Multiple positive solutions for equations involving critical Sobolev exponent in R N ∗

Multiple positive solutions for equations involving critical Sobolev exponent in R ^{N ∗}

C. O. Alves

Abstract This article concerns with the problem

−div(|∇u|^m−2∇u) =λhu^q+u^m∗−1, in R^N.

Using the Ekeland Variational Principle and the Mountain Pass Theorem, we show the existence of λ^∗ > 0 such that there are at least two non- negative solutions for eachλin (0, λ^∗).

1 Introduction

In this work, we study the existence of solutions for the problem (P)

−∆mu=λhu^q+u^m∗−1, R^N u≥0, u6= 0, u∈D^1,m(R^N)

where ∆mu= div (|∇u|^m−2∇u),λ >0,N > m≥2,m^∗=N m/(N−m), 0<

q < m−1,his a nonnegative function withL^Θ(R^N) with Θ = _{N m−(q+1)(N}^{N m} _−m), and

D^1,m(R^N) ={u∈L^m∗(R^N)| ∂u

∂xi ∈L^m(R^N)} endowed with the normkuk= R

|∇u|^m1/m

.

The case q= 0, m= 2 was studied by Tarantello [20], and a more general case withm≥2 by Cao, LI & Zhou [5]. In these two references, [5] and [20], it is proved that (P) has multiple solutions. In the casem= 2,h∈L^p(R^N) with p1 ≤p ≤p2 and 1 < q < 2^∗−1, Pan [18] proved the existence of a positive solution for (P). In the more general case, m≥2, h∈L^Θ(R^N), Gon¸calves &

Alves [10] proved the existence of a positive solution for (P).

∗1991 Mathematics Subject Classifications: 35J20, 35J25.

Key words and phrases: Mountain Pass Theorem, Ekeland Variational Principle.

c1997 Southwest Texas State University and University of North Texas.

Submitted April 22,1997. Published August 19, 1997.

1

By a solution to (P), we mean a functionu∈D^1,m(R^N),u≥0 and u6= 0 satisfying

Z

|∇u|^m−2∇u∇Φ =λ Z

hu^qΦ + Z

u^m∗−1Φ, ∀Φ∈D^1,m(R^N).

Hereafter, R

, D^1,m, L^p and |.|^p stand for R

R^N, D^1,m(R^N), L^p(R^N) and |.|^Lp respectively.

In the search of solutions we apply minimizing arguments to the energy functional

I(u) = 1 m

Z

|∇u|^m− λ q+ 1

Z

h u⁺q+1

− 1 m^∗

Z

u⁺m^∗

(1) associated to (P), whereu⁺(x) = max{u(x),0}. Note that the conditionh∈L^Θ implies thatI∈C¹ D^1,m,R

.

To show the existence of at least two critical points of the energy functional, we shall use the Ekeland Variational Principle [8], and the Mountain Pass Theo- rem of Ambrosetti & Rabinowitz [2] without the Palais-Smale condition. Using the Ekeland Variational Principle, we obtain a solutionu1 withI(u1)<0, and by the Mountain Pass Theorem we prove the existence of a second solutionu2

with I(u2) >0. Techniques for finding the solutions u1 and u2 are borrowed from Cao, Li & Zhou [5]. Then we combine these techniques with arguments developed by Chabrowski [6], Noussair, Swanson & Jianfu [17], Jianfu & Xip- ing [12], Azorero & Alonzo [9], Gon¸calves & Alves [10] and Alves, Gon¸calves &

Miyagaki [1] to obtain the following result

Theorem 1 There exists a constant λ^∗ > 0, such that (P) has at least two solutions,u1 andu2, satisfying

I(u1)<0< I(u2) ∀λ∈(0, λ^∗).

2 Preliminary Results

In this section we establish some results needed for the proof of Theorem 1.

Definition. A sequence{un} ⊂D^1,mis called a (P S)csequence, ifI(un)→c andI⁰(un)→0.

Lemma 1 If {un}is a (P S)c sequence, then {un}is bounded, and {u⁺_n} is a (P S)c sequence.

Proof. Using the hypothesis that {un} is a (P S)c sequence, there exist no

andM >0 such that I(un)− 1

m^∗I⁰(un)un≤M+kunk ∀n≥no. (2) Now, using (1) and the H¨older’s inequality, we have

I(un)− 1

m^∗I⁰(un)un≥ 1

N kunk^m+c1kunk^q+1 (3) where c1 is a constant depending of N, m, q,khk^Θ and Θ. It follows from (2) and (3) that{un} is bounded. Now, we shall show that{u⁺_n}is a also (P S)c

sequence. Since{un}is bounded, the sequenceu⁻_n =un−u⁺_n is also bounded.

