Multiplicity of Positive Solutions For a Quasilinear ...

Advanced Nonlinear Studies 5 (2005), 551–572

Multiplicity of Positive Solutions For a Quasilinear Problem in IRN Via Penalization Method Claudianor O. Alves∗ , Universidade Federal de Campina Grande ,

Departamento de Matem´ atica e Estat´ıstica

Cep:58109-970, Campina Grande - Pb, Brazil e-mail: [email protected]

Giovany M. Figueiredo† Departamento de Matem´ atica, Universidade Federal do Par´ a, Cep: 66075-110, Belém - Pa, Brazil e-mail: [email protected] Received in revised form 4 September 2005 Communicated by Ireneo Peral

Abstract The multiplicity and concentration of positive solutions are established for the equation −p ∆p u + V (z)|u|p−2 u = f (u) in IRN , where V is a positive continuous function and f ∈ C 1 is a function having subcritical and superlinear growth. 2000 Mathematics Subject Classifications: 35A15, 35H30, 35B40. Key words. Variational methods, Quasilinear problems, Behavior of solutions.

∗ Research † Research

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

551

552

1

C.O. Alves, G.M. Figueiredo

Introduction

In this paper we are concerned with the multiplicity and concentration of positive solutions for the problem   −p ∆p u + V (z)|u|p−2 u = f (u) in IRN u ∈ W 1,p (IRN ) with 1 0, ∀ z ∈ IRN , where > 0, ∆p u is the p-Laplacian operator, that is, ∆p u =

N X ∂ ∂u (|∇u|p−2 ), ∂x ∂x i i i=1

V is a continuous function satisfying: V (x) ≥ V0 = inf V (x) > 0 for all x ∈ IRN

(V1 )

V0 < min V (x),

(V2 )

x∈IRN

with x∈∂Ω

for some open and bounded domain Ω ⊂ IRN , and f ∈ C 1 (IR) is a function satisfying the following conditions: lim

|f (s)| = 0. |s|p−1

(f1 )

lim

|f (s)| =0 |s|q−1

(f2 )

s→0

There exists q ∈ (p, p∗ ) verifying

|s|→∞

where p∗ = NpN −p . There exists θ ∈ (p, q) such that 0 < θF (s) ≤ sf (s) for all s > 0 where F (s) =

Rs 0

(f3 )

f (t)dt.

The function s→

f (s) is increasing in (0, +∞). sp−1

(f4 )

There are constants C > 0 and σ ∈ (p, p∗ ) such that f 0 (s)s2 − (p − 1)f (s)s ≥ Csσ for all s ≥ 0.

(f5 )

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Multiplicity of positive solutions

For the case p = 2, equations of the kind −2 ∆u + V (z)u = f (u) in IRN

(P∗ )

arise in different models, for example, they are related to the existence of standing waves of the nonlinear Schrödinger equation i

∂Ψ = −2 ∆Ψ + (V (x) + E)Ψ − f (Ψ) for all x ∈ IRN , ∂t

(N LS)

when f (s) = |s|q−2 u, 2 < q < 2∗ = 2N/(N − 2). A standing wave of (N LS) is a solution of the form Ψ(x, t) = exp(−iEt/)u(x). In this case, u is a solution of (P∗ ). A result of multiplicity of positive solutions for (P∗ ), considering f (u) = |u|q−2 u with q ∈ (2, 2∗ ), can be found in [5], where Cingolani & Lazzo used Lusternik-Schnirelman category involving the set M = {x ∈ Ω / V (x) = V0 } and the following condition on V : V∞ = lim inf V (x) > V0 = inf V (x) > 0. |x|→∞

IRN

(V )

We recall that, if Y is a closed subset of a topological space X, the Lusternik-Schnirelman category denoted by catX Y is the least number of closed and contractible sets in X which cover Y . In [3] (see also [8]), Alves & Figueiredo showed that results similar to those found in [5] also hold for (P ) , assuming that condition (V ) holds, 2 ≤ p < N , and that f belongs to a large class which includes the model f (u) = |u|q−2 u with q ∈ (p, p∗ ). In [6], del Pino & Felmer proved that if the conditions (V1 ), (V2 ), (f1 ) − (f4 ) hold, problem (P∗ ) has a positive solution for small , which has a phenomenon of concentration close to a minimum point of potential V . In this paper, motivated by [5] and [6], we complete the study made in [6], in the sense that we are considering p-Laplacian operator and show the existence of multiple solutions. However, here we use a different approach from those explored in [6], because some estimates involving the p-Laplacian can not be obtained using the convergence in the C 2 sense and the uniqueness of positive solutions for problem (Pµ ) given by   −∆p u + µ|u|p−2 u = f (u) in IRN u ∈ W 1,p (IRN ) with 1 0, ∀ x ∈ IRN , which is an open problem for p-Laplacian operator for p 6= 2. To overcome these difficulties, we use some results of compactness on Nehari manifolds and Moser’s iteration method [12]. Our main result is the following:

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Theorem 1.1 Suppose that f satisfies (f1 ) − (f5 ) and that V satisfies (V1 ) − (V2 ). Then, for any δ > 0, there exists δ > 0 such that (P ) has at least catMδ (M ) positive solutions, for any 0 < < δ . Moreover, if u be a positive solution of (P ) and η ∈ IRN a global maximum point of u , then lim V (η ) = V0 .

→0

To prove Theorem 1.1, we will work with the problem below, which is equivalent to (P ) by change variable z = x   −∆p u + V (z)|u|p−2 u = f (u) in IRN u ∈ W 1,p (IRN ) with 1 0, ∀ z ∈ IR .

2

Technical lemmas

In this section we fix some notations and prove some lemmas which are key points in our arguments. Hereafter, we assume that f (s) = 0 for s < 0 and omit the symbol ”dx” in all integrals. Moreover, we recall that the weak solutions of (P˜ ) are related with critical points of the functional Z Z Z 1 1 I (u) = |∇u|p + V (x)|u|p − F (u) p IRN p IRN IRN which is well defined for u ∈ W where Z 1,p N p W = u ∈ W (IR ) : V (x)|u| < ∞ IRN

endowed with the norm kuk =

Z

p

Z

|∇u| +

IRN

V (x)|u|p

p1

.

IRN

Next, in relation to the problem (Aµ ), we use the functional Z Z Z 1 1 p p Eµ (u) = |∇u| + µ|u| − F (u), p IRN p IRN IRN which is well defined in Xµ = W 1,p (IRN ) endowed with the norm Z Z p1 kukXµ = |∇u|p + µ|u|p . IRN

IRN

Moreover, let us denote by cµ the minimax level of the Mountain Pass Theorem applied to Eµ and by Mµ the Nehari manifolds Z Z Z n o Mµ = u ∈ Xµ \ {0} : |∇u|p + µ|u|p = f (u)u . IRN

IRN

IRN

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2.1

An auxiliary problem

To establish the multiplicity of positive solutions, we will adapt for our case an argument explored by del Pino & Felmer [6] ( see also [4]), which consists of considering a modified problem. To this end, we need to fix some notations. (a) θ and afp−1 = Vk0 . Using Let θ be the number given in (f3 ), a, k > 0 satisfying k > θ−p the above numbers, let us define the function  f (s) if s ≤ a    fb(s) = V0 p−1 s if s > a.    k Moreover, we fix t0 , t1 ∈ (0, +∞) such that t0 < a < t1 and η ∈ C 1 ([t0 , t1 ]), satisfying η(s) ≤ fb(s) for all s ∈ [t0 , t1 ].

(η1 )

η(t0 ) = fb(t0 ) and η(t1 ) = fb(t1 ).

(η2 )

0

(η3 )

b0

0

b0

η (t0 ) = f (t0 ) and η (t1 ) = f (t1 ). The function s →

η(s) is nondecreasing for all s ∈ [t0 , t1 ]. sp−1

(η4 )

Using the functions η and fb, let us consider two new functions   fb(s) if s 6∈ [t0 , t1 ]   fe(s) =  η(s) if s ∈ [t0 , t1 ]   and g(x, s) = χΩ (x)f (s) + (1 − χΩ (x))fe(s), and the auxiliary problem   −∆p u + V (x)|u|p−2 u = g(x, u) 

u∈W

1,p

N

(IR ),

(P˜ )a

N

u > 0 in IR ,

where χΩ is the characteristic function related to the set Ω. It is easy to check that (f1 ) − (f5 ) imply that g is a Carathéodory function, for x ∈ IRN the function s → g(x, s) is of class C 1 , and that the conditions below hold uniformly in x ∈ IRN : g(x, s) = 0 for all s < 0. lim

|s|→0

g(x, s) =0 . |s|p−1

(g1 ) (g2 )

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lim

|s|→∞

g(x, s) = 0. |s|q−1

(g3 )

s

Z 0 ≤ θG(x, s) = θ

g(x, t)dt < g(x, s)s ∀x ∈ Ω and ∀s > 0.

