On the resonant Lane-Emden problem for the p-Laplacian

Grey Ercole Departamento de Matemática - ICEx, Universidade Federal de Minas Gerais,

Av. Antônio Carlos 6627, Caixa Postal 702, 30161-970, Belo Horizonte, MG, Brazil. E-mail: grey@mat.ufmg.br

July 1, 2013

Abstract

We study the positive solutions of the Lane-Emden problem ∆pu = λp jujq2 u in Ω, u = 0 on ∂Ω, whereΩ RN is a bounded and smooth domain, N 2, λp is the first eigenvalue of the p-Laplacian operator ∆p,p > 1, and q is close to p. We prove that any family of positive solutions of this problem converges in C1(Ω) tothe function θpep when q ! p, where ep is the positive and L∞-normalized first eigenfunction of the p-Laplacian

and θp := exp ep

pLp(Ω)

RΩ ep

pln ep

dx

. A consequence of this result is that the best constant of the immer-

sion W1,p0 (Ω) ,! Lq(Ω) is differentiable at q = p. Previous results on the asymptotic behavior (as q ! p) of

the positive solutions of the non-resonant Lane-Emden problem (i.e. with λp replaced by a positive λ 6= λp)are also generalized to the space C1(Ω) and to arbitrary families of these solutions. Moreover, if uλ,q denotes asolution of the non-resonant problem for an arbitrarily fixed λ > 0, we show how to obtain the first eigenpair ofthe p-Laplacian as the limit in C1(Ω), when q ! p, of a suitable scaling of the pair (λ, uλ,q). For computationalpurposes the advantage of this approach is that λ does not need to be close to λp. Finally, an explicit estimateinvolving L∞ and L1 norms of uλ,q is also derived using set level techniques. It is applied to any ground statefamily

vq

in order to produce an explicit upper bound for vq

∞ which is valid for q 2 [1, p + ε] where0 ε <

pN .

Keywords. asymptotic behavior, best constant, blow-up technique, first eigenpair, ground states, Lane-Emden,Picone’s inequality, p-Laplacian.

1 Introduction

Consider the Lane-Emden problem∆pu = λ jujq2 u in Ω,

u = 0 on ∂Ω,(1)

where λ > 0, Ω RN is a bounded and smooth domain, N 2, ∆pu := divjrujp2ru

is the p-Laplacian op-

erator with p > 1, and 1 q < p?, with p? denoting the Sobolev critical exponent defined by p? = Np/ (N p) ,if 1 < p < N, and p? = ∞, if p > N.

The author thanks the support of FAPEMIG and CNPq, Brazil.

1

If q = p, we have the p-Laplacian eigenvalue problem∆pu = λ jujp2 u in Ω

u = 0 on ∂Ω(2)

whose first eigenvalue λp is positive, simple, isolated and admits a first positive eigenfunction ep 2 C1,α Ωsatisfying

ep

L∞(Ω) = 1 in Ω. (We maintain this notation from now on.) Moreover, λp is also characterized bythe minimizing property

λp = min

(RΩ jrujp dxR

Ω jujp dx

: u 2 W1,p0 (Ω) n f0g

)=

RΩ

repp dxR

Ω

epp dx

. (3)

We recall that u 2 W1,p0 (Ω) is a weak solution of (1) if, and only if,Z

Ωjrujp2ru rϕdx = λ

ZΩjujq2 uϕdx for all ϕ 2 W1,p

0 (Ω) . (4)

This means that u is a critical point of the energy functional Jλ,q : W1,p0 (Ω)! R given by

Jλ,q (v) =1p

ZΩjrvjp dx λ

q

ZΩjvjq dx.

In the super-linear case p < q < p? the existence of at least one positive weak solution uλ,q of (1) withthe least energy Jλ,q among all possible nontrivial weak solutions follows from standard variational methods.Weak positive solutions satisfying this minimizing property are known as ground states. Non-uniqueness ofpositive weak solutions occurs for ring-shaped domains when q is close to p? (see [11, 12]) or when q > p andΩ is a sufficiently thin annulus (see [17]). On the other hand, when Ω is a ball, there exists only one positiveweak solution (see [1]). For the Laplacian (p = 2) and a general bounded domain, uniqueness happens if q issufficiently close to 2 (see [7, Lemma 1] and Remark 12).

In the sub-linear case 1 < q < p the existence of a positive weak solution follows from the sub- and super-solution method or from standard variational arguments concerning the global minimum of the energy func-tional Jλ,q in W1,p

0 (Ω). The uniqueness of such a weak positive solution follows from [9] where a more generalresult is proved.

In both cases a proof of existence by applying the subdifferential method can be found in [21], where onecan also find the proof of the boundedness (in the sup norm) of any positive weak solution of (1), a result whichimplies the C1,α-regularity up to the boundary by applying well-known estimates (see [10, 20, 24]).

With different goals, asymptotics of solutions of the Lane-Emden problem (1) has been studied by manyauthors since the 1990s. For example, in [12] for p < N, λ = 1 and q ! p?; or in [22] for p = N, λ = 1 andq ! ∞. Recently, in [13], the asymptotic behavior in W1,p

0 (Ω) of the positive ground state solutions vλ,q, asq ! p+, was described for all positive values of λ. In that paper vλ,q was obtained as the minimum of Jλ,q on the

positive Nehari manifold. More recently, the asymptotic behavior with q ! p in W1,p0 (Ω) was described in [3].

