Nonlinear homogeneous eigenvalue problem in R

Nonlinear homogeneous eigenvalue problem in R

: nonstandard variational approach

Pavel Dr´abek¹, Zakaria Moudan, Abdelfettah Touzani

Abstract. The nonlinear eigenvalue problem for p-Laplacian

8

<

:

−div(a(x)|∇u|^p−2∇u) =λg(x)|u|^p−2uinR^N, u >0 inR^N, lim

|x|→∞u(x) = 0,

is considered. We assume that 1< p < Nand thatgis indefinite weight function. The existence andC^1,α-regularity of the weak solution is proved.

Keywords: eigenvalue, the p-Laplacian, indefinite weight, regularity Classification: Primary 35P30, 35J70

1. Introduction

We consider the nonlinear eigenvalue problem (1.1)







−div(a(x)|∇u|^p−2∇u) =λg(x)|u|^p−2uin R^N, u >0 inR^N, lim

|x|→∞u(x) = 0,

where 1< p < N, g is a function that changes sign, i.e.g is an indefinite weight function,ais a positive and bounded function andλis a real parameter.

The aim of this work is to prove the existence andC^1,α regularity of the weak solution of (1.1). In comparison to similar results we use anonstandard variational approach— we do not minimize a Reyleigh-type quotient.

Let us note that this work was motivated by recent work [3] in which the following nonhomogeneous eigenvalue problem was considered:







−div(a(x)|∇u|^p−2∇u) =λf(x, u) in R^N, u >0 inR^N and lim

|x|→∞u(x) = 0,

wheref is a Carath´eodory function satisfying the condition 0≤f(x, t)≤g(x)|t|^γ withp < γ < p^∗ =_N^{N p}_−p andgsatisfying suitable integrability assumptions.

1 The author was supported by grant no. 201/94/0008 of the Grant Agency of the Czech Republic. This support is acknowledged

Modifying the approach from [3] we can deal with our problem and to get (λ, u) satisfying (1.1).

In this paper we will use the following notation: L^p :=L^p(R^N) denotes the usual Lebesgue space with the normk·k^p,W^1,p:=W^1,p(R^N) is the usual Sobolev space andD(R^N) :=C₀^∞(R^N) is the space of all functions with compact support inR^N with continuous derivatives of all orders.

2. Preliminaries, hypotheses and formulation of the main result We assume thata=a(x) is a measurable function such that

(2.1) 0< a₀≤a(x)∈L^∞,

gis an indefinite weight function satisfying:

(g1) there exists an open subset Ω6=∅ofR^N such that g(x)>0 a.e. in Ω;

(g₂) there exists a real numberδ,0< δ <∞such that g∈L

N p ∩L

N p+δ

.

Let us consider the function space: X :={u∈L^p∗;∇u∈(L^p)^N} equipped with the normkuk:= R

a(x)|∇u|^pdx¹_p

. (Here and henceforth the integrals are taken overR^N unless otherwise specified.) ThenX is a reflexive Banach space.

Using (2.1) and Sobolev inequality (see [1]) we conclude that there is a constant C₁ >0 such that

(2.2) kuk^p∗≤C₁kuk

holds for allu∈X.

Definition 2.1. Aweak solutionof (1.1) is a pair (λ, u) such thatλ >0,u∈X, u6= 0 and

(2.3)

Z

a(x)|∇u|^p−2∇u∇v dx=λ Z

g(x)|u|^p−2uv dx

for all v ∈ X. In this case u is called an eigenfunction corresponding to the eigenvalueλ >0.

Let us remark that under the assumptions (2.1) and (g₂) the integrals in (2.3) are well defined.

The main result of our paper is the following

Theorem 2.1. Let us assume(2.1),(g₁)and (g₂). Then the problem(1.1)has a positive eigenvalue λ > 0 and a corresponding eigenfunction u ∈ X, u > 0 in R^N and lim

|x|→∞u(x) = 0. Moreover, the eigenvalue λ is simple, isolated and unique in the following sense: if ˜λ6=λ is a positive eigenvalue of (1.1) andu˜ is a corresponding eigenfunction thenu˜ changes sign inR^N.

