# Ground State for the Schrödinger Operator with the Weighted Hardy Potential

Authors: J. Chabrowski; K. Tintarev
Journal: International Journal of Differential Equations
 (2011)

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## Abstract

We establish the existence of ground states onℝN 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.


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## Body

## 1. Introduction

In this paper, we investigate the existence of ground states of the Schrödinger operator associated with the quadratic form(1.1)QV(u)=∫RN(|∇u|2-ΛVV(x)u2)dx,u∈C∘∞(RN),N≥3,
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 V≥0, that the potential term ∫ℝNV(x)u2dx is continuous in D1,2(ℝN), where D1,2(ℝN) is the Sobolev space obtained as the completion of C∘∞(ℝN) with respect to the norm
(1.2)‖u‖D1,22=∫RN|∇u|2dx.
We are mainly interested in the case of the Hardy-type potential V(x)=m(x)/|x|2 with m∈L∞(ℝN). Assuming that V is positive on a set of positive measure, the constant ΛV is given by the variational problem
(1.3)ΛV=inf⁡u∈D1,2(RN),∫RNVu2dx=1∫RN|∇u|2dx,
and the continuity of ∫ℝNV(x)u2dx implies that ΛV>0. If problem (1.3) has a minimizer u, then it satisfies
(1.4)-Δu-ΛVV(x)u=0.
A solution of (1.4) is understood in the weak sense
(1.5)∫RN∇u∇ϕdx=ΛV∫RNV(x)uϕdx,
for every ϕ∈D1,2(ℝN).Since|u| is also a minimizer for ΛV, we may assume that u≥0 a.e. on ℝN. In particular, when V(x)=m(x)/|x|2 with m∈L∞(ℝN), then u>0 on ℝN by the Harnack inequality [1]. If the potential term is weakly continuous in D1,2(ℝN), for example, when V(x)=m(x)/|x|2 with m∈L∞(ℝN) and lim|x|→∞m(x)=limx→0m(x)=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(x)=1/|x|2 with the corresponding optimal constant ΛV=ΛN=((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.
LetΩ⊂ℝN be an open set, and let QV be as in (1.1). A sequence of nonnegative functions vk∈C∘∞(Ω) is said to be a null sequence for the functional QV if QV(vk)→0, as k→∞, and there exists a nonnegative function ψ∈C∘∞(Ω)  such that ∫Ωψvkdx=1 for each k.Let us recall that the capacity of a compact setE relative to an open set Ω⊂ℝN, with E⊂Ω, is given by
(1.6)cap(E,Ω)=inf⁡{∫Ω|∇u|2dx;u∈C∘∞(Ω),withu(x)≥1on⁡E}.
In the case Ω=ℝN, we use notation cap(E) (see [5]).We can now formulate the following “ground state alternative” (see [3, 4]).Theorem 1.2.
LetV be a measurable function bounded on every compact subset of Ω=ℝN-Z, where Z is a closed set of capacity zero, and assume that QV(u)≥0 for all u∈C∘∞(Ω). Then, if QV admits a null sequence vk, then the sequence vk converges weakly in Hloc1(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 solutionv of (1.4) is called a generalized ground state of the functional QV, if the functional admits a null sequence weakly convergent to v.IfV(x)=1/|x|2, then the functional QV has a ground state v(x)=|x|(2-N)/2 of infinite D1,2 norm, while (1.3) has no minimizer in D1,2(ℝN).It is important to note that the functionalQV with the optimal constant ΛV does not necessarily have a ground state. We quote the following statement from [4].Theorem 1.4.
LetV be a measurable function bounded on every compact subset of Ω=ℝN-Z, where Z is a closed set of capacity zero, and assume that QV(u)≥0 for all u∈C∘∞(Ω). Then either QV admits a null sequence, or there exists a function W, positive and continuous on Ω, such that
(1.7)QV(u)≥∫RNW(x)u2dx.For example, letm be a continuous function on ℝN-{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, ΛV=((N-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 W positive on ℝN-{0}.Obviously, ground states of finiteD1,2 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].Letf:ℝN→ℝ be a measurable function. We define the distribution function αf and a nonincreasing rearrangement f* of f in the following way
(1.8)αf(s)=|{x∈RN;|f(x)|>s}|,f*(t)=inf⁡{s>0;αf(s)≤t}.
We now set
(1.9)‖f‖(p,q)*={(∫0∞[t1/pf*(t)]qdtt)1/q,if1≤p,q<∞,sup⁡t>0⁡t1/pf*(t),if1≤p≤∞,q=∞.
The Lorentz space Lp,q(ℝN) is defined by
(1.10)Lp,q(RN)={f∈Lloc⁡1(RN);‖f‖(p,q)*<∞}.
The functional ∥f∥(p,q)* is only a quasinorm. To obtain a norm we replace f by f**(t)=(1/t)∫0tf*(s)dx in the definition of ∥f∥(p,q)*, that is, the norm is given by
(1.11)‖f‖(p,q)={(∫0∞[t1/pf**(t)]qdtt)1/q,if1≤p,q<∞,sup⁡t>0t1/pf**(t),if1≤p≤∞,q=∞.Lp,q(ℝN)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<∞LN/2,q(ℝN). This was extended in [14] to a larger class of weights ℱN/2 obtained as the completion of C∘∞(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 ifV∈LN/2,∞(ℝN), then the functional ∫ℝNV(x)u2dx is continuous on D1,2(ℝN), but not necessarily weakly continuous.The paper is organized as follows. In Section2, 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 QV∘ with V∘∈LN/2,∞(ℝN). We show that if QV∘ has ground state, then this property is stable under small perturbations of V∘. This is not true if QV∘ 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 QV∘ in the case where V∘(x)=m(x)/|x|2 with m∈L∞(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 “→” and weak convergence by “⇀”. The norms in the Lebesgue space Lp(Ω), 1≤p≤∞, are denoted by ∥u∥p.


