Existence and Multiplicity of Solution for a Class of ...

Advanced Nonlinear Studies 5 (2005), 73–87

Existence and Multiplicity of Solution for a Class of Quasilinear Equations Claudianor O. Alves∗ Universidade Federal de Campina Grande Departamento de matem´ atica e estat´ıstica Cep:58109-970, Campina Grande - PB, Brasil e-mail: [email protected] Received 16 October 2005 Communicated by Donato Fortunato

Abstract In this paper, we show the existence and multiplicity of solutions for the following class of quasilinear equations −∆p u + |u|p−2 u = f (u) in Ωλ , u(x) > 0 a.e in Ωλ , u = 0 on ∂Ωλ where Ωλ = λΩ, Ω is a bounded domain in IRN , λ is a positive parameter, 2 ≤ p < N , f is a continuous functions, and ∆p u = div(|∇u|p−2 ∇u) is the p-Laplacian operator. Here, we use variational method to get multiplicity of solution involving the Lusternick- Schnirelman category of Ω in itself.

2000 Mathematics Subject Classification. 35A17, 35H30, 35J65. Key words. Quasilinear problem, Variational methods, Positive solutions

∗ Research

Supported by IM-AGIMP and CNPq/PADCT 620017/2004-0

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C.O. Alvez

Introduction

In this paper we study the existence and multiplicity of solutions for the following class of quasilinear problem   −∆p u + |u|p−2 u = f (u), in Ωλ u > 0 in Ωλ (Pλ )  u = 0 on ∂Ωλ where Ωλ = λΩ, Ω ⊂ IRN is a smooth bounded domain, λ > 0, 2 ≤ p < N , ∆p u = div(|∇u|p−2 ∇u) and f : IR → IR is a function of C 1 class satisfying the following hypotheses: (t) The function tfp−1 is increasing in (0, +∞) and satisfies: lim sup |t|→∞

f (t) f (t) = 0 and lim p−1 = 0, s t→0 |t| |t|

(f1 )

for some s ∈ (0, N (p−1)+p ). N −p There exists θ > p such that Z 0 ≤ θF (t) < tf (t) ∀t > 0 where F (t) =

t

f (ξ)dξ.

(f2 )

0

There exist η ∈ (p − 1, N (p−1)+p ) and C > 0 such that N −p f 0 (t)t − (p − 1)f (t) ≥ Ctη ∀t > 0.

(f3 )

In the literature there are some works where the authors showed multiplicity of solutions for some problems related to (Pλ ) involving Lusternick- Schnirelman of Ω in itself, denoted by catΩ (Ω), see for example, for the case p=2, Benci & Cerami [6, 7, 8], Clap & Ding [10], Rey [13], and Bahri & Coron [5]. For p-Laplacian with p ≥ 2,we cite the papers of Alves & Ding [2, 3] and references therein. In [6], Benci & Cerami considered the problem   −∆u + γu = uq−1 , in Ω u > 0 in Ω (Pγ )  u = 0 on ∂Ω where Ω ⊂ IRN is a bounded domain with smooth boundary, γ is a positive parameter and q ∈ (2, 2∗ ). Using variational techniques, the authors showed that if γ is a large parameter, problem (Pγ ) has at least catΩ (Ω) of positive solutions. In [8], the same authors showed that similar results hold also to a larger class of nonlinearities. Using ³ the ´ results proved in [6], and considering the change variable v(x) = 1 γ 2−q u √xγ , it follows that the below problem   −∆v + v = v q−1 , in Ω√γ 0 v > 0 in Ω√γ (Pγ )  v = 0 on ∂Ω√γ

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Existence and multiplicity of solutions

√ has also at least catΩ√γ (Ω√γ ) of positive solutions, where Ω√γ = γΩ. Inspired by the work [6], we consider in this paper the class of problem (Pλ ), and show that some results established in [6] and [8] also hold for p-Laplacian operator and to a larger class of nonlinearities. Here, in the proof of some lemmas and propositions, we use different arguments of those found in [6] and [8], because the nonlinearity is not necessarily homogeneous and the p-Laplacian operator is not linear. To obtain our main result, we prove a compactness result on Nehari manifolds and the existence of radially symmetric ground state solution for (Pλ ) when Ωλ = Bλr (0). The main result in this work is the following Theorem 1.1 Assume (f1 ) − (f3 ) . Then, there exists λ∗ > 0 such that for λ > λ∗ , problem (Pλ ) has at least catΩ (Ω) of positive solutions.

