MULTIPLE SOLUTIONS FOR A PROBLEM WITH RESONANCE …

MULTIPLE SOLUTIONS FOR A PROBLEM WITH

RESONANCE INVOLVING THE p-LAPLACIAN

C. O. ALVES∗, P. C. CARRIAO∗∗ AND O. H. MIYAGAKI∗∗∗

Abstract. In this paper we will investigate the existence of multiple solu-tions for the problem

(P ) −∆pu + g(x, u) = λ1h(x) |u|p−2 u, in Ω, u ∈ H1,p0 (Ω)

where ∆pu = div(|∇u|p−2 ∇u

)is the p-Laplacian operator, Ω ⊆ IRN is a

bounded domain with smooth boundary, h and g are bounded functions,N ≥ 1 and 1 < p < ∞. Using the Mountain Pass Theorem and the EkelandVariational Principle, we will show the existence of at least three solutionsfor (P).

1. Introduction

In this paper, we will investigate the existence of multiple solutions forthe problem

−∆pu+ g(x, u) = λ1h(x)|u|p−2u, in Ω,

u = 0 on ∂Ω,(P)

where ∆pu =div(|∇u|p−2 ∇u) is the p-Laplacian operator, 1 < p < ∞,N ≥ 1, Ω is a bounded domain with smooth boundary,

g : Ω× IR → IR is bounded continuous functionsatisfying g(x, 0) = 0,(G1)

and its primitive denoted by

1991 Mathematics Subject Classification. Primary: 35A05, 35A15, 35J20.Key words and phrases. Radial solutions, Critical Sobolev exponents, Palais-Smale con-

dition, Mountain Pass Theorem.∗ Supported in part by Fapemig/ Brazil.∗∗ Supported in part by Fapemig and CNPq/Brazil.∗∗∗ Supported in part by Fapemig and CNPq/ Brazil.Received: March 18, 1998.

c©1996 Mancorp Publishing, Inc.

191

192 C. O. ALVES, P. C. CARRIAO AND O. H. MIYAGAKI

G(x, s) =s∫

0

g(x, t)dt is assumed to be bounded,(G2)

λ1 is the first eigenvalue of the following eigenvalue problem with weight −∆pu = λ1h(x)|u|p−2u, in Ω,u = 0 on ∂Ω,(PA)

where

(h) 0 ≤ h ∈ L∞(Ω) with h > 0 on subset of Ω with positive measure.

We recall that λ1 is simple, isolated and it is the unique eigenvalue withpositive eigenfunction Φ1 (see [1] or [2]). There are many papers treatingproblem (P ) with h = 1, among others, we would like to mention Lazer& Landesman [3], Ahmad, Lazer & Paul [4], De Figueiredo & Gossez [5],Amann, Ambrosetti & Mancini [6], Ambrosetti & Mancini [7], Thews [8],Bartolo, Benci & Fortunato [9], Ward [10], Arcoya & Canada [11], Costa &Silva [12], Fu [13], Goncalves & Miyagaki [14] when p = 2, and Boccardo,Drabek & Kucera [15], Anane & Gossez [16], Ambrosetti & Arcoya [17],Arcoya & Orsina [18], Fu & Sanches [19] when p = 2.We shall show in this paper, the existence of multiple solutions for problem(P ), by using similar arguments explored in [14] and [19]. Combining aversion of the Mountain Pass Theorem due to Ambrosetti & Rabinowitz(see [20] and [25]) and the Ekeland variational principle (see [21, Theorem4.1]), we will find nontrivial critical points of Euler- Lagrange functionalassociated to (P ) given by

I(u) =1p

∫Ω

|∇u|p − λ1p

∫Ωh |u|p +

∫ΩG(x, u) , u ∈ H1,p

0 (Ω),(1)

which are weak solutions of (P).Hereafter, we will denoted by ‖ ‖ and | |p the usual norms on the spacesH1,p

0 (Ω) and Lp(Ω) respectively, and by W the closed subspace

W =u ∈ H1,p

0 (Ω) /∫Ωhu |Φ1|p−2Φ1 = 0

.

We can easily prove that W is a complementary subspace of 〈Φ1〉. Thereforewe have the following direct sum (see e.g. Brezis [22])

H1,p0 (Ω) = 〈Φ1〉 ⊕ W.