Some these asymptotics on the non-resonant problem (that is, 0 < λ 6= λp) had already appeared in [14].However, up to our knowledge, only in [13] and [3] the resonant problem was dealt with, but the asymp-

totic behavior of its positive solutions was not fully determined. Indeed, although the families of solutions wereknown to have a subsequence converging in W1,p

0 (Ω) to a multiple of ep, this multiple was unknown; in prin-ciple, different multiples of ep could be obtained as limits of different subsequences these families. Moreover,

2

in the super-linear case, except for ground state families, nothing was known about the asymptotic behavior (asq ! p+) of other (eventually existing) families of positive solutions.

In the present paper we first consider the resonant Lane-Emden problem∆pu = λp jujq2 u in Ω,

u = 0 on ∂Ω,(5)

and an arbitrary family

uq

q2[1,p)[(p,p?) of positive solutions of this problem (not necessarily ground states,in the super-linear case). By using Picone’s inequality and blow-up arguments we prove our main result: theconvergence uq ! θpep in C1(Ω), as q ! p, where

θp := exp

RΩ ep

pln ep

dxRΩ ep

pdx

!.

As a consequence, we obtain the differentiability at q = p of the function q 2 [1, p?) 7! λq 2 R, where λq

denotes the minimum on W1,p0 (Ω)nf0g of the Rayleigh quotient Rq defined by Rq(u) := krukp

p / kukpq . (From

now on kvkr stands for the usual Lr norm of v.) Precisely, we prove that

limq!p

λq λp

q p= λp ln(θp

ep

p).

For this we use the fact that the function vq :=

λqλp

1qp wq is a positive weak solution of the resonant Lane-

Emden problem (5) for each q 2 [1, p) [ (p, p?), where wq denotes a positive and Lq-normalized minimizer ofthe Rayleigh quotient Rq. In the super-linear case p < q < p? the function vq is a ground state and in thesub-linear case 1 < q < p this function is the only positive solution of (5).

We emphasize that our results determine the exact asymptotic behavior of positive solutions, as q ! p, ofthe Lane-Emden problem (1) for any λ > 0. In fact, any family of positive solutions

uλ,q

q2[1,p)[(p,p?) of this

problem in the non-resonant case 0 < λ 6= λp is obtained, by scaling, from a family of positive solutions of theresonant case. Thus, as we will see, our main result implies that

limq!p

uλ,q

C1 =

0, if λ < λp∞, if λ > λp

and limq!p+

uλ,q

C1 =

∞, if λ < λp0, if λ > λp

(6)

(Here kvkC1 := kvk∞ + krvk∞ is the norm of a function v in C1(Ω).)A third consequence of our main result is that, for each λ > 0, 1 s ∞ and for any sequence qn ! p one

has:lim

qn!p

λ uλ,qn

qnps

= λp and

uλ,qn uλ,qn

s

!ep ep

s

the last convergence being in the C1(Ω) space.This result might be useful for numerical computation of the first eigenvalue of the p-Laplacian (see [5])

taking into account the following aspects: a) λ does not need to be close to λp; b) the sequence qn tending to pcan be arbitrarily chosen ; c) the normalization in the computational processes can be made by using any Ls-normwith s 1.

Finally, by using level set techniques we prove an explicit estimate involving L∞ and L1 norms of the solutionsuλ,q of (1), which is valid if q 2 [1, p+ p

N ). This estimate, which has independent interest, might be useful in acomputational approach of the Lane-Emden problem or even in the analysis of nodal solutions for this problem.

3

Moreover, we apply this estimate for any ground state family

vq

1qp+ε, with 0 ε < p

N , to obtain an explicit

estimate for vq

∞ which is uniform with respect to q 2 [1, p+ ε].This paper is organized as follows. In Section 2 we prove our main result on the asymptotic behavior for

the resonant case, as q ! p. Section 3 is dedicated to the consequences of our main result. At last, in Section4, we obtain, for each λ > 0, an estimate involving L∞ and L1 norms of the solution uλ,q. We also prove amonotonicity property for λq with respect to q and combine it with the previously derived estimate in order toobtain an explicit bound for ground state families which is valid for q in the relatively large interval [1, p+ ε],where 0 ε < p

N . (We do not impose any restriction on pN .)

2 Asymptotic behavior of the resonant problem

In this section we consider the resonant Lane-Emden problem∆pu = λp jujq2 u in Ω

u = 0 on ∂Ω.(7)

Our goal is to completely determine the asymptotic behavior of the weak positive solutions of this problem, asq ! p.

The weak solutions of (7) are the critical points of the energy functional Iq : W1,p0 (Ω) ! R defined by

Iq(u) :=1p

ZΩjrujp dx

λp

q

ZΩjujq dx.

Furthermore, a family

vq

q2[1,p)[(p,p?) of positive weak solutions of (7) is obtained from minimizers of theRayleigh quotient

Rq(u) :=

RΩ jrujp dxRΩ juj

q dx p

q

in W1,p0 (Ω) n f0g.

