Multiplicity results for quasi-linear problems

Nonlinear Analysis 68 (2008) 1802–1815 www.elsevier.com/locate/na

Multiplicity results for quasi-linear problems A. Ayoujil, A.R. El Amrouss ∗ University Mohamed I, Faculty of Sciences, Department of Mathematics, Oujda, Morocco Received 5 July 2006; accepted 11 January 2007

Abstract Applying the minimax arguments and Morse theory, we establish some results on the existence of multiple nontrivial solutions for a class of p-Laplacian elliptic equations. c 2007 Elsevier Ltd. All rights reserved.

MSC: 58E05; 35J65; 49B27 Keywords: p-Laplacian; Variational method; Critical group; Multiple solution; Morse theory

1. Introduction Let us consider the nonlinear elliptic problem  −1 p u = f (x, u) in Ω , (P) u=0 on ∂Ω , where Ω is a bounded domain in R N with smooth boundary ∂Ω , 1 p is the p-Laplacian operator defined by 1 p u := div(|∇u| p−2 ∇u), 1 < p < ∞. We assume that f : Ω × R → R is a Carath´eodory function satisfying the subcritical growth (f0 ) | f (x, t)| ≤ c(1 + |t|q−1 ),

∀t ∈ R, a.e. x ∈ Ω ,

for some c > 0, and 1 ≤ q < p ∗ where p ∗ =

Np N−p

if 1 < p < N and p ∗ = +∞ if N ≤ p.

1, p

We denote by λ1 the first eigenvalue of −1 p on W0 (Ω ). It is known that λ1 > 0 is a simple eigenvalue, which has an associated eigenfunction ϕ1 > 0 with kϕ1 k = 1. If f (x, 0) = 0 for a.e. x ∈ Ω the constant function u = 0 is a trivial solution of problem (P). In this case, the key point is proving the existence of nontrivial solutions for (P). For this purpose, R t we need to introduce some conditions on the behaviors of the perturbed function f (x, t) or its primitive F(x, t) = 0 f (x, s) ds near infinity and near zero. ∗ Corresponding author. Tel.: +212 56 50 03 43; fax: +212 56 50 06 03.

E-mail addresses: [email protected] (A. Ayoujil), [email protected] (A.R. El Amrouss). c 2007 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2007.01.013

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There are many papers devoted to the existence of multiple solutions of the above problem; see for example [3–6, 9,10] and the references therein. Most of them treated the problem (P) via direct variational methods or minimax arguments such as the well known saddle point theorem or the mountain pass theorem. In [3], a contribution was made for when p F(x,s) interacts asymptotically with the first eigenvalue. In [6], the |s| p authors treated the resonance near zero at the first eigenvalue from the right and the non-resonance condition at infinity below λ1 . Another contribution was made in [4], where the second author and co-workers established the stays asymptotically between the two first eigenvalues existence of at least one solution when the nonlinearity p F(x,s) |s| p of −1 p . For the asymptotically linear at both zero and infinity, the authors in [10] got the existence of at least two nontrivial solutions. More recently, in [9] the authors obtained the existence of multiple nontrivial solutions for the case lim|s|→∞ p F(x,s) < λ1 . |s| p As is well known, the Morse theory developed by Chang [2] or Mawhin and Willem [7] is very useful in studying the existence of multiple solutions for differential equations having the variational structure. Thus computation of critical groups may yield the existence and multiplicity of nontrivial solutions to our problem. We recall that the weak solutions of (P) are the critical points of the associated energy functional Φ, given by Z Z 1 p Φ(u) = |∇u| dx − F(x, u) dx, p Ω Ω 1, p

acting on the Sobolev space W0 (Ω ), which is a Banach space endowed with the norm Z

|∇.| p

k.k =

1

Ω

p

. 1, p

It is well known that under the subcritical growth condition (f0 ), the functional Φ is of class C 1 on W0 (Ω ). In this paper, we start by proving the existence of at least three nontrivial solutions via Morse theory and the following conditions. (f1 ) lim|t|→∞ (F(x, t) − λp1 |t| p ) = −∞ uniformly for a.e. x ∈ Ω . (f2 ) There exist µ ∈ (0, p), δ > 0 and α a non-positive constant, such that 0 < µF(x, t) ≤ t f (x, t),

for a.e. x ∈ Ω , 0 < |t| ≤ δ,

(1.1)

and µF(x, t) − t f (x, t) lim inf ≥ α > λ1 |t|→0 |t| p



µ −1 p



uniformly a.e. x ∈ Ω .