2

Preliminary results and notations

In this section we recall some results involving the situation when Ωλ = IRN and fix some notation used in this work. From now on, we will assume without loss of generality that 0 ∈ Ω. Moreover, let us fix a real number r > 0 such that the sets Ω+ and Ω− given by Ω+ = {x ∈ IRN ; d(x, Ω) ≤ r} and Ω− = {x ∈ Ω; d(x, ∂Ω) ≥ r} are homotopically equivalent to Ω. An important result that we will use in this work is related with the existence of a positive ground state solution to the following problem   −∆p u + |u|p−2 u = f (u), in IRN u > 0 in IRN (P∞ )  u ∈ W 1,p (IRN ) that is, with the existence of a positive function w ∈ W 1,p (IRN ) verifying 0 (w) = 0 I∞ (w) = c∞ and I∞

where I∞ (u) =

1 p

Z

Z (|∇u|p + |u|p )dx −

IRN

F (u)dx ∀u ∈ W 1,p (IRN ) IRN

and c∞ denotes the minimax level of Mountain Pass Theorem applied to I∞ . The theorem below shows the existence of ground state solution to (P∞ ) and its ´ & Miyagaki [1] ( see Coti - Zelati & Rabinowitz proof can be found in Alves, do O [9] for the case p = 2):

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Theorem 2.1 Problem (P∞ ) has a positive ground state solution. In what follows, let us denote also by bλ the minimax level obtained by the Mountain Pass Theorem applied to the energy functional Iλ,B : Wo1,p (Bλr ) → IR given by Z Z 1 Iλ,B (u) = (|∇u|p + |u|p )dx − F (u)dx p Bλr Bλr where Bλr = Bλr (0) and by Mλ,B the Nehari manifold related to Iλ,B given by o n 0 (u)u = 0 . Mλ,B = u ∈ Wo1,p (Bλr ) \ {0}; Iλ,B Using well know arguments, we have that bλ =

inf

u∈Mλ,B

Iλ,B (u).

By cλ , let us denote the minimax level of the functional Z Z 1 Iλ (u) = (|∇u|p + |u|p )dx − F (u)dx ∀u ∈ Wo1,p (Ωλ ). p Ωλ Ωλ Moreover, we also denote by Mλ and M∞ the Nehari manifold of the functionals Iλ and I∞ respectively.

3

A Compactness Result

In this section, let us establish a compactness result on Nehari manifolds involving minimizing sequence. Since that we intend to find positive solutions to (Pλ ), let us assume that f (t) = 0 ∀t ∈ (−∞, 0]. Theorem 3.1( Compactness theorem on Nehari manifold ) Let {un } ⊂ W 1,p (IRN ) be a sequence satisfying I∞ (un ) → c∞ and un ∈ M∞ . Then, un is strongly convergent or there exists yn ⊂ IRN with |yn | → ∞ such that the sequence vn (x) = un (x + yn ) is strongly convergent to a function v ∈ W 1,p (IRN ) verifying I∞ (v) = c∞ and v ∈ M∞ . Proof. Using well know arguments, we have that the sequence {un } is bounded in W 1,p (IRN ), hence for some subsequence, still denoted by un , we can assume that there exists u ∈ W 1,p (IRN ) satisfying un * u in W 1,p (IRN ).

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Using the Ekeland’s Variational Principle ( see [11] or [16] ) , we can assume that {un } verifies the following limits: 0 I∞ (un ) → c∞ and I∞ (un ) → 0.

(3.1)

Assuming (3.1), we will divide our study into two cases: u 6= 0 and u = 0. Case 1: u 6= 0. Since f has subcritical growth, it is easy to check that for some subsequence, still denoted by {un }, ∇un (x) → ∇u(x) a.e in IRN .