## 2. Existence of Minimizers

We consider the Hardy-type potentialV(x)=m(x)/|x|2 with m∈L∞(ℝN). 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. 						(2.1)m+(x)=lim⁡j∈N,j→∞m(γ+jx),(2.2)m-(x)=lim⁡j∈N,j→∞m(γ--jx).
Both functions m± satisfy m±(γ±x)=m±(x). We now define the following infima:
(2.3)Λm=inf⁡u∈D1,2(RN)-{0}∫RN|∇u|2dx∫RN(m(x)/|x|2)u2dx,
(we use the notation Λm instead of ΛV) and
(2.4)Λ±=inf⁡u∈D1,2(RN)-{0}∫RN|∇u|2dx∫RN(m±(x)/|x|2)u2dx.Lemma 2.1.
The following holds true(2.5)Λm≤min⁡(Λ+,Λ-).Proof.
Letu∈D1,2(ℝN)-{0}. Testing Λm with γ+-(N-2)/2u(γ+-jx) gives
(2.6)Λm≤∫RN|∇u|2dx∫RN(m(γ+jx)/|x|2)u2dx.
Letting j→∞ and using the Lebesgue dominated convergence theorem, we obtain
(2.7)Λm≤∫RN|∇u|2dx∫RN(m+(x)/|x|2)u2dx.
The inequality Λm≤Λ+ follows. The proof of the inequality Λm≤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∈ℝN;|x|≥R} for every R>0 and that the convergence in (2.2) is uniform on sets {x∈ℝN;|x|≤ρ} for every ρ>0. If Λm<min(Λ+,Λ+), then problem (2.3) has a minimizer.Proof.
Let{uk}⊂D1,2(ℝN) be a minimizing sequence for Λm, that is,
(2.8)∫RN|∇uk|2dx⟶Λm,∫RNm(x)|X|2uk2dx=1.
We can assume, up to a subsequence, that uk⇀w in D1,2(ℝN), L2(ℝN,dx/|x|2), and uk→w in Lloc2(ℝN) for some w∈D1,2(ℝN). Let vk=uk-w. We then have
(2.9)1=∫RNm(x)|x|2uk2dx=∫RNm(x)|x|2w2dx+∫RNm(x)|x|2vk2dx+o(1),(2.10)Λm=∫RN|∇uk|2dx+o(1)=∫RN|∇w|2dx+∫RN|∇vk|2dx+o(1).
We define a radial function χ+j∈C1(ℝN) 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 ok→∞(j)(1) to denote a quantity such that for each j∈ℕ, ok→∞(j)(1)→0 as k→∞. Thus,
(2.11)∫RNm(x)|x|2vk2dx=∫RNm(x)|x|2(vkχ-j)2dx+∫RNm(x)|x|2(vkχ+j)2dx+ok→∞(j)(1)=∫RNm(γ--jx)|x|2(vk-)2dx+∫RNm(γ+jx)|x|2(vk+)2dx+ok→∞(j)(1),
where
(2.12)vk-(x)=γ-(-(N-2)/2)jvk(γ--jx)χ-(γ--jx),vk+(x)=γ+((N-2)/2)jvk(γ+jx)χ+(γ+jx).
We now estimate the integrals involving vk- and vk+. We have
(2.13)|∫RNm(γ--jx)|x|2(vk-)2dx-∫RNm-(x)|x|2(vk-)2dx|≤|∫|x|<γ--jm(γ--jx)-m-(x)|x|2(vk-)2dx|+|∫γ--2j<|x|<γ+2jm(γ--jx)-m-(x)|x|2(vk-)2dx|=J1+J2.
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
(2.14)J2≤2‖m‖∞∫γ--2j<|x|<γ+2jvk2|x|2dx.
It is clear that J2 is a quantity of type ok→∞(j)(1). Therefore, we have
(2.15)|∫RNm(γ--jx)|x|2(vk-)2dx-∫RNm-(x)|x|2(vk-)2dx|≤ϵ+ok→∞(j)(1).
In a similar way, we obtain
(2.16)|∫RNm(γ+jx)|x|2(vk+)2dx-∫RNm+(x)|x|2(vk+)2dx|≤ϵ+ok→∞(j)(1)
for j sufficiently large. We now fix j∈ℕ so that (2.15) and (2.16) hold. Consequently, we have
(2.17)1≤∫RNm(x)|x|2w2dx+∫RNm-(x)|x|2(vk-)2dx+∫RNm+(x)|x|2(vk+)2dx+2ϵ+ok→∞(j)(1).
We now estimate ∫ℝN|∇vk|2dx in the following way
(2.18)∫RN|∇vk|2dx=∫RN|∇(vkχ-j+vkχ+j)|2dx=∫RN|∇(vkχ-j)|2dx+∫RN|∇(vkχ+j)|2dx+2∫RN∇(vkχ-j)∇(vkχ+j)dx=∫RN|∇vk-|2dx+∫RN|∇vk+|2dx+2∫RN|∇vk|2χ-jχ+jdx+2∫RNvk∇vk∇χ-jχ+jdx+2∫RNvk∇vkχ-j∇χ+jdx+2∫RNvk2∇χ-j∇χ+jdx≥∫RN|∇vk-|2dx+∫RN|∇vk+|2dx+2∫RNvk∇vk∇χ-jχ+jdx+2∫RNvk∇vkχ-j∇χ+jdx+2∫RNvk2∇χ-j∇χ+jdx.
Since vk→0 in Lloc2(ℝN), we obtain the following estimate
(2.19)∫RN|∇vk|2dx≥∫RN|∇vk-|2dx+∫RN|∇vk+|2dx+ok→∞(j)(1).