In fact, as it is well-known, the compactness of the immersion W1,p0 (Ω) ,! Lq(Ω) for 1 q < p? implies

that Rq : W1,p0 (Ω) n f0g ! R attains a positive minimum at a positive and Lq-normalized function wq 2

W1,p0 (Ω) \ C1,α Ω : wq

q = 1 and λq := min

nRq(u) : u 2 W1,p

0 (Ω) n f0go= Rq(wq). (8)

It is straightforward to verify that wq is a weak solution of∆pu = λq jujq2 u in Ω

u = 0 on ∂Ω

and hence that

vq =

λq

λp

1qp

wq (9)

is a positive weak solution of (7) for each q 2 [1, p) [ (p, p?).

4

Since wq

q = 1 one has vq

q =

λq

λp

1qp

. (10)

In the sub-linear case 1 q < p the function vq is the only critical point of Iq. Moreover, this function

minimizes the energy functional Iq on W1,p0 (Ω) n f0g, that is

Iq(vq) = minn

Iq(v) : v 2 W1,p0 (Ω) n f0g

o. (11)

This property can also be directly proved using (8) and (10). In fact, it is straightforward to verify that if v 2W1,p

0 (Ω) n f0g thenIq(v) min

t2RIq(tv) = Iq(tvv) Iq(vq)

where tv =λpR

Ω jvjq dx

1pqR

Ω jrvjp dx 1

pq .In the super-linear case 1 < p < q < p? the energy functional is not bounded from below. Indeed, for any

v 2 W1,p0 (Ω) n f0g one can verify that

limt!∞

Iq(tv) = ∞.

However, as it is well-known, the weak positive solution vq minimizes the energy functional Iq in the Neharimanifold

Nq :=

v 2 W1,p0 (Ω) n f0g :

ZΩjrvjp dx = λp

ZΩjvjq dx

.

Therefore, since any nontrivial solution of (7) belongs to Nq (take λ = λp and φ = u in (4)), it follows thatvq 2 Nq and also that vq is a ground state.

Remark 1 Since no general uniqueness result is known for the super-linear case, the existence of multiple ground statesfor (7) is possible, at least in principle, for each fixed q 2 (p, p?). However, all of them must have the same energy and alsothe same Lq norm. Moreover, if uq is an arbitrary nontrivial weak solution of (7), then

uq

q vq

q .

In the remaining of this section we denote by vq the function defined by (9) and by uq any positive solutionof the resonant Lane-Emden problem (7). Obviously, in the sub-linear case we must have uq = vq.

Lemma 2 Let uq 2 W1,p0 (Ω) be a positive weak solution of the resonant Lane-Emden problem (7) with q 2 [1, p) [

(p, p?). Then, uq

∞ A :=jΩj1 R

Ω

epp dx if 1 < q < p

1 if 1 < p < q < p?.

Proof. If 1 q < p uniqueness implies that uq = vq. Hence, (11) and the fact that 0 < ep 1 in Ω yield

λp

1q 1

p

ZΩ

uqq dx = Iq(uq)

Iq(ep)

= λp

ZΩ

epq

qepp

p

!dx λp

1q 1

p

ZΩ

epp dx.

5

Therefore, ZΩ

epp dx

ZΩ

uqq dx

and, since jΩj1 RΩ

epp dx 1, we obtain

jΩj1Z

Ω

epp dx

jΩj1

ZΩ

epp dx

1qjΩj1

ZΩ

uqq dx

1q uq

∞ .

If 1 < p < q < p? then uq

∞ 1 becauseZΩ

uqp dx 1

λp

ZΩ

ruqp dx =

ZΩ

uqq dx

uq qp

∞

ZΩ

uqp dx.

2

In the next lemma φp 2 W1,p0 (Ω) denotes the p-torsion function of Ω, that is, the solution of

∆pu = 1 in Ω,u = 0 on ∂Ω. (12)

(Classical results imply that φp > 0 in Ω and that φp 2 C1,β(Ω) for some 0 < β < 1.)

Lemma 3 For each 1 q < p, let uq 2 W1,p0 (Ω) be the positive weak solution of the Lane-Emden problem (7). It holds uq

pq∞ λp

φp p1

∞ . (13)

Proof. Since 8>><>>:∆puq = λpuq1

q λp uq q1

∞ = ∆p

λp uq q1

∞

1p1

φp

in Ω,

uq = 0 =

λp uq q1

∞

1p1

φp on ∂Ω

it follows from the comparison principle that uq

λp uq q1

∞

1p1

φp in Ω. Hence, we obtain (13) after passingto the maximum values. 2

Remark 4 It follows from Lemmas 2 and 3 that

1

λp φp

p1∞

lim infq!p

uq qp

∞ lim supq!p

uq qp

∞ 1.

which leads to following well-known lower bound to the first eigenvalue λp in terms of the p-torsion function of Ω :

1 φp p1

∞

λp. (14)

(More properties of the p-torsion function and some of its applications are given in [6, 16].)

6

In the sequel we prove an a priori L∞ boundedness result for an arbitrary family

uq

p<q<p? of positive weak

solutions of the super-linear Lane-Emden problem (7), if q is sufficiently close to p+. Our proof was motivatedby Lemma 2.1 of [15], where a Liouville-type theorem was proved for positive weak solutions of the inequality∆pw cwp1 in RN or in a half-space. It combines a blow-up argument with the following Picone’s inequality(see [2]), which is valid for all differentiable u 0 and v > 0 :

jrujp jrvjp2rv r

up

vp1

. (15)

Lemma 5 Let

uq

p<q<p? be a family of positive weak solutions of the (super-linear) Lane-Emden problem (7). Then,

lim supq!p+

uq qp

∞ < ∞. (16)

Proof. Let us suppose, by contradiction, that uqn

qnp∞ ! ∞ for some sequence qn ! p+. Let xn denote a

maximum point of uqn , so that uqn(xn) = uqn

∞ . Define

µn :=

λp uqn

qnp∞

1p , Ωn :=

nx 2 RN : µnx+ xn 2 Ω

oand

wn(x) := uqn

1∞ uqn(µnx+ xn); x 2 Ωn.

