A Liouville-type theorem for the p-Laplacian with potential term
Yehuda Pinchover
a,∗, Achilles Tertikas
b, Kyril Tintarev
caDepartment of Mathematics, Technion – Israel Institute of Technology, Haifa 32000, Israel bDepartment of Mathematics, University of Crete, 714 09 Heraklion, Greece cDepartment of Mathematics, Uppsala University, SE-751 06 Uppsala, Sweden
Received 27 July 2006; accepted 8 December 2006 Available online 13 June 2007
Abstract
In this paper we prove a sufficient condition, in terms of the behavior of a ground state of a singularp-Laplacian problem with a potential term, such that a nonzero subsolution of another such problem is also a ground state. Unlike in the linear case (p=2), this condition involves comparison of both the functions and of their gradients.
©2007 Elsevier Masson SAS. All rights reserved.
MSC:primary 35J10; secondary 35B05
Keywords:p-Laplacian; Ground state; Liouville theorem; Positive solution
1. Introduction
Positivity properties of quasilinear elliptic equations, in particular those with thep-Laplacian term in the principal part, have been extensively studied over the recent decades (see for example [2,3,6,10] and the references therein).
Fixp∈(1,∞), and a domainΩ⊆Rd. In this paper we use positivity properties of such equations to prove a general Liouville comparison principle for equations of the form
−p(u)+V|u|p−2u=0 inΩ,
wherep(u):= ∇ ·(|∇u|p−2∇u)is thep-Laplacian, andV ∈L∞loc(Ω;R)is a given potential. Throughout this paper we assume that
Q(u):=
Ω
|∇u|p+V|u|p
dx0 (1.1)
for allu∈C0∞(Ω).
* Corresponding author.
E-mail addresses:pincho@techunix.technion.ac.il (Y. Pinchover), tertikas@math.uoc.gr (A. Tertikas), kyril.tintarev@math.uu.se (K. Tintarev).
0294-1449/$ – see front matter ©2007 Elsevier Masson SAS. All rights reserved.
doi:10.1016/j.anihpc.2006.12.004
Definition 1.1.We say that a functionv∈Wloc1,p(Ω)is a (weak) solution of the equation 1
pQ(v):= −p(v)+V|v|p−2v=0 inΩ, (1.2)
if for everyϕ∈C0∞(Ω)
Ω
|∇v|p−2∇v· ∇ϕ+V|v|p−2vϕ
dx=0. (1.3)
We say that a real functionv∈Cloc1 (Ω)is asupersolution(resp.subsolution) of Eq. (1.2) if for every nonnegative ϕ∈C∞0 (Ω)
Ω
|∇v|p−2∇v· ∇ϕ+V|v|p−2vϕ
dx0 (resp.0). (1.4)
Remark 1.2.It is well known that any weak solution of (1.2) admits Hölder continuous first derivatives, and that any nonnegative solution of (1.2) satisfies the Harnack inequality [12,13,15].
Definition 1.3.We say that the functionalQhas aweighted spectral gap inΩif there is a positive continuous function W inΩsuch that
Q(u)
Ω
W|u|pdx ∀u∈C0∞(Ω). (1.5)
Definition 1.4.LetQbe a nonnegative functional onC0∞(Ω). We say that a sequence{uk} ⊂C0∞(Ω)of nonnegative functions is anull sequenceof the functionalQinΩ, if there exists an open setBΩ(i.e.,Bis compact inΩ) such that
B|uk|pdx=1, and
klim→∞Q(uk)= lim
k→∞
Ω
|∇uk|p+V|uk|p
dx=0. (1.6)
We say that a positive functionv∈Cloc1 (Ω)is aground stateof the functionalQinΩ ifv is anLploc(Ω)limit of a null sequence ofQ.
Remark 1.5.The requirement that{uk} ⊂C0∞(Ω), can clearly be weakened by assuming only that{uk} ⊂W01,p(Ω).
Also the requirement that
B|uk|pdx=1 can be replaced by
B|uk|pdx1, wherefkgk means that there exists a positive constantCsuch thatC−1gkfkCgkfor allk∈N.
The following theorem was proved in [10].