(1.2)

The main result reads as follows. Theorem 1.1. Under (f0 ), (f1 ) and (f2 ), the problem (P) has at least three nontrivial solutions. The second purpose of this paper is to show the existence of at least two nontrivial solutions of problem (P) under the following assumptions. (f3 ) λ1 . l(x) = lim inf|t|→∞

f (x,t) |t| p−2 t

≤ k(x) = lim sup|t|→∞

f (x,t) |t| p−2 t

≤ C uniformly a.e. x ∈ Ω , for some C > 0.

Here the relation l(x) . k(x) indicates that l(x) ≤ k(x) with strict inequality holding on a set of positive measure. (f4 ) lim sup|t|→0

p F(x,t) |t| p

≤ β < λ1 , uniformly a.e. x ∈ Ω , with β ∈ R.

We can state the following result. Theorem 1.2. Assume (f0 ), (f3 ) and (f4 ). Then the problem (P) has at least two nontrivial solutions, in which one is positive and one is negative. Remark 1. In Theorem 1.1 [9], the authors established the existence of at least three nontrivial solutions of problem (P), under (f0 ) and the following hypotheses.

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(f1 0 ) lim

|t|→∞

(f2

0)

p F(x, t) < λ1 . |t| p

For some ν ∈ (1, p), there are constants r, ar > 0 such that F(x, t) ≥ ar |t|ν ,

(f2

00 )

for |t| ≤ r.

p F(x, t) − t f (x, t) > 0, a.e. x ∈ Ω , ∀t 6= 0. 00

Note that our condition (f1 ) is weaker than (f1 0 ) and condition (f2 ) is a strong assumption. More, the existence of at least two nontrivial solutions of our problem are obtained in Theorem 1.2 [10], by assuming the following conditions. (f3 0 ) f ∈ C(Ω × R, R); f (x, 0) = 0, ∀x ∈ Ω . p f (x,t) (f4 0 ) lim|t|→0 |t|f (x,t) p−2 t = µ, lim|t|→∞ |t| p−2 t = λ and µ < λ1 < λ. Obviously, our second result extends this result. Our multiplicity results are not covered by the results mentioned in [4,6,9,10]. This paper is organized as follows. In Section 2, several technical lemmas are presented and proved. In Section 3, we will prove the existence of at least three nontrivial solutions by combining the minimax method and Morse theory. In Section 4, we will give the proof of Theorem 1.2 by using a minimax method such as the mountain pass theorem. 2. Preliminary results We consider the truncated problem  −1 p u = f ± (x, u) in Ω , (P± ) u=0 on ∂Ω , where f ± (x, t) =



f (x, t) 0

if ± t ≥ 0, otherwise.

We denote by u + := max(u, 0) and u − := max(−u, 0) the positive and negative parts of u. We need the following lemmas. 1, p

Lemma 2.1. The mappings u 7→ u ± are continuous on W0 (Ω ). 1, p

Proof. Since u ± = 12 (|u| ± u), it suffices to prove that the mapping u 7→ |u| is continuous on W0 (Ω ) i.e. u n → u 1, p implies |u n | → |u|. We have |u n | → |u| in L p (Ω ); by weak compactness |u n | * z in W0 (Ω ) for a subsequence. Hence z = |u| and |u n | * |u| for the whole sequence. On the other hand, we have ∇|u| = sgn(u)∇u a.e. and 1, p 1, p |||u||| = kuk. By uniform convexity of W0 (Ω ) it follows that |u n | → |u| in W0 (Ω ). Thus, the lemma follows.  Lemma 2.2. All solutions of (P+ ) (resp. (P− )) are positive (resp. negative) solutions of (P). 1, p