Corollary. Let the assumptions of Theorem2.1 be satisfied and moreover, let a∈C¹(R^N). Then the assertion of Theorem2.1holds withu∈C^1,α(B_R(0)), for anyR >0 andα=α(R)∈]0,1 [.

Remark 2.1. Similar results were proved in papers [2], [5] and [6]. However, different (more restrictive) assumptions on the weight function g and different methods were used in these papers. On the other hand, our result does not contain any information about “higher” eigenvalues of (1.1).

3. Proof of Theorem 2.1 and of Corollary

Proposition 3.1. Assume (2.1), (g₁) and (g₂). Then the problem (1.1) has a weak solution(λ, u), u∈X andλ >0, such thatu6≡0 andu≥0 inR^N. Proof: The proof follows the lines of Theorem 3.1 in [3]. Since the character of our problem is different from that considered in [3], we give the proof in detail here for the reader’s convenience. Let α∈ ]1, p[ be fixed and consider the following functional:

J=

R g(x)|u|^pdx kuk^α+kuk^p∗.

It is easy to see thatJ is well-defined for anyu∈X,u6≡0. Due to (2.2) and the H¨older inequality we have

(3.1)

Z

g(x)|u|^pdx≤ Z

|g(x)||u|^pdx≤ kgk^N

pkuk^pp^∗≤C₁^pkgk^N

pkuk^p. Sinceα < p < p^∗ then kuk^p≤ kuk^α+kuk^p∗ for allu∈X, so

J(u)≤

R g(x)|u|^pdx

kuk^p ≤C₁^pkgk^N

p

. Then there exists a constants₁ <∞(s₁ =C₁^pkgk^N

p

) such thatJ(u)≤s₁ holds for allu∈X,u6≡0. Thus s:= sup

u∈X u6=0

J(u) is a real number.

Lemma 3.1. There exist s₀ ∈ ]0, s[ and a sequence (un)^∞_n=1 ⊂ X, un ≥ 0 such that s₀ ≤ J(un) ≤ s holds for all n, and lim

n→∞J(un) = s. Furthermore, Rg|un|^pdx→R

g|u|^pdx, asn→ ∞andJ(u) =sfor someu∈X.

Proof of Lemma 3.1: Let Ω be from (g1) and choose ϕ₀ ∈ D(R^N) such that suppϕ₀ ⊂⊂ Ω and sup

x∈R^N

ϕ₀(x) > 0. Set s₀ = ¹₂J(ϕ0). Then s₀ =

12

R

Ω

g|ϕ0|^pdx

kϕ0k^α+kϕ0k^p∗ > 0 and s₀ < s. Let (un)^∞_n=1 ⊂ X, u_n 6= 0, be a sequence such thatJ(un)→sas n→ ∞. Sinces₀ < swe can choose (un)^∞_n=1 such that J(un)≥s₀ for alln and due to the equalityJ(u) =J(|u|) we may assume that u_n≥0. Then (3.1) implies that there existss₁ such that

s₀(kunk^α+kunk^p∗)≤s₁kunk^p

holds for alln; so we can find real numbers 0< δ₁< δ₂ such that

(3.2) δ₁≤ kunk ≤δ₂

hold for alln, and this implies that (un)^∞_n=1 is bounded inX. Due to the reflex- ivity ofX we may assume without loss of generality that for someu∈X we have u_n →u weakly in X and pointwise a.e. in R^N. (Remark that for any bounded open setB ⊂R^N we have for u∈X:

Z

B

|u|^p∗dx _p∗¹

≤ kuk^p∗≤C₁kuk

and

Z

B

|∇u|^pdx ¹_p

≤ k∇uk^p≤C₁^′kuk

and then

kukW^1,p(B)≤Ckuk

holds for allu∈X. The compact imbeddingW^1,p(B)֒→֒→L^p(B) then implies that unk →u in L^p(B) and hence pointwise a.e.). This implies that u≥0 a.e.

inR^N. Using the H¨older inequality, for all 0≤R≤ ∞and alln, we have

Z

|x|≥R

g(x)|un|^pdx ≤

Z

|x|≥R

|g(x)||un|^pdx

≤ Z

|x|≥R

|g(x)|^Np dx

_N^p Z

|x|≥R

|un|^p∗dx ^N_N^−p

≤C₂ Z

|x|≥R

|g(x)|^Np dx _N^p

,

whereC₂ is a constant independent ofR andn. The same holds also foru:

Z

|x|≥R

g(x)|u|^pdx ≤C₃

Z

|x|≥R

|g(x)|^Np dx _N^p

.