(3.2)

0 0 (u)(u) = 0, thus Using the fact that I∞ (un )(u) = 0, from (3.1) we have I∞

1 0 c∞ ≤ I∞ (u) = I∞ (u) − I∞ (u)(u), θ which implies c∞

³1

1´ ≤ − 2 θ

Z

Z p

p

(|∇u| + |u| )dx + IRN

IRN

1 [ f (u)(u) − F (u)]dx. θ

From Fatou’s Lemma Z n³ 1 1 ´ Z o 1 p p c∞ ≤ Γ ≤ lim inf − |∇un | + |un | + [ f (un )(un ) − F (un )] ≤ c∞ , n→∞ 2 θ IRN IRN θ ³ ´R R where Γ = 12 − θ1 IRN |∇u|p + |u|p + IRN [ θ1 f (u)(u) − F (u)], and thus Z Z p p lim (|∇un | + |un | )dx = (|∇u|p + |u|p )dx. (3.3) n→∞

IRN

IRN

Combining (3.2) with (3.3), un → u in W 1,p (IRN ). Case 2:

u = 0. In this case, there exist R, η > 0 and yn ⊂ IRN such that Z lim sup |un |p dx ≥ η. n→∞

BR (yn )

Because if this claim does not hold, we have Z lim sup sup |un |p dx = 0, n→+∞ y∈IRN

and by Lions [12],

BR (y)

Z lim

n→+∞

IRN

|un |s dx = 0 ∀s ∈ (p, p∗ ).

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Using the last limit and the fact that {un } ⊂ M∞ , it is easy to check that kun k → 0 as n → +∞; consequently I∞ (un ) → 0 as n → +∞.

(3.4)

On the other hand, I∞ (un ) → c∞ > 0 as n → +∞ which leads to a contradiction with (3.4). From the Sobolev imbedding, we have that |yn | → ∞. Defining vn (x) = un (x + yn ), we have 0 I∞ (vn ) → c∞ and I∞ (vn ) → 0. It is clear that {vn } is bounded in W 1,p (IRN ) and there exists v ∈ W 1,p (IRN ) with v 6= 0 such that vn * v in W 1,p (IRN ). Repeating the same arguments used in Case 1, it follows that vn → v in W 1,p (IRN ). Verification of (3.1): Using Ekeland’s Variational {wn } ∈ M∞ satisfying

Principle,

there

exists

a

sequence

0 0 wn = un + on (1), I∞ (wn ) → c∞ and I∞ (wn ) − γn E∞ (wn ) = on (1) 0 where γn is a real number and E∞ (w) = I∞ (w)w ∀w ∈ W 1,p (IRN ), . From (f3 ), there exists δ > 0 such that 0 |E∞ (wn )(wn )| ≥ δ ∀n ∈ IN.

In fact, using the definition of E∞ , it follows that Z Z 0 −E∞ (wn )(wn ) = [f 0 (wn )wn − (p − 1)f (wn )]wn dx ≥ C IRN

0 thus, assuming by contradiction that E∞ (wn )(wn ) → 0, we get Z wnη+1 dx → 0 IRN

and by interpolation

Z IRN

wns+1 dx → 0.

Form (f1 ), for each ² > 0 there exists C² > 0 such that |f (t)| ≤ ²|t| + C² |t|s ∀t ∈ (0, +∞),

IRN

wnη+1 dx

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thus

Z IRN

f (wn )wn → 0

and consequently wn → 0 in W 1,p (IRN ), but this is an absurd, because it is possible to show that there exists α > 0 such that kwk ≥ α ∀w ∈ M∞ . 0 Since I∞ (wn )wn = on (1), we have 0 γn E∞ (wn )wn = on (1)

thus γn = on (1), and hence 0 I∞ (wn ) → c∞ and I∞ (wn ) → 0.

Therefore, without loss generality, we can assume that 0 I∞ (un ) → c∞ and I∞ (un ) → 0.

Using the same type of arguments explored in the proof of (3.1), it is possible to show the next proposition, where we denote by kIλ0 (v)k∗ the norm of the derivative of the restriction of Iλ to Mλ at v, which is defined by kIλ0 (v)k∗ =

sup y ∈ T v Mλ kyk = 1

Iλ0 (v)(y).

