations of π‘‰βˆ˜. This is not true if π‘„π‘‰βˆ˜ does not have a ground state; rather it is stable under larger perturbation of π‘‰βˆ˜. The final Section is devoted to a higher integrability property of minimizers of π‘„π‘‰βˆ˜ in the case where π‘‰βˆ˜(π‘₯)=π‘š(π‘₯)/|π‘₯|2 with π‘šβˆˆπΏβˆž(ℝ𝑁). We also examine the behaviour of the principal eigenfunction around 0.

Throughout this paper, in a given Banach space, we denote strong convergence by β€œβ†’β€ and weak convergence by β€œβ‡€β€. The norms in the Lebesgue space 𝐿𝑝(Ξ©), 1β‰€π‘β‰€βˆž, are denoted by ‖𝑒‖𝑝.

2. Existence of Minimizers

We consider the Hardy-type potential 𝑉(π‘₯)=π‘š(π‘₯)/|π‘₯|2 with π‘šβˆˆπΏβˆž(ℝ𝑁). In Theorem 2.2 we formulate conditions on π‘š guaranteeing the existence of a principal eigenfunction. Let 𝛾+>1 and π›Ύβˆ’>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. π‘š+(π‘₯)=limπ‘—βˆˆβ„•,π‘—β†’βˆžπ‘šξ€·π›Ύπ‘—+π‘₯ξ€Έπ‘š,(2.1)βˆ’(π‘₯)=limπ‘—βˆˆβ„•,π‘—β†’βˆžπ‘šξ€·π›Ύβˆ’βˆ’π‘—π‘₯ξ€Έ.(2.2) Both functions π‘šΒ± satisfy π‘šΒ±(𝛾±π‘₯)=π‘šΒ±(π‘₯). We now define the following infima: Ξ›π‘š=infπ‘’βˆˆπ·1,2ξ€·β„π‘ξ€Έβˆ’{0}βˆ«β„π‘||||βˆ‡π‘’2𝑑π‘₯βˆ«β„π‘ξ€·π‘š(π‘₯)/|π‘₯|2𝑒2,𝑑π‘₯(2.3) (we use the notation Ξ›π‘š instead of Λ𝑉) and Λ±=infπ‘’βˆˆπ·1,2ξ€·β„π‘ξ€Έβˆ’{0}βˆ«β„π‘||||βˆ‡π‘’2𝑑π‘₯βˆ«β„π‘ξ€·π‘šΒ±(π‘₯)/|π‘₯|2𝑒2.𝑑π‘₯(2.4)

Lemma 2.1. The following holds true Ξ›π‘šξ€·Ξ›β‰€min+,Ξ›βˆ’ξ€Έ.(2.5)

Proof. Let π‘’βˆˆπ·1,2(ℝ𝑁)βˆ’{0}. Testing Ξ›π‘š with 𝛾+βˆ’(π‘βˆ’2)/2𝑒(𝛾+βˆ’π‘—π‘₯) gives Ξ›π‘šβ‰€βˆ«β„π‘||||βˆ‡π‘’2𝑑π‘₯βˆ«β„π‘ξ€·π‘šξ€·π›Ύπ‘—+π‘₯ξ€Έ/|π‘₯|2𝑒2.𝑑π‘₯(2.6) Letting π‘—β†’βˆž and using the Lebesgue dominated convergence theorem, we obtain Ξ›π‘šβ‰€βˆ«β„π‘||||βˆ‡π‘’2𝑑π‘₯βˆ«β„π‘ξ€·π‘š+(π‘₯)/|π‘₯|2𝑒2.𝑑π‘₯(2.7) The inequality Ξ›π‘šβ‰€Ξ›+ follows. The proof of the inequality Ξ›π‘šβ‰€Ξ›βˆ’ 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 {π‘₯βˆˆβ„π‘;|π‘₯|β‰₯𝑅} for every 𝑅>0 and that the convergence in (2.2) is uniform on sets {π‘₯βˆˆβ„π‘;|π‘₯|β‰€πœŒ} for every 𝜌>0. If Ξ›π‘š<π‘šπ‘–π‘›(Ξ›+,Ξ›+), then problem (2.3) has a minimizer.