Sinceg∈L

N

p, we have lim

R→∞

R

|x|≥R

|g|^Np dx= 0, which implies that, for anyε >0, there existsR_ε>0 such that

Z

|x|≥Rε

g(x)|u|^pdx ≤ε

and

Z

|x|≥Rε

g(x)|un|^pdx ≤ε hold for alln.

On the other hand, using the Rellich-Kondrachov theorem (see [1]), and the continuity of the Nemytskii operator we prove, forε >0 fixed that:

Z

|x|<Rε

g(x)|u_nk|^pdx→ Z

|x|<Rε

g(x)|u|^pdxas n→ ∞.

Indeed, let us consider the functionF(x, t) :=g(x)|t|^p, then

|F(x, t)|=|g||t|^p< |g|^Np+δ

N

p +δ +|t|^m

m p

,

for allt∈Rand a.e.xin Bε:={x∈R^N;|x|< Rε}, wherem:=p

N p +δ′

and the dash denotes the exponent conjugate.

Hence the Nemytskii operatorN_F associated withFis continuous fromL^m(Bε) inL¹(Bε). Note that

N p < N

p +δ implies N

p +d ′

<

N p

′

= p^∗ p and hencem < p^∗. Then from imbeddings

X ֒→W^1,p(Bε)֒→֒→L^M(Bε) we conclude thatN_F(un)→N_F(u) inL¹(Bε), i.e.

Z

|x|<Rε

g(x)|u_n|^pdx→ Z

|x|<Rε

g(x)|u|^pdx as n→ ∞.

Finally, Z

g(x)(|un|^p− |u|^p)dx ≤

Z

|x|≥Rε

g(x)|un|^pdx +

Z

|x|≥Rε

g(x)|u|^pdx + +

Z

Bε

g(x)(|un|^p− |u|^p)dx ≤3ε fornlarge enough, which implies that (3.3)

Z

g(x)|un|^pdx→ Z

g(x)|u|^pdx as n→ ∞. Since we haveJ(un)≥s₀ for alln, then

Z

g(x)|un|^pdx≥s₀(kunk^α+kunk^p∗).

Due to (3.2) we have, Z

g(x)|u_n|^pdx≥s₀(δ^α₁ +δ₁^p∗) and (3.3) implies

Z

g(x)|u^p|dx≥s₀(δ^α₁ +δ₁^p∗)>0

and, therefore,u6≡0 inR^N. From the uniform boundedness principle, we obtain kuk^α+kuk^p∗≤lim inf(kunk^α+kunk^p∗)

and so

s= lim supJ(un) = lim sup R

g(x)|un|^pdx kunk^α+kunk^p∗

= lim sup

1

ku_nk^α+ku_nk^p∗ Z

g(x)|u|^pdx

≤

Rg(x)|u|^pdx

lim inf(kunk^α+kunk^p∗) ≤

R g(x)|u|^pdx

kuk^α+kuk^p∗ =J(u)

and, consequentlyJ(u) =s. The lemma is proved.

Now, we prove thatuis an eigenfunction corresponding to a positive eigenvalue λ >0. Sinceu6≡0 in R^N then for any fixedv∈X we can find ε₀ =ε₀(v)>0

such that ku+εvk > 0 holds for allε ∈ ]−ε₀, ε₀[ . We consider the function η: ]−ε₀, ε₀[→Rdefined as follows:

η(ε) =J(u+εv).

The functionF(ε) =R

a(x)|∇u+ε∇v|^pdx=ku+εvk^p is differentiable and F^′(ε) =p

Z

a(x)|∇u+ε∇v|^p−2(∇u+ε∇v)∇v dx.

Since ku+εvk >0 on ]−ε₀, ε₀[ then the same is true for F(ε), i.e. F(ε) > 0 on ]−ε₀, ε₀[ , and, therefore, the function G(ε) := ku+εvk^α = (F(ε))^αp is differentiable and

G^′(ε) = α

pF^′(ε)(F(ε))^αp−1. Atε= 0 we have

G^′(0) =αkuk^α−p Z

a(x)|∇u|^p−2∇u∇v dx.

