quasilinear eigenvalue problem

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120 (1995) MATHEMATICA BOHEMICA No.2, 169{195

THE LEAST EIGENVALUES OF NONHOMOGENEOUS DEGENERATED QUASILINEAR EIGENVALUE PROBLEMS

PavelDr b ek,* Plzen

(Received January 6, 1994)

Summary. We prove the existence of the least positive eigenvalue with a corresponding nonnegative eigenfunction of the quasilinear eigenvalue problem

−div(a(x u)| ∇u| p ;2 ∇u) =b(x u)|u| p;2 u in u= 0 on@

where is a bounded domain,p>1 is a real number anda(x u),b(x u) satisfy appropriate

growth conditions. Moreover, the coecienta(x u) contains a degeneration or a singularity.

We work in a suitable weighted Sobolev space and prove the boundedness of the eigenfunc-tion inL

1(). The main tool is the investigation of the associated homogeneous eigenvalue

problem and an application of the Schauder xed point theorem.

Keywords: weighted Sobolev space, degenerated quasilinear partial dierential equations, weak solutions, eigenvalue problems, Schauder xed point theorem, boundedness of the solution

MSC 1991: 35J20, 35J70, 35B35, 35B45

1. Introduction

The aim of this paper is to prove the existence of the least positive eigenvalueλand

the corresponding nonnegative eigenfunctionuof the nonhomogeneous degenerated quasilinear eigenvalue problem

(1.1) − div(a(x, u)| ∇ u|p;2

∇ u) = λb(x, u)|u|p;2

uinΩ, u = 0 on∂Ω,

*The autor has been supported by the Grant Agency of the Czech Republic under Grant No. 201/94/0008

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whereΩis a bounded domain,p > 1is a real number anda(x, s), b(x, s) : Ω×R→R

are real functions satisfying appropriate growth conditions (see Section 4). Moreover, the function a(x, s) may contain a degeneration or a singularity. We work in a

suitableweighted Sobolev space W1p

0 (w, Ω)with the weight functionw > 0a.e. inΩ

(see Section 2) and prove that for a given R > 0 there exists the least λ > 0 and a corresponding u∈ W1p

0 (w, Ω)∩ L 1

(Ω) such that u>0 a.e. in Ω,u L

p

( )= R and

the equation in(1.1)is fulﬁlled in the weak sense(see Theorem 4.10). In fact, a more

general result (dealing with more general growth conditions imposed on b(x, s)) is

proved in Theorem 4.5.

This paper generalizes the result of Boccardo 5] and Dr bek, Kuera 6] (where

nondegenerated uniformly elliptic quasilinear operators were considered) and

com-pletes the papers on eigenvalues of p-Laplacian published by Anane 2], Barles 3],

Bhattacharya 4], Garca Azorero, Peral Alonso 9], Otani, Teshima 14] and oth-ers (where nondegenerated and homogeneous operators were considered). Let us

note that neither global results for nonlinear eigenvalue problems, nor Ljusternik-Schnirelmann theory can be used, since the operator in (1.1) is not (in general) a potential operator.

Thepaper is organizedas follows. In Section 2, which has apreliminary character,

we dene appropriate weighted Sobolev spaces and prove some useful imbeddings. We prove also a version of Friedrichs inequality in the weighted Sobolev space. More-over, an auxiliary assertion due to Stampacchia is proved and we present some conse-quences of Clarkson's inequality. In Section 3 we study thehomogeneous eigenvalue problemassociated with (1.1) (i.e. we consider the problem (1.1) witha(x, u) := a(x)

andb(x, u) := b(x)). We prove the existence of the least positive eigenvalue and the

corresponding nonnegative eigenfunction of this problem. We show that the eigen-function belongs to L1

(Ω). We also prove the simplicity of the least eigenvalue

and study some useful properties of the homogeneous operator associated with the principal part. The main result we prove in Section 4. The tools are an a apriori

estimate in L1

(Ω), the results for the homogeneous eigenvalue problem (namely

the continuous dependence of the least eigenvalue and the corresponding nonnega-tive eigenfunction of the homogeneous problem with respect toa(x), b(x)) and the

Schauder xed point theorem. Finally, Section 5 containsexamples which illustrate

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2. Preliminaries

2.1. Weighted Sobolev space. Let us suppose that Ω is an open bounded

subset of then-dimensional Euclidean space R n,

p > 1 is an arbitrary real number

andwis aweight function(i.e. positive and measurable) inΩ. Assume that

(2.1) w∈ L1 loc(Ω) and 1 w ∈ L 1 p;1 loc (Ω).

Let us dene the weigted Sobolev space W1p

(w, Ω) as the set of all real valued

functionsudened inΩfor which

(2.2) u1pw= Z |u|p dx + Z w| ∇ u|p dx 1 p <∞.

It follows from (2.1) thatW1p

(w, Ω)is areﬂexive Banach space and thatW1p 0 (w, Ω)

is well dened as the closure ofC1 0

(Ω)inW1p

(w, Ω)with respect to the norm·1pw

(see e.g. Kufner, Sndig 11]). Lets>

1

p ;1 be a real number. A simple application of the Hlder inequality yields

that thecontinuous imbedding

(2.3) W1p (w, Ω) → W1p 1 (Ω) holds provided 1 w ∈ L s (Ω)andp1= ps s + 1.

2.2. Compact imbeddings. It follows from (2.3) and from the Sobolev

imbed-ding theorem (see e.g. Adams 1], Kufner, John, Fuk 10]) that for s + 16ps <

n(s + 1)we have (2.4) W1p 0 (w, Ω) → W 1p1 0 (Ω) → L q (Ω), where 16q = np 1 n;p1 = np s n(s+1);ps, and for

ps>n(s + 1) the imbedding (2.4) holds

with arbitrary16q <∞.

Moreover, thecompact imbedding W1p

0 (w, Ω) →→ L r

(Ω)

holds provided16r < q.

An easy calculation yields thats > n

p implies q > p. In particular, we have (2.5) W1p 0 (w, Ω) →→ Lp+ (Ω)

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for06η < q− pprovided (2.6) 1 w ∈ L s (Ω)ands∈ n p, +∞ ∩ 1 p− 1, +∞ .

