ON ELLIPTIC EQUATIONS IN R N WITH CRITICAL EXPONENTS ∗

ON ELLIPTIC EQUATIONS IN R ^N WITH CRITICAL EXPONENTS ^∗

C.O. Alves, J.V. Goncalves, & O.H. Miyagaki

Abstract

In this note we use variational arguments –namely Ekeland’s Principle and the Mountain Pass Theorem– to study the equation

−∆u+a(x)u=λu^q+u^2∗−1 inR^N.

The main concern is overcoming compactness difficulties due both to the unboundedness of the domainR^N, and the presence of the critical expo- nent 2^∗= 2N/(N−2).

1 Introduction

In this note we use variational methods to explore existence of weak solutions for the problem

(∗)





−∆u+a(x)u=λu^q+u^2∗−1 in R^N Ra(x)u²<∞, R

|∇u|²<∞ u≥0, u6≡0

whereais a nonnegative L^∞_loc function,λ≥0, 0< q≤1 and 2^∗ is the critical exponent, 2^∗= 2N/(N−2), forN ≥3.

This problem has been explored by many authors including Br´ezis & Niren- berg [6], Ambrosetti-Br´ezis & Cerami [1], Guedda & Veron [9] (see also their references) for the case of elliptic equations in bounded domains. As far as unbounded domains are concerned we recall the work by Benci & Cerami [12], Noussair-Swanson & Jianfu [3], Jianfu & Xiping [14], Egnell [7], Azorero &

Alonso [4], Miyagaki [10].

∗1991 Mathematics Subject Classifications: 35J20, 35K20.

Key words and phrases: Elliptic equations, unbounded domains, critical exponents, variational methods.

c1996 Southwest Texas State University and University of North Texas.

Submitted: August 7, 1996. Published October 22, 1996.

Partially supported by CNPq/Brasil.

1

In this work we shall assume the following condition ona.

(a1) a(x)>0, x∈B_R^c_o and Z

B^c

Ro

1 a <∞.

Our main results are the following:

Theorem 1 Let 0 < q < 1and assume (a1). Then there exists λ^∗ > 0such that(∗)has a solution for 0< λ < λ^∗,N ≥3.

Theorem 2 Let N ≥ 4 and q = 1. Assume (a1) and a(x) = 0, x ∈ B2r₀

for somer0∈(0, R0/2). Then there is λ^∗ >0such that (∗)has a solution for 0< λ < λ^∗.

These theorems complement the results in [10] in the sense that here the function a is allowed to vanish on a ball BR_o. Actually in Theorem 2, we requirea to vanish onB2r0. In addition we consider the case 0< q ≤1 while in [10],a >0 is continuous and q∈(1,2^∗).

2 Preliminaries

Let

E=

u∈ D^1,2 | Z

au²<∞

with inner product and norm given by hu, vi=

Z

(∇u∇v+auv), kuk²= Z

|∇u|²+au² .

Recall thatD^1,2 is the closure ofCo^∞with respect to the gradient normkuk²1= R|∇u|². Moreover

D^1,2=n

u∈L^2∗ |∂iu∈L²o and the norm

kuk⁰ ≡ |u|L^2∗+|∇u|L²

is equivalent to theD^1,2 norm. In additionD^1,2→L^2∗.

The following lemma is a variant of a result by Willem & Omana [13] and by Costa [2].

Lemma 1 Assume (a1). Then E → L^s for 1 ≤ s ≤ 2^∗ and E ,→ L^s for 1≤s <2^∗.

We shall look for the critical points of the functional I(u) =1

2 Z

|∇u|²+a|u|²

− 1 q+ 1

Z

λu^q+1₊ − 1 2^∗

Z u²₊^∗ in the Hilbert spaceE.

Using standard techniques we can show that I ∈ C¹(E,R), and that its derivative is given by

hI⁰(u), vi= Z

(∇u∇v+auv)−λ Z

u^q+v−Z

u²+^∗−1v . Therefore, the critical points ofI are the weak solutions of (∗).