Proof. Let {π‘’π‘˜}βŠ‚π·1,2(ℝ𝑁) be a minimizing sequence for Ξ›π‘š, that is, ξ€œβ„π‘||βˆ‡π‘’π‘˜||2𝑑π‘₯βŸΆΞ›π‘š,ξ€œβ„π‘π‘š(π‘₯)||𝑋||2𝑒2π‘˜π‘‘π‘₯=1.(2.8) We can assume, up to a subsequence, that π‘’π‘˜β‡€π‘€ in 𝐷1,2(ℝ𝑁), 𝐿2(ℝ𝑁,𝑑π‘₯/|π‘₯|2), and π‘’π‘˜β†’π‘€ in 𝐿2loc(ℝ𝑁) for some π‘€βˆˆπ·1,2(ℝ𝑁). Let π‘£π‘˜=π‘’π‘˜βˆ’π‘€. We then have ξ€œ1=β„π‘π‘š(π‘₯)|π‘₯|2𝑒2π‘˜ξ€œπ‘‘π‘₯=β„π‘π‘š(π‘₯)|π‘₯|2𝑀2ξ€œπ‘‘π‘₯+β„π‘π‘š(π‘₯)|π‘₯|2𝑣2π‘˜Ξ›π‘‘π‘₯+π‘œ(1),(2.9)π‘š=ξ€œβ„π‘||βˆ‡π‘’π‘˜||2ξ€œπ‘‘π‘₯+π‘œ(1)=ℝ𝑁||||βˆ‡π‘€2ξ€œπ‘‘π‘₯+ℝ𝑁||βˆ‡π‘£π‘˜||2𝑑π‘₯+π‘œ(1).(2.10) We define a radial function πœ’π‘—+∈𝐢1(ℝ𝑁) such that 0β‰€πœ’π‘—+(π‘₯)≀1, πœ’π‘—+(π‘₯)=0 for |π‘₯|β‰€π›Ύβˆ’βˆ’2𝑗 and πœ’π‘—+(π‘₯)=1 for |π‘₯|>𝛾+2𝑗. Let πœ’π‘—βˆ’(π‘₯)=1βˆ’πœ’π‘—+(π‘₯). In what follows, we use π‘œ(𝑗)π‘˜β†’βˆž(1) to denote a quantity such that for each π‘—βˆˆβ„•, π‘œ(𝑗)π‘˜β†’βˆž(1)β†’0 as π‘˜β†’βˆž. Thus, ξ€œβ„π‘π‘š(π‘₯)|π‘₯|2𝑣2π‘˜ξ€œπ‘‘π‘₯=β„π‘π‘š(π‘₯)|π‘₯|2ξ€·π‘£π‘˜πœ’π‘—βˆ’ξ€Έ2ξ€œπ‘‘π‘₯+β„π‘π‘š(π‘₯)|π‘₯|2ξ€·π‘£π‘˜πœ’π‘—+ξ€Έ2𝑑π‘₯+π‘œ(𝑗)π‘˜β†’βˆž(=ξ€œ1)β„π‘π‘šξ€·π›Ύβˆ’βˆ’π‘—π‘₯ξ€Έ|π‘₯|2ξ€·π‘£βˆ’π‘˜ξ€Έ2ξ€œπ‘‘π‘₯+β„π‘π‘šξ€·π›Ύπ‘—+π‘₯ξ€Έ|π‘₯|2𝑣+π‘˜ξ€Έ2𝑑π‘₯+π‘œ(𝑗)π‘˜β†’βˆž(1),(2.11) where π‘£βˆ’π‘˜(π‘₯)=π›Ύβˆ’(βˆ’(π‘βˆ’2)/2)π‘—π‘£π‘˜ξ€·π›Ύβˆ’βˆ’π‘—π‘₯ξ€Έπœ’βˆ’ξ€·π›Ύβˆ’βˆ’π‘—π‘₯ξ€Έ,𝑣+π‘˜(π‘₯)=𝛾+((π‘βˆ’2)/2)π‘—π‘£π‘˜ξ€·π›Ύπ‘—+π‘₯ξ€Έπœ’+𝛾𝑗+π‘₯ξ€Έ.(2.12) We now estimate the integrals involving π‘£βˆ’π‘˜ and 𝑣+π‘˜. We have ||||ξ€œβ„π‘π‘šξ€·π›Ύβˆ’βˆ’π‘—π‘₯ξ€Έ|π‘₯|2ξ€·π‘£βˆ’π‘˜ξ€Έ2ξ€œπ‘‘π‘₯βˆ’β„π‘π‘šβˆ’(π‘₯)|π‘₯|2ξ€·π‘£βˆ’π‘˜ξ€Έ2||||≀||||ξ€œπ‘‘π‘₯|π‘₯|<π›Ύβˆ’βˆ’π‘—π‘šξ€·π›Ύβˆ’βˆ’π‘—π‘₯ξ€Έβˆ’π‘šβˆ’(π‘₯)|π‘₯|2ξ€·π‘£βˆ’π‘˜ξ€Έ2||||+||||ξ€œπ‘‘π‘₯π›Ύβˆ’βˆ’2𝑗<|π‘₯|<𝛾+2π‘—π‘šξ€·π›Ύβˆ’βˆ’π‘—π‘₯ξ€Έβˆ’π‘šβˆ’(π‘₯)|π‘₯|2ξ€·π‘£βˆ’π‘˜ξ€Έ2||||𝑑π‘₯=𝐽1+𝐽2.(2.13) By the uniform convergence of π‘š(π›Ύβˆ’βˆ’π‘—π‘₯) to π‘šβˆ’(π‘₯), we see that 𝐽1β‰€πœ– for 𝑗 sufficiently large uniformly in π‘˜. For 𝐽2 we have 𝐽2≀2β€–π‘šβ€–βˆžξ€œπ›Ύβˆ’βˆ’2𝑗<|π‘₯|<𝛾+2𝑗𝑣2π‘˜|π‘₯|2𝑑π‘₯.(2.14) It is clear that 𝐽2 is a quantity of type π‘œ(𝑗)π‘˜β†’βˆž(1). Therefore, we have ||||ξ€œβ„π‘π‘šξ€·π›Ύβˆ’βˆ’π‘—π‘₯ξ€Έ|π‘₯|2ξ€·π‘£βˆ’π‘˜ξ€Έ2ξ€œπ‘‘π‘₯βˆ’β„π‘π‘šβˆ’(π‘₯)|π‘₯|2ξ€·π‘£βˆ’π‘˜ξ€Έ2||||𝑑π‘₯β‰€πœ–+π‘œ(𝑗)π‘˜β†’βˆž(1).(2.15) In a similar way, we obtain |||||ξ€œβ„π‘π‘šξ€·π›Ύπ‘—+π‘₯ξ€Έ|π‘₯|2𝑣+π‘˜ξ€Έ2ξ€œπ‘‘π‘₯βˆ’β„π‘π‘š+(π‘₯)|π‘₯|2𝑣+π‘˜ξ€Έ2|||||𝑑π‘₯β‰€πœ–+π‘œ(𝑗)π‘˜β†’βˆž(1)(2.16) for 𝑗 sufficiently large. We now fix π‘—βˆˆβ„• so that (2.15) and (2.16) hold. Consequently, we have ξ€œ1β‰€β„π‘π‘š(π‘₯)|π‘₯|2𝑀2ξ€œπ‘‘π‘₯+β„π‘π‘šβˆ’(π‘₯)|π‘₯|2ξ€·π‘£βˆ’π‘˜ξ€Έ2ξ€œπ‘‘π‘₯+β„π‘π‘š+(π‘₯)|π‘₯|2𝑣+π‘˜ξ€Έ2𝑑π‘₯+2πœ–+π‘œ(𝑗)π‘˜β†’βˆž(1).(2.17) We now estimate βˆ«β„π‘|βˆ‡π‘£π‘˜|2𝑑π‘₯ in the following way ξ€œβ„π‘||βˆ‡π‘£π‘˜||2ξ€œπ‘‘π‘₯=ℝ𝑁||βˆ‡ξ€·π‘£π‘˜πœ’π‘—βˆ’+π‘£π‘˜πœ’π‘—+ξ€Έ||2=ξ€œπ‘‘π‘₯ℝ𝑁||βˆ‡ξ€·π‘£π‘˜πœ’π‘—βˆ’ξ€Έ||2ξ€œπ‘‘π‘₯+ℝ𝑁||βˆ‡ξ€·π‘£π‘˜πœ’π‘—+ξ€Έ||2ξ€œπ‘‘π‘₯+2β„π‘βˆ‡ξ€·π‘£π‘˜πœ’π‘—βˆ’ξ€Έβˆ‡ξ€·π‘£π‘˜πœ’π‘—+ξ€Έ=ξ€œπ‘‘π‘₯ℝ𝑁||βˆ‡π‘£βˆ’π‘˜||2ξ€œπ‘‘π‘₯+ℝ𝑁||βˆ‡π‘£+π‘˜||2ξ€œπ‘‘π‘₯+2ℝ𝑁||βˆ‡π‘£π‘˜||2πœ’π‘—βˆ’πœ’π‘—+ξ€œπ‘‘π‘₯+2β„π‘π‘£π‘˜βˆ‡π‘£π‘˜βˆ‡πœ’π‘—βˆ’πœ’π‘—+ξ€œπ‘‘π‘₯+2β„π‘π‘£π‘˜βˆ‡π‘£π‘˜πœ’π‘—βˆ’βˆ‡πœ’π‘—+ξ€œπ‘‘π‘₯+2ℝ𝑁𝑣2π‘˜βˆ‡πœ’π‘—βˆ’βˆ‡πœ’π‘—+β‰₯ξ€œπ‘‘π‘₯ℝ𝑁||βˆ‡π‘£βˆ’π‘˜||2ξ€œπ‘‘π‘₯+ℝ𝑁||βˆ‡π‘£+π‘˜||2ξ€œπ‘‘π‘₯+2β„π‘π‘£π‘˜βˆ‡π‘£π‘˜βˆ‡πœ’π‘—βˆ’πœ’π‘—+ξ€œπ‘‘π‘₯+2β„π‘π‘£π‘˜βˆ‡π‘£π‘˜πœ’π‘—βˆ’βˆ‡πœ’π‘—+ξ€œπ‘‘π‘₯+2ℝ𝑁𝑣2π‘˜βˆ‡πœ’π‘—βˆ’βˆ‡πœ’π‘—+𝑑π‘₯.(2.18) Since π‘£π‘˜β†’0 in 𝐿2loc(ℝ𝑁), we obtain the following estimate ξ€œβ„π‘||βˆ‡π‘£π‘˜||2ξ€œπ‘‘π‘₯β‰₯ℝ𝑁||βˆ‡π‘£βˆ’π‘˜||2ξ€œπ‘‘π‘₯+ℝ𝑁||βˆ‡π‘£+π‘˜||2𝑑π‘₯+π‘œ(𝑗)π‘˜β†’βˆž(1).(2.19) This, combined with (2.9), gives the following estimate Ξ›π‘šβ‰₯ξ€œβ„π‘||||βˆ‡π‘€2ξ€œπ‘‘π‘₯+ℝ𝑁||βˆ‡π‘£βˆ’π‘˜||2ξ€œπ‘‘π‘₯+ℝ𝑁||βˆ‡π‘£+π‘˜||2𝑑π‘₯+π‘œ(𝑗)π‘˜β†’βˆž(1)β‰₯Ξ›π‘šξ€œβ„π‘π‘š(π‘₯)|π‘₯|2𝑀2𝑑π‘₯+Ξ›βˆ’ξ€œβ„π‘π‘šβˆ’(π‘₯)|π‘₯|2ξ€·π‘£βˆ’π‘˜ξ€Έ2𝑑π‘₯+Ξ›+ξ€œβ„π‘π‘š+(π‘₯)|π‘₯|2𝑣+π‘˜ξ€Έ2𝑑π‘₯+π‘œ(𝑗)π‘˜β†’βˆž(1).(2.20) Let Ξ›βˆ—=min(Ξ›βˆ’,Ξ›+). We deduce from (2.17) and (2.20) that ξ€·Ξ›βˆ—βˆ’Ξ›π‘šξ€Έξ‚΅ξ€œβ„π‘π‘šβˆ’(π‘₯)|π‘₯|2ξ€·π‘£βˆ’π‘˜ξ€Έ2π‘šπ‘‘π‘₯++(π‘₯)|π‘₯|2𝑣+π‘˜ξ€Έ2𝑑π‘₯≀2πœ–Ξ›π‘š+π‘œ(𝑗)π‘˜β†’βˆž(1).(2.21) Letting π‘˜β†’βˆž, we obtain limsupπ‘˜β†’βˆžξ‚΅ξ€œβ„π‘π‘šβˆ’(π‘₯)|π‘₯|2ξ€·π‘£βˆ’π‘˜ξ€Έ2π‘šπ‘‘π‘₯++(π‘₯)|π‘₯|2𝑣+π‘˜ξ€Έ2≀𝑑π‘₯2πœ–Ξ›π‘šξ€·Ξ›βˆ—βˆ’Ξ›π‘šξ€Έ.(2.22) It then follows from (2.17) that ξ€œ1β‰€β„π‘π‘š(π‘₯)|π‘₯|2𝑀2𝑑π‘₯+2πœ–Ξ›π‘šξ€·Ξ›βˆ—βˆ’Ξ›π‘šξ€Έ.(2.23) Since πœ–>0 is arbitrary, we get βˆ«β„π‘(π‘š(π‘₯)/|π‘₯|2)𝑀2𝑑π‘₯=1, and the result follows.