2.3. Friedrichs inequality in weighted Sobolev spaces. In what follows we

will always assume that (2.6) is fullled. Let u∈ C1

0 (Ω). Then due toq > p and

the imbeddingW1p 1 0 (Ω) → L q (Ω)we have (2.7) Z |u|p dx 1 p 6c 1 Z |u|q dx 1 q 6c 2 Z [|u|p 1 +| ∇ u|p 1 ] dx 1 p 1 .

The Friedrichs inequality inW1p 1 0 (Ω)yields (2.8) Z [|u|p1 +| ∇ u|p1 ] dx 1 p 1 6c 3 Z | ∇ u|p1 dx 1 p 1 .

Using the Hlder inequality we obtain

(2.9) Z | ∇ u|p1 dx 1 p 1 = Z | ∇ u|p1 w p 1 p 1 w p 1 p dx 1 p 1 6 Z w| ∇ u|p dx 1 p Z 1 w p 1 p p p;p 1 dx p;p 1 p 1 p 1 6 Z 1 ws dx 1 ps Z w| ∇ u|p dx 1 p

(see Subsection 2.1 for the relation betweens,pandp1). It follows from (2.7){(2.9)

that Z |u|p dx6c 4 Z w| ∇ u|p dx

with a constantc4> 0independent ofu∈ C 1

0 (Ω). Hence the norm

uw= Z w| ∇ u|p dx 1 p on the spaceW1p

0 (w, Ω)is equivalent to the norm ·

1pwdened by (2.2).

2.4. Equivalent norms. Let us assume thatw˜is a weight function dened in Ωand satisfying inequalities

(2.10) c5w(x)

6w(x)˜ 6c 6w(x)

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for a.e.x∈ Ωwith some constantsc6 >c 5> 0. Then obviously W1p 0 ( ˜w, Ω) = W 1p 0 (w, Ω)

and the norms · w~and · w areequivalent onW 1p

0 (w, Ω). It follows from

Clark-son's inequality (see Kufner, John, Fuk 10]) thatW1p 0

(w, Ω)is auniformly convex Banach space with respect to the norm · w~ for any w˜satisfying (2.10).

2.5. Lemma. (cf. Murthy, Stampacchia [13]). Let ζ = ζ(t) be a nonnegative, nonincreasing function on a half line t>k

0 >0 such that (2.11) ζ(h)6c 7(h− k) ; (ζ(k)) for h > k>k 0. Then σ > 0, δ > 1 imply ζ(k0+ d) = 0, where d = c 1 7(ζ(k 0)) ;1 · 2 ;1.

Proof.

Let us dene a sequence (kn)by

(2.12) kn= kn;1+

d 2n

, n = 1, 2, . . . .

Substituting (2.12) into (2.11) we get by induction

ζ(kn) 6 ζ(k0) 2n ;1 → 0

forn→ ∞. Since lim

n!1

kn= k0+ dandζis nonincreasing, we obtainζ(k0+ d) = 0.

2.6. Lemma. Let p>2. Then

(2.13) |t 2| p − |t 1| p >p|t 1| p ;2 t 1( t 2− t 1) + |t 2− t 1| p 2p;1− 1

for all pointst 1and t 2in R n . Let 1 < p < 2. Then (2.14) |t 2| p − |t 1| p >p|t 1| p ;2 t 1( t 2− t 1) + 3p(p− 1) 16 |t 2− t 1| 2 (|t 1| + | t 2|) 2;p

for all pointst 1and t 2in R n .

Proof of this lemma is based on Clarkson's inequality and can be found in Lindqvist 12].

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2.7.

Remark

. It follows from (2.13) and (2.14) that the inequality (2.15) |t 2| p − |t 1| p > p|t 1| p;2 t 1( t 2− t 1)

holds for anyt 1, t 2 ∈ R n ,t 1 = t

2 and for any p > 1. Note that inequality (2.15) is

just a restating of the strict convexity of the mapping t → |t|

p and can be proved

independently of (2.13) and (2.14).

3. Homogeneous eigenvalue problem

3.1. Weak formulation. Let us suppose thatwis the weight function satisfying

(2.1) and (2.6). Leta(x),b(x)be measurable functions satisfying

(3.1) w(x)

c8

6a(x)6c 8w(x),

(3.2) 06b(x)

for a.e. x ∈ Ω with some constant c8 > 1, and b(x) ∈ L q q ;p(Ω) for p < q < q, b(x)∈ L1 (Ω)forq

= p (see Subsection 2.2 forq). Moreover, let

(3.3) meas{x ∈ Ω; b(x) > 0} > 0.

Further we will assume thatp < q

< q. The proofs in the forthcoming subsections

can be performed in the same way also in the caseq

= p.

Let us considerhomogeneous eigenvalue problem

(3.4) − div(a(x)| ∇ u|p;2

∇ u) = λb(x)|u|p;2

uinΩ, u = 0on∂Ω.

We will say that λ ∈ R is an eigenvalue and u ∈ W 1p

0 (w, Ω), u = 0, is the

correspondingeigenfunction of the eigenvalue problem (3.4) if

(3.5) Z a(x)| ∇ u|p;2 ∇ u ∇ ϕ dx = λ Z b(x)|u|p;2 uϕ dx

holds for anyϕ∈ W1p 0 (w, Ω).

3.2. Lemma. There exists the least (the ﬁrst) eigenvalue λ1 > 0 and at least

one corresponding eigenfunction u1

>0 a.e. in Ω(u

1≡ 0) of the eigenvalue problem

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Proof.

Set λ1= inf Z a(x)| ∇ v|p dx ; Z b(x)|v|p dx = 1 . Obviouslyλ1 >0. Let(v

n)be the minimizing sequence forλ1, i.e.

(3.6) Z b(x)|vn| p dx = 1and Z a(x)| ∇ vn| p dx = λ1+ δn,

with δn → 0+ for n → ∞. It follows from (3.6) that vna 6 c

9, with c9 > 0

independent ofn. The reexivity ofW1p

0 (w, Ω)(see Subsection 2.4) yields the weak

convergence vn u1 in W 1p

0 (w, Ω) for some u

1 (at least for some subsequence of

(vn)). The compact imbedding W 1p

0

(w, Ω) →→ Lq

(Ω) implies the strong

conver-gencevn → u1 in L q

(Ω). It follows from (3.2),(3.6), from the Minkowski and the

Hlder inequality that

1 = lim n!1 Z b(x)|vn| p dx 1 p 6 lim n!1 Z b(x)|vn− u1| p dx 1 p + Z b(x)|u1| p dx 1 p 6 lim n!1 Z (b(x)) q q ;pdx q ;p pq · Z |vn− u1| q dx 1 q + Z b(x)|u1| p dx 1 p = Z b(x)|u1| p dx 1 p , and analogously Z b(x)|u1| p dx 1 p 6 lim n!1 Z (b(x)) q q ;pdx q ;p pq · Z |u1− vn| q dx 1 q + lim n!1 Z b(x)|vn| p dx 1 p = 1. Hence Z b(x)|u1| p dx = 1.