Proof. Define Φ± : W0 (Ω ) → R, Z Z 1 Φ± (u) = |∇u| p dx − F± (x, u) dx, p Ω Ω Rt where F± (x, t) = 0 f ± (x, s) ds. It is well known that under subcritical growth condition on f , Φ± is well defined 1, p on W0 (Ω ), weakly lower semi-continuous and C 1 -functionals. Let u be a solution of (P+ ), or equivalently, u be a critical point of Φ+ . Taking v = u − in Z 0 hΦ+ (u), vi = (|∇u| p−2 ∇u∇v − f + (x, u)v) dx = 0, Ω

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shows that ku − k = 0, so u − = 0 and u = u + is also a critical point of Φ with critical value Φ(u) = Φ+ (u). Furthermore, by Anane [1], u ∈ L ∞ (Ω ) ∩ C 1 (Ω ). The maximum principle implies that either u > 0 or u ≡ 0. Similarly, nontrivial critical points of Φ− are negative solutions of (P).  For the proof of the main theorem, we need the following lemma. Lemma 2.3. Nontrivial local minimizers of Φ± are also minimizers of Φ. 1, p

Proof. For a local minimizer u 0 > 0 of Φ+ , we will show that for every sequence u n → u 0 in W0 (Ω ) we have Φ(u n ) ≥ Φ(u 0 ),

for sufficiently large n.

(2.1)

By (f0 ), we get |F(x, t)| ≤ C|t|q ,

∀t, for some C > 0.

− ± In view u n = u + n − u n and ∇u n = ±∇u n , we obtain Z Z Z Z 1 1 p − p Φ(u n ) = |∇u + | − F (x, u ) + |∇u | − F− (x, u n ) + n n n p Ω p Ω Ω Ω 1 − p − q ≥ Φ+ (u + n ) + ku n k − Cku n kq . p

From Lemma 2.1, we have + u+ n → u0 = u0

and

u− n →0

1, p

in W0 (Ω ).

(2.2)

So Φ(u n ) ≥ Φ+ (u 0 ) = Φ(u 0 )

for sufficiently large n.

Consequently, if u − n = 0, then (2.1) is proved. Otherwise, we will show that q

p − ku − n k ≥ pCku n kq ,

for a large n.

(2.3)

Indeed, letting Ωn = {x ∈ Ω : u n (x) < 0}, we claim that |Ωn | → 0,

as n → ∞.

(2.4)

In fact, given ε > 0, take a compact subset Ω ε of Ω such that |Ω \ Ω ε | < ε and let Ωnε = Ω ε ∩ Ωn . Then Z Z p p p ku n − u 0 k p ≥ |u n − u 0 | ≥ u 0 ≥ c p |Ωnε |, Ωnε

Ωnε

where c = minΩ ε u 0 > 0. Passing to the limit, we obtain |Ωnε | → 0,

as n → ∞.

Since Ωn ⊂ Ωnε ∪ (Ω \ Ω ε ) and ε > 0 is arbitrary, (2.4) follows. If (2.3) does not hold, setting zn =

u− n . ku − n kq 1, p

We claim that kz n k is bounded in W0 (Ω ). In fact, if q ≤ p, then q

p

p − − ku − n k ≤ Cku n kq ≤ Cku n kq .

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If p < q, take An =

ku − nk q

.

p ku − n kq

From (2.2), it follows that q p kz n k = An ku − n kq

−1

q

p ≤ Cku − nk

−1

≤ M,

for a large n.