Note that Bdn/µn Ωn where we are denoting by dn the distance from xn to the boundary ∂Ω and by Bdn/µnthe ball centered at x = 0 with radius dn/µn.

It follows that µn ! 0+, 0 < wn 1 = kwnk∞ = wn(0) in Ωn and∆pwn = wqn1

n in Ωn,wn = 0 on ∂Ωn.

(17)

By passing to a subsequence we can also suppose that xn ! x0 2 Ω and that uqn

qnp∞ is increasing. It is

well-known that Ωn tends either to RN or to a half-space if x0 2 Ω or x0 2 ∂Ω, respectively.Let BR be a ball with radius R sufficiently large satisfying 0 2 BR and

λR < 1, (18)

where λR denotes the first eigenvalue of the p-Laplacian with homogeneous Dirichlet conditions in BR.Now, let n0 be such that

BR Ωn for all n n0.

Since 0 wn 1 in BR, global Hölder regularity implies that there exist constants K > 0 and β 2 (0, 1),both depending on R (but independent of n n0), such that kwnkC1,β(BR)

K (see [20, Theorem 1]). Hence,

compactness of the immersion C1,β(BR) ,! C1(BR) implies that, up to a subsequence, wn ! w in C1(BR). Notethat w 0 in BR and w(0) = 1 (since wn(0) = 1).

Moreover, we have∆pw = wp1 in BR

in the weak sense. In fact, if φ 2 C∞0 (BR) is an arbitrary test function of BR then (17) yieldsZ

BRjrwnjp2rwn rφdx =

ZBR

wqn1n φdx.

7

Thus, after making n ! ∞ we obtainZBRjrwjp2rw rφdx =

ZBR

wp1φdx. (19)

We remark that the Strong Maximum Principle (see [25]) really implies that w > 0 in BR.Now, let eR 2 W1,p

0 (BR) \ C1(BR) be a positive first eingenfunction of the p-Laplacian for the ball BR. SinceC∞

0 (BR) is dense in W1,p0 (BR) the equality (19) is also valid for all φ 2 W1,p

0 (BR). In particular, it is valid forφ = ep

R/wp1 (see Remark 6 after this proof).It follows from Picone’s inequality that

ZBRjreRjp dx

ZBRjrwjp2rw r

ep

Rwp1

!dx. (20)

Hence, (19) yields

λR

ZBR

epRdx

ZBR

wp1 epR

wp1 dx =Z

BR

epRdx

that is, λR 1 which contradicts (18). 2

Remark 6 The quotient eR/w of C1 functions is well-defined in BR (since w > 0) and at the points of the boundary ∂BRwhere w is null. Indeed, since eR = 0 on ∂BR this fact is a consequence of the Hopf’s Boundary Lemma (see [25] again): ify 2 ∂BR is such that w(y) = 0 then any inward directional derivative of both eR and w is positive. Thus, L’Hôpital’s ruleimplies that

limx!yx2BR

eR(x)w(x)

> 0.

Lemma 7 Let

uq

q2[1,p)[(p,p?) be a family of positive solutions of the Lane-Emden problem (7) and define, for each

q 2 [1, p) [ (p, p?), the function Uq :=uq uq

∞. Then Uq converges to ep in C1(Ω) as q ! p. Moreover,

ZΩ

Upq Uq

q

q pdx !

ZΩ

eppln ep

dx as q ! p. (21)

Proof. It is easy to verify that (∆pUq = λp

uq qp

∞ Uq1q in Ω,

Uq = 0 on ∂Ω.(22)

Thus, it follows from Lemmas 2, 3 and 5 that

0 < C1 uq qp

∞ C2

for all q 2 [1, p)[ (p, p+ ε), for some ε > 0, being C1 and C2 constants that do not depend on q 2 [1, p)[ (p, p+ε).

Therefore, since the right-hand side of the equation in (22) is uniformly bounded with respect to q 2 [1, p) [(p, p+ ε) global Hölder regularity again implies that

Uq

C1,β(Ω) K, where K and 0 < β < 1 are also uniformwith respect to q 2 [1, p+ ε].

8

Hence, compactness of the immersion C1,α(Ω) ,! C1(Ω) implies that, up to a subsequence, Uq converges inC1(Ω) to a function U 0 (as q ! p) with kUk∞ = 1. We also have that λp

uq qp

∞ ! c 2 (λpC1, λpC2).Taking the limit q ! p in the weak formulation (4) with λ = λp

uq qp

∞ , we obtainZΩjrUjp2rU rϕdx = c

ZΩjUjp2 Uϕdx

for an arbitrary test function ϕ 2 W1,p0 (Ω). This proves that U is a nonnegative eigenfunction associated with

the eigenvalue c and such that kUk∞ = 1. But this fact necessarily implies that c = λp and U = ep. Thus,the uniqueness of the limits λp

uq qp

∞ ! λp and Uq ! ep show that these convergences do not depend onsubsequences. Therefore, we conclude that

uq qp

∞ ! 1 and that Uq ! ep in C1(Ω).In order to prove (21) we firstly observe thatU

pq Uq

q

q p

1jq pj max

0t1jtp tqj = 1

jq pj1p

pq

qqpjq pj = 1

p

pq

qqp

,

implying thatUp

q Uqq

q pis uniformly bounded with respect to q close to p with

lim supq!p

Upq Uq

q

q p

limq!p

1p

pq

qqp

=1

p exp(1). (23)

Now, by taking into account the convergence Uq ! ep in C1(Ω), (21) follows from Lebesgue’s dominatedconvergence theorem if we prove that

1Uqpq

q p!ln ep

as q ! p+ a.e. in Ω

andUpq

q 1q p

!ln ep

as q ! p a.e. in Ω.