Theorem 1.6.LetΩ⊆Rdbe a domain,V ∈L∞loc(Ω), andp∈(1,∞). Suppose that the functionalQis nonnegative onC0∞(Ω). Then
(a) Qhas either a weighted spectral gap or a ground state.
(b) IfQadmits a ground statev, thenv >0andvsatisfies(1.2).
(c) The functionalQadmits a ground state if and only if (1.2)admits a unique positive supersolution.
Example 1.7. Consider the functionalQ(u):=
Rd|∇u|pdx. It follows from [7, Theorem 2] that ifdp, thenQ admits a ground stateϕ=constant inRd. On the other hand, ifd > p, then
u(x):=
1+ |x|p/(p−1)(p−d)/p
, v(x)=constant
are two positive supersolutions of the equation−p(u)=0 inRd. Therefore, Theorem 1.6(c) implies that ifd > p, thenQhas a weighted spectral gap inRd. See also Example 3.2.
In a recent paper [9], Theorem 1.6 was used in order to prove, forp=2, the following Liouville-type statement.
Theorem 1.8.([9])LetΩ be a domain inRd,d1. Consider two strictly elliptic Schrödinger operators defined on Ωof the form
Pj:= −∇ ·(Aj∇)+Vj, j=0,1, (1.7)
whereVj∈Lploc(Ω;R)for somep > d/2, andAj:Ω→Rd2 are measurable symmetric matrices such that for any KΩthere existsμK>1such that
μ−K1IdAj(x)μKId ∀x∈K. (1.8)
(HereId is thed-dimensional identity matrix, and the matrix inequalityAB means thatB−Ais a nonnegative matrix onRd.)
Assume that the following assumptions hold true.
(i) The operatorP1admits a ground stateϕinΩ.
(ii) P00onC0∞(Ω), and there exists a real functionψ∈Hloc1 (Ω)such thatψ+=0, andP0ψ0inΩ, where u+(x):=max{0, u(x)}.
(iii) The following matrix inequality holds
(ψ+)2(x)A0(x)Cϕ2(x)A1(x) a.e. inΩ, (1.9)
whereC >0is a positive constant.
Then the operatorP0admits a ground state inΩ, andψis the corresponding ground state. In particular,ψis(up to a multiplicative constant)the unique positive supersolution of the equationP0u=0inΩ.
The purpose of this paper is to find an analog of Theorem 1.8 whenp=2. The main statement is as follows.
Theorem 1.9.LetΩbe a domain inRd,d1, and letp∈(1,∞). Forj=0,1, letVj∈L∞loc(Ω), and let Qj(u):=
Ω
∇u(x)p+Vj(x)u(x)p
dx u∈C0∞(Ω).
Assume that the following assumptions hold true.
(i) The functionalQ1admits a ground stateϕinΩ.
(ii) Q00onC∞0 (Ω), and the equationQ0(u)=0inΩadmits a subsolutionψ∈Wloc1,p(Ω)satisfyingψ+=0.
(iii) The following inequality holds almost everywhere inΩ
ψ+Cϕ, (1.10)
whereC >0is a positive constant.
(iv) The following inequality holds almost everywhere inΩ∩ {ψ >0}
|∇ψ|p−2C|∇ϕ|p−2, (1.11)
whereC >0is a positive constant.
Then the functionalQ0admits a ground state inΩ, andψis the ground state. In particular,ψis(up to a multiplicative constant)the unique positive supersolution of the equationQ0(u)=0inΩ.
Remark 1.10.Condition (1.11) is redundant forp=2. Forp=2 it is equivalent to the assumption that the following inequality holds inΩ:
|∇ψ+|C|∇ϕ| ifp >2,
|∇ψ+|C|∇ϕ| ifp <2, (1.12)
whereC >0 is a positive constant.
Remark 1.11. This theorem holds if, in addition to (1.10), one assumes instead of |∇ψ|p−2C|∇ϕ|p−2 in Ω∩ {ψ >0}(see (1.11)), that the following inequality holds true almost everywhere inΩ∩ {ψ >0}
ψ2|∇ψ|p−2Cϕ2|∇ϕ|p−2, (1.13)
where C >0 is a positive constant. This can be easily observed by repeating the proof of Theorem 1.9 with the equivalent energy functional represented in the form (2.14) instead of (2.13).