Proposition 3.1 The functional Iλ satisfies the Palais - Smale condition on Mλ , that is, if {un } ⊂ Mλ satisfies Iλ (un ) → c and kIλ0 (un )k∗ → 0 then there exists a subsequence, still denoted by {un }, which is strongly convergent in Wo1,p (Ωλ ). The next proposition shows an important property involving the critical points of Iλ on Mλ . Proposition 3.2 If u ∈ Mλ is a critical point of Iλ on Mλ , then u is a nontrivial critical point of Iλ in Wo1,p (Ωλ ). Proof. If u ∈ Mλ is a critical point of Iλ on Mλ , then u 6= 0 and there exists γ ∈ IR verifying Iλ0 (u) = γEλ0 (u)

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where Eλ (u) = Iλ0 (u)u. Since that Iλ0 (u)u = 0, we have γEλ0 (u)u = 0. Using the same type of arguments explored in the proof of (3.1), there exists η∗ > 0 such that Eλ0 (u)u ≤ −η∗ ∀u ∈ Mλ , so γ = 0 and thus Iλ0 (u) = 0.

4

Behavior of minimax levels

In this section, we study the behavior of some minimax levels with relation to the parameter λ. To this end, we need to make some definitions. For each x ∈ IRN and R > r > 0, let us denote by AR,r,x the following set AR,r,x = BR (x) \ B r (x). When x = 0, let us denote by AR,r the set AR,r,0 , that is, AR,r = AR,r,0 . For each u ∈ W 1,p (IRN ) with compact support, we consider R p N x|∇u| dx β(u) = RIR . |∇u|p dx IRN Moreover, for each x ∈ IRN , let us denote by a(R, r, λ, x) the following number n o cλ,x a(R, r, λ, x) = inf Jλ,x (u), β(u) = x, u ∈ M where

and

1 Jλ,x (u) = p

Z

Z p

p

(|∇u| + |u| )dx − AλR,λr,x

F (u)dx AλR,λr,x

n o 0 cλ,x = u ∈ Wo1,p (AλR,λr,x ) \ {0}; Jλ,x M (u)u = 0 .

Next, let us denote by a(R, r, λ) the number a(R, r, λ, 0) , Jλ the functional Jλ,0 , cλ the set M cλ,0 . and M Proposition 4.1 The number a(R, r, λ) satisfies lim inf a(R, r, λ) > c∞ . λ→∞

Proof. From definitions of a(R, r, λ) and c∞ , it follows that a(R, r, λ) ≥ c∞ .

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cλ verifying Assuming by contradiction that there exist λn → ∞ and un ∈ M n β(un ) = 0, and a(R, r, λn ) → c∞ , from Theorem 2, we have un (x) = wn (x) + Ψ(x − yn ) where {wn } ⊂ W 1,p (IRN ) is a sequence converging strongly to 0 in W 1,p (IRN ), {yn } ⊂ IRN is such that |yn | → ∞, and Ψ ∈ W 1,p (IRN ) is a positive function verifying 0 I∞ (Ψ) = c∞ and I∞ (Ψ) = 0. Since Iλ is rotationally invariant, we can assume that yn = (yn1 , 0, 0, ..., 0) and that yn1 < 0. Now we set

Z |∇Ψ|p dx.

M= IRN

Clearly M > 0. Since kwn k → 0, it follows that Z |∇(wn + Ψ(., yn ))|p dx → M, Brλn /2 (yn )

from which we obtain

Z |∇un |p dx → M, Θn

where Θn = Brλn /2 (yn ) ∩ [Bλn R (0) \ Bλn r (0)], and hence Z |∇un |p dx → 0,

(4.1)

Υn

where Υn = [Bλn R (0) \ Bλn r (0)] \ Brλn /2 (yn ). Since β(un ) = 0, we have Z Z Z 0= x1 |∇un |p dx = x1 |∇un |p dx + x1 |∇un |p dx Aλn R,λn r

Θn

Υn

thus

Z |∇un |p dx ≥ 0

−(rλn /2)(M + on (1)) + Rλn Υn

with on (1) → 0. Then

Z |∇un |p dx ≥ Υn

and this contradicts (4.1).

rM − on (1) 2R

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Proposition 4.2 The numbers bλ and cλ verify the following limits lim cλ = c∞ and

λ→∞

lim bλ = c∞ .

