Results. Math. Online Firstc© 2014 Springer BaselDOI 10.1007/s00025-014-0386-2 Results in Mathematics

Principal Eigenvalue of p-Laplacian Operatorin Exterior Domain

Maya Chhetri and Pavel Drabek

Abstract. We consider an eigenvalue problem of the form

−Δpu = λ K(x)|u|p−2u in Ωe

u(x) = 0 for ∂Ωu(x) → 0 as |x| → ∞,

⎫⎬

⎭

where Ω ⊂ IRN is a simply connected bounded domain, containing theorigin, with C2 boundary ∂Ω and Ωe := IRN \ Ω is the exterior domain,1 < p < N, Δpu := div(|∇u|p−2∇u) is the p-Laplacian operator and

K ∈ L∞(Ωe) ∩ LN/p(Ωe) is a positive function. Existence and propertiesof principal eigenvalue λ1 and its corresponding eigenfunction are estab-lished which are generally known in bounded domain or in IRN . We alsoestablish the decay rate of positive eigenfunction as |x| → ∞ as well asnear ∂Ω.

Mathematics Subject Classification (2010). 35J60, 35J62, 35J92.

Keywords. p-Laplacian, principal eigenvalue, exterior domain.

1. Introduction

Many mathematical models from fluid dynamics, population dynamics, glace-ology, nonlinear elasticity etc. are modeled using the boundary value problemof the form

−Δpu = f(x, u) in Du > 0 in D

u(x) = 0 on ∂D,

⎫⎬

⎭(1.1)

The second author was supported by the Grant Agency of Czech Republic, Project No.13-00863S.

M. Chhetri and P. Drabek Results. Math.

where D ⊂ IRN is a domain. In order to prove the existence and multiplic-ity results for (1.1), the properties of principal eigenvalue and correspondingpositive eigenfunction of the eigenvalue problem of the p-Laplacian play thekey role. These properties are well known if D is either a bounded domain orthe entire IRN . There is a vast literature devoted to this topic and we referreaders to [1,2,4,7,11,14] and references therein. However, when D is an ex-terior domain, to the best of our knowledge, similar results are not availablein the literature even for the case p = 2. The purpose of this paper is to fillthis gap by investigating the existence and properties of principal eigenvalueand corresponding eigenfunction in an exterior domain. We also establish thedecay rate of positive eigenfunction at infinity as well as near the compactboundary ∂D which are helpful, especially when using the method of upperand lower solutions.

In this paper, we consider an eigenvalue problem for the p-Laplacian inan exterior domain, namely

−Δpu = λK(x)|u|p−2u in Ωe

u(x) = 0 for ∂Ωu(x) → 0 as |x| → ∞,

⎫⎬

⎭(1.2)

where 1 < p < N, λ > 0 and Ωe := IRN \ Ω, where Ω ⊂ IRN is a simplyconnected bounded domain, containing the origin, with C2 boundary ∂Ω. Theweight function K : Ωe → (0,∞) satisfies K ∈ L∞(Ωe) ∩ LN/p(Ωe).

For the functional framework, we let X := C∞0 (Ωe), where the closure

is taken with respect to the norm ‖u‖X :=(∫

Ωe |∇u|p dx)1p . In particular, by

Sobolev inequality

‖u‖Lp∗ (Ωe) ≤ c‖∇u‖Lp(Ωe)

one has X ↪→ Lp∗(Ωe), where p∗ = Np

N−p .

Definition 1.1. We call ϕ ∈ X an eigenfunction associated with the eigenvalueλ of (1.2) if for all φ ∈ X

∫

Ωe

|∇ϕ|p−2∇ϕ · ∇φdx = λ

∫

Ωe

K(x)|ϕ|p−2ϕφdx. (1.3)

Throughout the paper Br := {x ∈ IRN : ‖x‖ < r} denotes the open ballof radius r and Ber := {x ∈ IRN : ‖x‖ > r}. We first prove the existence ofprincipal eigenvalue and eigenfunction.

Lemma 1.1. The value λ1 := min∫

Ωe |∇u|p dx over V := {u ∈ X :∫

Ωe K(x)|u|pdx = 1} is achieved at ϕ1 ∈ X with ϕ1 ≡ 0 and λ1 > 0.

Next we establish properties of λ1 and ϕ1.