Remark 1.12.Suppose that 1< p <2, and assume that the ground stateϕ >0 of the functionalQ1is such thatw=1 is a ground state of the functional
E1ϕ(w)=
Ω
ϕp|∇w|pdx, (1.14)
that is, there is a sequence{wk} ⊂C0∞(Ω)of nonnegative functions satisfyingE1ϕ(wk)→0, and
B|wk|p=1 for a fixedBΩ (this implies thatwk→1 inLploc(Ω)). In this case, the conclusion of Theorem 1.9 holds if there is a nonnegative subsolution ψ+ ofQ0(u)=0 satisfying (1.10) alone, without any assumption on the gradients (like (1.11) or (1.13)). This statement follows from the proof of Theorem 1.9 together with the trivial inequality
Ω
v2|∇w|2
w|∇v| +v|∇w|p−2
dx
Ω
vp|∇w|pdx
which actually holds pointwise. We use this observation in Example 3.2.
Remark 1.13. By Picone identity, a nonnegative functionalQcan be represented as the integral of a nonnegative LagrangianL. Although the expression forLcontains an indefinite term (see (2.2)), it admits a two-sided estimate by a simplified Lagrangian with nonnegative terms (see Lemma 2.2). We call the functional associated with this simplified Lagrangian thesimplified energy. It plays a crucial role in the proof of Theorem 1.9.
Remark 1.14.Condition (1.11) is essential whenp >2, and presumably also whenp <2. Whenp >2,Ω=Rdand V is radially symmetric, Proposition 4.2 shows that the simplified energy functional is not equivalent to either of its two terms that lead to conditions (1.10) and (1.11), respectively (see also Remark 4.1).
The outline of the paper is as follows. In Section 2 we study the representation ofQas a functional with a positive Lagrangian, and derive the equivalent simplified energy. Theorem 1.9 is proved in Section 3, and Section 4 is devoted to the irreducibility of the simplified energy to either of its terms. In Section 5, we study a connection between the ground states of the functionalQand of its linearization.
2. Picone identity
Letv >0,v∈Cloc1 (Ω), andu0,u∈C0∞(Ω). Denote R(u, v):= |∇u|p− ∇
up vp−1
· |∇v|p−2∇v, (2.1)
and
L(u, v):= |∇u|p+(p−1)up
vp|∇v|p−pup−1
vp−1∇u· |∇v|p−2∇v. (2.2)
Then the following(generalized) Picone identityholds [5,2,3]
R(u, v)=L(u, v). (2.3)
WriteL(u, v)=L1(u, v)+L2(u, v), where L1(u, v):= |∇u|p+(p−1)up
vp|∇v|p−pup−1
vp−1|∇u||∇v|p−1, (2.4)
and
L2(u, v):=pup−1
vp−1|∇v|p−2
|∇u||∇v| − ∇u· ∇v
0. (2.5)
From the obvious inequalitytp+(p−1)−pt0, we also have thatL1(u, v)0. Therefore,L(u, v)0 inΩ. Let v∈Cloc1 (Ω)be a positive solution (resp. subsolution) of (1.2). Using (2.3) and (1.3) (resp. (1.4)), we infer that for everyu∈C0∞(Ω),u0,
Q(u)=
Ω
L(u, v)dx, resp.Q(u)
Ω
L(u, v)dx. (2.6)
Let noww:=u/v, wherevis a positive solution of (1.2) andu∈C0∞(Ω),u0. Then 2.6 implies Q(vw)=
Ω
|v∇w+w∇v|p−wp|∇v|p−pwp−1v|∇v|p−2∇v· ∇w
dx. (2.7)
Similarly, ifvis a nonnegative subsolution of (1.2), then Q(vw)
Ω
|v∇w+w∇v|p−wp|∇v|p−pwp−1v|∇v|p−2∇v· ∇w
dx. (2.8)
A need to study the linearized operator arises at a certain step in this paper. This linearized operator is a Schrödinger operator of the form
P u:=
−∇ ·(A∇)+V
u inΩ. (2.9)
We assume thatV ∈L∞loc(Ω;R), and A:Ω →Rd2 is a measurable (symmetric) matrix valued function satisfying (1.8). We consider the quadratic form
a[u] :=
Ω
A∇u· ∇u+V|u|2
dx (2.10)
onC0∞(Ω)associated with the operatorP. We have the following version of Picone identity (see [10]).