In what follows, we denote π‘š(∞)=lim|π‘₯|β†’βˆžπ‘š(π‘₯), assuming that this limit exists. As a direct consequence of Theorem 2.2, we obtain the following result.

Theorem 2.3. Let π‘šβˆˆπΏβˆž(ℝ𝑁), and assume that π‘š is continuous at 0. Further, suppose that π‘š(∞)>0 and π‘š(0)>0. If Ξ›π‘š<Λ𝑁min(1/π‘š(∞),1/π‘š(0)), then there exists a minimizer for Ξ›π‘š.

Remark 2.4. Ξ›π‘š has a minimizer also in the following cases, corresponding formally to Ξ›+ or Ξ›βˆ’ taking the value +∞.(i)Let π‘š(0)=0 and π‘š(∞)>0. If Ξ›π‘š<Λ𝑁/π‘š(∞), then a minimizer for Ξ›1(π‘š) exists.(ii)Let π‘š(0)>0 and π‘š(∞)=0. If Ξ›π‘š<Λ𝑁/π‘š(0), then a minimizer for Ξ›1(π‘š) exists.(iii)If π‘š(0)=π‘š(∞)=0, π‘š(π‘₯)β‰₯0 and β‰’ 0 on ℝ𝑁, then Ξ›π‘š 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 π‘š satisfying conditions of Theorems 2.2 and 2.3. In general, functions satisfying this condition have large local maxima.

Example 2.5. Let π‘šπ΄(⎧βŽͺ⎨βŽͺβŽ©π‘šπ‘₯)=1(π‘₯),for0<|π‘₯|<1,π΄π‘š2π‘š(π‘₯),for1≀|π‘₯|≀2,3(π‘₯),for2<|π‘₯|,(2.24) where 𝐴>0 is a constant to be chosen later and π‘š1∢𝐡(0,1)βˆ’{0}β†’[0,∞), π‘š2∢(1≀|π‘₯|≀2)β†’[0,∞), and π‘š3βˆΆβ„N⧡𝐡(0,2)β†’[0,∞) are continuous bounded functions satisfying the following conditions: π‘š1(π‘₯)=0 for |π‘₯|=1, π‘š2(π‘₯)=0 for |π‘₯|=1, π‘š2(π‘₯)=0 for |π‘₯|=2, and π‘š2(π‘₯)>0 for 1<|π‘₯|<2,π‘š3(π‘₯)=0 for |π‘₯|=2. Further we assume that π‘š3||π‘₯(π‘₯)=π‘Ž+1||||π‘₯2||||π‘₯+β‹―+π‘βˆ’1||||π‘₯𝑁||𝑏+|π‘₯|2,(2.25) for |π‘₯|β‰₯𝑅>2, where π‘Ž>0, 𝑏>0 and 𝑅 constants. A function π‘š1(π‘₯) for small 𝛿>0 is given by π‘š1||π‘₯(π‘₯)=1||||π‘₯+β‹―+𝑁|||π‘₯|,(2.26) for 0<|π‘₯|≀𝛿<1. We have limπ‘—β†’βˆžπ‘šπ΄ξ€·π›Ύπ‘—+π‘₯ξ€Έ=limπ‘—β†’βˆžπ›Ύ+βˆ’2𝑗||π‘₯π‘Ž+1||||π‘₯2||||π‘₯+β‹―+π‘βˆ’1||||π‘₯𝑁||𝛾+βˆ’2𝑗𝑏+|π‘₯|2=||π‘₯1||||π‘₯2||||π‘₯+β‹―+π‘βˆ’1||||π‘₯𝑁|||π‘₯|2=π‘š+(π‘₯),limπ‘—β†’βˆžπ‘šπ΄ξ€·π›Ύβˆ’βˆ’π‘—π‘₯ξ€Έ=||π‘₯1||||π‘₯+β‹―+𝑁|||π‘₯|=π‘šβˆ’(π‘₯).(2.27) Both limits are uniform. Since π‘šβˆ’ and π‘š+ are bounded, Ξ›βˆ’ and Ξ›+ are positive and finite. We have Ξ›π‘š=inf𝐷1,2ξ€·β„π‘ξ€Έβˆ’{0}βˆ«β„π‘||||βˆ‡π‘’2𝑑π‘₯βˆ«β„π‘ξ€·π‘šπ΄(π‘₯)/|π‘₯|2𝑒2≀1𝑑π‘₯𝐴inf𝐷1,2ξ€·β„π‘ξ€Έβˆ’{0}βˆ«β„π‘||||βˆ‡π‘’2𝑑π‘₯∫1≀|π‘₯|≀2ξ€·π‘š2(π‘₯)/|π‘₯|2𝑒2Λ𝑑π‘₯<π‘šπ‘–π‘›βˆ’,Ξ›+ξ€Έ,(2.28) for 𝐴 large. By Theorem 2.2, Ξ›π‘š with π‘š=π‘šπ΄ has a minimizer.