This, combined with (2.9), gives the following estimate
(2.20)Λm≥∫RN|∇w|2dx+∫RN|∇vk-|2dx+∫RN|∇vk+|2dx+ok→∞(j)(1)≥Λm∫RNm(x)|x|2w2dx+Λ-∫RNm-(x)|x|2(vk-)2dx+Λ+∫RNm+(x)|x|2(vk+)2dx+ok→∞(j)(1).
Let Λ*=min(Λ-,Λ+). We deduce from (2.17) and (2.20) that
(2.21)(Λ*-Λm)(∫RNm-(x)|x|2(vk-)2dx+m+(x)|x|2(vk+)2dx)≤2ϵΛm+ok→∞(j)(1).
Letting k→∞, we obtain
(2.22)limsup⁡k→∞(∫RNm-(x)|x|2(vk-)2dx+m+(x)|x|2(vk+)2dx)≤2ϵΛm(Λ*-Λm).
It then follows from (2.17) that
(2.23)1≤∫RNm(x)|x|2w2dx+2ϵΛm(Λ*-Λm).
Since ϵ>0 is arbitrary, we get ∫ℝN(m(x)/|x|2)w2dx=1, and the result follows.In what follows, we denotem(∞)=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.
Letm∈L∞(ℝN), and assume that m is continuous at 0. Further, suppose that m(∞)>0 and m(0)>0. If Λm<ΛNmin(1/m(∞),1/m(0)), then there exists a minimizer for Λm.Remark 2.4.
Λm has a minimizer also in the following cases, corresponding formally to Λ+ or Λ- taking the value +∞.(i)
Letm(0)=0 and m(∞)>0. If Λm<ΛN/m(∞), then a minimizer for Λ1(m) exists.(ii)
Letm(0)>0 and m(∞)=0. If Λm<ΛN/m(0), then a minimizer for Λ1(m) exists.(iii)
Ifm(0)=m(∞)=0, m(x)≥0 and ≢ 0 on ℝN, then Λm has a minimizer.We point out that Theorem2.3 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 functionsm satisfying conditions of Theorems 2.2 and 2.3. In general, functions satisfying this condition have large local maxima.Example 2.5.
Let(2.24)mA(x)={m1(x),for0<|x|<1,Am2(x),for1≤|x|≤2,m3(x),for2<|x|,
where A>0 is a constant to be chosen later and m1:B(0,1)̅-{0}→[0,∞), m2:(1≤|x|≤2)→[0,∞), and m3:ℝN∖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
(2.25)m3(x)=a+|x1||x2|+⋯+|xN-1||xN|b+|x|2,
for |x|≥R>2, where a>0, b>0 and R constants. A function m1(x) for small δ>0 is given by
(2.26)m1(x)=|x1|+⋯+|xN||x|,
for 0<|x|≤δ<1. We have
(2.27)lim⁡j→∞mA(γ+jx)=lim⁡j→∞γ+-2ja+|x1||x2|+⋯+|xN-1||xN|γ+-2jb+|x|2=|x1||x2|+⋯+|xN-1||xN||x|2=m+(x),lim⁡j→∞mA(γ--jx)=|x1|+⋯+|xN||x|=m-(x).
Both limits are uniform. Since m- and m+ are bounded, Λ- and Λ+ are positive and finite. We have
(2.28)Λm=inf⁡D1,2(RN)-{0}∫RN|∇u|2dx∫RN(mA(x)/|x|2)u2dx≤1Ainf⁡D1,2(RN)-{0}∫RN|∇u|2dx∫1≤|x|≤2(m2(x)/|x|2)u2dx<min(Λ-,Λ+),
for A large. By Theorem 2.2, Λm with m=mA has a minimizer.Example 2.6.
Consider a sequence of functions of the formmk(x)=BMk(x)+Af(x),k=1,2,…, where A>0, B>0 are constants and Mk and f are continuous functions satisfying the following conditions:  (a)
Mk(0)=1, Mk(x)>0 on ℝN, Mk(∞)=0, for k=1,2,…,(b)
Mk(x)=k on 1<|x|<2 for k=1,2,…,(c)
f(x)≥0 on ℝN, f(0)=0 and f(∞)=1. Then mk(0)=B and mk(∞)=A for k=1,2,…. We show that for k sufficiently large mk satisfies the conditions of Theorem 2.3. Let u(x)=exp(-|x|) (one can take any other function from D1,2(ℝN) which is ≢0 on (1<|x|<2)). Thus
(2.29)Λmk≤∫RN|∇(exp⁡(-|x|))|2dx∫RN((BMk(x)+Af(x))/|x|2)exp⁡(-2|x|)dx≤∫RN|∇(exp⁡(-|x|))|2dxB∫RN(Mk(x)/|x|2)exp⁡(-2|x|)dx⟶0,
as k→∞. So we can find k∘≥1 so that
(2.30)Λmk<ΛNmin⁡(1A,1B)fork≥k∘.In Proposition2.7, we described a class of weight functions m satisfying conditions of Theorem 2.3.Proposition 2.7.
Letm∈C(ℝN). Suppose that m(x)≥0, m(0)>0, and m(∞)>0. Assume that there exists a ball B(xM,r) such that m(x)≥m(xM)>0 for x∈B(xM,r) and 0∉B(xM,r)¯. If
(2.31)m(0)m(xM),m(∞)m(xM)<r2(N-2)22(r+|xM|)2(N+1)(N+2).
Then Λm<ΛNmin(1/m(0),1/m(∞)). (Hence, there exists a minimizer for Λm.)Proof.
Letu∈H∘1(B(xM,r))-{0}. Then
(2.32)∫B(xM,r)m(x)|x|2u2dx≥m(xM)∫B(xM)u2|x|2dx≥m(xM)(r+|xM|)2∫B(xM,r)u2dx.