So, let K Ω compact and 0 < δ < minK

ep. Then

0 < minK

ep δ ep δ Uq ep + δ in K

for all q sufficiently close to p. Hence, in K one has

ln(ep + δ) lim infq!p+

1Uqpq

q p lim sup

q!p+

1Uqpq

q p ln(ep δ), (24)

since

limq!p+

1 (ep + δ)qp

q p= ln(ep + δ)

and

limq!p+

1 (ep δ)qp

q p= ln(ep δ).

9

Therefore, making δ ! 0+ in (24) we conclude that

limq!p+

1Uqpq

q p= ln ep =

ln ep in K.

Analogously we prove that

limq!p

Upqq 1q p

=ln ep

in K.

2

From now on

θp := exp

RΩ ep

pln ep

dxRΩ ep

pdx

!.

Lemma 8 Let

uq

q2[1,p)[(p,p?) be a family of positive weak solutions of the Lane-Emden problem (7). Then,

lim supq!p

uq

∞ θp lim infq!p+

uq

∞ . (25)

Proof. Let Uq =uq uq

∞as in Lemma 7. Applying Picone’s inequality to Uq and ep one has

ZΩ

rUqp dx

ZΩ

repp2rep r

Up

q

ep1p

!dx. (26)

(Hopf’s boundary lemma again implies that Upq /ep1

p 2 W1,p0 (Ω).) Therefore, it follows from (22) that

λp uq qp

∞

ZΩ

Uqq dx λp

ZΩ

ep1p

Upq

ep1p

dx = λp

ZΩ

Upq dx

and from this we obtain uq qp

∞ 1q p

ZΩ

Uqq dx

ZΩ

Upq Uq

q

q pdx if p < q < p? (27)

and uq qp

∞ 1q p

ZΩ

Uqq dx

ZΩ

Upq Uq

q

q pdx if 1 < q < p. (28)

Case q ! p+. Let us suppose, by contradiction, that there exist L < θp and a sequence qn ! p+ such that uqn

∞ L. Then (27) and Lemma 7 yield

ZΩ

eppln ep

dx = limZ

Ω

Upqn Uqn

qn

qn pdx lim

Lqnp 1qn p

ZΩ

Uqnqn dx = ln L

ZΩ

eppdx,

that is, θp L, thus reaching a contradiction. We have proved the second inequality in (25).

10

Case q ! p. Analogously, if we suppose that there exist L > θp and a sequence qn ! p such that uqn

∞ L,

then we obtain from (27) and Lemma 7 that

ln LZ

Ωep

pdx limLqnp 1

qn p

ZΩ

Uqnqn dx lim

ZΩ

Upqn Uqn

qn

qn pdx =

ZΩ

eppln ep

dx

and hence L θp. This proves the first inequality in (25). 2

Lemma 9 Let

uq

q2[1,p)[(p,p?) be a family of positive weak solutions of the Lane-Emden problem (7). Then,

lim supq!p+

uq

∞ θp lim infq!p

uq

∞ . (29)

Proof. By applying Picone’s inequality again, but interchanging Uq with ep in (26), the lemma follows similarly.In fact, we obtain uq

qp∞ 1

q p

ZΩ

Uqq (ep/Uq)

pdx Z

Ω

Upq Uq

q

q p(ep/Uq)

pdx if p < q < p? (30)

and uq qp

∞ 1q p

ZΩ

Uqq (ep/Uq)

pdx Z

Ω

Upq Uq

q

q p(ep/Uq)

pdx if 1 q < p. (31)

Note that the uniform convergence Uq ! ep in Ω together with the Hopf’s boundary lemma guarantee thatep/Uq ! 1 uniformly in Ω. Thus, it follows thatZ

ΩUq

q (ep/Uq)pdx !

ZΩ

eppdx, as q ! p

and ZΩ

Upq Uq

q

q p(ep/Uq)

pdx !Z

Ωep

pln ep

dx, as q ! p

according to (21) and (23).Thus, if we suppose that there exist L > θp and qn ! p+ such that

uqn

∞ L, the uniform convergence

Uqn ! ep together with (30) imply that

ln L = limLqnp 1

qn pR

Ω eppln ep

dxRΩ ep

pdx(32)

and hence we arrive at the contradiction L θp. Therefore, the first inequality in (29) holds.On the other hand, if we assume, by contradiction again, the existence of L < θp and qn ! p such that uqn

∞ L then it follows from (31) that

ln LZ

Ωep

pdx = limLqnp 1

qn p

ZΩ

Uqnqn (ep/Uqn)

pdx

limZ

Ω

Upqn Uqn

qn

qn p(ep/Uqn)

pdx =Z

Ωep

pln ep

dx.