Lemma 2.1.Letψbe a (real valued) solution of the equationP ψ=0inΩ. Then for anyv∈C0∞(Ω)we have a[ψ v] =
Ω
ψ2A∇v· ∇vdx. (2.11)
Moreover, if ψ∈Hloc1 (Ω) is a nonnegative subsolution of the equation P ψ=0 inΩ, then for any nonnegative v∈C0∞(Ω)we have
a[ψ v]
Ω
ψ2A∇v· ∇vdx. (2.12)
So, in the linear case, the quadratic form induces a convenient weighted (Dirichlet-type) norm v2:=
Ω
ψ2A∇v· ∇vdx onC0∞(Ω).
Recall that in the quasilinear case (p=2), the LagrangianLin Picone’s identity and (2.7) contain indefinite terms.
Therefore, it is more convenient to replace identity (2.7) by two-sided inequalities with a simpler expression which we call thesimplified energy.
Lemma 2.2.Letv∈Cloc1 (Ω)be a positive solution of (1.2)and letw∈C01(Ω)be a nonnegative function. Then Q(vw)
Ω
v2|∇w|2
w|∇v| +v|∇w|p−2
dx. (2.13)
Moreover, for allp=2 Q(vw)C
Ω
|∇w|2
w|∇v|vp−22 +v
p
p−2|∇w|p−2
dx (2.14)
In particular, forp >2we have Q(vw)
Ω
vp|∇w|p+v2|∇v|p−2wp−2|∇w|2
dx. (2.15)
Ifvis only a nonnegative subsolution of (1.2), then Q(vw)C
Ω∩{v>0}
v2|∇w|2
w|∇v| +v|∇w|p−2
dx. (2.16)
Ifp=2, then Q(vw)C
Ω∩{v>0}
|∇w|2
w|∇v|vp−22 +vpp−2|∇w|p−2
dx. (2.17)
Moreover, forp >2we have Q(vw)C
Ω
vp|∇w|p+v2|∇v|p−2wp−2|∇w|2
dx. (2.18)
Proof. Let 1< p <∞. We need the following elementary algebraic vector inequality (cf. [4,14])
|a+b|p− |a|p−p|a|p−2a·b |b|2
|a| + |b|p−2
(2.19) for alla, b∈Rd.
Indeed, lett= |b|/|a|andθ=(a·b)/(|a||b|). Note that for−1θ1
tlim→∞
|t2+2θ t+1|p/2−1−p θ t
t2(1+t )p−2 =1 uniformly, (2.20)
and
tlim→0+
|t2+2θ t+1|p/2−1−p θ t t2(1+t )p−2 =p
2
1+(p−2)θ2
> Cp>0. (2.21)
Finally, we claim that fort >0 and−1θ1 we have
f (t, θ ):=t2+2θ t+1p/2−1−p θ t >0. (2.22) Indeed, sets:=(t2+2θ t+1)1/20, then
f (t, θ )=
sp+(p−1)−ps +p
t2+2θ t+11/2
−1−θ t .
Clearly, fors0 we have,g(s):= [sp+(p−1)−ps]0, andg(s)=0 if and only ifs=1, which holds if and only ift= −2θ.
On the other hand, let h(t, θ ):=p
t2+2θ t+11/2
−1−θ t .
Thenh(t, θ )0, andh(t, θ )=0 if and only ifθ= ±1. Note that ifθ= −1 andt= −2θ, then we havef (2,−1)= 2p >0. Thus,f (t, θ ) >0 for allt >0 and−1θ1.
Therefore, for 1< p <∞, relations (2.20)–(2.22) imply t2+2θ t+1p/2−1−p θ tt2(1+t )p−2. Thus, (2.19) holds true for alla, b∈Rd.