Example 2.6. Consider a sequence of functions of the form π‘šπ‘˜(π‘₯)=π΅π‘€π‘˜(π‘₯)+𝐴𝑓(π‘₯),π‘˜=1,2,…, where 𝐴>0, 𝐡>0 are constants and π‘€π‘˜ and 𝑓 are continuous functions satisfying the following conditions: (a)π‘€π‘˜(0)=1, π‘€π‘˜(π‘₯)>0 on ℝ𝑁, π‘€π‘˜(∞)=0, for π‘˜=1,2,…, (b)π‘€π‘˜(π‘₯)=π‘˜ on 1<|π‘₯|<2 for π‘˜=1,2,…, (c)𝑓(π‘₯)β‰₯0 on ℝ𝑁, 𝑓(0)=0 and 𝑓(∞)=1. Then π‘šπ‘˜(0)=𝐡 and π‘šπ‘˜(∞)=𝐴 for π‘˜=1,2,…. We show that for π‘˜ sufficiently large π‘šπ‘˜ satisfies the conditions of Theorem 2.3. Let 𝑒(π‘₯)=exp(βˆ’|π‘₯|) (one can take any other function from 𝐷1,2(ℝ𝑁) which is β‰’0 on (1<|π‘₯|<2)). Thus Ξ›π‘šπ‘˜β‰€βˆ«β„π‘||||βˆ‡(exp(βˆ’|π‘₯|))2𝑑π‘₯βˆ«β„π‘ξ€·ξ€·π΅π‘€π‘˜ξ€Έ(π‘₯)+𝐴𝑓(π‘₯)/|π‘₯|2ξ€Έβ‰€βˆ«exp(βˆ’2|π‘₯|)𝑑π‘₯ℝ𝑁||||βˆ‡(exp(βˆ’|π‘₯|))2𝑑π‘₯π΅βˆ«β„π‘ξ€·π‘€π‘˜(π‘₯)/|π‘₯|2ξ€Έexp(βˆ’2|π‘₯|)𝑑π‘₯⟢0,(2.29) as π‘˜β†’βˆž. So we can find π‘˜βˆ˜β‰₯1 so that Ξ›π‘šπ‘˜<Λ𝑁1min𝐴,1𝐡forπ‘˜β‰₯π‘˜βˆ˜.(2.30)

In Proposition 2.7, we described a class of weight functions π‘š satisfying conditions of Theorem 2.3.

Proposition 2.7. Let π‘šβˆˆπΆ(ℝ𝑁). Suppose that π‘š(π‘₯)β‰₯0, π‘š(0)>0, and π‘š(∞)>0. Assume that there exists a ball 𝐡(π‘₯𝑀,π‘Ÿ) such that π‘š(π‘₯)β‰₯π‘š(π‘₯𝑀)>0 for π‘₯∈𝐡(π‘₯𝑀,π‘Ÿ) and 0βˆ‰π΅(π‘₯𝑀,π‘Ÿ). If π‘š(0)π‘šξ€·π‘₯𝑀,π‘š(∞)π‘šξ€·π‘₯𝑀<π‘Ÿ2(π‘βˆ’2)22ξ€·||π‘₯π‘Ÿ+𝑀||ξ€Έ2.(𝑁+1)(𝑁+2)(2.31) Then Ξ›π‘š<Λ𝑁min(1/π‘š(0),1/π‘š(∞)). (Hence, there exists a minimizer for Ξ›π‘š.)

Proof. Let π‘’βˆˆπ»1∘(𝐡(π‘₯𝑀,π‘Ÿ))βˆ’{0}. Then ξ€œπ΅(π‘₯𝑀,π‘Ÿ)π‘š(π‘₯)|π‘₯|2𝑒2ξ€·π‘₯𝑑π‘₯β‰₯π‘šπ‘€ξ€Έξ€œπ΅(π‘₯𝑀)𝑒2|π‘₯|2π‘šξ€·π‘₯𝑑π‘₯β‰₯𝑀||π‘₯π‘Ÿ+𝑀||ξ€Έ2ξ€œπ΅(π‘₯𝑀,π‘Ÿ)𝑒2𝑑π‘₯.(2.32) Hence, ∫𝐡(π‘₯𝑀,π‘Ÿ)||||βˆ‡π‘’2𝑑π‘₯∫𝐡(π‘₯𝑀,π‘Ÿ)ξ€·π‘š(π‘₯)/|π‘₯|2≀||π‘₯𝑑π‘₯π‘Ÿ+𝑀||ξ€Έ2∫𝐡(π‘₯𝑀,π‘Ÿ)||||βˆ‡π‘’2𝑑π‘₯π‘šξ€·π‘₯π‘€ξ€Έβˆ«π΅(π‘₯𝑀,π‘Ÿ)𝑒2.𝑑π‘₯(2.33) Since 𝐻1∘(𝐡(π‘₯𝑀,π‘Ÿ))βˆ’{0}βŠ‚{u∈𝐷1,2(β„π‘βˆ«);ℝ𝑁(π‘š(π‘₯)/|π‘₯|2)𝑒2𝑑π‘₯>0}, we deduce from the above inequality that Ξ›π‘šβ‰€ξ€·||π‘₯π‘Ÿ+𝑀||ξ€Έ2π‘šξ€·π‘₯π‘€ξ€Έπœ†π·1𝐡π‘₯𝑀,,π‘Ÿξ€Έξ€Έ(2.34) where πœ†π·1(𝐡(π‘₯𝑀,π‘Ÿ)) denotes the first eigenvalue for β€œβˆ’Ξ”β€ in 𝐡(π‘₯𝑀,π‘Ÿ) with the Dirichlet boundary conditions. We now estimate πœ†π·1=πœ†π·1(𝐡(π‘₯𝑀,π‘Ÿ)). We test πœ†π·1 with 𝑣(π‘₯)=π‘Ÿβˆ’|π‘₯βˆ’π‘₯𝑀| for π‘₯∈𝐡(π‘₯𝑀,π‘Ÿ). We have ξ€œπ΅(π‘₯𝑀,π‘Ÿ)𝑣2ξ€œπ‘‘π‘₯=𝐡(0,π‘Ÿ)(π‘Ÿβˆ’|π‘₯|)2𝑑π‘₯=πœ”π‘ξ€œπ‘Ÿ0(π‘Ÿβˆ’π‘ )2π‘ π‘βˆ’1𝑑𝑠=2πœ”π‘π‘Ÿπ‘+2,ξ€œπ‘(𝑁+1)(𝑁+2)𝐡(π‘₯𝑀,π‘Ÿ)||||βˆ‡π‘£2πœ”π‘‘π‘₯=π‘π‘Ÿπ‘π‘.(2.35) Hence πœ†π·1β‰€βˆ«π΅(π‘₯𝑀,π‘Ÿ)||||βˆ‡π‘£2𝑑π‘₯∫𝐡(π‘₯𝑀,π‘Ÿ)𝑣2=𝑑π‘₯(𝑁+1)(𝑁+2)2π‘Ÿ2.(2.36) Combining this with (2.34), we derive Ξ›π‘šβ‰€ξ€·||π‘₯(𝑁+1)(𝑁+2)π‘Ÿ+𝑀||ξ€Έ22π‘Ÿ2π‘šξ€·π‘₯𝑀.(2.37) Therefore Ξ›π‘š<Λ𝑁min(1/(π‘š(0)),1/(π‘š(∞))) 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 π‘š(π‘₯) is a continuous bounded and nonnegative function such that π‘š(π‘₯)β‰€π‘š(0) on ℝ𝑁 and π‘š(0)>0 (or π‘š(π‘₯)β‰€π‘š(∞) on ℝ𝑁, π‘š(∞)>0), then Ξ›π‘š does not have a minimizer. Indeed, suppose that π‘š(π‘₯)β‰€π‘š(0) on ℝ𝑁 and that Ξ›π‘š has a minimizer 𝑒. Then, by the Hardy inequality, we obtain Λ𝑁β‰₯βˆ«π‘š(0)ℝ𝑁||||βˆ‡π‘’2𝑑π‘₯βˆ«β„π‘ξ€·π‘š(π‘₯)/|π‘₯|2𝑒2β‰₯βˆ«π‘‘π‘₯ℝ𝑁||||βˆ‡π‘’2𝑑π‘₯βˆ«π‘š(0)ℝ𝑁𝑒2/|π‘₯|2ξ€Έβ‰₯Λ𝑑π‘₯𝑁.π‘š(0)(2.38) So 𝑒 is a minimizer for Λ𝑁, which is impossible.