Theorem 1.1. Let λ1 and ϕ1 be as in Lemma 1.1. Then λ1 is the principaleigenvalue of (1.2) and ϕ1 is the corresponding eigenfunction. Let R > 0 besuch that Ω ⊂ BR. Then the following holds:

Principal Eigenvalue of p-Laplacian Operator

(a) (Regularity of ϕ1). There exists α ∈ (0, 1) such that ϕ1 ∈ C1,α(Ωe ∩BR).(b) ϕ1 > 0 or ϕ1 < 0 in Ωe.(c) (Simplicity of λ1). There exists unique 0 < ϕ1 ∈ V such that λ1 =

min∫

Ωe |∇ϕ1|p dx.(d) The eigenvalue λ1 is isolated and any eigenfunction ϕλ associated with an

eigenvalue λ = λ1 changes sign, that is, ϕ+λ = 0 and ϕ−

λ = 0 in Ωe.(e) For all u ∈ X

∫

Ωe

|∇u|p dx ≥ λ1

∫

Ωe

K(x)|u|p dx,

and equality holds if and only if u = tϕ1 for some t ∈ IR.(f) (Decay near infinity). Let l > p and C0 > 0 be such that 0 < K(x) < C0

|x|lfor all x ∈ Ωe. Then there exist 0 < C1 < C2 such that for all x ∈ BeR

C1

|x|N−pp−1

≤ ϕ1(x) ≤ C2

|x|N−pp−1

.

(g) (Estimate near ∂Ω). There exist 0 < C1 < C2 such that for all x ∈ BR∩Ωe

C1dist(x, ∂Ω) < ϕ1(x) < C2dist(x, ∂Ω).

(h) (Gradient estimate near infinity). Let Ω := B1 and K(x) = K(|x|) beradially symmetric function. Then ϕ1(x) = ϕ1(|x|) is radially symmetric.Moreover, there exist constants 0 < C1 < C2 and r∗ > 0 sufficiently largesuch that for all x ∈ Ber∗ , we have

C1

|x|N−1p−1

≤ |∇ϕ1(x)| ≤ C2(log |x|) N−pN(p−1)

|x|N−1p−1

.

2. Proof of Results

Proof of Lemma 1.1. Let {vn} ⊂ X be a minimizing sequence for λ1, that is,vn ∈ V and

∫

Ωe |∇vn|p dx = λ1 + δn, where δn ↘ 0. Then {vn} is boundedin X. Since X is reflexive, there exists ϕ1 ∈ X such that vn ⇀ ϕ1 (up to asubsequence).

Next we show that∫

Ωe

K(x)|vn|p dx →∫

Ωe

K(x)|ϕ1|p dx. (2.4)

M. Chhetri and P. Drabek Results. Math.

Indeed, take R > 0 such that Ω ⊂ BR. Then H˙older inequality followed bySobolev inequality yields

∫

|x|≥R

K(x)|vn|p dx ≤

⎛

⎜⎝

∫

|x|≥R

|K(x)|Np dx

⎞

⎟⎠

pN

⎛

⎜⎝

∫

|x|≥R

|vn|p∗

⎞

⎟⎠

pp∗

≤ C‖vn‖pX

⎛

⎜⎝

∫

|x|≥R

|K(x)|Np dx

⎞

⎟⎠

pN

≤ C(λ1 + δn)

⎛

⎜⎝

∫

|x|≥R

|K(x)|Np dx

⎞

⎟⎠

pN

.

Since K ∈ LN/p(Ωe), it is easy to see that∫

|x|≥R |K(x)|Np → 0 as R → ∞.

Therefore ∫

|x|≥R

K(x)|vn|p dx ≤ ε(R) → 0 as R → ∞. (2.5)

Observe that Ωe∩BR is bounded. Then applying Rellich-Kondrachov theorem[9, Theorem 1.2.28], we get

∫

Ωe∩BR

K(x)|vn|p dx →∫

Ωe∩BR

K(x)|ϕ1|p dx as n → ∞. (2.6)

Combining (2.5) and (2.6) yields (2.4), as desired.Then by the weak lower semicontinuity of the mapping u �→ ∫

Ωe |∇u|p dxand (2.4), we can infer that

∫

Ωe

|∇ϕ1|p dx = λ1 and ϕ1 ∈ V. (2.7)

It then follows from (2.7) that λ1 > 0 and ϕ1 ≡ 0. �

Proof of Theorem 1.1. From the variational characterization in Lemma 1.1and from the Method of Lagrange Multipliers it follows that λ1 is an eigenvalueand ϕ1 is associated eigenfunction. By [12, Lemma 7.6], |ϕ| ∈ X and |∇ϕ| =∇|ϕ| for any ϕ ∈ X. Therefore, if ϕ1 ∈ X is a minimizer for