The following auxiliary result concerns the geometry ofI.

Lemma 2 Ifasatisfies(a1)and if0< q≤1then there existsλ^∗>0such that if0< λ < λ^∗ then

(i) I(u)≥r, kuk=ρ, for some r, ρ >0 If in additionφ≥0, φ6≡0, andφ∈E then

(ii) I(tφ)→ −∞ as t→ ∞

(iii) I(tφ)<0, for small t >0, and0< q <1.

3 Proofs

For the sake of completeness, we present a proof of Lemma 3, which is based on the proof in [2].

Proof of Lemma 3. At first letR > Ro. Then we have Z

B_R^c

|u|= Z

B^c_R

a^1/2|u| a^1/2 ≤ Z

B^c_R

1 a

!1/2 Z

B_R^c

a|u|²

!1/2

≤Ckuk which shows that

|u|L¹≤Ckuk, u∈E.

Now using the interpolation inequality

|u|^s≤ |u|^α1|u|¹r^−α, α+1−α

r = 1

s, 1≤s≤r≤2^∗, 0≤α≤1 and the embeddingE→L^2∗, we infer thatE→L^s, 1≤s≤2^∗.

On the other hand, for sufficiently largeR >0 we have Z

B_R^c

1 a< . So ifun*0 inE, then for largen

Z

B_R^c |un| ≤C Z

B_R^c

1 a ≤ . Using compact Sobolev embeddings we also have

un→0 inL¹(BR)

so thatun → 0 in L¹. Using again the interpolation inequality stated above, one concludes the proof of Lemma 3.

Proof of Lemma 4.

Verification of(i). From the continuous embedding in Lemma 3, we have I(u) ≥ ¹₂kuk²−^λc_q+1^q+1kuk^q+1−^S−2₂∗^∗/2kuk^2∗

≥ kuk^q+1

1

2kuk^2−(q+1)−^λcq+1^q+1 −^S−2^2∗/2∗ kuk^2∗−(q+1) whereS is the best constant for the embeddingD^1,2→L^2∗, that is

S= inf

( R|∇u|²

R |u|^2∗2/2^∗ | u∈ D^1,2, u6≡0 )

.

Letting

Q(t)≡ 1

2t^2−(q+1)−S^−2∗/2

2^∗ t^2∗−(q+1), t≥0, there isρ >0 such that

maxt≥0 Q(t) =Q(ρ)>0.

Takingkuk=ρand λ^∗= _c^q+1

q+1Q(ρ) we get (i).

Verification of(ii). Takingφ6≡ 0,φ≥0, φ∈E we have I(tφ) = t²

2kφk²− t^q+1 q+ 1λ

Z

φ^q+1−t^2∗ 2^∗

Z φ^2∗ which gives (ii).

Verification of(iii). It is clear from the expression ofI(tφ) above taking into account that 0< q <1.

Proof of Theorem 1. By the proof of lemma 4,Iis bounded from below on Bρ. By the Ekeland Principle [8], there existsu∈Bρ such that

I(u)≤inf

B¯_ρ I+ and

I(u)< I(u) +ku−uk, u6≡ u. Now since 0< q <1 it follows that

I(tφ)<0, for smallt >0, φ6≡ 0, andφ∈ Co^∞. Again by Lemma 4

∂Binf_ρI≥r >0 and inf

B_ρ

I <0.

Choose >0 such that

0< < inf

∂B_ρI−inf

B_ρ

I.

Hence

I(u)< inf

∂Bρ

I so that

u∈Bρ. Hence letting

F(u)≡I(u) +ku−uk we notice thatuis a point of minimum of F onBρand so

I(u+δv)−I(u)

δ +kvk ≥ 0

which by passing to the limit asδ→0 gives that hI⁰(u), vi+kvk ≥0

and hencekI⁰(u)k ≤. Therefore, there is a sequence un∈Bρ such that I(un)→c^∗≡inf

Bρ

I <0 and I⁰(un)→0.