We now construct a ground state with infinite 𝐷1,2 norm.

Theorem 2.8. Let 𝛾>1, and assume that the function π‘šβˆˆπΏβˆž(ℝ𝑁) satisfies π‘š(𝛾π‘₯)=π‘š(π‘₯)forπ‘₯βˆˆβ„π‘.(2.39) Then the form 𝑄𝑉 with 𝑉(π‘₯)=π‘š(π‘₯)/|π‘₯|2 and Λ𝑉=Ξ›βˆ˜ (see (2.41) below) admits a ground state 𝑣 satisfying 𝑣(𝛾π‘₯)=𝛾(2βˆ’π‘)/2𝑣(π‘₯)forπ‘₯βˆˆβ„π‘.(2.40) The function 𝑣 is uniquely defined by its values on 𝐴𝛾={π‘₯βˆˆβ„π‘;1<|π‘₯|<𝛾}, and, moreover, the function 𝑣|𝐴𝛾 is a minimizer for the problem Ξ›βˆ˜βŽ§βŽͺ⎨βŽͺ⎩∫=inf𝐴𝛾||||βˆ‡π‘£2𝑑π‘₯βˆ«π΄π›Ύξ€·π‘š(π‘₯)/|π‘₯|2𝑒2𝑑π‘₯;π‘’βˆˆπ»1ξ€·π΄π›Ύξ€Έβˆ’{0},𝑒(𝛾π‘₯)=𝛾(2βˆ’π‘)/2⎫βŽͺ⎬βŽͺ⎭.𝑒(π‘₯)for|π‘₯|=1(2.41)

Proof. The problem (2.41) is a compact variational problem that has a minimizer 𝑣 which satisfies βˆ’Ξ”π‘£=Ξ›βˆ˜π‘š(π‘₯)|π‘₯|2𝑣,π‘₯βˆˆπ΄π›Ύ,(2.42) with the Neumann boundary conditions. Since the test functions satisfy 𝑒(𝛾π‘₯)=𝛾(2βˆ’π‘)/2𝑒(π‘₯) for |π‘₯|=1, one has πœ•π‘£πœ•π‘Ÿ(𝛾π‘₯)=π›Ύβˆ’π‘/2πœ•π‘£πœ•π‘Ÿ(π‘₯)for|π‘₯|=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 𝐴𝛾 to β„π‘βˆ’{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 𝐢1(β„π‘βˆ’{0}) and satisfies βˆ’Ξ”π‘£=Ξ›βˆ˜π‘š(π‘₯)|π‘₯|2𝑣,(2.44) in a weak sense. From this and the Harnack inequality on bounded subsets of β„π‘βˆ’{0} it follows that 𝑣 is positive on β„π‘βˆ’{0} and subsequently there exists a constant 𝐢>0 such that πΆβˆ’1|π‘₯|(2βˆ’π‘)/2≀𝑣(π‘₯)≀𝐢|π‘₯|(2βˆ’π‘)/2.(2.45) We can now explain the choice of the exponent (2βˆ’π‘)/2 in the constraint 𝑒(𝛾π‘₯)=𝛾(2βˆ’π‘)/2𝑒(π‘₯) 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 |π‘₯|=𝛾𝑗,π‘—βˆˆβ„•. Finally, we show that 𝑣 is a ground state for the corresponding quadratic form 𝑄 with 𝑉(π‘₯)=Ξ›βˆ˜π‘š(π‘₯)/|π‘₯|2. Using the ground state formula (2.10), from [20] and (2.45), we have π‘€π‘˜(π‘₯)=|π‘₯|1/π‘˜ for |π‘₯|≀1 and π‘€π‘˜(π‘₯)=|π‘₯|βˆ’1/π‘˜ for |π‘₯|β‰₯1, π‘„ξ€·π‘£π‘€π‘˜ξ€Έ=ξ€œβ„π‘π‘£2||βˆ‡π‘€π‘˜||2ξ€œπ‘‘π‘₯≀𝐢ℝ𝑁|π‘₯|2βˆ’π‘||βˆ‡π‘€π‘˜||2≀𝐢𝑑π‘₯π‘˜2ξ€œ10π‘Ÿβˆ’1+(2/π‘˜)πΆπ‘‘π‘Ÿ+π‘˜2ξ€œβˆž1π‘Ÿβˆ’1βˆ’(2/π‘˜)πΆπ‘‘π‘Ÿβ‰€π‘˜βŸΆ0,(2.46) as π‘˜β†’βˆž. Since π‘£π‘€π‘˜β†’π‘£ uniformly on compact sets, this implies that 𝑣 is a ground state for 𝑄. By (2.45) and the Sobolev inequality, π‘£βˆ‰π·1,2(ℝ𝑁).

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 𝐷1,2 norm. Then, we investigate potentials that do not give rise to a ground state with finite 𝐷1,2 norm.

We need the following existence result.

Proposition 3.1. Let π‘‰βˆ˜βˆˆπΏπ‘/2,∞(ℝ𝑁) be positive on a set of positive measure, and let Ξ›βˆ˜=infπ‘’βˆˆπ·1,2ℝ𝑁,βˆ«β„π‘π‘‰βˆ˜π‘’2𝑑π‘₯=1ξ€œβ„π‘||||βˆ‡π‘’2𝑑π‘₯.(3.1) Assume that 𝑉1βˆˆπΏπ‘/2,∞(ℝ𝑁) is positive on a set of positive measure and that the functional βˆ«β„π‘(𝑉1(π‘₯)βˆ’π‘‰βˆ˜(π‘₯))𝑒2𝑑π‘₯ is weakly continuous in 𝐷1,2(ℝ𝑁), and let Ξ›1=infπ‘’βˆˆπ·1,2ℝ𝑁,βˆ«β„π‘π‘‰1𝑒2𝑑π‘₯=1ξ€œβ„π‘||||βˆ‡π‘’2𝑑π‘₯.(3.2) If Ξ›1<Ξ›βˆ˜, then there exists a minimizer for Ξ›1.