We will be denoted by λ2, the following real number

λ2 = infu∈W

∫Ω

|∇u|p ;∫Ωh |u|p = 1

,

and we remind that this value is the second eigenvalue of the p-Laplacian(see [23] or [24]).From simplicity and isolation of λ1 (see [1] or [2]), we have 0 < λ1 < λ2 andby definition of λ2 it follows that∫

Ωh |w|p ≤ 1

λ2

∫Ω

|∇w|p , ∀w ∈ W.

MULTIPLE SOLUTIONS FOR A PROBLEM WITH RESONANCE 193

In this work, we will impose the following condition

g(x, t)t → 0, as |t| → ∞, ∀x ∈ Ω,(G3)

which appeared in [7] for p = 2 and [17] for the general case p > 1. Thiscondition together with the assumptions on the limits

T (x) = lim inf|t|→∞

G(x, t) and S(x) = lim sup|t|→∞

G(x, t), ∀x ∈ Ω,

imply that problem (P ) is in the class of the strongly resonance problem inthe sense of Bartolo-Benci & Fortunato [9].The following condition means a nonresonance with higher eigenvalues

G(x, t) ≥(λ1−λ2

p

)h(x) |t|p , ∀x ∈ Ω, ∀t ∈ IR.(G4)

In addition to (G3) which is a behaviour of g at infinity, we assume a condi-tion of the behaviour of G at origin

there exist 0 < δ and 0 < m < λ1 such thatG(x, t) ≥ m

p h(x) |t|p , for |t| < δ, ∀x ∈ Ω.(G5)

Our main result is the following:

Theorem 1. Assume conditions (h), (G1)-(G5). Then, problem (P) has atleast three solutions u1, u2 and u3, with

I(u1), I(u2) < 0 and I(u3) > 0,

provided that the following conditions hold

there exist t−, t+ ∈ IR with t− < 0 < t+ such that∫ΩG(x, t±Φ1) ≤ ∫

Ω T (x)dx < 0,(G6)

and ∫ΩS(x)dx ≤ 0.(G7)

Remark 1. Theorem 1 improves in some sense the main result proved in[14], since the proof given in [14] works only in Hilbert space framework.

2. Preliminary Results

In this section, we will state and prove some results required in the proofof Theorem 1. We recall that I : H1,p

0 (Ω) →IR is said to satisfy Palais-Smale condition at the level c ∈IR ((PS)c in short), if any sequence un ⊂H1,p

0 (Ω) such that

I(un) → c and I ′(un) → 0,

possesses a convergent subsequence in H1,p0 (Ω).

Lemma 1. Assume (h), (G1) and (G2). Then I satisfies the (PS)c condi-tion ∀ c < ∫

Ω T (x)dx.


Proof. We are going to adapt some arguments used in [16, p.1148]. First ofall, we shall show that un is bounded. Assume that un is unbounded,therefore, up to subsequence, we have

‖un‖ → ∞.

Letting

vn =un

‖un‖ ,(*n)

we can assume that there exists v ∈ H1,p0 (Ω) such that

vn v in H1,p0 (Ω)

andvn → v in Ls(Ω), for 1 ≤ s < p∗ =

Np

N − p.

Now, we will show that v = 0 and that there exists γ ∈IR such that

v(x) = γΦ1(x), ∀x ∈ Ω.

From (1) and choosing tn = ‖un‖ , we obtainI ′(un)un

tpn=

∫Ω

|∇vn|p − λ1

∫Ωh |vn|p + 1

tpn

∫Ωg(x, un)un.(2)

Using (G1) together with the fact that

limn→∞

I ′(un)un

tpn= 0,

we get ∫Ωh |v|p = 1

λ1(3)

and therefore v = 0.Using the weak convergence vn v, we know that

‖v‖ ≤ 1.(4)

By (3) and (4), it follows that v is an eigenfunction for λ1. Then there existsγ ∈IR such that

v(x) = γΦ1(x), ∀x ∈ Ω.(5)

In particular,un

‖un‖ → γΦ1, ∀x ∈ Ω,

which implies|un(x)| → ∞, ∀x ∈ Ω,

and by (G2) and Fatou’s lemma, we have

(6) lim infn→∞

∫ΩG(x, un(x))dx ≥

∫Ωlim infn→∞ G(x, un(x))dx ≥

∫ΩT (x)dx.

Now, using the inequality

c+ on(1) = I(un) ≥∫ΩG(x, un(x))dx,(7)


we have by (6) that

c ≥∫ΩT (x)dx,

which contradicts the hypothesis on the level c, then un is bounded.Let u ∈ H1,p

0 (Ω) be a function such that un u, using a similar argumentsexplored in [18], we can conclude that

un → u in H1,p0 (Ω),

and Lemma 1 follows.