(g4 )i

0

0 < pG(x, s) ≤ g(x, s)s ≤

1 V (x)sp ∀x 6∈ Ω and ∀s > 0. k

(g4 )ii

The function s→

g(x, s) is nondecreasing for each x ∈ IRN and for all s > 0. sp−1

(g5 )

Remark 2.1 Note that if u is a positive solution of (P˜ )a with u(x) ≤ t0 for every x ∈ IRN \Ω, then u is also a positive solution of (P˜ ).

3

Existence of solution of (P˜ )a

In this section, we use variational techniques related to the energy functional J : W −→ IR given by Z Z Z 1 1 |∇u|p + V (x)|u|p − G(x, u). J (u) = p IRN p IRN IRN Let us denote by N the Nehari manifold associated to J given by Z Z Z n p p N = u ∈ W \ {0} : |∇u| + V (x)|u| = IRN

IRN

o g(x, u)u .

IRN

It is easy to check that there exists r > 0 such that kuk ≥ r > 0, for all u ∈ N and ∀ > 0.

(3.1)

Hereafter, we recall that a sequence (un ) ⊂ W is called a (P S)d sequence for J , at level d ∈ IR, if J (un ) → d and J0 (un ) → 0. Our first lemma in this subsection shows that the functional J has the Mountain Pass Geometry and its proof uses well-know arguments.

Lemma 3.1 The functional J satisfies the following conditions: (i) There exist α, ρ > 0 such that: J (u) ≥ α with kuk = ρ. (ii) There exists e ∈ Bρc (0) such that J (e) < 0.

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By Lemma 3.1 together with a result found in Willem [14, Theorem 1.15], there exists {un } ⊂ W satisfying J (un ) → ρ and J0 (un ) → 0 where ρ is the minimax level of the Mountain Pass applied to J , which has the following characterization: ρ = inf sup J (tu). u∈W \{0} t≥0

The next lemma follows from standard arguments and we omit its proof. Lemma 3.2 Let {vn } be a (P S)d sequence for J . Then {vn } is bounded in W . Lemma 3.3 Let {vn } be a (P S)d sequence for J . Then for each ζ > 0, there exists R = R(ζ) > 0 such that Z lim sup [|∇vn |p + V (x)|vn |p ] < ζ. n→∞

IRN \BR

Proof. For R > 0, let ηR ∈ C ∞ (IRN ) be such that ηR (x) = 0 if x ∈ BR/2 (0) and ηR (x) = 1 if x 6∈ BR (0), with 0 ≤ ηR (x) ≤ 1 and |∇ηR | ≤ C R , where C is a constant independent of R. Note that Z Z p p ηR |∇un | + V (x)|un | = J0 (un )un ηR + g(x, un )un ηR IRN IRN Z − un |∇un |p−2 ∇un ∇ηR . IRN

Since {ηR vn } is bounded in W , it follows that J0 (un )un ηR = on (1). Thus fixing R > 0 such that Ω = {x ∈ IRN ; x ∈ Ω} ⊂ BR/2 (0) and using (g4 )ii , we get the estimate Z C |∇vn |p + V (x)|vn |p ≤ kvn kp−1 + on (1). R IRN \BR (0) Fixing ζ > 0 and passing to the limit in the last inequality , it follows that e C p p lim sup |∇vn | + V (x)|vn | ≤ 0, from Sobolev’s compact imbedding W 1,p (BR (0)) ,→ Ls (BR (0)), and by Lebesgue’s Theorem, we can deduce that Z Z g(x, un )un − lim n→∞

BR (0)

p − 1 ≤ s < p∗ ,

BR (0)

g(x, u)u = 0.

(3.3)

Using the growth condition on g and again the Sobolev’s imbedding, we get Z Z Z g(x, un )un ≤ C1 |un |p + C2 |un |q+1 . IRN \BR (0)

IRN \BR

IRN \BR

Consequently by Lemma 3.3, for R > 0 sufficiently large, Z lim sup g(x, un )un ≤ C1 ζ + C2 ζ (q+1)/p , n→∞

and since g(x, u)u ∈ L1 (IRN ), we can assume Z g(x, u)u < ζ.