11

Since this implies that L θp we obtain a contradiction, proving thus the second inequality in (29). 2

It is worth mentioning that in the Laplacian case p = 2 the self-adjointness of this operator produceslimq!2

uq

∞ = θ2 directly. Such an argument has already appeared in [8], where the asymptotic behavior of

positive solutions of a logistical type problem for the Laplacian was studied. In fact,

λ2 uq q2

∞

ZΩ

Uq1q e2dx =

ZΩrUq re2dx =

ZΩre2 rUqdx = λ2

ZΩ

e2Uqdx

leads to uq q2

∞ 1q 2

ZΩ

Uq1q e2dx =

ZΩ

1Uq2q

q 2e2Uqdx.

Thus, if uqn

∞ ! L then

ln L = limn

uqn

qn2∞ 1

qn 2= lim

n

RΩ

1Uqn2qn

qn2 e2Uqn dxRΩ Uqn1

qn e2dx= ke2k2

2

ZΩ

e22 jln e2j dx

proving that uq

∞ ! θ2.

Theorem 10 Let

uq

q2[1,p)[(p,p?) be a family of positive weak solutions of the Lane-Emden problem (7). Then,

limq!p

uq = θpep,

the convergence being in C1(Ω).

Proof. Lemmas 8 and 9 imply thatlimq!p

up

∞ ! θp. (33)

Thus, the right-hand side of (7) is bounded for all q sufficiently close to p. This fact, combined with the globalHölder regularity ensures that uq is uniformly bounded in C1,β(Ω) (with respect to q) for some 0 < β < 1. Weconclude, as in the proof of Lemma 7, that uq converges in C1(Ω) to a positive weak solution u 2 C1(Ω) \W1,p

0 (Ω) of the eigenvalue problem (2), when q ! p. Thus, u = kep for some k > 0. But, according to (33)k = θp, implying that the limit function is always θpep (that is, it does not depend on subsequences). Therefore,limq!p uq = θpep in C1(Ω). 2

Remark 11 The estimate(λp

ξp

∞)1 lim inf

q!p

uq q

q

where ξp is the first eigenfunction normalized by the W1,p0 norm, was proved in [3]. Since lim inf

q!p

uq q

q =

θp ep

p

p

(as consequence of Theorem 10) and (λp ξp

∞)1 =

ep p

p <

θp ep

p

p, we see this estimate is not sharp.

12

Remark 12 It was proved by Dancer in [7] that there exists δ > 0 such that for each q 2 (2, 2+ δ) the problem∆u = jujq2 u in Ω,

u = 0 on ∂Ω,

has a unique positive weak solution. It seems difficult to adapt his proof for the p-Laplacian, if 1 < p 6= 2, since it usesdecisively the linearity of the Laplacian. However, our Theorem 10 allow ones to conjecture that Dancer’s result might stillbe extended for the p-Laplacian, 1 < p 6= 2.

3 Applications

A consequence of Theorem 10 is the differentiability of the function q 2 [1, p?) 7! λq at q = p, where λq isdefined by (8). We remark that this function is, in fact, differentiable almost everywhere since it can be writtenas a product of two monotone functions (see Lemma 20 in the last section).

Corollary 13 It holds

limq!p

λq λp

q p= λp ln(θp

ep

p). (34)

Proof. We recall that for each q 2 [1, p) [ (p, p?) the function

vq =

λq

λp

1qp

wq

is a positive weak solution of the resonant Lane-Emden problem (7), where wq 2 W1,p0 (Ω) \ C1,α(Ω) satisfies wq

q = 1 and Rq(wq) = λq.

Thus,

limq!p

vq

q = limq!p

λq

λp

1qp

= exp

limq!p

ln λq ln λp

q p

.

On the other hand, it follows from Theorem 10 that

limq!p

vq

q = θp ep

p .

Therefore,

limq!p

ln λq ln λp

q p= ln(θp

ep

p)

what means that ln λq is differentiable at q = p and ddqln λq

q=p = ln(θp

ep

p). But this is equivalent to

differentiability of λq at q = p with ddqλq

q=p given by (34). 2

Another consequence of Theorem 10 is the complete description, in the C1(Ω) space, of the asymptotic be-havior for the positive solutions of the non-resonant problem (0 < λ 6= λp):

∆pu = λ jujq2 u in Ωu = 0 on ∂Ω.

(35)

13

Corollary 14 Let

uλ,q

q2[1,p)[(p,p?) be a family of positive solutions of (35). Then

limq!p

uλ,q

C1 =

0 if λ < λp∞ if λ > λp

and limq!p+

uλ,q

C1 =

∞ if λ < λp0 if λ > λp.

Proof. A simple scaling argument shows that

uλ,q :=

λ

λp

1pq

uq, (36)

where uq is a positive solution of the resonant Lane-Emden problem (7). Thus, it follows from Lemma 2 that

uλ,q

C1 =

λ

λp

1pq uq

C1

λ

λp

1pq uq

∞

λ

λp

1pq

A,

for some positive constant A which does not depend on q close to p. Thus, uλ,q

C1 ! ∞ when q ! p and

λ > λp or when q ! p+ and λ < λp.Since uq is uniformly bounded in C1,β(Ω) with respect to q close to p, the continuity of the immersion

C1,β(Ω) ,! C1(Ω) implies that uq

C1 K for some positive constant K that does not depend on q close top. Hence, when q ! p and λ < λp or when q ! p+ and λ > λp, we have

uλ,q

C1 =

λ

λp

1pq uq

C1 K

λ

λp

1pq! 0.