Hence,
(2.33)∫B(xM,r)|∇u|2dx∫B(xM,r)(m(x)/|x|2)dx≤(r+|xM|)2∫B(xM,r)|∇u|2dxm(xM)∫B(xM,r)u2dx.
Since H∘1(B(xM,r))-{0}⊂{u∈D1,2(ℝN);∫ℝN(m(x)/|x|2)u2dx>0}, we deduce from the above inequality that
(2.34)Λm≤(r+|xM|)2m(xM)λ1D(B(xM,r)),
where λ1D(B(xM,r)) denotes the first eigenvalue for “-Δ” in B(xM,r) with the Dirichlet boundary conditions. We now estimate λ1D=λ1D(B(xM,r)). We test λ1D with v(x)=r-|x-xM| for x∈B(xM,r). We have
(2.35)∫B(xM,r)v2dx=∫B(0,r)(r-|x|)2dx=ωN∫0r(r-s)2sN-1ds=2ωNrN+2N(N+1)(N+2),∫B(xM,r)|∇v|2dx=ωNrNN.
Hence
(2.36)λ1D≤∫B(xM,r)|∇v|2dx∫B(xM,r)v2dx=(N+1)(N+2)2r2.
Combining this with (2.34), we derive
(2.37)Λm≤(N+1)(N+2)(r+|xM|)22r2m(xM).
Therefore Λm<ΛNmin(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.Ifm(x) is a continuous bounded and nonnegative function such that m(x)≤m(0) on ℝN and m(0)>0 (or m(x)≤m(∞) on ℝN, m(∞)>0), then Λm does not have a minimizer. Indeed, suppose that m(x)≤m(0) on ℝN and that Λm has a minimizer u. Then, by the Hardy inequality, we obtain
(2.38)ΛNm(0)≥∫RN|∇u|2dx∫RN(m(x)/|x|2)u2dx≥∫RN|∇u|2dxm(0)∫RN(u2/|x|2)dx≥ΛNm(0).
So u is a minimizer for ΛN, which is impossible.We now construct a ground state with infiniteD1,2 norm.Theorem 2.8.
Letγ>1, and assume that the function m∈L∞(ℝN) satisfies
(2.39)m(γx)=m(x)forx∈RN.
Then the form QV with V(x)=m(x)/|x|2 and ΛV=Λ∘ (see (2.41) below) admits a ground state v satisfying
(2.40)v(γx)=γ(2-N)/2v(x)forx∈RN.
The function v is uniquely defined by its values on Aγ={x∈ℝN;1<|x|<γ}, and, moreover, the function v|Aγ is a minimizer for the problem
(2.41)Λ∘=inf⁡{∫Aγ|∇v|2dx∫Aγ(m(x)/|x|2)u2dx;u∈H1(Aγ)-{0},u(γx)=γ(2-N)/2u(x)for|x|=1}.Proof.
The problem (2.41) is a compact variational problem that has a minimizer v which satisfies
(2.42)-Δv=Λ∘m(x)|x|2v,x∈Aγ,
with the Neumann boundary conditions. Since the test functions satisfy u(γx)=γ(2-N)/2u(x) for |x|=1, one has
(2.43)∂v∂r(γx)=γ-N/2∂v∂r(x)for|x|=1.
Note that |v| is also a minimizer, so we may assume that v is nonnegative. We now extend the function v from Aγ to ℝN-{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(ℝN-{0}) and satisfies
(2.44)-Δv=Λ∘m(x)|x|2v,
in a weak sense. From this and the Harnack inequality on bounded subsets of ℝN-{0} it follows that v is positive on ℝN-{0} and subsequently there exists a constant C>0 such that
(2.45)C-1|x|(2-N)/2≤v(x)≤C|x|(2-N)/2.
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|=γj,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 wk(x)=|x|1/k for |x|≤1 and wk(x)=|x|-1/k for |x|≥1,
(2.46)Q(vwk)=∫RNv2|∇wk|2dx≤C∫RN|x|2-N|∇wk|2dx≤Ck2∫01r-1+(2/k)dr+Ck2∫1∞r-1-(2/k)dr≤Ck⟶0,
as k→∞. Since vwk→v uniformly on compact sets, this implies that v is a ground state for Q. By (2.45) and the Sobolev inequality, v∉D1,2(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 finiteD1,2 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.
LetV∘∈LN/2,∞(ℝN) be positive on a set of positive measure, and let
(3.1)Λ∘=inf⁡u∈D1,2(RN),∫RNV∘u2dx=1∫RN|∇u|2dx.
Assume that V1∈LN/2,∞(ℝN) is positive on a set of positive measure and that the functional ∫ℝN(V1(x)-V∘(x))u2dx is weakly continuous in D1,2(ℝN), and let
(3.2)Λ1=inf⁡u∈D1,2(RN),∫RNV1u2dx=1∫RN|∇u|2dx.
If Λ1<Λ∘, then there exists a minimizer for Λ1.Proof.
Let{uk}⊂D1,2(ℝN) be a minimizing sequence for (3.2), that is, ∫ℝNV1(x)uk2dx=1 and ∫ℝN|∇uk|2dx→Λ1. We may assume that, up to a subsequence, uk⇀w in D1,2(ℝN) and L2(ℝN,V1(x)dx). Let vk=uk-w. Then,
(3.3)1=∫RNV1(x)uk2dx=∫RNV1(x)vk2dx+∫RNV1(x)w2dx+o(1)=∫RNV1(x)w2dx+∫RN(V1(x)-V∘(x))vk2dx+∫RNV∘(x)vk2dx+o(1)=∫RNV∘(x)vk2dx+∫RNV1(x)w2dx+o(1).