In particular, u1 ≡ 0. The property of the weakly convergent sequence (vn) in

W1p 0 (w, Ω)yields λ1 6 Z a(x)| ∇ u1| p dx =u1 p a 6lim inf n!1 vn p a = lim inf n!1 Z a(x)| ∇ vn| p dx = lim inf n!1 (λ1+ δn) = λ1,

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i.e. (3.7) λ1= Z a(x)| ∇ u1| p dx.

It follows from (3.7) thatλ1> 0 and it is easy to see thatλ1 is the least eigenvalue

of (3.4) with the corresponding eigenfunctionu1. Moreover, ifuis an eigenfunction

corresponding toλ1then|u|is also an eigenfunction corresponding toλ1. Hence we

can suppose thatu1

>0a.e. inΩ.

3.3.

Remark

. It follows from the proof of Lemma 3.2 that vn u1 in

W1p

0 (w, Ω)andv

na → u1a. The uniform convexity of W 1p

0 (w, Ω)(see

Subsec-tion 2.4) then implies thestrong convergence vn→ u1in W 1p

0 (w, Ω).

3.4. Lemma. Let u ∈ W1p 0

(w, Ω), u > 0 a.e. in Ω, be the eigenfunction

cor-responding to the ﬁrst eigenvalue λ1 > 0 of the eigenvalue problem (3.4). Then

u∈ Lr

(Ω) for any 16r <∞.

Proof.

The assertion of lemma is fullled automatically if ps>n(s + 1)(see

Subsection 2.2). Let us suppose thatps < n(s + 1). ForM > 0 dene vM(x) = inf{u(x), M} ∈ W 1p 0 (w, Ω)∩ L 1 (Ω). Let us chooseϕ = vp+1 M (κ >0)in (3.8) Z a(x)| ∇ u|p;2 ∇ u ∇ ϕ dx = λ1 Z b(x)|u|p;2 uϕ dx. Obviouslyϕ∈ W1p 0 (w, Ω)∩ L 1

(Ω). It follows from (3.8) that

(3.9) (κp + 1) Z a(x)vp M| ∇ v M| p dx = λ1 Z b(x)up;1 vp+1 M dx.

Due to (3.1) and the imbeddingW1p 0 (w, Ω) → Lq (Ω)we have (3.10) (κp + 1) Z a(x)vp M| ∇ v M| p dx > κp + 1 c8 Z w(x)vp M| ∇ v M| p dx = κp + 1 c8(κ + 1) p Z w(x)| ∇(v+1 M )| p dx>c 9 Z (v+1 M ) q dx p q .

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Hence it follows from (3.2), (3.8), (3.9), (3.10) and the Hlder inequality that (3.11) Z v(+1)q M dx p q 6c 10 Z b(x)up;1 vp+1 M dx 6c 10 Z b(x) q q ;pdx q ;p q · Z u(+1)q dx p q . Sinceu∈ Lr

(Ω) for any16r6q(see Subsection 2.2), we can chooseκin (3.11) in

the following way:

(3.12) (κ + 1)q

= q.

Then substituting (3.12) into (3.11) we obtain

(3.13) Z v(+1 )q M dx p q 6c 11 Z uq dx p q 6c 12, i.e.vM ∈ L q 0 (Ω), q0

= (κ + 1)q, for anyM > 0. We haveu(x) = lim

M!1

vM(x), x∈ Ω.

Then the Fatou lemma and (3.13) yield

Z uq 0 dx p q 6lim inf M! 1 Z vq 0 Mdx p q 6c 12, i.e.u∈ Lq 0 (Ω), where q0 = q q q.

Repeating the same argument we can chooseκ in (3.11) as(κ + 1)q

= q0 and get u ∈ Lq 00 (Ω), q00 = q(q q ) 2, etc. Since

q > q∗, the bootstrap argument implies the

assertion of lemma.

3.5. Lemma. Let u ∈ W1p

0 (w, Ω), u

> 0 a.e. in Ω be the eigenfunction

cor-responding to the ﬁrst eigenvalue λ1 > 0 of the eigenvalue problem (3.4). Then

u∈ L1

(Ω).

Proof.

Letk>0be a real number. Set

ϕ(x) = sup{u(x), k} − k in (3.8). We obtain Z (u>k) a(x)| ∇ ϕ|p dx = λ1 Z (u>k) b(x)(ϕ + k)p;1 ϕ dx,

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i.e. (3.14) Z (u>k) w(x)| ∇ ϕ|p dx6λ 1c8 Z (u>k ) b(x)(ϕ + k)p;1 ϕ dx. Let us choose r > max (p− 1)qq p(q− q ), q .

Due to the homogeneity of (3.8) and Lemma 3.4 we can assume without loss of generality that uL r ( )= ˜R > 0. The imbeddingW1p 0 (w, Ω) → L q (Ω)implies (3.15) Z (u>k) w(x)| ∇ ϕ|p dx>c 13ϕ p L q ( ) .

Sincer > q, the Hlder inequality yields

(3.16) Z (u>k) b(x)(ϕ + k)p ;1 ϕ dx 6 Z (u>k) b(x) q q ;pdx q ;p q Z (u>k) (ϕ + k) p;1 p q ϕ q p dx p q 6c 14 Z (u>k) (ϕ + k)q dx p;1 q Z (u >k ) ϕq dx 1 q 6c 14 Z ur dx p;1 r ; meas Ω(u > k) p;1 q (1; q r ) × Z ϕq dx 1 q ; meas Ω(u > k) 1 q (1; q q ) .

It follows from (3.14){(3.16) that (3.17) ϕp;1 L q ( ) 6c 15( ˜R) ; meas Ω(u > k) p;1 q (1; q r )+ 1 q (1; q q ) .

On the other hand, forh > k we obtain

(3.18) ϕ p ;1 L q () = Z (u >k ) |u − k|q dx p;1 q > Z (u >h) |u − k|q dx p;1 q >(h− k) p;1 ; meas Ω(u > h) p;1 q .