The same remains true for the functionH(ε) =ku+εvk^p∗. Hence H^′(0) =p^∗kuk^p∗−p

Z

a(x)|∇u|^p−2∇u· ∇v dx.

Thus η is differentiable on ]−ε₀, ε₀[ . Since 0 is a maximum of η, we have η^′(0) = 0, which implies that

Z

a(x)|∇u|^p−2∇u∇v dx=λ Z

g(x)|u|^p−2v dx

holds for allv∈X, where

λ= p(kuk^α+kuk^p∗) (p^∗kuk^p∗−p+αkuk^α−p)R

g(x)|u|^pdx.

Proposition 3.2. Letu∈X be a weak solution for(1.1)such thatu6≡0,u≥0 a.e. inR^N. Thenu∈L^r for allp^∗ ≤r≤ ∞.

Proof: We use Nash-Moser bootstrap iterations similarly as in [3]. ForM >0 definev_M(x) = inf{u(x), M}and let choosev=v^kp+1_M (for somek >0) as a test function in (2.3). Then it is easy to see thatv∈X∩L^∞and that

Z

a(x)|∇u|^p−2∇u∇(v^κp+1_M )dx=λ Z

g(x)|u|^p−2uv^κp+1_M dx.

On one hand, due to (2.2) we have

(3.4) Z

a(x)|∇u|^p−2∇u∇(v_M^kp+1)dx= (kp+ 1) Z

a(x)|∇u|^p−2∇u∇v_Mv^kp_Mdx

≥(kp+ 1) Z

a(x)|∇v_M|^p v^kp_Mdx= kp+ 1 (k+ 1)^p

Z

a(x)|∇(v^k+1_M )|^pdx

≥ 1 C₁^p

kp+ 1 (k+ 1)^p

Z

v^(k+1)p_M ^∗dx _p∗^p

.

On the other hand,

(3.5)

Z

g(x)|u|^p−2uv^kp+1_M dx= Z

g(x)u^p−1v^kp+1_M dx

≤ Z

|g(x)|u^p−1v^kp+1_M dx≤ Z

|g(x)|u^(1+k)pdx

≤ kgk₍^N

p+δ)

Z

u^(k+1)qdx ^p_q

,

whereq=p

N p +δ′

. From (3.4) and (3.5) we obtain 1

C₁^p

kp+ 1 (k+ 1)^p

Z

v_M^(k+1)p∗dx _p∗^p

≤λkgkN p+δ

Z

u^(k+1)qdx ^p_q

.

Then there exists a constantC₃>0,C₃=λC₁^pkgkN p+δ

such that Z

v_M^(k+1)p∗dx _p∗^p

≤C₃(k+ 1)^p (kp+ 1)

Z

u^(k+1)qdx ^p_q

,

i.e.

(3.6) kv_Mk(k+1)p^∗ ≤C

1 k+1

4

"

k+ 1 (kp+ 1)^p1

# ¹

k+1

kuk(k+1)q,

where C₄ =C

1 p

3 >0. Sinceu∈X, it follows from (2.2) thatu∈L^p∗. Then we can choosek =k₁ in (3.6) such that (k1+ 1)q=p^∗ i.e. k₁ = ^p_q^∗ −1. Then we have

kv_Mk(k1+1)p^∗≤C

1 k1 +1

4

"

k₁+ 1 (k₁p+ 1)^1p

#_k¹

1+1

kuk^p∗.

But, lim

M→∞v_M(x) =u(x) and the Fatou lemma implies kuk(k1+1)p^∗ ≤C

1 k1+1

4

"

k₁+ 1 (k₁p+ 1)^p1

#_k¹

1+1

kuk^p∗.

Then u∈ L^(k1+1)p∗, and we can choose k = k₂ in (3.6) such that (k₂ + 1)q = (k1+ 1)p^∗ i.e.k₂ =^(p_q^∗2⁾² −1. Repeating the same argument we get

kuk(k2+1)p^∗≤C

1 k2 +1

4

"

k₂+ 1 (k₂p+ 1)^1p

#_k¹

2+1

kuk(k1+1)p^∗.