Proof. Let {π‘’π‘˜}βŠ‚π·1,2(ℝ𝑁) be a minimizing sequence for (3.2), that is, βˆ«β„π‘π‘‰1(π‘₯)𝑒2π‘˜π‘‘π‘₯=1 and βˆ«β„π‘|βˆ‡π‘’π‘˜|2𝑑π‘₯β†’Ξ›1. We may assume that, up to a subsequence, π‘’π‘˜β‡€π‘€ in 𝐷1,2(ℝ𝑁) and 𝐿2(ℝ𝑁,𝑉1(π‘₯)𝑑π‘₯). Let π‘£π‘˜=π‘’π‘˜βˆ’π‘€. Then, ξ€œ1=ℝ𝑁𝑉1(π‘₯)𝑒2π‘˜ξ€œπ‘‘π‘₯=ℝ𝑁𝑉1(π‘₯)𝑣2π‘˜ξ€œπ‘‘π‘₯+ℝ𝑁𝑉1(π‘₯)𝑀2ξ€œπ‘‘π‘₯+π‘œ(1)=ℝ𝑁𝑉1(π‘₯)𝑀2+ξ€œπ‘‘π‘₯ℝ𝑁𝑉1(π‘₯)βˆ’π‘‰βˆ˜ξ€Έπ‘£(π‘₯)2π‘˜ξ€œπ‘‘π‘₯+β„π‘π‘‰βˆ˜(π‘₯)𝑣2π‘˜=ξ€œπ‘‘π‘₯+π‘œ(1)β„π‘π‘‰βˆ˜(π‘₯)𝑣2π‘˜ξ€œπ‘‘π‘₯+ℝ𝑁𝑉1(π‘₯)𝑀2𝑑π‘₯+π‘œ(1).(3.3) Let βˆ«π‘‘=ℝ𝑁𝑉1(π‘₯)𝑀2𝑑π‘₯. Then βˆ«β„π‘π‘‰βˆ˜(π‘₯)𝑣2π‘˜π‘‘π‘₯β†’1βˆ’π‘‘. Assuming that 𝑑<1 we get Ξ›1=ξ€œβ„π‘||βˆ‡π‘£π‘˜||2ξ€œπ‘‘π‘₯+ℝ𝑁||||βˆ‡π‘€2𝑑π‘₯+π‘œ(1)β‰₯Ξ›βˆ˜(1βˆ’π‘‘)+Ξ›1𝑑+π‘œ(1).(3.4) From this, we deduce that Ξ›1β‰₯Ξ›βˆ˜ which is impossible. Hence, βˆ«β„π‘π‘‰1(π‘₯)𝑀2𝑑π‘₯=1. From this and the lower semicontinuity of the norm with respect to weak convergence, we derive that 𝑀 is a minimizer and π‘’π‘˜β†’π‘€ in 𝐷1,2(ℝ𝑁).
Proposition 3.1 is related to [19, Theorem  1.7] which asserts that a potential of the form 𝑉(π‘₯)=(1/|π‘₯|2)+𝑔(π‘₯), 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 𝐷1,2(ℝ𝑁) (see [12]) and the potential 𝑔 admits a principal eigenfunction.

Remark 3.2. (i) If 𝑉1>π‘‰βˆ˜, then Ξ›1β‰€Ξ›βˆ˜, but not necessarily Ξ›1<Ξ›βˆ˜.
(ii) If, in Proposition 3.1, assumption Ξ›1<Ξ›βˆ˜ is replaced by Ξ›βˆ˜<Ξ›1, then Ξ›βˆ˜ is attained.

Example 3.3. Let 𝑀 be a continuous function ℝ𝑁 such that 𝑀β‰₯0, β‰’0 on ℝ𝑁 and 𝑀(0)=𝑀(∞)=0. Define π‘šπ΄,𝐡(π‘₯)=𝐡𝑀(π‘₯)+𝐴, where 𝐴>0 and 𝐡>0 are constants. Let 𝑉1(π‘₯)=π‘šπ΄,𝐡(π‘₯)/|π‘₯|2 and π‘‰βˆ˜(π‘₯)=𝐴/|π‘₯|2. The functional βˆ«β„π‘(𝑉1(π‘₯)βˆ’π‘‰βˆ˜(π‘₯))𝑒2𝑑π‘₯=βˆ«β„π‘(𝐡𝑀(π‘₯)/|π‘₯|2)𝑒2𝑑π‘₯ is weakly continuous in 𝐷1,2(ℝ𝑁). It is easy to show that for every 𝐴>0 there exists 𝐡∘>0 such that Ξ›1<Ξ›βˆ˜ for 𝐡>𝐡∘. By Proposition 3.1  Λ1 has a minimizer for 𝐡>𝐡∘.

We now give a sufficient condition for the inequality Ξ›1<Ξ›βˆ˜.

Theorem 3.4. Suppose that 𝑉1 and π‘‰βˆ˜ satisfy assumptions of Proposition 3.1. Moreover, assume that the quadratic form π‘„π‘‰βˆ˜ has a positive ground state 𝑣, possibly with infinite 𝐷1,2 norm, and that if {π‘£π‘˜}βŠ‚πΆβˆžβˆ˜(ℝ𝑁) is a null sequence corresponding to Ξ›βˆ˜, then limsupπ‘˜β†’βˆžξ€œβ„π‘ξ€·π‘‰1(π‘₯)βˆ’π‘‰βˆ˜(𝑣π‘₯)2π‘˜π‘‘π‘₯>0.(3.5) Then Ξ›1<Ξ›βˆ˜ and Ξ›1 has a minimizer.

Proof. It suffices to show that the inequality ξ€œβ„π‘||||βˆ‡π‘’2𝑑π‘₯βˆ’Ξ›βˆ˜ξ€œβ„π‘π‘‰1(π‘₯)𝑒2𝑑π‘₯β‰₯0(3.6) fails for some π‘’βˆˆπ·1,2(ℝ𝑁). We have ξ€œβ„π‘||βˆ‡π‘£π‘˜||2𝑑π‘₯βˆ’Ξ›βˆ˜ξ€œβ„π‘π‘‰1(π‘₯)𝑣2π‘˜π‘‘π‘₯=π‘„π‘‰βˆ˜ξ€·π‘£π‘˜ξ€Έβˆ’Ξ›βˆ˜ξ€œβ„π‘ξ€·π‘‰1(π‘₯)βˆ’π‘‰βˆ˜(𝑣π‘₯)2π‘˜π‘‘π‘₯=π‘œ(1)βˆ’Ξ›βˆ˜ξ€œβ„π‘ξ€·π‘‰1(π‘₯)βˆ’π‘‰βˆ˜ξ€Έπ‘£(π‘₯)2π‘˜π‘‘π‘₯<0,(3.7) for sufficiently large π‘˜, which completes the proof of the theorem.

Note that the conditions of Theorem 3.4 are satisfied if, in particular, 𝑉1β‰₯π‘‰βˆ˜ on ℝ𝑁, with the strict inequality on a set of positive measure. Indeed, the sequence {π‘£π‘˜} converges weakly in 𝐻1loc(ℝ𝑁) to 𝑣>0, and the condition limsupπ‘˜β†’βˆžβˆ«β„π‘(𝑉1(π‘₯)βˆ’π‘‰βˆ˜(π‘₯))𝑣2π‘˜π‘‘π‘₯>0 follows from the Fatou lemma.

The situation becomes different if π‘„π‘‰βˆ˜ 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 π‘‰βˆ˜ satisfies the conditions of Proposition 3.1 and that (1.7) holds. (This occurs under conditions of Theorem 1.4 if π‘„π‘‰βˆ˜ has no ground state.) Let π‘Š be as in (1.7). Then, for every π‘‘βˆˆ(0,1/Ξ›βˆ˜), the functional π‘„π‘‰βˆ˜+π‘‘π‘Š has no ground state and Ξ›π‘‰βˆ˜+π‘‘π‘Š=Ξ›π‘‰βˆ˜. Furthermore, if the functional βˆ«β„π‘π‘Š(π‘₯)𝑒2𝑑π‘₯ is weakly continuous in 𝐷1,2(ℝ𝑁), the same conclusion holds for βˆ’βˆž<𝑑<0.