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Set ζ(t) = meas Ω(u > t). Then ζ(t) is a nonnegative and nonincreasing function

and it follows from (3.17), (3.18) that

ζ(h) p;1 q 6 c15( ˜R) (h− k)p ;1 ; ζ(k) p;1 q (1; q r )+ 1 q (1; q q ) , i.e. ζ(h)6˜c 15( ˜R)(h− k) ;q ; ζ(k) , where σ = q, δ = q p− 1 h p− 1 q 1−q r + 1 q 1−q q i .

Due to the choice ofr we have δ > 1. It follows from Lemma 2.5 that there exists d = d(r, q, ˜R, meas Ω) > 0such thatζ(d) = 0. Henceu(x)6dfor a.e. x∈ Ω.

3.6. Proposition. There exists precisely one nonnegative eigenfunction u1,

u1 L

q

() = 1, corresponding to the ﬁrst eigenvalue λ

1 > 0 of the eigenvalue

problem (3.4).

Proof.

Due to the variational characterization of λ1 the function u ∈

W1p 0

(w, Ω)is an eigenfunction corresponding toλ1 if and only if Z a(x)|∇u|p dx− λ1 Z b(x)|u|p dx = 0 = inf v2W 1p 0 (w ) Z a(x)|∇v|p dx− λ1 Z b(x)|v|p dx .

This imlies that if u1, u2 ∈ W 1p

0

(w, Ω) are two eigenfunctions corresponding to λ1

then also v1(x) = max x2 {u1(x), u2(x)}, v2(x) = min x2 {u1(x), u2(x)}

are eigenfunctions corresponding toλ1provided thatv2≡ 0. Indeed, we havev1, v2∈

W1p 0 (w, Ω)and Z a(x)|∇v1| p dx− λ1 Z b(x)|v1| p dx + Z a(x)|∇v2| p dx− λ1 Z b(x)|v2| p dx = Z a(x)|∇u1| p dx− λ1 Z b(x)|u1| p dx + Z a(x)|∇u2| p dx− λ1 Z b(x)|u2| p dx. Hence Z a(x)|∇v1| p dx− λ1 Z b(x)|v1| p dx = Z a(x)|∇v2| p dx− λ1 Z b(x)|v2| p dx = 0.

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Let u1

> 0 and u 2

> 0 be two eigenfunctions corresponding to λ

1 such that u1≡ u2, min x2 {u1(x), u2(x)} ≡ 0and u1 L q ()=u 2 L q ()= 1. Denote v3(x) = k1v2(x) = k1min x2

{u1(x), u2(x)}, wherek1 > 0is chosen in such a

way that v3 L q ()= 1. Thenv3∈ W 1p

0 (w, Ω)is again an eigenfunction corresponding toλ

1 such thatv3≡

u1. Moreover,

{x ∈ Ω; u1(x) = 0} ⊆ {x ∈ Ω; v3(x) = 0}.

Setv5(x) = k2v4(x) = k2max x2

{u1(x), v3(x)}, where k2> 0is chosen such that

v5 L q ()= 1. Thenv5∈ W 1p 0 (w, Ω)is an eigenfunction corresponding toλ

1such thatv5≡ u1and

{x ∈ Ω; v5(x) = 0} = {x ∈ Ω; u1(x) = 0}.

Let, now,u1

>0andu 2

>0be two eigenfunctions corresponding toλ

1 such that u1≡ u2,u1 L q ( )=u 2 L q ( )= 1and min x2 {u1(x), u2(x)} ≡ 0.

Denoteu˜1= k3max{u1(x), u2(x)}, where0 < k3< 1is chosen such that

˜u1 L

q

( )= 1,

andu˜2= k4max{u1(x), ˜u1(x)}, where 0 < k4< 1is such that

˜u2 L

q

( )= 1.

Thenu˜1andu˜2 are eigenfunctions corresponding toλ1 such thatu˜1≡ ˜u2and

{x ∈ Ω; ˜u1= 0} = {x ∈ Ω; ˜u2= 0}.

We will prove the assertion of proposition via contradiction. Due to the argument presented above we assume that u>0andv >0are eigenfunctions corresponding

toλ1 such that (3.19) uL q ( )=vL q ()= 1, u≡ v,

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and vanishing inΩon the same set (almost everywhere in the sense of the Lebesgue measure). Then (3.20) Z a(x)| ∇ u|p;2 ∇ u ∇ ϕ dx = λ1 Z b(x)|u|p ;2 uϕ dx for anyϕ∈ W1p 0 (w, Ω), and (3.21) Z a(x)| ∇ v|p;2 ∇ v ∇ ψ dx = λ1 Z b(x)|v|p;2 vψ dx for anyψ∈ W1p 0 (w, Ω). Forε > 0set u"= u + εandv"= v + ε. Substitute ϕ =u p "− v p " up;1 " into (3.20) and ψ = v p "− u p " vp;1 "

into (3.21). Since u" v " ,v" u " ∈ L1 (Ω)and ∇ ϕ = 1 + (p− 1) v" u" p ∇ u − p v" u" p;1 ∇ v, ∇ ψ = 1 + (p− 1) u" v" p ∇ v − p u" v" p;1 ∇ u, we haveϕ, ψ∈ W1p

0 (w, Ω). Adding (3.20) and (3.21) (with ϕandψ chosen above)

we obtain Z a(x) 1 + (p− 1) v" u" p | ∇ u|p + 1 + (p− 1) u" v" p | ∇ v|p dx − Z a(x) p v" u" p;1 | ∇ u|p;2 ∇ u ∇ v + p u" v" p;1 | ∇ v|p;2 ∇ v ∇ u dx = λ1 Z b(x) h u u" p;1 − v v" p ;1 i (up "− v p ") dx.

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Since| ∇ log u"| = j

r u j

u

" , the last equality is equivalent to

∇ log u"(∇ log v"− ∇ log u") dx

− Z a(x)pup "| ∇ log v "| p ;2

∇ log v"(∇ log u"− ∇ log v") dx

= λ1 Z b(x) h u u" p;1 − v v" p ;1 i (up "− v p ") dx.