Since of courseun is bounded,

un* u^∗ in E

and

un →u^∗ a.e. in R^N. Now passing to the limit in

o(1) = Z

(∇un∇φ+aunφ)−λ Z

u^q_n+φ− Z

u²_n+^∗−1φ, φ∈E we infer thatI⁰(u^∗) = 0 showing thatu^∗ is a solution of problem (∗).

In order to show thatu^∗6≡ 0, we follow the arguments in [6]. Assume that u^∗≡0 and that

kunk²→`≥0.

UsingI⁰(un)→0 we have

kunk²− Z

u²_n+^∗ =o(1) so thatR

u²n+^∗ →`and from the expression c^∗+o(1) =λ

1 2− 1

q+ 1 Z

u^q+1_n+ + 1 N

Z u²_n+^∗ we infer that

c^∗= ` N which is impossible.

Proof of Theorem 2. By Lemma 4 and the Mountain Pass Theorem, there exists a sequenceun in E such that

I(un)→c and I⁰(un)→0 where

c= inf

γ∈Γ max

0≤t≤1I(γ(t)), c≥r and

Γ ={γ∈C([0,1], X) | γ(0) = 0, γ(1) =e} wheree∈E satisfiesI(e)≤0.

Claim. There ise≡eλ such that 0< c < _N¹S^N2,0< λ < λ^∗. From the expression

hI⁰(un), uni −2^∗I(un) =

1−2^∗ 2

kunk²+λ 2^∗

2 −1 Z

u²_n+

one shows, by takingλ^∗>0 smaller than the one found in lemma 4, thatun is bounded. So that, passing to subsequences,

un * uin E andun →ua.e. inR^N for someu∈ E.

Remark. I(un+)→c andI⁰(un+)→0.

Indeed,un− is also bounded so that o(1) =hI⁰(un), un−i=

Z

(|∇un−|²+au²_n−). Moreover, ifφ∈E then

hI⁰(un+), φi = hun+, φi −λ Z

un+φ− Z

u²_n+^∗−1φ

= hI⁰(un), φi − hun−, φi so that,I⁰(un+)→0.On the other hand,

I(un)−1 2 Z

(|∇un−|²+au²_n−)

= 1

2 Z

(|∇un+|²+au²_n+)−λ 2 Z

u²_n+− 1 2^∗

Z u²_n+^∗

= I(un+),

which givesI(un+)→c. So we may assume thatun≥0 and thusu≥0.

Now as in the proof of Theorem 1 one shows thatusatisfies the equation in (∗). Again arguing as in [6] we assume thatu≡0. Then

kunk²→` for some`≥0 and using the facts that

I(un)→c, I⁰(un)→0

we infer thatc≥ N¹S^N/2, contradicting 0< c < _N¹S^N/2given by the Claim.

Proof of the Claim. (Arguments adapted from [6].) Consider the cut-off functionφ∈ Co^∞such that

φ≡1 on Br0, φ≡0 on R^N\B2r0. Now consider the function

w(x) = [N(N−2)]^(N−2)/4

(+|x|²)^(N−2)/2 , x∈R^N, >0 which satisfies

−∆w=w^2∗−1 inR^N. It is well known (see e.g. Talenti [5], Aubin [11] ) that

kwk²1 =|w|²2^∗∗ = S^N/2.

Let

ψ=φw

and letv∈ Co^∞ given by

v= ψ

Rψ2∗^1/2∗. Now it can be shown (see e.g. [6], [10]) thatX≡R

|∇v|² satisfies X≤S+O(^(N−2)/2).

Moreover there is somet>0 such that

maxt≥0 I(tv) =I(tv)

and d

dtI(tv)|^t=t= 0.

which gives

0< t< X^1/(2∗−2)≡t0.