By induction

kuk(kn+1)p^∗≤C

1 kn+1

4

"

kn+ 1 (knp+ 1)^1p

# ¹

kn+1

kuk(k_n−1+1)p^∗

holds for alln∈N, wherekn=_p∗

q

n

−1. Then

kuk(kn+1)p^∗≤C

n

P

j=1 1 kj+1

4

n

Y

j=1





kj+ 1 (kjp+ 1)^1p





1 kj+1

kuk^p∗.

But lim

y→∞

"

y+1 (yp+1)^1p

#√¹ y+1

= 1 and

"

y+1 (yp+1)^1p

#√¹ y+1

>1 for ally >0. Then there exists a constantC₅ >0 such that

1<

"

kn+ 1 (knp+ 1)

1 p

#√ ¹ kn+1

< C₅ holds for alln∈N, and therefore,

kuk(kn+1)p^∗ ≤C

n

P

j=1 1 kj+1

4 C

n

P

j=1

√1 kj+1

5 kukp^∗,

since 





1

kj+1 =_q

p^∗

j

,_p^q_∗ <1,

√_k1

j+1 =q_q

p^∗

j

,q_q

p^∗ <1.

Then we conclude that there exists a constantC₆>0 such that kuk(kn+1)p^∗≤C₆kukp^∗

holds for alln∈N. Sincekn→ ∞asn→ ∞, we getu∈L^∞and by interpolation u∈L^r for allr∈h _{N p}

N−p,∞i

. This completes the proof of Proposition 3.2.

Proposition 3.3. Letu∈X,u≥0andu6≡0be a weak solution of (1.1). Then u >0 inR^N and

|x|→∞lim u(x) = 0.

Proof: The positivity ofufollows from the weak Harnack type inequality proved in [10, Theorem 1.1]. More precisely, due to Proposition 3.2 we have u∈ L^∞, and using Theorem 1.1 of [10] there exists a constantC_R>0 such that

(3.7) max

K(R)u(x)≤C_R min

K(R)u(x),

where K(R) denotes the cube in R^N of side R and center 0 whose sides are parallel to the coordinate axes. Let D ⊂ R^N be such that|D| 6= 0 and u≡ 0 a.e. inD. Then there exists R₀ >0 such that |D∩K(R₀)| 6= 0 (otherwiseD =

S

R∈Q

(D∩K(R)) will be of measure zero). Thus 0≤max

K(R)u(x)≤C_R min

K(R)u(x) = 0 holds for all R > R₀ which implies u ≡ 0 in K(R). Hence u = 0 a.e. in R^N, a contradiction. Thusu >0 inR^N. Finally, letBr(x) denote the ball centered at x∈R^N with radiusr >0. Then by Theorem 1 of [8], for someC=C(N, p)>0 we obtain an estimate:

kukL^∞(B1(x))≤C kukL^p∗(B2(x))

independently ofx∈R^N. Hence the decay ofufollows.

Proposition 3.4. The value of λ >0 (andu >0) is independent of the choice of α ∈ ]1, p[ at the beginning of the proof of Proposition 3.1. Namely, λ is simple, isolated and if ˜λis a positive eigenvalue of(1.1) andu˜is corresponding eigenfunction thenu˜changes sign in R^N.

Proof: The simplicity ofλfollows from the proof of Lemma 3.1 in [7] adapted for Ω = R^N. Remaining two facts (i.e. λis isolated and unique positive eigen- value having eigenfunctions which do not change sign) follow from the proof of Lemma 2.3 in [4]. In particular, this implies thatλanduare independent ofα.

The assertion of Theorem 2.1 follows now from Propositions 3.1–3.4. The assertion of Corollary follows directly from the regularity result [9].

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of AMS.

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Univ. Carolinae36(1995), 519–527.

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P. Dr´abek:

Department of Mathematics, University of West Bohemia, Univerzitn´ı 8, P.O. Box 314, 306 14 Plzeˇn, Czech Republic

Zakaria Moudan, Abdelfettah Touzani:

Laboratoire d’Analyse Non lineaire (LANOLIF), Departement de Mathematiques et Informatique, Universite Sidi Mohamed Ben Abdellah, Faculte des Sciences Dhar-Mahraz, B.P. 1796 Fes Atlas, Fes – Maroc

(Received October 10, 1996)