Letp>2. We use (2.13) in order to estimate the left hand side of (3.22) (we rst

sett

1=∇ log u", t

2=∇ log v"and then t 1=∇ log v", t 2=∇ log u"). We obtain (3.23) λ1 Z b(x) u u" p;1 − v v" p;1 ](up "− v p ") dx > 1 2p;1− 1 Z

a(x)| ∇ log u"− ∇ log v"| p (up "+ v p ") dx = 1 2p;1− 1 Z a(x) 1 vp " + 1 up " |v"∇ u − u"∇ v| p dx>0.

Let 1 < p < 2. We use (2.14) in order to estimate the left hand side of (3.22)

(similarly as above) obtaining

(3.24) λ1 Z b(x) u u" p;1 − v v" p;1 (up "− v p ") dx > 3p(p− 1) 16 Z a(x) 1 up " + 1 vp " |v"∇ u − u"∇ v| 2 (v"| ∇ u| + u"| ∇ v|) 2;p dx>0. We haveu, v∈ L1

(Ω)(see Lemma 3.5) and

(3.25) u

u"

→ 1, v v"

→ 1 (ε → 0+)

a.e. inΩwhereu > 0andv > 0, respectively

(3.26) u

u"

= 0, v v"

= 0(for anyε > 0)

elsewhere (sinceuandv vanish on the same set inΩ). Hence it follows from (3.25),

(3.26) and the Lebesgue theorem that for anyp,1 < p <∞, λ1 Z b(x) u u" p;1 − v v" p ;1 (up "− v p ") dx→ 0 (ε → 0 +).

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This together with (3.23), (3.24) and the Fatou lemma implies

|v ∇ u − u ∇ v| = 0a.e. inΩ

for any1 < p <∞. Hence there exists a constantk > 0such thatu = kv a.e. inΩ.

But (3.19) yieldsk = 1, i.e. u = va.e. inΩ, which is a contradiction.

The proof of Proposition 3.6 follows the lines of the proof of Lemma 3.1 in Lindqvist 12] for the nondegenerate case(a(x)≡ 1 inΩ).

3.7. Lemma. Let J : W1p 0 (w, Ω)−→ [W 1p 0 (w, Ω)] be an operator deﬁned by J(u), ϕ = Z a(x)| ∇ u|p;2 ∇ u ∇ ϕ dx for any u, ϕ ∈ W1p

0 (w, Ω) (here ·, · denotes the duality between [W 1p

0 (w, Ω)]

and W1p

0 (w, Ω)). Then J is surjective and J ;1 : [W1p 0 (w, Ω)] −→ W1p 0 (w, Ω) is

bounded and continuous.

Proof.

The operatorJ is bounded, strictly monotone, continuous and coercive.

Then it follows from the Browder theorem (see e.g. Fuk, Kufner 8]) that J is

surjective. It follows from the Hlder inequality that (3.27) J(v) − J(u), v − u >(v p;1 a − u p;1 a )(v a− ua) for anyu, v∈ W1p 0

(w, Ω). The boundedness ofJ;1follows immediately from (3.27).

Let us suppose to the contrary that J;1 is not continuous. Then there exists a

sequence(fn)such thatfn→ f in [W 1p 0 (w, Ω)] and J;1 (fn)− J ;1 (f )a >δ for someδ > 0. Denoteun= J ;1 (fn), u = J ;1

(f ). It follows from (3.27) that fn· una >f n, un = J(un), un >u n p a, i.e. un p ;1 a 6f n

( · denotes the norm in the dual space[W 1p

0 (w, Ω)]

). Then(un)is bounded in

W1p

0 (w, Ω)and we can assume that there existsu˜∈ W 1p 0 (w, Ω) such thatu n ˜u inW1p 0 (w, Ω). Hence we have

(3.28) J(un)− J(˜u), un− ˜u =

=J(un)− J(u), un− ˜u + J(u) − J(˜u), un− ˜u −→ 0

since J (un) → J(u) in [W 1p

0

(w, Ω)]. It follows from (3.27) (where we set

v = un, u = ˜u) and (3.28) that una → ˜ua. The uniform convexity of W

1p

0

(w, Ω)

equipped with the norm · a (see Subsection 2.4) implies un → ˜u in W 1p

0 (w, Ω).

This convergence together with the convergence J (un) → J(u) in [W 1p

0 (w, Ω)]

impliesu = u˜ which is a contradiction. The continuity ofJ;1 is proved.

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4. Nonhomogeneous eigenvalue problem

4.1. Weak formulation. In this section we will consider thenonhomogeneous eigenvalue problem

(4.1) −div(a(x, u)| ∇ u|p;2

∇ u) = λb(x, u)|u|p;2

u inΩ, u = 0 on∂Ω.

Let g : [0,∞) → [1, ∞) be a nondecreasing function, α(x) ∈ L

q q ;p(Ω) for q > q > p, α(x) ∈ L1 (Ω) for q

= p(for q, q see Subsection 3.1),

β > 0 a constant.

We assume that a(x, s), b(x, s)are Carathodory functions (i.e. continuous ins for

a.e.x∈ Ωand measurable inxfor alls∈R)and

(4.2) w(x) c8 6a(x, s)6c 8g(|s|)w(x), (4.3) 06b(x, s)6α(x) + β|s| q ;p

hold for a.e. x∈ Ωand for alls∈R.

Moreover, assume that

(4.4) meas{x ∈ Ω; b(x, v(x)) > 0} > 0

for anyv∈ Lq

(Ω), v≡ 0. (Note that the condition (4.4) is fullled e.g. ifb(x, s) > 0

for a.e.x∈ Ωand for alls= 0.)

We will say that λ ∈ R is an eigenvalue and u ∈ W 1p

0 (w, Ω), u ≡ 0, is the

correspondingeigenfunction of the eigenvalue problem(4.1)if

(4.5) Z a(x, u(x))| ∇ u|p;2 ∇ u ∇ ϕ dx = λ Z b(x, u(x))|u|p;2 uϕ dx

holds for anyϕ∈ W1p 0 (w, Ω).

4.2. Proposition (apriori estimate). Let u ∈ L1

(Ω),uL q

() = R > 0,

u>0 be any eigenfunction of (4.1) corresponding to the eigenvalue λ. Then there

exists d(R) > 0 (independent of g) such thatuL 1

( )

6d(R).

Proof.

Chooseϕ = up+1 in (4.5) with

κ>0. We obtain (κp + 1) Z a(x, u(x))up | ∇ u|p dx = λ Z b(x, u(x))u(+1)p dx, i.e.