Let t=∫ℝNV1(x)w2dx. Then ∫ℝNV∘(x)vk2dx→1-t. Assuming that t<1 we get
(3.4)Λ1=∫RN|∇vk|2dx+∫RN|∇w|2dx+o(1)≥Λ∘(1-t)+Λ1t+o(1).
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 uk→w in D1,2(ℝN).
Proposition3.1 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 D1,2(ℝN) (see [12]) and the potential g admits a principal eigenfunction.Remark 3.2.
(i) IfV1>V∘, then Λ1≤Λ∘, but not necessarily Λ1<Λ∘.
(ii) If, in Proposition3.1, assumption Λ1<Λ∘ is replaced by Λ∘<Λ1, then Λ∘ is attained.Example 3.3.
LetM be a continuous function ℝN such that M≥0, ≢0 on ℝN and M(0)=M(∞)=0. Define mA,B(x)=BM(x)+A, where A>0 and B>0 are constants. Let V1(x)=mA,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 D1,2(ℝN). 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 inequalityΛ1<Λ∘.Theorem 3.4.
Suppose thatV1 and V∘ satisfy assumptions of Proposition 3.1. Moreover, assume that the quadratic form QV∘ has a positive ground state v, possibly with infinite D1,2 norm, and that if {vk}⊂C∘∞(ℝN) is a null sequence corresponding to Λ∘, then
(3.5)limsup⁡k→∞∫RN(V1(x)-V∘(x))vk2dx>0.
Then Λ1<Λ∘ and Λ1 has a minimizer.Proof.
It suffices to show that the inequality(3.6)∫RN|∇u|2dx-Λ∘∫RNV1(x)u2dx≥0
fails for some u∈D1,2(ℝN). We have
(3.7)∫RN|∇vk|2dx-Λ∘∫RNV1(x)vk2dx=QV∘(vk)-Λ∘∫RN(V1(x)-V∘(x))vk2dx=o(1)-Λ∘∫RN(V1(x)-V∘(x))vk2dx<0,
for sufficiently large k, which completes the proof of the theorem.Note that the conditions of Theorem3.4 are satisfied if, in particular, V1≥V∘ on ℝN, with the strict inequality on a set of positive measure. Indeed, the sequence {vk} converges weakly in Hloc1(ℝN) to v>0, and the condition limsupk→∞∫ℝN(V1(x)-V∘(x))vk2dx>0 follows from the Fatou lemma.The situation becomes different ifQV∘ 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 thatV∘ satisfies the conditions of Proposition 3.1 and that (1.7) holds. (This occurs under conditions of Theorem 1.4 if QV∘ has no ground state.) Let W be as in (1.7). Then, for every t∈(0,1/Λ∘), the functional QV∘+tW has no ground state and ΛV∘+tW=ΛV∘. Furthermore, if the functional ∫ℝNW(x)u2dx is weakly continuous in D1,2(ℝN), the same conclusion holds for -∞<t<0.Proof.
First, we observe that the constantsΛ∘ 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
(3.8)∫RN|∇u|2dx-Λ∘∫RN(V∘(x)+tW(x))u2dx≥0,
for t∈(0,1/Λ∘) which implies Λ1≥Λ∘. Let vk∈C∘∞(ℝN-Z) satisfy QV1(vk)→0. Then
(3.9)(1-Λ∘t)∫RNWvk2dx≤QV1(vk)⟶0,
which implies that, up to subsequence, vk→0 a.e. If vk were a null sequence, it would converge in Hloc1(ℝN) and it would have a limit zero. Therefore, QV1 admits no null sequence and consequently no ground state. Assume now that the functional ∫ℝNW(x)u2dx is weakly continuous in D1,2(ℝN). Let {wk}⊂D1,2(ℝN) be a minimizing sequence for Λ∘. If {wk} has a subsequence weakly convergent in D1,2(ℝN) to some w≠0, then it is easy to see that |w| would be a minimizer for Λ∘ and thus a ground state for QΛ∘. Therefore, wk⇀0. By the weak continuity of ∫ℝNW(x)u2dx, we get
(3.10)∫RNV1(x)wk2dx=∫RNV∘(x)wk2dx+o(1)=1+o(1),
and thus
(3.11)Λ1≤∫RN|∇wk|2dx=Λ∘+o(1).
This yields Λ1≤Λ∘. Then,
(3.12)∫RN|∇u|2dx-Λ1∫RNV1(x)u2dx≥Λ1Λ∘(∫RN|∇u|2dx-Λ∘∫RNV1(x)u2dx)=Λ1Λ∘(QV∘(u)-tΛ∘∫RNW(x)u2dx)≥Λ1∫RN(Λ∘-1-t)W(x)u2dx.
Since t<0, this implies that QV1 has no ground state.Theorem3.5 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 D1,2(ℝN) norm.Theorem 3.6.
Assume thatV∘ satisfies conditions of Proposition 3.1 and that W∈L2,∞(ℝN) is such that the functional ∫ℝNW(x)u2dx is weakly continuous in D1,2(ℝN). Then, for every λ∈(0,Λ∘) there exists σ∈ℝ such that QV∘+σW has a ground state of finite D1,2(ℝN) norm corresponding to the energy constant (3.2).Proof.
Assume without loss of generality thatW is positive on a set of positive measure. Let 0<λ<Λ∘ and consider
(3.13)σ=inf⁡u∈D1,2(RN),∫RNW(x)u2dx=1λ-1(∫RN|∇u|2dx-λ∫RNV∘(x)u2dx).