L q (Ω ),

Therefore, for a subsequence, z n → z in where kzkq = 1 and z ≥ 0. But Ωµ = {x ∈ Ω : z(x) ≥ µ} has a positive measure for all sufficiently small µ > 0 and Z Z q q kz n − zkq ≥ |z n − z| = z q ≥ µq (|Ωµ | − |Ωn |). Ωµ \Ωn

Ωµ \Ωn

Letting n → +∞, one has 0 ≥ |Ωµ |, a contradiction. Similarly, a nontrivial local minimizer of Φ− is a minimizer of Φ. Now, we will show that the critical groups of Φ at zero are trivial. To proceed, some concepts are needed. Let X be a Banach space and Φ ∈ C 1 (X, R). For γ ∈ R and c ∈ R, we set Φ γ = {u ∈ X : Φ(u) ≤ γ }, K = {u ∈ X : Φ 0 (u) = 0}, K c = {u ∈ X : Φ(u) = c, Φ 0 (u) = 0}. Denote by Hq (A, B) the q-th homology group of the topological pair (A, B) with integer coefficient. The critical groups of Φ at an isolated critical point u ∈ K c are defined by Cq (Φ, u) = Hq (Φ c ∩ U, Φ c ∩ U \ {u}),

q ∈ Z,

where U is a closed neighborhood of u. Moreover, it is known that Cq (Φ, u) is independent of the choice of U due to the excision property of homology. We refer the readers to [2,7] for more information. Recall that in the case when Φ satisfies the Palais–Smale condition and for [a, b] ⊂ R ∪ {∞} the critical set 1, p

K ab = {u ∈ W0 (Ω ); a ≤ Φ(u) ≤ b, Φ 0 (u) = 0} is finite, we have the following Morse relations between the Morse critical groups and homological characterization of subset sets: M Hq (Φ a , Φ b ) = Cq (Φ, u).  K ab

Lemma 2.4. Assume (f0 ) and (f2 ). Cq (Φ, 0) ∼ = 0, ∀q ∈ Z. 1, p

Proof. Let Bρ = {u ∈ W0 (Ω ), kuk ≤ ρ}, ρ > 0 which is to be chosen later. The idea of the proof is to construct a retraction of Bρ \ {0} to Bρ ∩ Φ 0 \ {0} and to prove that Bρ ∩ Φ 0 is contractible in itself. For this purpose, we need to analyze the local properties of Φ near zero. Thus, some technical affirmations must be proved. Note that the following claim has been proved in case p = 2, [8, Lemma 1.1]. Claim 1. Under (f0 ) and (f2 ), zero is a local maximum for the functional Φ(su), s ∈ R, for u 6= 0. In fact, it follows from the first condition of (f2 ) that there exists a constant c0 > 0 such that F(x, t) ≥ c0 |t|µ ,

for x ∈ Ω , |t| ≤ δ.

(2.5)

Using (f0 ) and (2.5), we get F(x, t) ≥ c0 |t|µ − c1 |t|q , for some q ∈ ( p,

p∗ )

and c1 > 0.

x ∈ Ω, t ∈ R

(2.6)

A. Ayoujil, A.R. El Amrouss / Nonlinear Analysis 68 (2008) 1802–1815

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1, p

Then, for u ∈ W0 (Ω ), u 6= 0 and s > 0, we have Z Z 1 F(x, su) dx |∇u| p dx − Φ(su) = s p p Ω Ω Z sp kuk p − (c0 |su|µ − c1 |su|q ) dx ≤ p Ω ≤

sp q q kuk p − c0 s µ kukµ µ + c1 s kukq . p

(2.7)

Since µ < p < q, there exists a s0 = s0 (u) > 0 such that Φ(su) < 0,

for all 0 < s < s0 .

(2.8)

Claim 2. There exists ρ > 0 such that d Φ(su)|s=1 > 0, ds

(2.9)

1, p

for every u ∈ W0 (Ω ) with Φ(u) = 0 and 0 < kuk ≤ ρ. 1, p

Indeed, let u ∈ W0 (Ω ) be such that Φ(u) = 0. In turn, for (f2 ) and (f0 ) respectively, we have for ε > 0 sufficiently small that there exists r := r (ε) > 0 such that µF(x, u) − f (x, u)u ≥ (α − ε)|u| p ,

a.e. x ∈ Ω and |u| ≤ r,

and µF(x, u) − f (x, u)u ≥ −cε |u|q ,

a.e. x ∈ Ω and |u| > r,

for some q ∈ ( p, p ∗ ) and cε > 0. Define Ωr (u) = {x ∈ Ω : |u| > r }

and

Ω r (u) = {x ∈ Ω : |u| ≤ r }.