Proof. First, we observe that the constants Ξ›βˆ˜ and Ξ›1 corresponding to π‘‰βˆ˜ and 𝑉1=π‘‰βˆ˜+π‘‘π‘Š, respectively, are equal. Indeed, since 𝑉1>π‘‰βˆ˜, one has Ξ›1β‰€Ξ›βˆ˜ by monotonicity. On the other hand, it follows from (1.7) that ξ€œβ„π‘||||βˆ‡π‘’2𝑑π‘₯βˆ’Ξ›βˆ˜ξ€œβ„π‘ξ€·π‘‰βˆ˜(𝑒π‘₯)+π‘‘π‘Š(π‘₯)2𝑑π‘₯β‰₯0,(3.8) for π‘‘βˆˆ(0,1/Ξ›βˆ˜) which implies Ξ›1β‰₯Ξ›βˆ˜. Let π‘£π‘˜βˆˆπΆβˆžβˆ˜(β„π‘βˆ’π‘) satisfy 𝑄𝑉1(π‘£π‘˜)β†’0. Then ξ€·1βˆ’Ξ›βˆ˜π‘‘ξ€Έξ€œβ„π‘π‘Šπ‘£2π‘˜π‘‘π‘₯≀𝑄𝑉1ξ€·π‘£π‘˜ξ€ΈβŸΆ0,(3.9) which implies that, up to subsequence, π‘£π‘˜β†’0 a.e. If π‘£π‘˜ were a null sequence, it would converge in 𝐻1loc(ℝ𝑁) and it would have a limit zero. Therefore, 𝑄𝑉1 admits no null sequence and consequently no ground state. Assume now that the functional βˆ«β„π‘π‘Š(π‘₯)𝑒2𝑑π‘₯ is weakly continuous in 𝐷1,2(ℝ𝑁). Let {π‘€π‘˜}βŠ‚π·1,2(ℝ𝑁) be a minimizing sequence for Ξ›βˆ˜. If {π‘€π‘˜} has a subsequence weakly convergent in 𝐷1,2(ℝ𝑁) to some 𝑀≠0, then it is easy to see that |𝑀| would be a minimizer for Ξ›βˆ˜ and thus a ground state for π‘„Ξ›βˆ˜. Therefore, π‘€π‘˜β‡€0. By the weak continuity of βˆ«β„π‘π‘Š(π‘₯)𝑒2𝑑π‘₯, we get ξ€œβ„π‘π‘‰1(π‘₯)𝑀2π‘˜ξ€œπ‘‘π‘₯=β„π‘π‘‰βˆ˜(π‘₯)𝑀2π‘˜π‘‘π‘₯+π‘œ(1)=1+π‘œ(1),(3.10) and thus Ξ›1β‰€ξ€œβ„π‘||βˆ‡π‘€π‘˜||2𝑑π‘₯=Ξ›βˆ˜+π‘œ(1).(3.11) This yields Ξ›1β‰€Ξ›βˆ˜. Then, ξ€œβ„π‘||||βˆ‡π‘’2𝑑π‘₯βˆ’Ξ›1ξ€œβ„π‘π‘‰1(π‘₯)𝑒2β‰₯Λ𝑑π‘₯1Ξ›βˆ˜ξ‚΅ξ€œβ„π‘||||βˆ‡π‘’2𝑑π‘₯βˆ’Ξ›βˆ˜ξ€œβ„π‘π‘‰1(π‘₯)𝑒2ξ‚Ά=Λ𝑑π‘₯1Ξ›βˆ˜ξ‚΅π‘„π‘‰βˆ˜(𝑒)βˆ’π‘‘Ξ›βˆ˜ξ€œβ„π‘π‘Š(π‘₯)𝑒2𝑑π‘₯β‰₯Ξ›1ξ€œβ„π‘ξ€·Ξ›βˆ˜βˆ’1ξ€Έβˆ’π‘‘π‘Š(π‘₯)𝑒2𝑑π‘₯.(3.12) Since 𝑑<0, this implies that 𝑄𝑉1 has no ground state.

Theorem 3.5 concerns with small perturbations of a potential that does not change the constant Ξ› 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 𝐷1,2(ℝ𝑁) norm.

Theorem 3.6. Assume that π‘‰βˆ˜ satisfies conditions of Proposition 3.1 and that π‘ŠβˆˆπΏ2,∞(ℝ𝑁) is such that the functional βˆ«β„π‘π‘Š(π‘₯)𝑒2𝑑π‘₯ is weakly continuous in 𝐷1,2(ℝ𝑁). Then, for every πœ†βˆˆ(0,Ξ›βˆ˜) there exists πœŽβˆˆβ„ such that π‘„π‘‰βˆ˜+πœŽπ‘Š has a ground state of finite 𝐷1,2(ℝ𝑁) norm corresponding to the energy constant (3.2).

Proof. Assume without loss of generality that π‘Š is positive on a set of positive measure. Let 0<πœ†<Ξ›βˆ˜ and consider 𝜎=infπ‘’βˆˆπ·1,2ℝ𝑁,βˆ«β„π‘π‘Š(π‘₯)𝑒2𝑑π‘₯=1πœ†βˆ’1ξ‚΅ξ€œβ„π‘||||βˆ‡π‘’2ξ€œπ‘‘π‘₯βˆ’πœ†β„π‘π‘‰βˆ˜(π‘₯)𝑒2ξ‚Ά.𝑑π‘₯(3.13) Since (βˆ«β„π‘|βˆ‡π‘’|2βˆ«π‘‘π‘₯βˆ’πœ†β„π‘π‘‰βˆ˜(π‘₯)𝑒2𝑑π‘₯)1/2 defines an equivalent norm on 𝐷1,2(ℝ𝑁), it is easy to show that there exists a minimizer for 𝜎. It is clear that this minimizer is also a ground state of π‘„π‘‰βˆ˜+πœŽπ‘Š corresponding to the optimal constant πœ†.

If we assume additionally that π‘Š is positive on a set of positive measure, then it is easy to show that 𝜎 is a continuous decreasing function of πœ† with limπœ†β†’0𝜎(πœ†)=+∞ and 𝜎∘=limπœ†β†’Ξ›βˆ˜πœŽ(πœ†)β‰₯0. In particular, if (1.7) holds with a weight π‘Šβˆ˜ satisfying π‘Šβˆ˜β‰₯π›Όπ‘Š, then 𝜎∘β‰₯𝛼. In other words, given π‘‰βˆ˜ and π‘Š as in Theorem 3.6, the potential π‘‰βˆ˜+πœŽπ‘Š admits a ground state whenever 𝜎β‰₯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(π‘₯)=π‘š(π‘₯)/|π‘₯|2, where π‘š(π‘₯) is continuous and π‘š(0)>0 and π‘š(∞)>0. The corresponding ground state, if it exists, is denoted by πœ™1, which is chosen to be positive on ℝ𝑁. Obviously the ground state πœ™1 satisfies Δ𝑒=Ξ›π‘šπ‘š(π‘₯)|π‘₯|2𝑒inℝ𝑁(4.1) in a weak sense.

We need the following extension of the Hardy inequality: let Ξ©βŠ‚β„π‘ be a bounded domain and 0∈Ω, then for every 𝛿>0, there exists a constant 𝐴(𝛿,Ξ©)>0 such that ξ€œΞ©π‘’2|π‘₯|2ξ‚΅1𝑑π‘₯β‰€Ξ›π‘ξ‚Άξ€œ+𝛿Ω||||βˆ‡π‘’2ξ€œπ‘‘π‘₯+𝐴(𝛿,Ξ©)Ω𝑒2𝑑π‘₯,(4.2) for every π‘’βˆˆπ»1(Ξ©) (see [21]).