3. Existence of two solutions (Ekeland’s principle)

We will denote by Q± the following sets

Q+ = tΦ1 + w, t ≥ 0 and w ∈ Wand

Q− = tΦ1 + w, t ≤ 0 and w ∈ W .It is easy to see that

∂Q+ = ∂Q− =W.

Lemma 2. If conditions (h),(G2) and (G6) hold, then functional I is bound-ed from below on H1,p

0 (Ω). Moreover, the infimum is negative on Q+ andQ−.

Proof. From condition (G2), its easy to see that I is bounded from below onH1,p

0 (Ω).Using condition (G6), we have

I(t±Φ1) =∫ΩG(x, t±Φ1) ≤

∫ΩT (x)dx < 0,

thereforeinf

u∈Q±I(u) < 0.

Remark 2. Using condition (G4) and the definition of the number λ2, weremark that

I(w) ≥ 1p

∫Ω

|∇w|p − λ2p

∫Ωh(x) |w|p ≥ 0, ∀w ∈ W.

Therefore Lemma 2 implies that if the infimum of I on Q± is achieved by,for example, u±

0 ∈ Q±, we can assume that

u±0 ∈ Q±\ W.(8)

This fact is very important when we are working with Ekeland’s variationalprinciple.

Theorem 2. If conditions (h),(G1),(G2),(G4) and (G6) hold, then thereexist u1 ∈ Q+ and u2 ∈ Q− solutions of (P), such that

I(u1), I(u2) < 0.


Proof. From the proof of Lemma 2 we can conclude that

infu∈Q±

I(u) ≤∫ΩG(x, t±Φ1) ≤

∫ΩT (x)dx < 0.

Ifinf

u∈Q±I(u) =

∫ΩG(x, t±Φ1) = I(t±Φ1) ≤

∫ΩT (x)dx < 0,

occurs we can take u1 = t+Φ1 and u2 = t−Φ1. Otherwise if

infu∈Q±

I(u) <∫ΩG(x, t±Φ1) ≤

∫ΩT (x)dx,

holds using the Ekeland’s variational principle and the same argument ex-plored in [14], we can show that there exist sequences un ⊂ Q+ andvn ⊂ Q− satisfying

I(un) → infu∈Q+

I(u) and I ′(un) → 0,

andI(vn) → inf

u∈Q−I(u) and I ′(vn) → 0.

By Lemma 1, there exist u1 and u2 such that

un → u1 and vn → u2 in H1,p0 (Ω).

Therefore, u1 and u2 are solutions of (P ) verifying

I(u1) = infu∈Q+

I(u) < 0 and I(u2) = infu∈Q−

I(u) < 0,

which implies from Remark 2 that u1 ∈ Q+ and u2 ∈ Q−. This completesthe proof of Theorem 2.

4. Existence of a third solution (Mountain Pass)

Using condition (G5) and arguing as in [14], we can easily show that

G(x, t) ≥ m

ph(x) |t|p − C |t|σ , ∀x ∈ Ω, t ∈ IR(9)

where p < σ < p∗ and C is a constant independent of x.By (9), we have that

I(u) ≥ m

pλ1

∫Ω

|∇u|p − C

∫Ω

|u|σ ,

and then

I(u) ≥ m

pλ1‖u‖p + o (‖u‖) , as ‖u‖ → 0.(10)

Using (G6), we obtainI(t±Φ1) < 0,

which together with (10) imply that there exist r, ρ > 0 and e = t+Φ1 suchthat

I(u) ≥ r > 0, for ‖u‖ ≤ ρ and I(e) < 0.


Therefore, using a version of the Mountain Pass Theorem without a sortof Palais-Smale condition [25, Theorem 6], there exists a sequence un ⊂H1,p

0 (Ω) satisfying

I(un) → c ≥ r > 0 and∥∥I ′(un)

∥∥H1,p

0 (Ω)∗ (1 + ‖un‖) → 0.(11)

Remark 3. We recall the sequence obtained in (11) was introduced by Ce-rami in [26].

Theorem 3. If conditions (h),(G1) -(G3) and (G5)-(G7) hold, then prob-lem (P) has a solution u3, with

I(u3) > 0.