Set nowa:=w|∇v|,b:=v|∇w|. Then we obtain (2.13) and (2.16) by applying (2.19) to (2.7) and (2.8), respec- tively. 2
The following Allegretto–Piepenbrink-type theorem was proved in [10].
Theorem 2.3.([10, Theorem 2.3])Let Q be a functional of the form(1.1). Then the following assertions are equivalent (i) The functionalQis nonnegative onC0∞(Ω).
(ii) Eq.(1.2)admits a global positive solution.
(iii) Eq.(1.2)admits a global positive supersolution.
The next lemma is well known forp=2 (see for example [1, Lemma 2.9]).
Lemma 2.4.Letv∈Cloc1 (Ω)be a subsolution of Eq.(1.2). Thenv+is also a subsolution of(1.2).
Proof. Fixϕ∈C0∞(Ω),ϕ0. As in [1, Lemma 2.9], define forε >0 vε:=
v2+ε21/2
, and ϕε:=vε+v 2vε
ϕ.
Thenvε→ |v|,∇vε→ ∇|v|, andϕε→(sgnv+)ϕasε→0. An elementary computation shows that
∇vε· ∇ϕ∇v· ∇ vϕ
vε
,
and therefore,
∇
vε+v 2
· ∇ϕ∇v· ∇ϕε. Since
Ω
|∇v|p−2∇v· ∇ϕε+V|v|p−2vϕε
dx0, (2.23)
it follows that
Ω
|∇v|p−2∇
vε+v 2
· ∇ϕ+V|v|p−2vϕε
dx0. (2.24)
Lettingε→0 we obtain
Ω
|∇v+|p−2∇v+· ∇ϕ+V|v+|p−2v+ϕ
dx0. 2 (2.25)
3. Proof of the main result
Proof of Theorem 1.9. By Lemma 2.4, we may assume thatψ0.
Let{uk}be a null sequence forQ1, that isQ1(uk)→0 and, for some nonempty open setBΩ,
Bupkdx=1.
Without loss of generality, we may assume thatB⊂suppψ. Letwk:=uk/ϕ. From (2.13) it follows that with some C >0
Ω
ϕ2|∇wk|2
wk|∇ϕ| +ϕ|∇wk|p−2
dxCQ1(uk)→0.
Fixα, β∈R+, then the functionf:R2+→R+defined by f (s, t ):=α2t2
βs1/(p−2)+αtp−2
is nondecreasing monotone function in each variable separately. Hence, assumptions (1.10) and (1.11) imply that
Ω
ψ2|∇wk|2
wk|∇ψ| +ψ|∇wk|p−2
dx→0.
Together with (2.13) this implies that Q0(ψ wk)→0. On the other hand, sincewk→1 inLploc(Ω), it follows that ψ wk →ψ inLploc(Ω). Consequently,
Bϕpwpkdx=1 implies that
Bψpwkpdx1. In light of Remark 1.5, we conclude thatψis a ground state ofQ0. 2
Example 3.1.Assume that 1dp2,p >1,Ω=Rd, and consider the functionalQ1(u):=
Rd|∇u|pdx. By Example 1.7, the functionalQ1admits a ground stateϕ=constant inRd.
LetQ0be a functional of the form (1.1) satisfyingQ00 onC0∞(Rd). Letψ∈Wloc1,p(Rd),ψ+=0 be a subso- lution of the equationQ0(u)=0 inRd, such thatψ+∈L∞(Rd). It follows from Theorem 1.9 thatψ is the ground state of Q0 inRd. In particular,ψis (up to a multiplicative constant) the unique positive supersolution and unique bounded solution of the equationQ0(u)=0 inRd. Note that there is no assumption on the behavior of the potential V0at infinity. This result generalizes some striking Liouville theorems for Schrödinger operators onRd that hold for d=1,2 andp=2 (see [9, Theorems 1.4–1.6]).