(3.5)

IRN \BR

From (3.3), (3.4) and (3.5) Z lim sup n→∞

Z g(x, un )un −

IRN

IRN

g(x, u)u < ζ, ∀ ζ

and (3.2) is proved. Defining R Pn =R IRN h|∇vn |p−2 ∇vn − |∇v|p−2 ∇v, ∇vn − ∇vi+ V (x)[|vn |p−2 vn − |v|p−2 v, vn − v] IRN we have Pn = J0 (vn )(vn ) +

Z IRN

g(x, vn )vn − J0 (vn )(v) −

Z g(x, vn )v + on (1) IRN

and since J0 (vn )(vn ) = J0 (vn )(v) = on (1) and

(3.4)

IRN \BR

Z

Z g(x, vn )vn −

IRN

g(x, vn )v = on (1) IRN

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it follows that Pn → 0 when n → ∞. From the last limit together with some estimates found in [7] ( see also [11] ), we have that vn → v in W and the proof is finished. Proposition 3.2 The functional J restricted to N satisfies the (P S) condition. Proof. Let {vn } ⊂ N be such that J (vn ) → c and kJ0 (vn )|N k∗ = on (1). Then there exists {λn } ⊂ IR verifying J0 (vn ) = λn φ0 (vn ) + on (1), where φ : W → IR is given by Z Z p φ (u) = |∇u| + IRN

p

(3.6)

Z

V (x)|u| −

IRN

g(x, u)u. IRN

Since IRN

= [Ω ∪ {vn < t0 }] ∪ [(IRN \Ω ) ∩ {t0 ≤ vn ≤ t1 }] ∪ [(IRN \Ω ) ∩ {vn > t1 }],

from (f5 ) and (η4 ) φ0 (vn )vn

Z

σ

≤ −C

Z

|vn | ≤ −C Ω ∪{vn 0 and J0 (u ) = 0. − N Since J0 (u )(u− ) = 0, it follows that ku k = 0, and thus u ≥ 0 in R . By Li 1,α Gongbao [10, Theorem 1.11], u ∈ Cloc (IRN ), and by Harnack’s inequality [13], we have u > 0 in RN .

Now we use the characterization of ρ given by ρ =

inf

sup J (tu) = inf J (u)

u∈W \{0} t≥0

u∈N

which is more suitable for our purpose.

3.1

Multiplicity of solutions to (P˜ )a

Hereafter, we let M denote the subset of Ω given by M = {x ∈ Ω : V (x) = V0 }, and for each δ > 0, Mδ the set given by Mδ = {x ∈ IRN : d(x, M ) ≤ δ}. Moreover, w will denote a ground state solution of problem (AV0 ) and η˜ a smooth non increasing cut-off function, defined in [0, ∞) satisfying η˜(s) = 1 if 0 ≤ s ≤

δ and η˜(s) = 0 if s ≥ δ. 2

For any y ∈ M , let us define a function Ψ,y ∈ W as x − y Ψ,y (x) = η˜(| x − y |)w ,

561


and denote by t > 0 the unique positive number verifying max J (tΨ,y ) = J (t Ψ,y ). t≥0

From the above definitions, we can consider a function Φ : M → N given by Φ (y) = t Ψ,y .

Lemma 3.4 The function Φ verifies the following limit: lim J (Φ (y)) = cV0 , unif ormly in y ∈ M.

→0

Proof. Suppose, by way of contradiction, that the lemma does not hold. Then, there exist δ0 > 0, {yn } ⊂ M and n , n → 0, such that | Jn (Φn (yn )) − cV0 |≥ δ0 .

(3.7)

n Considering the change of variable : z = n x−y , if z ∈ B δ (0), it follows that n z ∈ n n Bδ (0) and n z + yn ∈ Bδ (yn ) ⊂ Mδ ⊂ Ω. Since G ≡ F in Ω, we deduce

Jn (Φn (yn ))

=

tp n

Z

p

p

IRN

|∇˜ η (|n z|)w(z)| +

tp n p

Z IRN

V (n z + yn )|˜ η (|n z|)w(z)|

p

(3.8)

Z − IRN

F (tn η˜(|n z|)w(z)).

Claim 1. The sequence {tn } verifies tn → 1 as n → 0. Indeed, from the definition of tn , it follows that Z f (tn η˜(|n z|)w(z))|˜ η (|n z|)w(z)|p . kΨn ,yn kpn = (tn η˜(|n z|)w(z))p−1 IRN

(3.9)

Since η˜ ≡ 1 in Bδ/2 (0) and Bδ/2 (0) ⊂ Bδ/2n (0) for n sufficiently large, we get from (3.9) Z f (tn w(z)) kΨn ,yn kpn ≥ |w(z)|p . p−1 (t w(z)) Bδ/2 (0) n Denoting by z ∈ IRN a vector such that w(z) = minB¯δ/2 (0) w(z), we get the inequality kΨn ,yn kpn ≥

f (tn w(z)) (tn w(z))p−1

Z

|w(z)|p .