Proof. Let un ⊂ H1,p0 (Ω) be the sequence obtained in (11); then arguing

as in Lemma 1, if un is unbounded, we can assume that|un(x)| → ∞, ∀x ∈ Ω.(12)

Using (11), we have

on(1) = I ′(un)(un) = ‖un‖p − λ1 |un|pp +∫Ωg(x, un)undx,

and then

0 ≤ ‖un‖p − λ1 |un|pp ≤ −∫Ω

|g(x, un)un|+ on(1).

Combining (12), (G3) with the inequality above, we conclude that

‖un‖p − λ1 |un|pp → 0.

Now, using the equality

c+ on(1) = I(un) =1p

[‖un‖p − λ1 |un|pp

]+

∫ΩG(x, un(x))dx,

together with Fatou Lemma and (G7) we obtain

c ≤ lim supn→∞

∫ΩG(x, un(x))dx ≤

∫ΩS(x)dx ≤ 0,

which is a contradiction , because c > 0 by (11). Then un is bounded.Let u3 ∈ H1,p

0 (Ω) be such that

un u3.(13)

By a similar argument explored in [18], we have that

un → u3 in H1,p0 (Ω),(14)

and consequently

I(u3) = c ≥ r > 0 and I ′(u3) = 0,

which shows that u3 is a solution of problem (P).

5. Proof of Theorem 1

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


6. Example

Making Ω = (0, 1), p = 2, and h = 1, we shall give an elementary exampleof a nonlinearity g verifying the set of assumptions.We recall that λn = n2, n = 1, 2, ... are eigenvalues of (PA) and Φ1 = sinπxis the first eigenvalue of (PA).Let g : Ω×IR→IR defined by

g(x, s) = R(x)g1(s),

where g1 :IR→IR is given by

g1(s) =

s, for 0 ≤ s ≤ 1,2− s, for 1 < s ≤ 5,s − 8 for 5 < s ≤ 8 +

√302 ,

8 +√30− s, for 8 +

√302 < s ≤ 8 +

√30,

0 for s ≥ 8 +√30,

−g(−s) for s ≤ 0,

and R : Ω →IR is defined by

R(x) =

4x+ 1, for 0 ≤ x ≤ 1

2 ,−4x+ 5, for 1

2 ≤ x ≤ 1.

Then

G1(s) =s∫

0

g1(t)dt and G(x, s) =s∫

0

g(x, t)dt = R(x)G1(s)

and

S(x) = T (x) = −R(x)2

.

By the definition of g, it is easy to see that it verifies the conditions (Gi) fori= 6.Thus, we shall prove that G satisfies (G6), for t+ = 8.Indeed, observe that

G(x, 8Φ1(x)) = G(1− x, 8Φ1(1− x)), x ∈ Ω,

that is, the function above is symmetric with respect to x = 12 .

Then,

1∫0

G(x, 8Φ1(x)) dx = 2

12∫0

R(x)G1(8Φ1(x))dx

= 2

12∫0

4xG1(8Φ1(x))dx+ 2

12∫0

G1(8Φ1(x))dx

≡ I1 + I2.


Now, we shall estimate each integrals Ij , (j = 1, 2). Since G1(2 +√2) = 0,

choosing α0 ∈IR such that 8Φ1(α0) = 2+√2, which satisfies 0 < α0 <

16 , we

obtain

I1 ≤ 2α0∫0

4xG1(8Φ1(x))dx ≤ 8α0 <43.

On the other hand,

I2 = 2(

16∫

0

+

13∫

16

+

12∫

13

)G1(8Φ1(x))dx

≤ 2(

16∫

0

G1(2)dx+

13∫

16

G1(4)dx+

12∫

13

G1(6)dx)

≤ −2.Therefore,

1∫0

G(x, 8Φ1(x)) dx = I1 + I2 < −23<

1∫0

T (x)dx.

Analogously for t− = −8. This proves that G satisfies (G6).

Acknowledgments. The authors are grateful to the referee for his helpfulremarks. Also, the first and the third author wish to thank the Departa-mento de Matematica, Universidade Federal de Minas Gerais, for the warmhospitality.

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C. O. AlvesDepartamento de matematica e EstatısticaUniversidade Federal da Paraıba58109-970 Campina Grande (PB)-BRAZIL

E-mail address: [email protected]

P. C. CarriaoDepartamento de MatematicaUniversidade Federal de Minas Gerais31270-010 Belo Horizonte (MG)-BRAZIL


O. H. MiyagakiDepartamento de MatematicaUniversidade Federal de Vicosa36571-000 Vicosa (MG)- BRAZIL


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