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(4.6) κp + 1 (κ + 1)p

a(x, u(x))| ∇(u+1

)|p

dx = λ

b(x, u(x))u(+1)p

dx.

It follows from (4.2) and the imbeddingW1p

0 (w, Ω) → L q (Ω)that (4.7) Z

a(x, u(x))| ∇(u+1

)|p dx> 1 c8 Z w(x)| ∇(u+1 )|p dx >c 16( Z u(+1)q dx) p q

withc16> 0independent ofκ, Randg.

Applying the Hlder inequality, (4.3) and the Minkowski inequality we obtain

(4.8) Z b(x, u(x))u(+1)p dx 6 Z b(x, u(x)) q q ;p dx q ;p q Z u(+1 )q dx p q 6 " Z α(x) q q ;pdx q ;p q + β Z uq dx q ;p q # Z u(+1)q dx p q .

It follows from (4.6), (4.7) and (4.8) that (4.9) Z u(+1)q dx 6c 17 (κ + 1)q (κp + 1) q p h α L q q ;p ( ) + βRq ;p i q p · Z u(+1)q dx q q ,

withc17> 0independent ofκ, Randg. Letj be a nonnegative integer. Substitute

κ = q j ;(q ) j (q ) j into (4.9): (4.10) Z u q j+1 (q ) j dx6c 17 h q j (q ) j i q h q j ;(q ) j (q ) j p + 1 i q p × α L q q ;p ( ) + βRq ;p q p Z u q j ( q ) j;1 dx q q . Since lim j!1 qj+1 (q )j =∞,

there exists the leastj0 such that

r = q j 0 +1 (q )j0 > max n (p− 1)qq p(q− q ), q o .

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It follows from (4.9), (4.10) (settingj = j0, j0− 1, · · · , 1) that Z ur dx 1 r 6R(R),˜ where_{R > 0}˜ is independent of_g.

Now we set a(x) := a(x, u(x))and b(x) := b(x, u(x)) in the proof of Lemma 3.5.

Following the lines of this proof we obtain

uL 1

()

6d(R),

where d = d(R) is independent ofg. This completes the proof of Proposition 4.2.

4.3. Truncation in the principal part. Let R > 0 and d = d(R) > 0be as

above. We dene (4.11) ˜a(x, s) = 8 > < > : a(x, s) forx∈ Ω, |s|6d(R), a(x, d(R)) forx∈ Ω, s > d(R), a(x,−d(R)) forx∈ Ω, s < −d(R).

Let us consider the nonhomogeneous eigenvalue problem (4.12) −div(˜a(x, u)| ∇ u|p;2

∇ u) = λb(x, u)|u|p;2

uin Ω, u = 0on∂Ω.

Then it follows from Proposition 4.2 thatu∈ W1p

0 (w, Ω), uL q

()= R, u >0is

an eigenfunction of (4.12)if and only if it is an eigenfunction of (4.1). 4.4. Application of the ﬁxed point theorem. For a given v ∈ Lq

(Ω) set av(x) = ˜a(x, v(x)), bv(x) = b(x, v(x)). It follows from (4.2), (4.3), (4.4) and (4.11)

thatav(x)andbv(x)full (3.1), (3.2), (3.3) for any xedv∈ L q

(Ω). Let us consider

the homogeneous eigenvalue problem (4.13) − div(av(x)| ∇ u| p;2 ∇ u) = λbv(x)|u| p;2 uinΩ, u = 0on∂Ω

for any xed v ∈ Lq

(Ω). Due to the results of Section 3 there exists the least

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that uv >0a.e. in Ω, u v ∈ L 1 (Ω) and uv L q

( ) = R. Hence we can dene the

operator S : Lq (Ω)→ Lq (Ω)

which associates with v ∈ Lq

(Ω) the rst nonnegative eigenfunction uv of (4.13)

such thatuv L

q

( )= R.

Let us assume for a moment thatS is acompact operator. Since it maps the ball BR = {u ∈ L q (Ω),uL q ( )

6 R} into itself it follows from the Schauder xed

point theorem (see e.g. Fuk, Kufner 8]) that S has aﬁxed point u∈ BR. Hence

there existsλu> 0such that

− div(au(x)| ∇ u| p;2 ∇ u) = λubu(x)|u| p;2 uinΩ, u = 0 on∂Ω,

and it follows from the considerations in Subsection 4.3 that λu > 0 is the least

eigenvalue of (4.1) andu∈ L1

(Ω), u>0a.e. inΩ, is the corresponding eigenfunction

satisfyinguL q

()= R.

Themain result of this paper follows from the considerations presented above. 4.5. Theorem. Let the assumptions from Subsection 4.1 be fulﬁlled. Then for a given real number R > 0 there exists the least eigenvalue λ > 0 and the corresponding eigenfunction u∈ W1p

0

(w, Ω)∩ L1

(Ω) of the nonhomogeneous eigenvalue problem (4.1) such that u>0 a.e. in Ω andu

L q

()= R.

In the forthcoming subsections it remains to prove the compactness of the operator

S in order to justify our assumption in Subsection 4.4.

4.6. The Nemytskii operators. Let us dene the Nemytskii operators G1: u→ |u|

p ;2

u, G2: u→ |u| p

, G3: u→ b(x, u(x)).

ThenGiis a bounded and continuous operator fromL q (Ω)intoL q p;1(Ω)fori = 1, from Lq (Ω)into L q

p(Ω) for i = 2, and from L q (Ω) into L q q ;p(Ω) for i = 3 (see

e.g. Vajnberg 15], Fuk, Kufner 8]). The Nemytskii operator

G4: (u, z1, . . . , zn)→ ˜a(x, u(x))(z 2 1(x) + . . . + z 2 n(x)) p;1 2

is bounded and continuous from Lq (Ω) × Lp (w, Ω) × . . . × Lp (w, Ω) into L p p;1(w ; 1

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4.7. Lemma. Let z, zn∈ W 1p 0 (w, Ω) and Z av(x)| ∇ z| p ;2 ∇ z ∇ ϕ dx = Z f (x)ϕ(x) dx, Z avn(x)| ∇ zn| p ;2 ∇ zn∇ ψ dx = Z fn(x)ψ(x) dx for any ϕ, ψ∈ W1p 0 (w, Ω) and let vn→ v in L q (Ω), fn→ f in [W 1p 0 (w, Ω)] . Then zn→ z in W 1p 0 (w, Ω).