Then

I⁰(un)u⁻_n →0 and we conclude that

u⁻_n→0. (4) From (4) we achieve that

kunk=u⁺_n+on(1). (5) Therefore, by (4) and (5)

I(un) =I(u⁺_n) +on(1) and

I⁰(un) =I⁰(u⁺_n) +on(1),

which consequently implies that{u⁺_n}is a (P S)c sequence. ²

From Lemma 1, it follows that any (P S)c sequence can be considered as a sequence of nonnegative functions.

Lemma 2 If {un}is a (P S)c sequence with un * u inD^1,m, then I⁰(u) = 0, and there exists a constantM >0depending ofN, m, q,|h|Θ andΘ, such that

I(u)≥ −M λ^Θ

Proof. If{un}is a (P S)c sequence withun* u, using arguments similar to those found in [10], [12] and [17], we can obtain a subsequence, still denoted by un, satisfying

un(x) → u(x) a.e. in R^N (6)

∇un(x) → ∇u(x) a.e. in R^N (7)

u(x) ≥ 0 a.e. in R^N. (8)

From (6), (7) and using the hypothesis that{un}is bounded inD^1,m, we get

I⁰(u) = 0, (9)

which in impliesI⁰(u)u= 0, and kuk^m=λ

Z

hu^q+1+ Z

u^m∗.

Consequently

I(u) =λ 1

m− 1

q+ 1 Z

hu^q+1+ 1 N

Z u^m∗.

Using H¨older and Young Inequalities, we obtain I(u)≥ −1

N |u|^mm^∗∗−M λ^Θ+ 1

N |u|^mm^∗∗=−M λ^Θ

whereM=M(N, m, q,Θ,khkΘ). ²

For the remaining of this article, we will denote by S the best Sobolev constant for the imbeddingD^1,m,→L^m∗.

Lemma 3 Let{un} ⊂D^1,m be a(P S)c sequence with c < 1

NS^N/m−M λ^Θ,

whereM >0is the constant given in Lemma 2. Then, there exists a subsequence {unj}that converges strongly in D^1,m.

Proof By Lemmas 1 and 2, there is a subsequence, still denoted by{un}and a functionu∈D^1,m such thatun* u. Letwn =un−u. Then by a lemma in Brezis & Lieb [3], we have

kwnk^m = kunk^m− kuk^m+on(1) (10) kwnk^mm^∗∗ = |un|^mm^∗∗− |u|^mm^∗∗+on(1). (11) Using the Lebesgue theorem (see Kavian [13]), it follows that

Z

hu^q+1_n −→

Z

hu^q+1. (12)

From (10), (11) and (12), we obtain

kwnk^m=|wn|^mm^∗∗+on(1) (13)

and 1

mkwnk^m− 1

m^∗ |wn|^mm^∗∗=c−I(u) +on(1). (14)

Using the hypothesis that{wn}is bounded inD^1,m, there existsl≥0 such that kwnk^m→l≥0. (15) From (13) and (15), we have

|wn|^mm^∗∗→l, (16) and using the best Sobolev constantS and recalling that

kwnk^m≥S Z

|wn|^m∗ m/m^∗

, (17)

we deduce from (15), (16) and (17) that

l≥Sl^m/m∗. (18)

Now, we claim thatl= 0. Indeed, ifl >0,from (18)

l≥S^N/m. (19)

By (14), (15) and (16), we have 1

Nl=c−I(u). (20)

From (19), (20) and Lemma 2 we get c≥ 1

NS^N/m−M λ^Θ,

but this result contradicts the hypothesis. Thus,l= 0 and we conclude that un→u in D^1,m.

3 Existence of a first solution (Local Minimiza- tion)

Theorem 2 There exists a constantλ^∗₁>0such that for 0< λ < λ^∗₁ Problem (P) has a weak solution u1 with I(u1)<0.

Proof. Using arguments similar to those developed in [5], we have I(u)≥

1 m−

kuk^m+o(kuk^m)−C()λ^{m/(m−(q+1))},

whereC() is a constant depending on >0. The last inequality implies that for small, there exist constantsγ, ρandλ^∗₁>0 such that

I(u)≥γ >0, kuk=ρ , and 0< λ < λ^∗₁.

Using the Ekeland Variational Principle, for the complete metric spaceBρ(0) withd(u, v) =ku−vk, we can prove that there exists a (P S)γ_osequence{un} ⊂ Bρ(0) with

γo= inf{I(u)|u∈Bρ(0)}.

Choosing a nonnegative function Φ ∈ D^1,m\{0}, we have that I(tΦ) < 0 for smallt >0 and consequentlyγo<0.