∫

Ωe |∇u|p dxover V , so is |ϕ1| ∈ X. Hence, without loss of generality, we may assume thatϕ1 ≥ 0 almost everywhere in Ωe. Moreover, it also follows from the variationalcharacterization that λ1 is the smallest (principal) eigenvalue of (1.2).(a) Repeating the argument of [8, Lemma 4.5] for ϕ1 ∈ X one obtains that

there exists C > 0 such that ‖ϕ1‖L∞(Ωe) ≤ C‖ϕ1‖Lp∗ (Ωe). Therefore ϕ1 ∈L∞(Ωe). Then it follows from [13, Theorem 1] or [5, Theorem 2] thatfor any R > 0 there exists α ∈ (0, 1) such that ϕ1 ∈ C1,α(Ωe ∩ BR).

Principal Eigenvalue of p-Laplacian Operator

The boundary regularity of ϕ1 then follows from [13, Theorem 1], that is,ϕ1 ∈ C1,α(Ωe ∩BR).

(b) Harnack’s inequality [16, Theorem 1.1] applied to ϕ1 implies that ϕ1 > 0everywhere in Ωe.

(c) The following argument is taken from [6, pp. 229-230]. Let ϕ, ψ ∈ V beminimizers of

∫

Ωe |∇u|p dx. We may assume, by (b), that ϕ > 0 and ψ > 0in Ωe. We construct the function w := η

1p where η := ϕp+ψp

2 . Then

∫

Ωe

K(x)wp dx =12

⎛

⎝

∫

Ωe

K(x)ϕp dx+∫

Ωe

K(x)ψp dx

⎞

⎠ = 1,

that is, w ∈ V . We also have, pointwise in Ωe,

|∇w|p = η1−p∣∣∣∣12(ϕp−1∇ϕ+ ψp−1∇ψ)

∣∣∣∣

p

= η

∣∣∣∣12

(ϕp

η

∇ϕϕ

+ψp

η

∇ψψ

)∣∣∣∣

p

= η

∣∣∣∣z

∇ϕϕ

+ (1 − z)∇ψψ

∣∣∣∣

p

; z =ϕp

ϕp + ψp∈ (0, 1)

≤ η

(

z

∣∣∣∣∇ϕϕ

∣∣∣∣

p

+ (1 − z)∣∣∣∣∇ψψ

∣∣∣∣

p)

=12

(

ϕp∣∣∣∣∇ϕϕ

∣∣∣∣

p

+ ψp∣∣∣∣∇ψψ

∣∣∣∣

p)

=12

(|∇ϕ|p + |∇ψ|p) . (2.8)

Integrating over Ωe yields∫

Ωe

|∇w|p dx ≤ 12

∫

Ωe

(|∇ϕ|p + |∇ψ|p) dx. (2.9)

But equality must hold in (2.9) since ϕ and ψ both minimize∫

Ωe |∇u|p dxover V . Hence equality must hold in (2.8) almost everywhere in Ωe. Thestrict convexity of the function t �→ |t|p then implies that ∇ϕ

ϕ = ∇ψψ almost

everywhere in Ωe. But this yields ∇(ϕψ

)= 0 almost everywhere in Ωe.

Since Ωe is connected and ϕ, ψ are C1,α functions (by (b)), there existsa constant C > 0 such that ϕ = Cψ everywhere in Ωe. However, sinceϕ, ψ ∈ V , we eventually obtain C = 1. This proves the claim.

(d) The proof of this part is identical as in [8, Lemma 4.6].(e) This part follows from the variational characterization of λ1 (see Lemma

1.1) and the uniqueness of the eigenfunction ϕ1 (part (c)).