(3.10)

Bδ/2 (0)

Supposing, by contradiction, that there exists a subsequence, which we also denote by (tn ), verifying tn → ∞. From (f3 ) and passing to the limit of n → ∞ in (3.10), we have lim kΨn ,yn kpn = ∞,

n→∞

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which is an absurd, because it easy to check that lim kΨn ,yn kpn = kwkpV0 .

(3.11)

n→∞

Hence, (tn ) is a bounded sequence and we may assume, without loss of generality, that tn → t0 with t0 ≥ 0. Note that t0 > 0, because if t0 = 0, from growth of g, by limitation of kΨn ,yn kpn and (3.9), we can prove that lim kΨn ,yn kpn = 0

n→∞

obtaining again a contradiction with (3.11). On the other hand, using Lebesgue’s Theorem we have Z Z lim f (˜ η (|n z|)w(z))˜ η (|n z|)w(z) = n→∞

IRN

f (w)w

(3.12)

IRN

hence, passing to the limit of n → ∞ in (3.9), we obtain the equality Z Z Z f (t0 w)w . |∇w|p + V0 |w|p = tp−1 IRN IRN IRN 0 Since w belongs to the Nehari’s manifolds defined by functional EV0 , it follows that t0 = 1 and thus lim Jn (Φn ,yn ) = EV0 (w) = cV0 ,

n→∞

which leads to a contradiction with (3.7) and the lemma is proved.

For any δ > 0, let ρ = ρ(δ) > 0 be such that Mδ ⊂ Bρ (0), and let χ : IRN → IRN be defined as χ(x) = x for | x |≤ ρ and χ(x) = ρx/|x| for |x| ≥ ρ. In what follows, let us consider β : N → IRN given by R p N χ(x) | u(x) | . β(u) = IR R p | u(x) | IRN The next lemma shows an important property involving the function β, which is one of the key points in our arguments to establish the existence of multiple solutions by means of Lusternik-Schnirelman category, and its proof follows using the same type of arguments explored in [8] ( see also [3] ). Lemma 3.5 The function β verifies the following limit: lim β(Φ (y)) = y, unif ormly in y ∈ M.

→0

The result below is related to the existence of a minimum to Eµ on the Nehari manifolds and its proof can be found in [1], [3] or [8].

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Lemma 3.6 (A compactness Lemma) Let {un } ⊂ Mµ be such that Eµ (un ) → cµ . Then, a) {un } has a subsequence strongly convergent in W 1,p (IRN ) or b) there exists a sequence {˜ yn } ⊂ IRN such that, up to a subsequence, vn (x) = un (x + yñ ) converges strongly in W 1,p (IRN ). In particular there exists a minimizer for cµ . Proposition 3.3 Let n → 0 and {un } ⊂ Nn be such that Jn (un ) → cV0 . Then there exists a sequence {˜ yn } ⊂ IRN such that vn (x) = un (x + yñ ) has a convergent subsequence in W 1,p (IRN ). Moreover, up to a subsequence, yn → y ∈ M , where yn = n yñ . Proof. Arguing as in Alves & Soares [2, Lemma 4], there exist a sequence {˜ yn } ⊂ IRN and constants R and γ satisfying Z | un |p ≥ γ > 0. lim inf n→∞

BR (˜ yn )

Considering vn (x) = un (x + yñ ), we have, up to a subsequence, vn * v 6≡ 0 in W 1,p (IRN ). Thus, fixing tn > 0 such that vñ = tn vn ∈ MV0 , it follows EV0 (˜ vn ) → cV0 and (˜ vn ) ⊂ MV0 , and hence, up to a subsequence, vñ * v˜ weakly in W 1,p (IRN ) and tn → t0 > 0. Therefore, v˜ 6≡ 0 and from Lemma 3.6, vñ → v˜ in W 1,p (IRN ) showing that vn → v in W 1,p (IRN ) with v 6≡ 0. To conclude the proof of the proposition, we consider yn = n yñ . Our goal is to show that {yn } has a subsequence, still denoted by {yn }, satisfying yn → y for y ∈ M . First of all, we claim that {yn } is bounded. Indeed, suppose that there exists a subsequence, still denoted by {yn }, verifying |yn | → ∞. Note that Z Z Z |∇vn |p + V0 |vn |p ≤ g(n z + yn , vn )vn , IRN

IRN

IRN

thus fixing R > 0 such that BR (0) ⊃ Ω, we have Z Z Z Z p p e |∇vn | + V0 |vn | ≤ f (vn )vn + IRN

IRN

IRN \BR/n (0)

BR/n (0)

Since vn → v in W 1,p (IRN ) and V0 p−1 v , fe(vn ) ≤ k n it follows that Z

p

Z

|∇vn | + IRN

IRN

|vn |p = on (1)

f (vn )vn .