Since (∫ℝN|∇u|2dx-λ∫ℝNV∘(x)u2dx)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 QV∘+σW corresponding to the optimal constant λ.If we assume additionally thatW is positive on a set of positive measure, then it is easy to show that σ is a continuous decreasing function of λ with limλ→0σ(λ)=+∞ and σ∘=limλ→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-typeV(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 ℝN. Obviously the ground state ϕ1 satisfies
(4.1)Δu=Λmm(x)|x|2uinRN
in a weak sense.We need the following extension of the Hardy inequality: letΩ⊂ℝN be a bounded domain and 0∈Ω¯, then for every δ>0, there exists a constant A(δ,Ω)>0 such that
(4.2)∫Ωu2|x|2dx≤(1ΛN+δ)∫Ω|∇u|2dx+A(δ,Ω)∫Ωu2dx,
for every u∈H1(Ω) (see [21]).Proposition 4.1.
Let(4.3)Λm<ΛNmin⁡(1m(0),1m(∞)).
Then ϕ1∈L2*(1+δ)(B(0,r)) for some δ>0 and r>0.Proof.
LetΦ∈C1(ℝN) be such that Φ(x)=1 on B(0,r), Φ(x)=0onℝN-B(0,2r), 0≤Φ(x)≤1 on ℝN and |∇Φ(x)|≤2/r. For simplicity, we set λ=Λm, u=ϕ1. We define v=Φ2umin(u,L)p-2=Φ2uuLp-2, where L>0 and p>2. Testing (4.1) with v, we get
(4.4)∫RN(|∇u|2uLp-2Φ2+(p-2)∇u∇uLuLp-2Φ2+2∇u∇ΦuuLp-2Φ)dx=λ∫RNm(x)|x|2u2uLp-2Φ2dx.
Applying the Young inequality to the third term on the left side, we get
(4.5)(1-η)∫RN|∇u|2uLp-2Φ2dx+(p-2)∫RN∇u∇uLuLp-2Φ2dx≤λ∫RNm(x)|x|2u2uLp-2Φ2dx+C(η)∫RNu2uLp-2|∇Φ|2dx,
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:
(4.6)(1-η)∫RN|∇u|2uLp-2Φ2dx+(1-η)(p-2)∫RN∇u∇uLuLp-2Φ2dx≤λ∫RNm(x)|x|2u2uLp-2Φ2dx+C(η)∫RNu2uLp-2|∇Φ|2dx.
Multiplying this inequality by (p+2)/4 and noting that (p+2)/4>1, we get
(4.7)(1-η)[∫RN|∇u|2uLp-2Φ2dx+p2-44∫RN∇u∇uLuLp-2Φ2dx]≤λ(p+2)4∫RNm(x)|x|2u2uLp-2Φ2dx+C(η)(p+2)4∫RNu2uLp-2|∇Φ|2dx.
We now observe that
(4.8)∫RN|∇(uuL(p/2)-1)|2Φ2dx=∫RN|∇u|2uLp-2Φ2dx+p2-44∫RN|∇uL|2uLp-2Φ2dx.
Hence, (4.7) takes the form
(4.9)(1-η)∫RN|∇(uuL(p/2)-1)|2Φ2dx≤λ(p+2)4∫RNm(x)|x|2u2uLp-2Φ2dx+C(η)(p+2)4∫RNu2uLp-2|∇Φ|2dx.
Since λm(0)/ΛN<1, we can choose ϵ1>0 so that (λ/ΛN)(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):
(4.10)λ(p+2)4∫RNm(x)|x|2u2uLp-2Φ2dx≤λ(p+2)4∫B(0,r1)m(0)+ϵ1|x|2u2uLp-2dx+λ(p+2)‖m‖∞4r12∫B(0,2r)u2uLp-2dx.
Applying the Hardy inequality (4.2), we get
(4.11)λ(p+2)4∫B(0,r)m(x)|x|2u2uLp-2dx≤λ(p+2)4(m(0)+ϵ1)(1ΛN+ϵ)∫B(0,r1)|∇(uuL(p/2)-1)|2dx+(λ(p+2)4A(B(0,r1),ϵ)+λ(p+2)‖m‖∞4r12)∫B(0,2r)(uuLp/2-1)2dx,
for every ϵ>0. Inserting this estimate into (4.9), we obtain
(4.12)(1-η-λ(p+2)4(m(0)+ϵ1)(1ΛN+ϵ))×∫B(0,r)|∇(uuL(p/2)-1)|2dx≤C1∫B(0,2r)(uuL(p/2)-1)2dx,
where C1=(λ(p+2)/4)A(B(0,r1),ϵ)+((λ(p+2)∥m∥∞)/(4r12))+(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
(4.13)λ(1+δ4)(m(0)+ϵ1)(1ΛN+ϵ)=λΛN(1+δ4)(m(0)+ϵ1)+λϵ(1+δ4)(m(0)+ϵ1)<1.
We point out that we have used here the inequality (λ/ΛN)(m(0)+ϵ1)<1. With this choice of ϵ and δ, we now choose η>0 so small that
(4.14)C2:=1-η-λ(1+δ4)(m(0)+ϵ1)(1ΛN+ϵ)>0.
Finally, we apply the Sobolev inequality in H1(B(0,r)) and deduce
(4.15)SC2(∫B(0,r)|uuL(p/2)-1|2*dx)2/2*≤(C1+C2)∫B(0,2r)(uuL(p/2)-1)2dx,
where S denotes the best Sobolev constant of the embedding of H1(B(0,r)) into L2*(B(0,r)). Letting L→∞ we deduce that u∈L2*(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 onℝN∖B(0,R). Although this will not be used in the sequel, we add it for the sake of completeness. We denote by D1,2(ℝN∖B(0,R)) the Sobolev space defined by(4.16)D1,2(RN∖B(0,R))={u:∇u∈L2(RN∖B(0,R))andu∈L2*(RN∖B(0,R))}.Lemma 4.2.