λ→∞

Proof. Here, we will prove only the first limit, because the second limit follows from the same type of reasoning. Let Φ be a function in Co∞ (IRN ) defined by Φ(x) = 1 in B1 (0), Φ(x) = 0 in c B2 (0) and 0 ≤ Φ(x) ≤ 1 ∀x ∈ IRN . For each R > 0, let us consider the function ΦR (x) = Φ(x/R) and wR (x) = ΦR (x)w(x), where w is a ground state solution of problem (P∞ ). Since that 0 ∈ Ω, there exists λ∗ > 0 such that B2R (0) ⊂ Ωλ for λ ≥ λ∗ . Let tR > 0 such that tR wR ∈ Mλ Then cλ ≤ Iλ (tR wR ) ∀λ ≥ λ∗ . Taking the limit when λ → ∞, we obtain lim sup cλ ≤ I∞ (tR wR ), λ→∞

and using the fact that w is a ground state, we have the following limit Claim 1:

lim tR = 1.

R→∞

In fact, from definition of tR , we have Z Z ³ ´ p |∇wR |p + wR dx = IRN

Thus, for R > 1 Z ³ ´ Z |∇wR |p + |wR |p ≥ IRN

B1 (0)

IRN

f (tR wR ) p p−1 wR dx. tp−1 R wR

f (tR w) p p−1 p−1 w ≥ tR w

(4.2)

Z

f (tR a) p p−1 p−1 a B1 (0) tR a

where a = min w(x). |x|≤1

Note that {tR } is bounded, because if there exists Rn → ∞ with tRn → ∞, we have Z Z ³ ´ Z f (tR w) p f (tR a) p w ≥ |∇wR |p + |wR |p ≥ p−1 p−1 p−1 p−1 a . N t w t IR B1 (0) R B1 (0) R a From (f2 )

f (tRn a) → ∞, (tRn a)p−1

thus

Z lim

n→∞

IRN

³

´ |∇wRn |p + |wRn |p = +∞.

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But, the last limit can not hold, because by Lebesgue’s Theorem Z Z ³ ´ ³ ´ p p lim |∇wRn | + |wRn | → |∇w|p + |w|p < ∞. n→∞

IRN

(4.3)

IRN

Therefore tR is bounded. Using the same type of arguments, there is no Rn → +∞ with tRn → 0 as n → +∞, because by (f1 ) |f (t)| ≤ ²|t|p−1 + C² |t|s so

¯ f (t w ) ¯ ²|t w |p−1 + C |t w |s ¯ Rn Rn ¯ Rn Rn ² R n Rn . ¯ p−1 p−1 ¯ ≤ p−1 p−1 tRn wRn tRn wRn

Since wRn → w in W 1,p (IRN ), from Lebesgue Theorem Z lim sup n→∞

IRN

that is,

¯ f (t w ) ¯ ¯ Rn Rn p ¯ ¯ p−1 p−1 wRn ¯ ≤ ²C ∀² > 0 tRn wRn

Z lim

n→∞

IRN

f (tRn wRn ) p p−1 wRn = 0. tp−1 Rn wRn

Using (4.2) and the last limit, we get wRn → 0 in W 1,p (IRN ), which leads to a contradiction with (4.3). Thus, there exists R0 , δ > 0 such that tR > δ for R ≥ R0 . Fixing Rn → ∞ with tRn → t0 , it follows of (4.2) Z Z ³ ´ f (t0 w) p |∇w|p + wp dx = p−1 p−1 w dx. N N t w IR IR 0 From the fact that f (t)/|t|p−1 is increasing in (0, ∞), the last equality implies that t0 = 1 and Claim 1 is proved. From Claim 1, I∞ (tR wR ) → I∞ (w) = c∞ as R → ∞, and thus lim sup cλ ≤ c∞ .

(4.4)

λ→∞

On the other hand, using the definition of cλ and c∞ , we get the inequality cλ ≥ c∞ ∀λ > 0 which implies lim inf cλ ≥ c∞ . λ→∞

(4.5)

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From (4.4) and (4.5), lim cλ = c∞ .

λ→∞

b > 0 such that : Proposition 4.3 There exists λ b Iλ (u) ≤ bλ and u ∈ Mλ ⇒ β(u) ∈ λΩ+ r ∀λ ≥ λ. Proof. Assume that there exists λn → ∞, un ∈ Mλn and Iλn (un ) ≤ bλn with xn = β(un ) ∈ / λn Ω+ r . Fixing R > diamΩ, Aλn R,λn r,xn ⊃ Ωλn , and a(R, r, λn , xn ) ≤ bλn . Using the fact that a(R, r, λn , xn ) = a(R, r, λn ), we have a(R, r, λn ) ≤ bλn .