M. Chhetri and P. Drabek Results. Math.

(f) The hypotheses of [15, Theorem 1] are satisfied for arbitrary γ1 >Np since

K(x) ∈ L∞(Ωe) ∩ LN/p(Ωe) and ‖ϕ1‖L∞(Ωe) ≤ C‖ϕ1‖Lp∗ (Ωe), (see (a)).Then Serrin’s estimate [15, Theorem 1] yields

‖ϕ1‖L∞(B1(x)) ≤(‖ϕ1‖Lp∗ (B2(x)) + ‖λ1Kϕ

p−11 ‖Lγ1 (B2(x))

)

for all x ∈ BeR, where Br(x) := {y ∈ IRN : ‖y − x‖ < r} ⊂ Ωe for r = 1, 2.Therefore ‖ϕ1‖L∞(B1(x)) → 0 as |x| → ∞ and thus ϕ1(x) → 0 uniformlyas |x| → ∞. We can now apply [3, Theorem C(ii)] to get a constantC1 > 0 such that ϕ1(x) ≥ C1

|x|N−pp−1

for x ∈ BeR. Similarly, by applying

[3, Theorem 4] (with u0 = 0), we get C2 > 0 such that ϕ1(x) ≤ C2

|x|N−pp−1

for

x ∈ BeR.(g) The compactness of the boundary ∂Ω together with the regularity up

to the boundary, ϕ1 ∈ C1,α(Ωe), imply that there exists C2 > 0 suchthat ϕ1(x) ≤ C2dist(x, ∂Ω). The existence of C1 > 0 and that ϕ1(x) ≥C1dist(x, ∂Ω) follows from [17, Theorem 2] and the compactness of ∂Ω.

(h) Let ϕ1 > 0 satisfy (2.7). Rotation invariance of both integrals∫

Be1|∇u|p dx

and∫

Be1K(x)|u|p dx implies that ϕ1 also satisfies (2.7), where ϕ1 is a

rotation of ϕ1 about the origin. But the uniqueness result (part (c)) forcesϕ1 = ϕ1, that is, ϕ1 is radially symmetric. We denote r := |x| and writeϕ1(x) = ϕ1(r). The regularity argument for second order ODE, (see [10,p.110]) implies that

− (rN−1|ϕ′

1(r)|p−2ϕ′1(r)

)′= rN−1K(r)ϕp−1

1 (r) (2.10)

holds almost everywhere in (1,∞). It follows from (2.10), Rolle’s theorem,positivity of ϕ1 on (1,∞) and the boundary condition ϕ1(1) = ϕ1(∞) = 0that there exists a unique r0 ∈ (1,∞) such that ϕ′

1(r0) = 0. Integrating(2.10) from r0 to r > r0, we get

|ϕ′1(r)| = λ

1p−11 r

1−Np−1

⎛

⎝

r∫

r0

sN−1K(s)ϕp−11 (s) ds

⎞

⎠

1p−1

. (2.11)

Fix r1 > r0 and denote C1 := λ1

p−11

(∫ r1r0sN−1K(s)ϕp−1

1 (s) ds) 1

p−1. Then

(2.11) implies |ϕ′1(r)| ≥ C1

rN−1p−1

.

On the other hand, using the decay rate of ϕ1 near infinity and theHolder inequality with exponents N

p and (Np )′ = NN−p , we get

|ϕ′1(r)| = λ

1p−11 r

1−Np−1

⎛

⎝

r∫

r0

sN−1K(s)ϕp−11 (s) ds

⎞

⎠

1p−1

Principal Eigenvalue of p-Laplacian Operator

≤ C2λ1

p−11 r

1−Np−1

⎛

⎝

r∫

r0

sN−1K(s)sp−N ds

⎞

⎠

1p−1

≤ C2λ1

p−11 r

1−Np−1

⎛

⎝

r∫

r0

sp(N−1)

N K(s)sp−N

N ds

⎞

⎠

1p−1

≤ C2λ1

p−11 r

1−Np−1

⎛

⎝

r∫

r0

sN−1KNp (s) ds

⎞

⎠

pN(p−1)

⎛

⎝

r∫

r0

s−1 ds

⎞

⎠

N−pN(p−1)

≤ C2λ1

p−11 r

1−Np−1 ‖K‖

LNp (Be

1)(log r − log r0)

N−pN(p−1)

≤ C2(log r)

N−pN(p−1)

rN−1p−1

,

where C2 := C2λ1

p−11 ‖K‖

LNp (Be

1).

This completes the proof. �

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M. Chhetri and P. Drabek Results. Math.

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Maya ChhetriDepartment of Mathematics and StatisticsThe University of North Carolina at GreensboroGreensboro, NC 27402, USAe-mail: maya@uncg.edu

Pavel DrabekDepartment of Mathematics and NTIS, Faculty of Applied SciencesUniversity of West BohemiaUniverzitnı 22, 30614 Plzen, Czech Republice-mail: pdrabek@kma.zcu.cz

Received: January 31, 2014.

Accepted: April 16, 2014.