Takingλ^∗₁ >0,such that 0< 1

NS^N/m−M λ^Θ ∀λ∈(0, λ^∗₁)

from Lemma 3, we obtain a subsequence{un_j} ⊂ {un}and u1 ∈D^1,m, such that

un_j →u in D^1,m. Therefore,

I⁰(u1) = 0 and I(u1) =γo<0,

which completes this proof. ²

4 Existence of a second solution (Mountain Pass)

In this section, we shall use arguments similar to those explored by Cao, Li &

Zhou [5], Chabrowski [6], Noussair, Swanson & Jianfu [17], Jianfu & Xiping [12]

and Gon¸calves & Alves [10] to obtain the following

Theorem 3 There exists a constantλ^∗₂ >0such that for 0< λ < λ^∗₂ Problem (P) has a weak solutionu2 with I(u2)>0.

Proof. By arguments found in [5] and [10], we can prove that there exists δ1 > 0 such that for all λ ∈ (0, δ1), the functional I has the Mountain Pass Geometry, that is:

(i) There exist positive constantsr, ρwithI(u)≥r >0 for kuk=ρ (ii) There existse∈D^1,m withI(e)<0 andkek> ρ .

Then by [16], there exists a (P S)γ₁ sequence{vn}with γ1= inf

g∈Γ max

t∈[0,1]I(g(t)) where

Γ ={g∈C([0,1], D^1,m)|g(0) = 0 and g(1) =e}.

Using the next claim, which is a variant of a result found in [5], we can complete the proof of this theorem.

Claim. There existsλ^∗₂>0 such that for the constantM given by Lemma 2, 0< γ1< 1

NS^N/m−M λ^Θ ∀λ∈(0, λ^∗₂).

Assuming this claim, by Lemma 3 there exists a subsequence {vn_j} ⊂ {vn} and a functionu2∈D^1,m such thatvn_j →u2. Therefore,

I⁰(u2) = 0 and I(u2) =γ1>0.

Which concludes the present proof. ²

Verification of the above claim. Forx∈R^N, let

Ψ(x) =

N

N−m m−1

m−1(N−m)/m²

h

1 +|x|^m/(m−1)i

N−m m

.

Then it is well known that (see [7] or [19])

kΨk^m=|Ψ|^mm^∗∗=S^N/m. (21) Letδ2>0 such that

1

NS^N/m−M λ^Θ>0 ∀λ∈(0, δ2). Then from (1) and (21), we have

I(tΨ)≤t^m mS^N/m, and there existsto∈(0,1) with

sup

0≤t≤to

I(tΨ)< 1

NS^N/m−M λ^Θ ∀λ∈(0, δ2). Moreover, from (1) and (21), we have

I(tΨ) = t^m

m −t^m∗ m^∗

S^N/m−λt^q+1 q+ 1

Z

hΨ^q+1, and remarking that

t^m m −t^m∗

m^∗

≤ 1

N ∀t≥0, we obtain

I(tΨ)≤ 1

NS^N/m−λt^q+1 q+ 1

Z

hΨ^q+1;

therefore,

sup

t≥to

I(tΨ)≤ 1

NS^N/m−λt^q+1₀ q+ 1

Z

hΨ^q+1.

Now, takingλ >0 such that

−λt^q+1₀ q+ 1

Z

hΨ^q+1<−M λ^Θ

that is,

0< λ < t^q+1₀ R hΨ^q+1 M(q+ 1)

!1/(Θ−1)

=δ3

we deduce that sup

t≥to

I(tΨ)< 1

NS^N/m−M λ^Θ ∀λ∈(0, δ3). Choosingλ^∗₂ = min{δ1, δ2, δ3}, we have

sup

t≥0

I(tΨ)< 1

NS^N/m−M λ^Θ ∀λ∈(0, λ^∗₂). and consequently

0< γ1< 1

NS^N/m−M λ^Θ ∀λ∈(0, λ^∗₂) which proves the claim.

Proof of Theorem 1. Theorem 1 is an immediate consequence of Theorems 2 and 3.

Remark. Using Lemma 3 and the same arguments explored by Azorero &

Alonzo, in the case 0 < q < p [9], we can easily show that for small λ the following problem has infinitely many solutions with negative energy levels.

(P)_∗ −∆mu =λh|u|^q−1u+|u|^m∗−2u, in R^N u∈D^1,m

This result is obtained using the concept and properties of genus, and working with a truncation of the energy functional associated with (P)_∗.

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C. O. Alves

Universidade Federal da Para´ıba - PB - Brazil E-mail address: [email protected]