(4.6)

Taking the limit of n → ∞ in (4.6) and using Proposition 4.2, we get lim inf a(R, r, λn ) ≤ c∞ n→∞

finding a contradiction to Proposition 4.1. Proposition 4.4 The functional Iλ,B has a ground state solution uλ,r , which is radially symmetric on the origin. Proof. In this proof we denote by I the functional Iλ,B . It is well know that the functional I has a positive ground state solution v. Thus I(v) = bλ and I 0 (v) = 0. If v ∗ is the Schwartz symmetrization of v, we have that v ∗ ∈ Wo1,p (Bλr (0)) and satisfies Z Z |∇v ∗ |p dx ≤ |∇v|p dx. (4.7) Bλr (0)

Bλr (0)

00

Since F (t) ≥ ∀t ≥ 0, F is a convex function and consequently Z Z ∗ F (αv )dx = F (αv)dx ∀α > 0. Bλr (0)

Bλr (0)

Using the fact that v ∈ Mλ,B , we have I 0 (v)v = 0

(4.8)

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and I(v) = max I(tv). t≥0

From (f1 ) − (f2 ), it follows that there exists a unique t∗ > 0 such that t∗ v ∗ ∈ Mλ,B . Thus by (4.7)-(4.8) bλ ≤ I(t∗ v ∗ ) ≤ I(t∗ v) ≤ max I(tv) = I(v) = bλ , t≥0

that is, I(t∗ v ∗ ) = bλ and t∗ v ∗ ∈ Mλ,B . From the last equality, t∗ v ∗ is a critical point of I on Mλ,B , so t∗ v ∗ is a critical point of I in Wo1,p (Bλr ), and thus I(t∗ v ∗ ) = bλ and I 0 (t∗ v ∗ ) = 0.

Remark 4.1 In what follows, we denote by uλ,r the ground state solution t∗ v ∗ given in Proposition 6, i.e., uλ,r = t∗ v ∗ . For λ > 0 and r > 0 ( see introduction ), we define the operator Ψr : λΩ− → Wo1,p (Ωλ ) given by ½ uλ,r (|x − y|) ∀x ∈ Bλr (y), [Ψr (y)](x) = 0 ∀x ∈ Ωλ \ Bλr (y) where uλ,r is a positive function, radially symmetric about the origin, such that uλ,r ∈ Mλ,B . Note that for every y ∈ λΩ− , β(Ψr (y)) = y. In the next result, we denote by Iλbλ the following set n o Iλbλ = u ∈ Mλ ; Iλ (u) ≤ bλ . b we have Proposition 4.5 For λ ≥ λ, catIλbλ ≥ catΩλ (Ωλ ). Proof.

Assume that

Iλbλ = A1 ∪ ..... ∪ An ,

where Aj ,j = 1, ..., n is closed and contractible in Iλbλ , that is, there exists hj ∈ C([0, 1] × Aj , Iλbλ ) such that hj (0, u) = u and hj (1, u) = w for all u ∈ Aj ,

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C.O. Alvez

where w ∈ Aj is fixed. Consider Bj = Ψ−1 r (Aj ), 1 ≤ j ≤ n. The sets Bj are closed and λΩ− = B1 ∪ .... ∪ Bn . Using the deformation gj : [0, 1] × Bj → λΩ+ given by gj (t, y) = β(hj (t, Ψr (y))) we can conclude by Proposition 4.4 that Bj is contractible in λΩ+ . It follows that catΩλ (Ωλ ) = catλΩ+ (λΩ− ) ≥ n. Proof of Theorem 1.1 First of all, let us remember that Iλ satisfies the Palais-Smale condition on Mλ , thus applying the Lusternik - Schnirelman theory and Proposition 4.5, we have that Iλ on Mλ has at least catΩλ (Ωλ ) critical points whose energy is less than bλ for λ ≥ λ. Moreover, all solutions obtained are positive by maximum principle ( see [15] and [14] ) .

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