For everyδ>0, there exists a constant A=A(δ,R)>0 such that
(4.17)∫|x|≥Ru2|x|2dx≤(1ΛN+δ)∫|x|≥R|∇u|2dx+A∫R≤|x|≤R+1u2dx,
for every u∈D1,2(ℝN∖B(0,R)).Proof.
LetΦ∈C1(ℝN) be such that Φ(x)=0 onB(0,R)̅, Φ(x)=1 on ℝN∖B(0,R+1), 0≤Φ(x)≤1 on ℝN∖B(0,R)and |∇Φ(x)|≤2/R on ℝN. Then, uΦ∈D1,2(ℝN), and, by the Hardy and Young inequalities, we have
(4.18)∫|x|≥Ru2|x|2dx=∫|x|≥R(uΦ)2|x|2dx+∫|x|≥R(1-Φ2)u2|x|2dx≤ΛN-1∫|x|≥R|∇(uΦ)|2dx+1R2∫R≤|x|≤R+1u2dx≤ΛN-1∫|x|≥R|∇u|2Φ2dx+ΛN-1∫|x|≥Ru2|∇Φ|2dx+2ΛN-1∫|x|≥RuΦ∇u∇Φdx+1R2∫R≤|x|≤R+1u2dx≤(ΛN-1+δ)∫|x|≥R|∇u|2dx+(ΛN-1+C(δ))∫|x|≥Ru2|∇Φ|2dx+1R2∫R≤|x|≤R+1u2dx,
and the result follows with A(δ,R)=(4/R2)(ΛN-1+C(δ))+1/R2.Proposition 4.3.
Suppose thatm(∞)>0 and Λm<ΛNmin(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∈L2*(1+δ)(ℝN∖B(0,R)).Proof.
We modify the argument used in the proof of Proposition4.1. Since Λm<(ΛN/m(∞)), there exist ϵ>0 and R>0 such that (Λm/ΛN)(m(∞)+ϵ)<1 and m(x)<(m(∞)+ϵ) for |x|≥R. Let Ψ∈C1(ℝN) be such that Ψ(x)=0 on B(0,R), Ψ(x)=1 on ℝN-B(0,R+1), 0≤Ψ(x)≤1 on ℝN, and |∇Ψ(x)|≤(2/R) on ℝN. Let λ=Λm, u=ϕ1, and v=uuLp-2Ψ2, where L>1, p>2, and uL=min(u,L). It is clear that v∈D1,2(ℝN). Testing (4.1) with v and applying the Young inequality, we obtain
(4.19)(1-η)∫RN|∇u|2uLp-2Ψ2dx+(p-2)∫RN∇u∇uLuLP-2Ψ2dx≤λ∫RNm(x)|x|2u2uLp-2Ψ2dx+C(η)∫RNu2uLp-2|∇Ψ|2dx.
From this, as in the proof of Proposition 4.1, we derive that
(4.20)(1-η)∫RN|∇(uuL(p/2)-1)|2Ψ2dx≤λ(p+2)4∫RNm(x)|x|2u2uLp-2Ψ2dx+C(η)(p+2)4∫RNu2uLp-2|∇Ψ|2dx.
We now estimate the first integral on the right side of (4.20). Using Lemma 4.2, we have, for every ϵ1>0,
(4.21)∫RNm(x)|x|2u2uLp-2Ψ2dx≤(m(∞)+ϵ)∫|x|≥R+1(uuL(p/2)-1)2|x|2dx+(m(∞)+ϵ)×∫R≤|x|≤R+1(uuL(p/2)-1)2|x|2dx≤(ΛN-1+ϵ1)(m(∞)+ϵ)×∫|x|≥R+1|∇(uuL(p/2)-1)|2dx+A(ϵ1,R)(m(∞)+ϵ)∫R+1≤|x|≤R+2(uuL(p/2)-1)2dx+(m(∞)+ϵ)R2×∫R≤|x|≤R+1(uuL(p/2)-1)2dx
Inserting this into (4.20), we obtain
(4.22)[1-η-λ(p+2)4(ΛN-1+ϵ1)(m(∞)+ϵ)]∫|x|≥R+1|∇(uuL(p/2)-1)|2dx≤C1(δ,ϵ1,R)∫R≤|x|≤R+2(uuL(p/2)-1)2dx,
where
(4.23)C1(δ,ϵ1,R):=λ(p+2)4(m(∞)+ϵ)A(ϵ1,R)+λ(p+2)4R2(m(∞)+ϵ)+C(η)(p+2)R2.
We now set p=2+δ. We choose δ>0 and ϵ1>0 such that
(4.24)λ(1+δ4)(ΛN-1+ϵ1)(m(∞)+ϵ)<1.
Then we choose η>0 small enough to guarantee the inequality
(4.25)C2:=1-η-λ(1+δ4)(ΛN-1+ϵ1)(m(∞)+ϵ)>0.
Having chosen ϵ1 and δ, we apply the Sobolev inequality to deduce from (4.22)
(4.26)SC2(∫|x|≥R+1|uuL(p/2)-1|2*dx)2/2*≤C1∫R≤|x|≤R+1(uuL(p/2)-1)2dx,
where S is the best Sobolev constant for the embedding of D1,2(ℝN-B(0,R+1)) into L2*(ℝN-B(0,R+1)). Letting L→∞, the result follows.Continuing  with the above notationsλ=Λm, u=ϕ1, we put u=|x|-sv, with s>0 to be chosen later. We have
(4.27)div⁡(|x|-2s∇v)=-λ|x|-2-sm(x)u+u(-s2|x|-s-2+sN|x|-s-2-2s|x|-s-2).