Example 3.2.Letd >1,d=p, andΩ:=Rd\ {0}be the punctured space. Letc∗p,d:= |(p−d)/p|pbe the Hardy constant, and consider the functional
Q(u):=
Ω
|∇u|p−c∗p,d|u|p
|x|p
dx, u∈C0∞(Ω). (3.1)
By Hardy’s inequality,Qis nonnegative onC0∞(Ω). The proof of Theorem 1.3 in [11] shows thatQadmits a null sequence. It can be easily checked that the function v(r):= |r|(p−d)/p is a positive solution of the corresponding radial equation:
−|v|p−2
(p−1)v+d−1 r v
−c∗p,d|v|p−2v
rp =0, r∈(0,∞).
Therefore,ϕ(x):= |x|(p−d)/pis the ground state of the equation
−p(u)−c∗p,d|u|p−2u
|x|p =0 inΩ. (3.2)
Note thatϕ /∈Wloc1,p(Rd)forp=d. In particular,ϕis not a positive supersolution of the equationp(u)=0 inRd. LetQ0be a functional of the form (1.1) satisfyingQ00 onC0∞(Ω). Letψ∈Wloc1,p(Ω), 1< p <∞,p=d, ψ+=0, be a subsolution of the equationQ0(u)=0 inΩ satisfying
ψ+(x)C|x|(p−d)/p, x∈Ω. (3.3)
Whenp >2, we require in addition that the following inequality is satisfied
ψ+(x)2∇ψ+(x)p−2C|x|2−d, x∈Ω. (3.4)
It follows from Theorem 1.9, Remark 1.11 and Remark 1.12 thatψis the ground state ofQ0inΩ. The reason why (3.4) is stated only forp >2 hinges on the fact that forp2,
C−1Q0(ϕw)E1ϕ(w)=
Ω
|x|p−d|∇w|pdx, (3.5)
and for allp >1 the functionalE1ϕ has a ground state1. The null sequence convergent to this ground state is given by [11], relation (2.2) withR→ ∞. Therefore, Remark 1.12 applies.
Next, we present a family of functionalsQ0for which the conditions of Example 3.2 are satisfied.
Example 3.3.Letd2, 1< p < d,α0, andΩ:=Rd\ {0}. Let Wα(x):= −
d−p p
pα dp/(d−p)+ |x|p−1p (α+ |x|pp−1)p
.
Note that ifα=0 this is the Hardy potential as in the Example 3.2. IfQ0is the functional (1.1) with the potential V :=Wα, then
ψα(x):=
α+ |x|pp−1−(d−p)(pp2−1)
is a solution ofQ0(u)=0 inΩ, and thereforeQ00 onC0∞(Ω). Moreover, one can use the calculations of Exam- ple 3.2 to show thatψα is a ground state ofQ0. Indeed, we note first thatψ=ψαsatisfies (3.3). Ifd−d1< p < d, then ψαsatisfies also (3.4) and therefore, it is a ground state in this case. In the remaining casepdd−12, Example 3.2 concludes thatψα is a ground state from the property of the functional (3.5).
4. The simplified energy
In this section we give examples showing that none of the terms in the simplified energy (2.15) forp >2 is dominated by the other, so that (2.15) cannot be further simplified. In particular, neither condition (1.10) nor condition (1.11) in Theorem 1.9 can be omitted.
Letp >2, and fixv >0,v∈Cloc1 (Ω). Forw∈C0∞(Ω)we denote Ev1(w):=
Ω
vp|∇w|pdx, (4.1)
and
Ev2(w):=
Ω
v2|∇v|p−2wp−2|∇w|2dx. (4.2)
Remark 4.1. Suppose that V =0 in Ω, and assume that Qhas a weighted spectral gap in Ω. In this case, the constant functionv=1is a positive solution of (1.2) which is not a ground state inΩ. Clearly,E2v=0 onC∞0 (Ω).
Therefore, the inequalityQ(vu)CE2v(u)foru∈C0∞(Ω)is generally false, or in other words, the first term in the simplified energy (2.15) is necessary. The above assumptions are satisfied ifQis thep-Laplacian operator, and either int(Rd\Ω)= ∅, orΩ=Rdandp < d(see Example 1.7).
In the following proposition we restrict our consideration to the caseΩ=Rdand a radial positive solutionv.
Proposition 4.2.LetΩ=Rdandp >2.