2

Our results generalize those in [3] and in [13] to C1 norm. Note that in the super-linear case, our results arereally more general than those in [13] since they do apply to arbitrary families of positive solutions and not onlyfor ground states, as in [13].

A third consequence of Theorem 10 is that it provides a theoretical method for obtaining approximations forfirst eigenpairs of the p-Laplacian by solving a non-resonant problem (35) with λ > 0 arbitrary and q close to p.In fact, we have the following corollary.

Corollary 15 For 1 s ∞ and λ > 0 fixed let Uλ,q :=uλ,q uλ,q

s

and µλ,q := λ uλ,q

qps . Then, as q ! p :

µλ,q ! λp and Uλ,q !ep ep

sin C1(Ω).

Proof. It follows from (36) that

Uλ,q =uq uq

sand µλ,q := λ

uλ,q qp

s = λλp

λ

uq qp

s = λp uq qp

s .

But, since uq ! θpep in C1(Ω) as q ! p, it follows that Uλ,q =uq

kuqks! ep

kepksin C1(Ω). Moreover,

µλ,q =

RΩ

rUλ,qp dxR

Ω

Uλ,qq dx

!R

Ω

repp dxR

Ω

epp dx

= λp, as q ! p,

14

since ∆pUλ,q = µλ,qUq1λ,q in Ω. 2

Corollary 15 provides a method for obtaining numerical approximations of the first eigenpair (λp, ep

kepks).

In fact, in a first step one can compute a numerical solution of problem (35) with q close to p and hence, afterLs-normalization one obtains approximations for λp and ep

kepkssimultaneously.

Of course, a numerical solution of the nonlinear problem (35), for some λ > 0 fixed, is easier to obtain thandirectly compute the first eigenpair of the p-Laplacian (by solving the corresponding eigenvalue problem). Aspreviously mentioned, the advantage here is that λ can be chosen arbitrarily in computational implementationsof (35) and does not need to be close to λp. A similar approach was recently used in [5], where the iterative sub-and super-solution method was applied to compute the positive solutions of the sub-linear problem.

We emphasize that this approach is now well-supported for the super-linear case after our results, since itdoes apply to any family of positive solutions. In fact, the previously known results were valid only to groundstate family and thus the application of this method would be hardly possible, because of the necessity of provingthat a numerical solution of the super-linear problem is in fact a ground state.

Now, we show that the quotient

Λq := λ

uλ,q q

q uλ,q p

p

also converges to λp as q ! p. Moreover, we estimate the convergence order in the approximation of Λq by µλ,q.

Proposition 16 We have

(i) λp 6 Λq.

(ii) limq!p

Λq = λp.

(iii) If q is sufficiently close to p, then Λq

µλ,q 1

K jq pj

for some positive constant K which does not depend on q.

Proof. (i) follows from (36) since

Λq = λp

uq q

q uq p

p

=

ruq p

p uq p

p

λp,

and then (ii) follows from Theorem 10.Now, (iii) follows since Λq

µλ,q 1

= Uq

qq Uq p

p

1

=

1 Uq p

p

ZΩ

Uq

q Upq

dx 1 Uq

pp

ZΩ

max06t61

jtq tpj dx K jp qj ,

15

taking into account that Uq

pp !

ep p

p as q ! p. 2

Remark 17 Since µλ,q λp in the sub-linear case (see [5]), the rate of convergence of both µλ,q and Λq to λp is at leastO(p q).

4 Uniform explicit estimates

In this last section we present an estimate involving L∞ and Ls norms of uλ,q whose proof is inspired byarguments based on level set techniques developed in [19] (see also [4, Theor. 2]). We also verify that the

function q ! jΩjpq λq is decreasing for q 2 [1, p?). By combining these results we obtain an explict bound for the

ground state family

vq

1qp+εwhere 0 < ε < p

N .

From now on jDj :=Z

Ddx denotes the volume of the set D RN .

Lemma 18 Let D RN be a bounded and smooth domain. Then,

ZDjujpdx jDj

pN

CN,p

ZDjrujp dx (37)

for all u 2 W1,p0 (D) n f0g, where

CN,p = NωpNN

p

p 1

p1(38)

and ωN is the volume of the unit ball in RN .

Proof. We need only to verify that CN,p jDjpN is a lower bound for λp(D). Let λp(D) and λp(D) denote,

respectively, the first eigenvalue of the p-Laplacian in D and D, where D denotes the ball in RN centered atthe origin and such that jDj = jDj . It is well-known (see [18]) that λp(D) λp(D). It follows from (14) that Φp

1p∞ is a lower bound for λp(D), where Φp is the p-torsion function of D (as in (12) with Ω replaced by

D). It is straightforward to verify that

Φp(jxj) =p 1

pN 1

p1

R

pp1D jxj

pp1

, jxj RD

where RD := (jDj /ωN)1/N is the radius of the ball D. Thus, one has Φp 1p

∞ =Φp(0)

1p

=

p 1

pN 1

p1 Rp

p1D

1p= CN,p jDj

pN λp(D) λp(D).