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showing that vn → 0 in W 1,p (IRN ), leading to a contradiction. Hence {yn } is bounded and, up to a subsequence, yn → y ∈ IRN . Arguing as above, if y 6∈ Ω , we will obtain again vn → 0 in W 1,p (IRN ), thus y ∈ Ω. If V (y) = V0 , we have y 6∈ ∂Ω and consequently y ∈ M . Supposing, by contradiction, that V (y) > V0 , we have Z Z Z 1 cV0 = EV0 (e v) < |∇e v |p + V (y)|e v |p − F (e v ). p IRN IRN IRN Using again the fact that ven → ve in W 1,p (IRN ), from Fatou’s Lemma Z Z Z 1 p p cV0 < lim inf |∇e vn | + V (n z + yn )|e vn | − F (e vn ) n→∞ p IRN IRN IRN that is, cV0 < lim inf Jn (tn un ) ≤ lim inf Jn (un ) = cV0 , n→∞

n→∞

obtaining a contradiction.

Let h : IR+ → IR+ is a positive function satisfying h() → 0 as → 0 and e = {u ∈ N : J (u) ≤ cV + h()}. N 0

Lemma 3.7 Let δ > 0 and Mδ = {x ∈ IRN : dist(x, M ) ≤ δ}. Then lim sup inf β(u) − y = 0. →0

e u∈N

y∈Mδ

e Proof. Let {n } ⊂ IR be a sequence with n → 0. For each n, there exists {un } ⊂ N such that sup inf β(u) − y = inf β(un ) − y + on (1). y∈Mδ y∈Mδ e u∈N n

Thus, it is sufficient to find a sequence {yn } ⊂ Mδ verifying lim β(un ) − yn = 0. n→∞

(3.13)

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e ⊂ N and thus, In order to obtain the sequence {yn }, we recall firstly that {un } ⊂ N n n ρn ≤ Jn (un ) ≤ cV0 + h(n ). Once that ρn ≥ cV0 , we get Jn (un ) → cV0 . From Proposition 3.3, there exists a sequence {yñ } ⊂ IRN such that yn = n yñ ∈ Mδ for n sufficiently large. Defining vn (x) = un (z + yñ ), by a direct calculus R p N [χ(n z + yn ) − yn ] | vn (z) | R = yn + on (1), β(un ) = yn + IR p | vn (z) | IRN because vn → v in W 1,p (IRN ). Therefore the sequence {yn } verifies (3.13). Theorem 3.2 For any δ > 0, there exists δ > 0 such that (P˜ )a has at least catMδ (M ) positive solutions, for any 0 < < δ . Proof. Fixing a small > 0, by Lemmas 3.4 and 3.7, we have that β ◦ Φ is homotopic to the inclusion map id : M → Mδ , which implies that e ) ≥ catM (M ). catNe (N δ Since functional J satisfies the (P S)c condition for c ∈ (cV0 , cV0 + h()), by means of the Lusternik-Schnirelman theory of critical points (see [9], [14]) we can conclude that J has ˜ ) on N . Therefore, from the last inequality, J has at least catM (M ) at least catN˜ (N δ critical points on N . Consequently by Corollary 3.1, J has at least catMδ (M ) critical points on W .

4

Multiplicity of solutions for the original problem

Here, we use the Moser iteration technique [12] to prove that each solution found in Theorem 3.2 is a solution of (P˜ ) for small .

Lemma 4.1 Let {xn } ⊂ Ωn and {n } be sequences with n → 0 as n → ∞. If vn (x) = un (x + xn ) where un is a solution of (Pñ )a given by Theorem 3.2, then {vn } converges uniformly on compact subsets of IRN . Proof. Note that vn verifies the following problem:   −∆p vn + Vn (x)vnp−1 = g(n x + xn , vn ) (Pn )  vn ∈ W 1,p (IRN ) and vn > 0 in IRN

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where xn = n xn and Vn (x) = V (n x + xn ). For each n ∈ IN and L > 0, define vn if vn ≤ L vL,n = L if vn ≥ L, p(β−1)

zL,n = vL,n

vn

β−1 and wL,n = vn vL,n

with β > 1 to be determined later. Taking zL,n as a test function, we obtain Z Z p(β−1) pβ−p−1 vL,n |∇vn |p = −p(β − 1) vL,n vn |∇vn |p−2 ∇vn ∇vL,n N N IR IR Z Z p(β−1) p(β−1) − Vn |vn |p vL,n , g(n x + xn , vn )vn vL,n + IRN

IRN

and then Z IRN

p(β−1)

vL,n

|∇vn |p ≤

Z

p(β−1)

IRN

g(n x + xn , vn )vn vL,n

Z

p(β−1)

− IRN

V0 |vn |p vL,n

.