• There exists a positive continuous radial functionϕonRdsuch that the simplified energyEϕ1(w)+Eϕ2(w)has a weighted spectral gap, but there exist a sequence{uk} ⊂C0∞(Rd)withuk0, and an open setBRdsuch that E1ϕ(uk)→0and
B|uk|pdx=1.
• Moreover, there exists a positive radial continuous functionψ onRd such that the simplified energyE1ψ(w)+ E2ψ(w)has a weighted spectral gap, but there exist a sequence{vk} ⊂C∞0 (Rd)withvk 0, and an open set BRdsuch thatE2ψ(vk)→0and
B|vk|pdx=1.
Proof. Step 1. LetQbe a functional of the form (1.1) onRdwith a radial potentialV. By the proof of Theorem 2.3 (see [10, Theorem 2.3]), it follows that aradialfunctionalQis nonnegative onC0∞(Rd)if and only if the equation Q(u)=0 admits a positiveradial solution in Rd. On the other hand, from the standard rearrangement argument it is evident that Qhas a weighted spectral gap if and only if it has a radial weighted spectral gap with a radial, decreasing and fast decaying weightW. Therefore,Qhas a weighted spectral gap if and only if there exists a positive continuous radial potentialW such that the Euler–Lagrange equation for the functionalQ(u)−
RdW|u|pdx has a
positiveradialsolution. But such a solution is a (radial) supersolution of the equationQ(u)=0 inRd which is not a solution. Therefore, it is sufficient to consider the restrictions ofQandQ to radial functions. (So, in fact, we are dealing with a one-dimensional problem.)
Step 2. We establish for a radial functionϕ a criterion for the existence of null sequences forE1ϕ andEϕ2 using a change of variable. First, for a positive continuous functionϕon[0,∞)defineρ1(r)by
ρ1(r):=
r
1
ϕp(s)sd−1−1/(p−1)
ds. (4.3)
Assume further thatpd, thenρ1is well-defined on(0,∞)andρ1(0)= −∞. Since dr
dρ1
=
ϕprd−11/(p−1)
,
it follows that for a radially symmetricw∈C0∞(Rd)we have
E1ϕ(w)=
Rd
ϕp|∇w|pdx=Cd M1
−∞
w(ρ1)pdρ1, (4.4)
where
M1=M1ϕ:=
∞
1
ϕprd−1−1/(p−1)
dr. (4.5)
Recall that from Example 1.7 it follows that forp >1 thep-Laplacian on(a, b)admits a ground state if and only if (a, b)=R. Therefore,E1ϕhas a null sequence if and only ifM1ϕ= ∞andpd(cf. [8, Theorem 3.1]).
Consider now the functionalE2ϕ. The substitutionu:=wp/2implies that E2ϕ(w)=
Rd
ϕ2|∇ϕ|p−2wp−2|∇w|2dx=(2/p)2
Rd
ϕ2|∇ϕ|p−2|∇u|2dx. (4.6)
Let
ρ2(r):=
r
1
ϕ(s)−2ϕ(s)2−ps1−dds, (4.7)
and assume further thatd2, thenρ2is well-defined function on(0,∞)andρ2(0)= −∞.
Using spherical coordinates for radialw, and then the substitutionρ2, we obtain
Rd
ϕ2|∇ϕ|p−2wp−2|∇w|2dx=Cd M2
−∞
|u(ρ2)|2dρ2,
where
M2=M2ϕ= ∞
1
ϕ−2|ϕ|2−pr1−ddr. (4.8)
Therefore,E2ϕhas a null sequence (forp >2 andd2) if and only ifM2ϕ= ∞(cf. [8, Theorem 3.1]).
Therefore, in order to prove the proposition it is sufficient to find two positive radial functionsϕandψsatisfying M1ϕ= ∞andM2ϕ<∞, whileM1ψ<∞andM2ψ= ∞.
Step 3. Let us simplify now (4.5) and (4.8) in order to investigate whenMjϕare finite or infinite for a specificϕand j=1,2. Without loss of generality, we assume that the integration in (4.5) and (4.8) is fromr0to∞, wherer01.