Proposition 4.1. Let Ξ›π‘š<Λ𝑁1min,1π‘š(0)ξ‚Άπ‘š(∞).(4.3) Then πœ™1∈𝐿2βˆ—(1+𝛿)(𝐡(0,π‘Ÿ)) for some 𝛿>0 and π‘Ÿ>0.

Proof. Let Φ∈𝐢1(ℝ𝑁) be such that Ξ¦(π‘₯)=1 on 𝐡(0,π‘Ÿ), Ξ¦(π‘₯)=0onβ„π‘βˆ’π΅(0,2π‘Ÿ), 0≀Φ(π‘₯)≀1 on ℝ𝑁 and |βˆ‡Ξ¦(π‘₯)|≀2/π‘Ÿ. For simplicity, we set πœ†=Ξ›π‘š, 𝑒=πœ™1. We define 𝑣=Ξ¦2𝑒min(𝑒,𝐿)π‘βˆ’2=Ξ¦2π‘’π‘’πΏπ‘βˆ’2, where 𝐿>0 and 𝑝>2. Testing (4.1) with 𝑣, we get ξ€œβ„π‘ξ‚€||||βˆ‡π‘’2π‘’πΏπ‘βˆ’2Ξ¦2+(π‘βˆ’2)βˆ‡π‘’βˆ‡π‘’πΏπ‘’πΏπ‘βˆ’2Ξ¦2+2βˆ‡π‘’βˆ‡Ξ¦π‘’π‘’πΏπ‘βˆ’2Ξ¦ξ‚ξ€œπ‘‘π‘₯=πœ†β„π‘π‘š(π‘₯)|π‘₯|2𝑒2π‘’πΏπ‘βˆ’2Ξ¦2𝑑π‘₯.(4.4) Applying the Young inequality to the third term on the left side, we get (ξ€œ1βˆ’πœ‚)ℝ𝑁||||βˆ‡π‘’2π‘’πΏπ‘βˆ’2Ξ¦2ξ€œπ‘‘π‘₯+(π‘βˆ’2)β„π‘βˆ‡π‘’βˆ‡π‘’πΏπ‘’πΏπ‘βˆ’2Ξ¦2ξ€œπ‘‘π‘₯β‰€πœ†β„π‘π‘š(π‘₯)|π‘₯|2𝑒2π‘’πΏπ‘βˆ’2Ξ¦2ξ€œπ‘‘π‘₯+𝐢(πœ‚)ℝ𝑁𝑒2π‘’πΏπ‘βˆ’2||||βˆ‡Ξ¦2𝑑π‘₯,(4.5) where πœ‚>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: (ξ€œ1βˆ’πœ‚)ℝ𝑁||||βˆ‡π‘’2π‘’πΏπ‘βˆ’2Ξ¦2ξ€œπ‘‘π‘₯+(1βˆ’πœ‚)(π‘βˆ’2)β„π‘βˆ‡π‘’βˆ‡π‘’πΏπ‘’πΏπ‘βˆ’2Ξ¦2ξ€œπ‘‘π‘₯β‰€πœ†β„π‘π‘š(π‘₯)|π‘₯|2𝑒2π‘’πΏπ‘βˆ’2Ξ¦2ξ€œπ‘‘π‘₯+𝐢(πœ‚)ℝ𝑁𝑒2π‘’πΏπ‘βˆ’2||||βˆ‡Ξ¦2𝑑π‘₯.(4.6) Multiplying this inequality by (𝑝+2)/4 and noting that (𝑝+2)/4>1, we get ξ‚Έξ€œ(1βˆ’πœ‚)ℝ𝑁||||βˆ‡π‘’2π‘’πΏπ‘βˆ’2Ξ¦2𝑝𝑑π‘₯+2βˆ’44ξ€œβ„π‘βˆ‡π‘’βˆ‡π‘’πΏπ‘’πΏπ‘βˆ’2Ξ¦2≀𝑑π‘₯πœ†(𝑝+2)4ξ€œβ„π‘π‘š(π‘₯)|π‘₯|2𝑒2π‘’πΏπ‘βˆ’2Ξ¦2+𝑑π‘₯𝐢(πœ‚)(𝑝+2)4ξ€œβ„π‘π‘’2π‘’πΏπ‘βˆ’2||||βˆ‡Ξ¦2𝑑π‘₯.(4.7) We now observe that ξ€œβ„π‘|||βˆ‡ξ‚€π‘’π‘’πΏ(𝑝/2)βˆ’1|||2Ξ¦2ξ€œπ‘‘π‘₯=ℝ𝑁||||βˆ‡π‘’2π‘’πΏπ‘βˆ’2Ξ¦2𝑝𝑑π‘₯+2βˆ’44ξ€œβ„π‘||βˆ‡π‘’πΏ||2π‘’πΏπ‘βˆ’2Ξ¦2𝑑π‘₯.(4.8) Hence, (4.7) takes the form ξ€œ(1βˆ’πœ‚)ℝ𝑁|||βˆ‡ξ‚€π‘’π‘’πΏ(𝑝/2)βˆ’1|||2Ξ¦2𝑑π‘₯β‰€πœ†(𝑝+2)4ξ€œβ„π‘π‘š(π‘₯)|π‘₯|2𝑒2π‘’πΏπ‘βˆ’2Ξ¦2+𝑑π‘₯𝐢(πœ‚)(𝑝+2)4ξ€œβ„π‘π‘’2π‘’πΏπ‘βˆ’2||||βˆ‡Ξ¦2𝑑π‘₯.(4.9) Since πœ†π‘š(0)/Λ𝑁<1, we can choose πœ–1>0 so that (πœ†/Λ𝑁)(π‘š(0)+πœ–1)<1. By the continuity of π‘š, there exists 0<π‘Ÿ1<π‘Ÿ such that π‘š(π‘₯)β‰€π‘š(0)+πœ–1 for π‘₯∈𝐡(0,π‘Ÿ1). This is now used to estimate the first integral on the right side of (4.9): πœ†(𝑝+2)4ξ€œβ„π‘π‘š(π‘₯)|π‘₯|2𝑒2π‘’πΏπ‘βˆ’2Ξ¦2πœ†π‘‘π‘₯≀(𝑝+2)4ξ€œπ΅(0,π‘Ÿ1)π‘š(0)+πœ–1|π‘₯|2𝑒2π‘’πΏπ‘βˆ’2+𝑑π‘₯πœ†(𝑝+2)β€–π‘šβ€–βˆž4π‘Ÿ21ξ€œπ΅(0,2π‘Ÿ)𝑒2π‘’πΏπ‘βˆ’2𝑑π‘₯.(4.10) Applying the Hardy inequality (4.2), we get πœ†(𝑝+2)4ξ€œπ΅(0,π‘Ÿ)π‘š(π‘₯)|π‘₯|2𝑒2π‘’πΏπ‘βˆ’2𝑑π‘₯β‰€πœ†(𝑝+2)4ξ€·π‘š(0)+πœ–1ξ€Έξ‚΅1Ξ›π‘ξ‚Άξ€œ+πœ–π΅(0,π‘Ÿ1)|||βˆ‡ξ‚€π‘’π‘’πΏ(𝑝/2)βˆ’1|||2+𝑑π‘₯πœ†(𝑝+2)