Proof.

Dene operators J, Jn: W 1p 0 (w, Ω)→ [W 1p 0 (w, Ω)] by J(u), ϕ = Z av(x)| ∇ u| p ;2 ∇ u ∇ ϕ dx, Jn(u), ψ = Z avn(x)| ∇ u| p;2 ∇ u ∇ ψ dx for anyϕ, ψ, u∈ W1p 0 (w, Ω). HenceJ (z) = f andJ n(zn) = fn.

Letn∈N be xed. Consider the equation

Jn(u) = h. It follows that Z av n(x)| ∇ u| p dx = Z h(x)u(x) dx, up w 6c 18huw, (4.14) J;1 n (h) w 6c 18h 1 p;1 for anyh∈ [W1p 0 (w, Ω)], where

c1 8> 0is independent ofnandh. Analogously

(4.15) J;1 (h)w 6c 18h 1 p;1

(cf. Lemma 3.7). Applying Lemma 3.7 fora(x) := av(x)we obtain continuity ofJ ;1

(withJ dened in this subsection).

Assume that(un)is a sequence satisfyingun → z inW 1p

0 (w, Ω). It follows from

the continuity of the Nemytskii operatorG4that

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6 sup k'k w 61 Z w(x); 1 p;1 a v n (x)| ∇ un| p;2 ∇ un− av(x)| ∇ z| p;2 ∇ z p p;1 dx p × Z w(x)| ∇ ϕ|p ) dx 1 p + sup k'k w 61 Z w(x); p;1 p av(x)| ∇ z| p ;2 ∇ z − av(x)| ∇ un| p ;2 ∇ un p p;1 dx p;1 p × Z w(x)| ∇ ϕ|p dx 1 p → 0 forn→ ∞. Set un = J ;1

(fn). Then the assumptions of lemma and the continuity of J ;1

imply

(4.17) un→ z inW

1p

0 (w, Ω).

The relations (4.14){(4.17) and the continuity ofJ;1 now yield

zn− zw 6J ;1 n (f n)− J ;1 (fn)w+J ;1 (fn)− J ;1 (f )w 6J ;1 n (J n− J)J ;1 (fn)w+J ;1 (fn)− J ;1 (f )w 6c 1 8Jn(un)− J(un) 1 p;1 +J ;1 (fn)− J ;1 (f )w→ 0

forn→ ∞, which completes the proof.

4.8. Proposition. The operator S : Lq

(Ω)→ Lq

(Ω) deﬁned in Subsection 4.4 is compact.

Proof.

We prove thatSis a continuous operator fromLq

(Ω)intoW1p 0 (w, Ω).

The assertion then follows from the compact imbeddingW1p 0

(w, Ω) →→ Lq

(Ω)(see

Subsection 2.2). Let uvn= S(vn), uv= S(v). Suppose to the contrary that vn→ v

inLq (Ω)and (4.18) uv n− u vw >δ

for someδ > 0. We have

(4.19) Z av(x)| ∇ uv| p;2 ∇ uv∇ ϕ dx = λv Z bv(x)|uv| p;2 uvϕ dx, (4.20) Z av n(x)| ∇ u v n| p;2 ∇ uv n∇ ψ dx = λ v n Z bv n(x)|u v n| p;2 uv nψ dx

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for any ϕ, ψ ∈ W1p

0 (w, Ω). It follows from Lemma 3.7 that for any v n ∈ L q (Ω) there existszn ∈ W 1p 0 (w, Ω) such that (4.21) Z av n(x)| ∇ z n| p;2 ∇ zn∇ ϕ dx = λv Z bv(x)|uv| p;2 uvϕ dx for anyϕ∈ W1p 0 (w, Ω). Lemma 4.7 yieldsz n → uv in W 1p

0 (w, Ω) (and hence also

in Lq

(Ω)). Applying the Hlder inequality, (4.3) and the Minkowski inequality, we

forn→ ∞. Applying the Hlder inequality, (4.3), the Minkowski inequality and the

continuity of the Nemytskii operatorsG2,G3 we obtain

(4.23) Z h b(x, vn(x))|zn| p − b(x, v(x))|uv| p i dx 6 Z b(x, vn(x)) |zn| p − |uv| p dx + Z b(x, vn(x))− b(x, v(x)) |uv| p dx 6 " Z α(x) q q ;pdx q ;p q + β Z |vn(x)| q dx q ;p q # × Z |z n| p − |uv| p q p dx p q + Z b(x, vn(x))− b(x, v(x)) q q ;p dx q ;p q Z |uv| q dx p q → 0

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forn→ ∞. It follows from the variational characterization ofλv n, (4.19){(4.23) that λv n 6 R av n(x)| ∇ z n| p dx R bvn(x)|zn| p dx = λv R bv(x)|uv| p;2 uvzndx R bvn(x)|zn| p dx → λv R bv(x)|uv| p dx R bv(x)|uv| p dx = λv. Hence (4.24) lim sup λvn 6λ v.

Applying the Hlder inequality, the Minkowski inequality and the assumptions (4.2), (4.3) we obtain from (4.20) (withψ = uvn):

1 c8 uv n p w 6 Z av n(x)| ∇ u v n| p dx = λv n Z bv n(x)|u v n| p dx 6λ vn Z |α(x)| q q ;pdx q ;p q +β Z |vn(x)| q dx q ;p q (4.25) × Z |uv n| q dx p q .

It follows from the assumption uv n L q ( ) = R, from v n → v in L q (Ω) and from(4.25) that (4.26) uv n w 6 const

for anyn∈N. Due to (4.26) we have

(4.27) uvn uin W

1p

0 (w, Ω)

(at least for some subsequence) for some u ∈ W1p

0 (w, Ω) and hence u

n → u in

Lq

(Ω).