We now consider the above equation in a small ball B(0,r). Since
(4.28)λ=Λm<ΛNmin⁡(1m(0),1m(∞))≤ΛNm(0),
there exists r>0 (small enough) such that λmaxx∈B(0,r)m(x)<ΛN. Let s=ΛN-ΛN-λm¯r with m¯r=maxx∈B(0,r)m(x), then
(4.29)-div⁡(|x|-2s∇v)≤0inB(0,r).
Let m̲r=minx∈B(0,r)m(x), and set s=ΛN-ΛN-λm̲r. Then
(4.30)-div⁡(|x|-2s∇v)≥0inB(0,r).Proposition 4.4.
Letm(0)>0 and
(4.31)Λm<ΛNmin⁡(1m(0),1m(∞)).
Then, there exists r>0 such that
(4.32)M1|x|-(ΛN-ΛN-λm̲r)≤ϕ1(x)≤M2|x|-(ΛN-ΛN-λm¯r),
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 ℝN∖B(0,ρ), and |∇η(x)|≤2/(ρ-r) on ℝN, where 0<r<ρ. We use as a test function in (4.29) w=η2vvl2(t-1)=η2vmin(v,l)2(t-1), where l,t>1. Substituting into (4.29), we obtain
(4.33)∫RN|x|-2s(2ηvvl2(t-1)∇v∇η+η2vl2(t-1)|∇v|2+2(t-1)η2vl2(t-1)|∇vl|2)dx≤0,
where s=ΛN-ΛN-λm¯r. By the Young inequality, for every ϵ>0, there exists C(ϵ)>0 such that
(4.34)2∫RN|x|-2sηvvl2(t-1)∇η∇vdx≤ϵ∫RN|x|-2sη2vl2(t-1)|∇v|2dx+C(ϵ)∫RN|x|-2s|∇η|2v2vl2(t-1)dx.
Taking ϵ=1/2, we derive from (4.33) that
(4.35)∫RN|x|-2s(η2vl2(t-1)|∇v|2+2(t-1)η2vl2(t-1)|∇vl|2)dx≤C∫RN|x|-2s|∇η|2v2vl2(t-1)dx,
where C>0 is a constant independent of l. To proceed further we use the Caffarelli-Kohn-Nirenberg inequality [24]:
(4.36)(∫B(0,ρ)|x|-bp|w|pdx)2/p≤Ca,b∫B(0,ρ)|x|-2a|∇w|2dx,
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 Ca,b>0 is a constant depending on a and b. We choose
(4.37)a=b=ΛN-ΛN-λm¯r<N-22.
In this case we have p=2*. We then deduce from (4.35) and (4.36) with w=ηvvlt-1 that(4.38)(∫RN|x|-2*s|ηvvlt-1|2*dx)2/2*≤Ca,b∫RN|x|-2s|∇(ηvvlt-1)|2dx≤2Ca,b∫RN|x|-2s(|∇η|2v2vl2(t-1)+η2vl2(t-1)|∇v|2+(t-1)2η2vl2(t-1)|∇vl|2)dx≤Ct∫RN|x|-2*s|∇η|2v2vl2(t-1)dx.
We now observe that(4.39)∫RN|x|-2*s|η|2*v2vl2*t-2dx≤∫RN|x|-2*s|ηvvlt-1|2*dx.
Indeed, to show this, we need to check that v2vl2*t-2≤vl2*(t-1)v2* on suppη. This can be verified by considering the cases vl=l and vl=v. The above inequality allows us to rewrite (4.38) as
(4.40)(∫RN|x|-2*s|η|2*v2vl2*t-2dx)2/2*≤Ct∫RN|x|-2*s|∇η|2v2vl2(t-1)dx.
Due to the properties of the function η, the above inequality becomes
(4.41)(∫B(0,r)|x|-2*sv2vl2*t-2dx)2/2*≤Ct(ρ-r)2∫B(0,ρ)|x|-2*sv2vl2(t-1)dx.
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 tj=t*(2*/2)j, j=0,1,…. Setting t=tj in (4.41), we obtain
(4.42)(∫B(0,r)|x|-2*sv2vl2tj+1-2dx)1/2tj+1≤(Ctj(ρ-r)2)1/2tj(∫B(0,ρ)|x|-2*sv2vl2tj-2dx)1/2tj.
We put rj=ρ∘(1+ρ∘j), j=0,1,… with ρ∘ small. Substituting in the last inequality ρ=rj, r=rj+1, we obtain(4.43)(∫B(0,rj+1)|x|-2*sv2vl2tj+1-2dx)1/2tj+1≤(Ctj(ρ∘-ρ∘2)2ρ∘2j)1/2tj(∫B(0,rj)|x|-2*sv2vl2tj-2dx)1/2tj.
Iterating gives
(4.44)(∫B(0,rj+1)|x|-2*sv2vl2tj+1-2dx)1/2tj+1≤(Cρ∘-ρ∘2)∑j=0∞1/tjρ∘-∑j=0∞j/tj∏j=0∞tjj/2tj(∫B(0,r∘)|x|-2*sv2vl2t*-2dx)1/2t*.
We now notice that infinite sums and the infinite product in the above inequality are finite. Since 2*<2t*<(1+δ∘)2*, we have
(4.45)∫B(0,r∘)|x|-2*sv2vl2t*-2dx≤∫B(0,r∘)|x|(2t*-2*)s|u|2t*dx≤r∘(2t*-2*)s∫B(0,r∘)|u|2*t*dx<∞.
We now deduce from (4.44) and (4.45) that(4.46)‖vl‖L2tj+1(B(0,ρ∘))≤‖vl‖L2tj+1(B(0,rj+1))≤r∘(2*s)/2tj+1(∫B(0,rj+1)|x|-2*sv2vltj+1-2dx)1/2tj+1≤C,whereC>0 is a constant independent of l and j. Letting tj→∞, we get ∥vl∥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.



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*Source: 358087-2011-09-15.xml*