Since for each ξ > 0 there exists Cξ > 0 such that |g(x, s)| ≤ ξ|s|p−1 + Cξ |s|q−1 ∀x ∈ IRN and s ∈ IR, fixing ξ sufficiently small , we have Z Z p(β−1) p vL,n |∇vn | ≤ C IRN

p(β−1)

IRN

vnq vL,n

.

From Sobolev imbedding and Hölder inequalities |wL,n |pp∗ ≤ Cβ p where p
0, where for each n ∈ IN , un is a solution of (Pñ )a given by Theorem 3.2. Then, lim V (xn ) = V0

n→∞

where xn = n xn . Proof. Up to a subsequence, xn → x ∈ Ω. e ⊂ N , Arguing as in Lemma 3.7, we have un ∈ N n n Jn (un ) → cV0 ,

J0n (un )un = 0 ∀ n ∈ IN

and kun kn ≤ C,

∀ n ∈ IN

, for some C > 0.

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Considering vn (z) = un (z+xn ), we have kvn kW 1,p (IRN ) ≤ C and vn * v in W 1,p (IRN ). Recalling that vn (0) = un (xn ) ≥ b > 0, by Lemma 4.1 v 6≡ 0. Fixing tn > 0 verifying ven = tn vn ∈ MV0 , and repeating the same arguments explored in proof of Claim 1 ( see Section 3.1 ), for some subsequence, still denoted by {tn }, we can assume that tn → to > 0. Moreover, cV0 ≤ EV0 (e vn )

≤

tpn p

Z

|∇vn |p + IRN

tpn p

Z

V (n z + xn )|vn |p −

Z

IRN

G(n z + xn , tn vn ) IRN

that is cV0 ≤ EV0 (e vn ) ≤ Jn (tn vn ) ≤ Jn (un ) = cV0 + on (1). Thus, EV0 (e vn ) → cV0 and {e vn } ⊂ MV0 . Considering v˜ = to v 6= 0, and repeating the same arguments explored in Alves [1, Thorem 2] , we get W 1,p (IRN ) and EV0 (e v ) = cV0 .

ven → ve in

(4.15)

Once that, V is continuous, we have lim V (xn ) = V (x) ≥ V0 ,

n→∞

therefore if V (x) > V0 , we get Z Z Z 1 1 v |p − F (e v) cV0 = EV0 (e v) < |∇e v |p + V (x)|e p IRN p IRN IRN and by (4.15) and Fatou’s Lemma Z Z Z 1 1 p p |∇e vn | + V (n z + xn )|e vn | − F (e vn ) cV0 < lim inf n→∞ p IRN p IRN IRN p Z Z Z tn tpn p p ≤ lim inf |∇vn | + V (n z + xn )|vn | − G(n z + x, tn vn ) n→∞ p IRN p IRN IRN = lim inf Jn (tn un ) ≤ lim inf Jn (un ) = cV0 , n→∞

n→∞

which leads a absurd. Thus, lim V (xn ) = V0 and the lemma is proved. n→∞

e is a solution of (P )a }, then there exists Lemma 4.3 If m∗ = sup{max u : u ∈ N ∂Ω

> 0 such that m∗ < ∞ for all ∈ (0, ); and, moreover, lim→0 m∗ = 0.

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Proof. First of all, note that if there exists some > 0 verifying m∗ = ∞, then for fixed e such that b > 0, there exists a solution u of (P )a in N max u ≥ b > 0. ∂Ω

Suppose, by contradiction, that there exists {n } ⊂ IR+ with n → 0 and m∗n = ∞. Then, there exists a sequence {xn } ⊂ ∂Ωn satisfying un (xn ) ≥ b > 0. Thus, by Lemma 4.2, we have lim V (xn ) = V0 ,

n→∞

where xn = n xn and {xn } ⊂ ∂Ω. Hence, up to a subsequence, we have xn → x in ∂Ω and V (x) = V0 , which does not make sense by (V2 ). Hence, there exists > 0 such that m∗ < ∞, for all ∈ (0, ). Suppose, by contradiction, that there exists δ > 0 and a sequence {n } ⊂ IR+ satisfying m∗n ≥ δ > 0. e such that Thus, there exists un a solution of (Pñ )a in N n m∗n −

δ < max u ≤ m∗n . 2 ∂Ωn n

Hence, δ δ δ = δ − ≤ m∗n − < max un ∂Ω 2 2 2 and then there exists a sequence {xn } ⊂ ∂Ωn such that un (xn ) ≥

δ . 2

Repeating the above arguments, we will again get an absurdity. Thus, the proof is finished.