The Hlder inequality, the Minkowski inequality, (4.3) and the continuity of the Nemytskii operatorsG1andG3 imply

Z [b(x, vn(x))|uvn| p ;2 uvn− b(x, v(x))|u| p ;2 u]ϕ dx (4.28) 6 Z b(x, vn(x))− b(x, v(x)) |uv n| p;2 uv nϕ dx + Z b(x, v(x)) |uv n| p;2 uv n− |u| p ;2 u ϕ dx 6

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6 Z b(x, vn(x))− b(x, v(x)) q q ;p dx q ;p q Z |uv n| q dx p;1 q × Z |ϕ|q dx 1 q + " Z |α(x)| q q ;pdx q ;p q + β Z |v(x)| q q ;pdx q ;p q # × Z |u vn| p ;2 uvn− |u| p;2 u q p;1 dx p;1 q Z |ϕ|q dx 1 q → 0 for anyϕ∈ W1p

0 (w, Ω). Passing to suitable subsequences we can assume that

(4.29) λv

n→ λ ∈ [0, λ v]

(see (4.24)). Letu∈ W1p

0 (w, Ω)be the unique solution of

(4.30) Z av(x)| ∇ u| p;2 ∇ u ∇ ϕ dx = λ Z bv(x)|u| p;2 uϕ dx for any ϕ∈ W1p

0 (w, Ω) (Lemma 3.7 guarantees the existence of u). It follows from

(4.28){(4.30) and from Lemma 4.7 that

(4.31) uv

n → uin W 1p

0 (w, Ω).

Now, (4.27), (4.31) implyu = uanduvn→ uin W 1p 0 (w, Ω). Hence we have λv >λ = R av(x)| ∇ u| p dx R bv(x)|u| p dx > inf ~ u6=0 ~ u2W 1p 0 (w) R av(x)| ∇ ˜u| p dx R bv(x)|˜u| p dx = R av(x)| ∇ uv| p dx R bv(x)|uv| p dx = λv.

This implies thatλ = λv andu = uv (see the uniqueness ofuv

>0, u v L q ()= R in Section 3).

In particular, this means that

uv n→ u

v inW 1p

0 (w, Ω),

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4.9.

Remark

. The proofs in Section 4 can be performed in the same way working with L1 (Ω) instead of L q q ;p(Ω) in the caseq

= p. Hence we obtain the

followingspecial version of Theorem 4.5.

4.10. Theorem. Let (4.2)–(4.4) be fulﬁlled with α(x) ∈ L1

(Ω) and q

= p. Then for a given real number R > 0 there exists the least eigenvalue λ > 0 and the corresponding eigenfunction u∈ W1p

0

(w, Ω)∪ L1

(Ω) of (4.1) such that u>0 a.e. in

Ω anduL p

()= R.

4.11.

Remark

. Since the eigenvalue problem (4.13) is homogeneous, we can

dene the operator_{S : L}˜ q

(Ω)−→ Lq

(Ω)which associates withv∈ Lq

(Ω)the rst

nonpositive eigenfunction−uvof (4.13) such that−uv L

q

()= R. It is clear from

the above considerations that_S˜has the_{same properties} as _S dened in Subsection

4.4. Hence repeating the same arguments as in Subsections 4.2{4.4, 4.6{4.8 we prove the followingdual version of Theorem 4.5.

4.12. Theorem. Let the assumptions of Theorem 4.5 be fulﬁlled. Then for a given real number R > 0 there exists the least eigenvalue ˜λ > 0 and the corresponding eigenfunction ˜u∈ W1p

0 (w, Ω)∩ L 1

(Ω) of the nonhomogeneous eigenvalue problem (4.1) such that ˜u60 a.e. in Ω and˜u

L q

()= R.

4.13.

Remark

. Letλand ˜λbe the least eigenvalues guaranteed by Theorem

4.5 and 4.12, respectively, for a given xedR > 0. Thenλ= ˜λmay hold due to the

fact that the eigenvalue problem (4.1) is not homogeneous in general.

5. Examples

5.1.

Example

. Let Ω be a bounded domain in R n,

p > 1, w(x) be positive

and measurable in Ωsatisfyingw(x)∈ L1 lo c (Ω), 1 w (x) ∈ L s (Ω)fors > max{n p , 1 p ;1}.

Consider the eigenvalue problem

(5.1) − div(w(x)eu 2 | ∇ u|p;2 ∇ u) = λ|u|p;2 uin Ω, u = 0on∂Ω.

In this case we have

a(x, s) = w(x)es 2

, b(x, s)≡ 1

for a.e.x∈ Ωand for alls∈R.

It follows from Theorem 4.10 thatfor any given real number R > 0 there exists the least eigenvalue λ > 0 and the corresponding eigenfunction u∈ W1p

0 (w, Ω)∩ L 1

(Ω) of (5.1)such that u>0 a.e. in Ω andu

L p

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5.2.

Example

. Let us consider forΩthe plane domainΩ = (−1, 1) × (−1, 1) (i.e.Ω⊂R 2 ). Forx = (x1, x2)∈ Ωset w(x) = 8 > < > : 1, x1 60, x 2(1− x 1) , x1> 0, x2> 0, |x2| (1− x1) , x1> 0, x2< 0

withν,µ, γreal numbers. Consider the eigenvalue problem

(5.2) − div(w(x)(1 + u4

)| ∇ u)|2

∇ u) = λu9in

Ω, u = 0on∂Ω.

In this case we havep = 4,

a(x, s) = w(x)(1 + s4

), b(x, s) = s6

for a.e. x∈ Ωand for alls∈R. Thus the principal part of the dierential operator

has adegeneration(orsingularity) which is concentrated on a partΓ1of the boundary

∂Ω,

Γ1={x = (x1, x2) ; x1= 1, x2∈ (−1, 1)},

as well as on a segmentΓ2 in the interior ofΩ,

Γ2={x = (x1, x2) ; x1∈ (0, 1), x2= 0}.

Condition (2.1) indicates that we have to choose ν andµ from the interval(−1, 3)

with no condition onγ. Let us assume that

(5.3) ν, µ∈ − 1,4 3 , γ∈ − ∞,4 3 .

It follows from (5.3) that 1 w(x) ∈ L

3

4(Ω) andq = 12(see Subsection 2.2). Hence the

growth condition (4.3) is fullled e.g. withq

= 10. Applying Theorem 4.5 we have

the following assertion.

Let us assume (5.3). Then for a given real number R > 0 there exists the least eigenvalue λ > 0 and the corresponding eigenfunction u ∈ W14

0

(w, Ω)∩ L1

(Ω) of

(5.2)such that u>0 a.e. in Ω andu L

1 0

()= R.

Note that forν,µandγ positivewe have adegenerationof the same extent atΓ1

andΓ2. On the other hand, thesingularity can occur in a limited extent atΓ2 (for

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Author's address: Pavel Drbek, Department of Mathematics, University of West Bo-hemia, P. O. Box 314, 30614 Plze, Czech Republic, e-mail:[email protected].