Notice thata= 0 onB2r0 andv= 0 onR^N\B2r0. Moreovert≥d0≡d0(r0) for some d0 > 0. Otherwise sinceX is bounded, if t → 0, thenI(tv)→ 0 contradicting

I(tv) = max

t≥0 I(tv)≥r >0 given by lemma 4 (i). On the other hand

I(tv) = t² 2

Z

|∇v|²−t^2∗ 2^∗ −λt²

2 Z

v²

≤ t²

2t²₀^∗−2−t^2∗ 2^∗

−λt² 2

Z

B2r0

v². Now recalling that as a function oft,

t²

2t²₀^∗−2−t^2∗ 2^∗

increases on the interval (0, t0) we get I(tv) ≤ t²₀^∗

1 2− 1

2^∗

−λt² 2

Z

B2r0

v²

≤ 1

Nt²0^∗−λd²₀ 2

Z

B_2r0

v²

≤ 1 N

h

S+O

^(N−2)/2i2^∗/(2^∗−2)

−λd²₀ 2

Z

B_2r0

v²

= 1

N h

S+O

^(N−2)/2 iN/2

−λd²₀ 2

Z

B2r0

v².

Using the inequality

(b+c)^α≤b^α+α(b+c)^α−1c b, c≥0, α≥1 withb=S,c=O ^(N−2)/2

andα=N/2 we get I(tv)≤ 1

NS^N/2+O(^(N−2)/2)−c0λ Z

B_2r0

v².

Therefore,

I(tv)≤ 1

NS^N/2+^(N−2)/2 (

M−c0λ^(2−N)/2 Z

B2r0

v² )

, wherec0=d²₀/2 and M is a positive constant.

We shall show that ^(N−2)/2

(

M−c0λ^(N−2)/2 Z

B_2r0

v² )

<0, for small >0. So that

I(tv)< 1 NS^N/2 and hence

0< c < 1 NS^N/2. Noticing that

d1≤ Z

B_2r0

ψ^2∗≤d2, for some d1, d2>0, (see [6]), it follows by a change of variables that

I(tv)≤ 1

NS^N/2+^(N−2)/2 (

M−c0λ^(4−N)/2

Z r0^−1/2 0

s^N−1ds (1 +s²)^N−2

) .

We are going to consider separately the casesN = 4 andN ≥5.

Case N = 4. We have I(tv) ≤ 1

4S²+ (

M−c0λ

Z r₀^−1/2 0

s³ds (1 +s²)²

)

≤ 1

4S²+n

M−c0λln(r0^−1/2)o . Now since

c0λln(r0^−1/2)→ ∞as→0 we infer that

I(tv)<1

4S², for small >0.

Case N≥5. Noticing that c0λ^(4−N)/2

Z r₀^−1/2 0

s^N−1ds

(1 +s²)^N−2 → ∞as→0 we infer that

I(tv)< 1

NS^N/2for small >0, which concludes the proof of this claim.

References

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[12] V. Benci & G. Cerami,Existence of positive solutions of the equation−∆u+

a(x)u=u^{(n+2)/(n−2)}in Rⁿ, J. Funct. Anal. 88(1990) 90-117.

[13] W. Omana & M. Willem, Homoclinic orbits for a class of Hamiltonian systems, Diff. Int. Equations 5(1992) 1115-1120.

[14] Y. Jianfu & Z. Xiping, On the existence of nontrivial solution of a quasi- linear elliptic boundary value problem for unbounded domains, Acta Math.

Sci. 7(1987) 341-359.

C.O. Alves

Dep. Mat. Univ. Fed. Paraiba

58109-970 - Campina Grande(PB), Brasil E-mail [email protected]

J.V. Goncalves

Dep. Mat. Univ. Bras´ılia 70.910-900 Brasilia(DF), Brasil E-mail [email protected] O.H. Miyagaki

Dep. Mat. Univ. Fed. Vi¸cosa 36570-000 Vi¸cosa(MG), Brasil E-mail [email protected]