4.1

Proof of Theorem 1.1:

• Multiplicity of positive solutions By Theorem 3.2, we have that the problem (P˜ )a has at least catMδ (M ) positive solutions for each ∈ (0, ). Let u be one of these solutions of (P )a . By Lemma 4.3, m∗ < t0 , for all ∈ (0, ¯); hence (u − t0 )+ (x) = 0 for a neighborhood of ∂Ω . Hence, (u − t0 )+ ∈ W01,p (IRN \Ω ) and the function (u − t0 )∗+ ∈ W 1,p (IRN ), where

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(u − t0 )∗+ (x) = 0 if x ∈ Ω and (u − t0 )∗+ (x) = (u − t0 )+ (x) if x ∈ IRN \Ω . Using (u − t0 )∗+ as function test, we have g(x, u ) p−2 |∇(u − + ((u − t0 )∗+ )2 V (x)|u | − u IRN \Ω IRN \Ω Z g(x, u ) p−2 t0 (u − t0 )∗+ = 0. + V (x)|u | − u IRN \Ω Z

t0 )∗+ |p

Z

The last equality together with (g4 )ii imply (u − t0 )∗+ = 0, a.e in x ∈ IRN \Ω and the result follows by Remark 2.1. • Concentration of positive solutions f ), In order to study the concentration behavior of positive solutions u˜ of problem (P we use the Moser iteration technique [12] and adapt for our case some arguments found in Li Gongbao [10]. The next two results are important points to prove the concentration of solutions and their proof can be found in [3]

Lemma 4.4 Let vn be a solution for problem   −∆p vn + Vn (x)|vn |p−2 vn = f (vn ) in IRN vn ∈ W 1,p (IRN ) with 1 0, ∀ x ∈ IRN ,

(∗)

where Vn (x) = V (n x + n yñ ), V satisfies (V1 ), f satisfies (f1 ) − (f5 ), {˜ yn } is given in Proposition 3.3 and vn → v in W 1,p (IRN ) with v 6≡ 0. Then vn ∈ L∞ (IRN ) and there exists C > 0 such that kvn kL∞ (IRN ) ≤ C for all n ∈ IN . Furthermore lim vn (x) = 0 uniformly in n.

|x|→∞

Lemma 4.5 Let Pn be a global maximum of vn . Then there exists δ > 0 such that vn (Pn ) ≥ δ ∀n ∈ IN . Proposition 4.1 The family of solutions obtained in Theorem 3.2 verifies the following property: If un is a positive solutions of (Pñ ) and zn is a global maximum point, then lim V (n zn ) = V0 . n→∞

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Proof. If un is a solution of problem (Pñ ), then vn (x) = un (x + yñ ) is a solution of problem   −∆p vn + Vn (x)|vn |p−2 vn = f (vn ) in IRN vn ∈ W 1,p (IRN ) with 1 0, ∀ x ∈ IRN , with Vn (x) = V (n x + n yñ ) and {yñ } ⊂ IRN given in Proposition 3.3. Moreover, up to a subsequence, vn → v in W 1,p (IRN ) and yn → y in M , where yn = n yñ . Consider pn a point where the global maximum of vn holds. By Lemmas 4.4 and 4.5, we have that pn ∈ BR (0) for some R > 0. Thus, if z is a point where the global maximum of un holds, we have that z = pn + yñ and therefore n zn = n pn + n yñ = n pn + yn . Since (pn ) is bounded, it follows that lim V (n zn ) = V0 .

n→∞

• Conclusion of the proof of Theorem 1.1 If u is a positive solution of (P˜ ), the function w (x) = u (x/) is a positive solution of (P ). Thus, if η and z are global maximum points of w and u respectively, we have the equality η = z , and consequently, lim V (η ) = V0 .

→0

Acknowledgments. The authors would like to thank the anonymous referee for valuable comments and suggestions made in a very detail report.

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