2

16

Theorem 19 Let uλ,q 2 W1,p0 (Ω) be a positive weak solution of the Lane-Emden problem

∆pu = λ jujq2 u in Ω,u = 0 on ∂Ω,

where λ > 0 and 1 q < p(N+1N ). Then, one has

uλ,q

∞

λNp KN,p

uλ,q

1

pp+N(pq)

, (39)

where

KN,p := C N

pN,p

p+ N(p 1)

p

p+N(p1)p

(40)

and CN,p is defined by (38).

Proof. Let us denote uλ,q simply by u. For each 0 < t < kuk∞ , define

At := fx 2 Ω : u > tg .

The function

(u t)+ = max fu t, 0g =

u t, if u > t,0, if u t,

belongs to W1,p0 (Ω). Classical results [10, 20, 24] guarantee that uλ,q 2 C1,α(Ω) for some 0 < α < 1. Therefore,

we have ZAtjrujp dx = λ

ZAt

uq1 (u t) dx λ kukq1∞

ZAt(u t) dx. (41)

(Note that At is open and therefore r(u t)+ = ru in At.)Now, we estimate

RAtjrujp dx from below. For this, we apply Hölder’s inequality and (37) from Lemma 18

with D = At to obtainZAt(u t) dx

p jAtjp1

ZAt(u t)p dx jAtjp1jAtj

pN

CN,p

ZAtjrujp dx.

Thus,

CN,pjAtjpN+1p

ZAt(u t) dx

pZ

Atjrujp dx

what yields, taking into account (41),

CN,pjAtjpN+1p

ZAt(u t) dx

p λ kukq1

∞

ZAt(u t) dx.

Hence, we obtain ZAt(u t) dx

p1 λ

CN,pkukq1

∞ jAtjp+N(p1)

N .

17

This last inequality can be rewritten as

ZAt(u t) dx

N(p1)p+N(p1)

λ

CN,pkukq1

∞

Np+N(p1)

jAtj . (42)

By defining

f (t) :=Z

At(u t) dx,

it follows from Cavalieri’s Principle that

f (t) =Z ∞

tjAsj ds

and therefore f 0 (t) = jAtj . Thus, (42) can be rewritten as

1

λ

CN,pkukq1

∞

Np+N(p1)

f (t)N(p1)

p+N(p1) f 0 (t) . (43)

Integration of (43) yields

t (λNp KN,p)

pp+N(p1) (kuk∞)

N(q1)p+N(p1)

f (0)

pp+N(p1) f (t)

pp+N(p1)

(λ

Np KN,p)

pp+N(p1) (kuk∞)

N(q1)p+N(p1) (kuk1)

pp+N(p1)

where

KN,p := C N

pN,p

p+ N(p 1)

p

p+N(p1)p

.

Making t ! kuk∞ we obtain

kuk∞ (λNp KN,p)

pp+N(p1) (kuk∞)

N(q1)p+N(p1) (kuk1)

pp+N(p1)

and hence, by noting that

0 N(q 1)p+ N(p 1)

< 1 () 1 q < p

N + 1N

we obtain

(kuk∞)p+N(pq)p+N(p1)

λ

Np KN,p kuk1

pp+N(p1)

from which (39) follows. 2

It is interesting to note that the well-known Sobolev inequality (see [23]) could be used instead of (37) toprove a similar estimate to (39). However, it imposes the restriction 1 < p < N. On the other hand (37) does notrequire any relation between p and N.

Let us remark that the estimate (39) might, in principle, be used in the analysis of nodal solutions of theLane-Emden problem. In fact, it is valid in each nodal domain and does not depend on the Lebesgue volume ofit.

18

Lemma 20 The function q 2 [1, p?)! jΩjpq λq 2 R is non-increasing. In particular, any ground state vq satisfies vq

q jΩj

1q . (44)

Proof. Let u 2 W1,p0 (Ω)nf0g and 1 q1 < q2 < p? (q2 p? if p? < ∞). Hölder inequality implies that

jΩj1

ZΩjujq1 dx

pq1 jΩj

pq1

"ZΩjujq2 dx

q1q2 jΩj1

q1q2

# pq1

=

jΩj1

ZΩjujq2 dx

pq2

.

It follows thatjΩj

pq2 λq2 jΩj

pq2 Rq2(u) jΩj

pq1 Rq1(u),

what yields

jΩjp

q2 p

q1 λq2 Rq1(u) for all u 2 W1,p0 (Ω) nf0g.

Therefore,

jΩjp

q2 p

q1 λq2 λq1 = infnRq1(u) : u 2 W1,p

0 (Ω) nf0go

.

The estimate (44) follows directly from the monotonicity of jΩjpq λq since

vq

q =

λq

λp

1qp

. 2

Theorem 21 Let

vq

1qp+εa ground state family, where 0 ε < p

N . It holds vq

∞ CN,p,ε; for all 1 q p+ ε

for some positive constant CN,p,ε depending only on N, p and ε.

Proof. Hölder inequality combined with (44) gives vq

1 jΩj1 1

q vq

q jΩj1 1

q jΩj1q = jΩj ; 1 q < p?.

Thus, for 1 q p+ ε < p(N+1N ) (39) and the monotonicity of q 7! jΩj

pq λq yield

vq

∞

λNp

q KN,p vq

1

pp+N(pq)

CN,p,ε := max1qp+ε

jΩj

Np (p

pq ) λ

Np

1 KN,p jΩj p

p+N(pq).

2

19

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21