We set firstϕ(r):=r1−d/pη(r), where 2< p < d. Then (4.5) becomes, up to a constant multiple, M1=
∞
r0
ϕprd−1−1/(p−1)
dr= ∞
r0
η−p/(p−1)r−1dr. (4.9)
Set now η(r):=
t(p−1)/(p−2)(logt )γ(p−2)/p
, wheret:=logr, andγ >0.
Then we have M1=
∞ r0
η−p/(p−1)r−1dr= ∞ t0
t(p−1)/(p−2)(logt )γ(2−p)/(p−1)
dt= ∞ t0
(logt )γ (2−p)/(p−1)
t dt.
On the other hand, M2=
∞ r0
ϕ−2|ϕ|2−pr1−ddr= ∞ r0
p−d p +rη
η
2−pη−pr−1dr. (4.10)
Denote M˜2:=
∞ r0
η−pr−1dr= ∞ t0
t1−p(logt )γ (2−p)dt. (4.11)
Sincer|η|/η1, it follows from (4.10) that there existC >0 such that
C−1M˜2M2CM˜2. (4.12)
Consequently, for 0< γ(p−1)/(p−2)and 2< p < d, we haveM1ϕ= ∞andM2ϕ<∞.
On the other hand, fixβ ∈R,β=0, and let ψ:(1,∞)→ [1,∞), be a smooth monotone function such that ψ (r)rβ, and such thatψsatisfies for anyn=1,2, . . . ,
ψ(r)= e−r, r∈ [2n+1/4,2n+3/4],
|β|rβ−1, r∈ [2n+1,2n+2]. Therefore, ifβ > (p−d)/p, thenM1ψ<∞.
Consider nowM2ψ, and recall that 2< pd. Consequently, there existε >0 andC >0 such that the integrand of M2ψsatisfies
ψ−2|ψ|2−pr1−dCeεr
on a set of infinite measure. Hence,M2ψ= ∞. 2
Remark 4.3.The proof above takes into account that (4.5), (4.8) both converge or both diverge whenϕ(r)is of the formrα(logr)β, and the differentiation occurs only with respect toγforϕ(r)=r1−d/p(logr)(p−1)/p(log logr)γ. 5. Application: ground state of the linearized functional
We consider the linearized problem associated with the functional Q0. Let ϕ be a positive solution of the equationQ(u)=0 inΩ, and let
a[u] :=
Ω
|∇ϕ|p−2|∇u|2+V (x)ϕp−2u2
dx. (5.1)
Proposition 5.1.Letϕbe a positive solution of the equationQ(u)=0inΩ satisfying∇ϕ=0.
1. Ifp >2andϕis a ground state ofQ, thenϕis a ground state ofa.
2. Ifp <2andϕis a ground state ofa, thenϕis a ground state ofQ.
Proof. Consider first the casep >2. Assume thatϕis a ground state ofQ. Let{uk}be a null sequence of nonnegative functions, and letwk:=uk/ϕ. By (2.15),
Ω
ϕ2|∇ϕ|p−2wpk−2|∇wk|2dx→0. (5.2)
Setvk:=wkp/2. Then
Ω
ϕ2|∇ϕ|p−2|∇vk|2dx→0 (5.3)
which by (2.11) yields a[ϕvk] →0.
Taking into account Remark 1.5, we conclude that{ϕvk}is a null sequence fora.
The casep <2 is similar. If{zk}is a null sequence of nonnegative functions for the forma, then (5.3) is satisfied withvk:=zk/ϕ. This implies (5.2) withwk=v2/pk , which by (2.13) yieldsQ(uk)→0 withuk=ϕwk. Therefore, {uk}is a null sequence forQand the proposition is proved. 2
Acknowledgements
Parts of this research were done while K.T. was visiting the Technion, Y.P. was visiting University of Crete and Uppsala University and A.T. was visiting Uppsala University. Y.P. and A.T. acknowledge partial support by the RTN European network Fronts–Singularities, HPRN-CT-2002-00274. The visit to Technion was supported by the Fund for the Promotion of Research at the Technion and the visits to Uppsala were supported by the Swedish Research Council.
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