Then, since Φ(u) = 0 and by the Poincar´e’s inequality, we write d Φ(su) = hΦ 0 (su), ui|s=1 ds s=1 Z Z = |∇u| p dx − f (x, u)u dx, Ω Ω  Z Z µ = 1− |∇u| p dx + (µF(x, u) − f (x, u)u) dx p Ω Ω r (u) Z + (µF(x, u) − f (x, u)u) dx, Ωr (u)   Z Z µ kuk p + (α − ε) |u| p dx − cε |u|q dx, ≥ 1− r p Ω (u) Ωr (u) ≥ θkuk p − Cε kukq , for some Cε > 0, where θ = (1 − µp + λα1 − λε1 ). Since p < q, the inequality (2.9) follows for ε small enough such that θ > 0. 1, p

Claim 3. For all u ∈ W0 (Ω ) with Φ(u) ≤ 0 and kuk ≤ ρ, we have Φ(su) ≤ 0,

for all s ∈ (0, 1).

(2.10)

Indeed, given kuk ≤ ρ with Φ(u) ≤ 0, assume by contradiction that there exists some s0 ∈ (0, 1] such that Φ(s0 u) > 0. Thus, by the continuity of Φ, there exists an s1 ∈ (s0 , 1] such that Φ(s1 u) = 0. Choose s2 ∈ (s0 , 1] such

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that s2 = min{s ∈ [s0 , 1]; Φ(su) = 0}. It is easy to see that Φ(su) ≥ 0 for each s ∈ [s0 , s2 ]. Taking u 1 = s2 u, it is clear that d d Φ(su) − Φ(s2 u) ≥ 0 implies that Φ(su) = Φ(su 1 ) ≤ 0. ds ds s=s2 s=1 This is a contradiction with (2.9). The proof of the claim is completed. Let us fix ρ > 0 such that zero is the unique critical point of Φ in Bρ . First, by taking the mapping h : [0, 1] × (Bρ ∩ Φ 0 ) → Bρ ∩ Φ 0 as h(s, u) = (1 − s)u, we have that Bρ ∩ Φ 0 is contractible in itself. Now, we prove that (Bρ ∩ Φ 0 ) \ {0} is contractible in itself too. For this purpose, define a mapping T : Bρ \ {0} → (0, 1] by T (u) = 1,

for u ∈ (Bρ ∩ Φ 0 ) \ {0},

T (u) = s,

for u ∈ Bρ \ Φ 0 with Φ(su) = 0, s < 1.

(2.11)

From the relations (2.8)–(2.10), the mapping T is well defined and if Φ(u) > 0 then there exists an unique T (u) ∈ (0, 1) such that Φ(su) < 0,

∀s ∈ (0, T (u)),

Φ(T (u)u) = 0, Φ(su) > 0, ∀s ∈ (T (u), 1).

(2.12)

Thus, using (2.9) and (2.12) and the Implicit Function Theorem we get that the mapping T is continuous. Next, we define a mapping η : Bρ \ {0} → (Bρ ∩ Φ 0 ) \ {0} by η(u) = T (u)u, u ∈ Bρ \ {0} with Φ(u) ≥ 0, η(u) = u, u ∈ Bρ \ {0} with Φ(u) < 0.

(2.13)

Since T (u) = 1 as Φ(u) = 0, the continuity of η follows from the continuity of T . Obviously, η(u) = u for u ∈ (Bρ ∩ Φ 0 ) \ {0}. Thus, η is a retraction of Bρ \ {0} to (Bρ ∩ Φ 0 ) \ {0}. Since 1, p W0 (Ω ) is infinite dimensional, Bρ \ {0} is contractible in itself. By the fact that retracts of contractible space are also contractible, (Bρ ∩ Φ 0 ) \ {0} is contractible in itself. From the homology exact sequence, one has Hq (Bρ ∩ Φ 0 , (Bρ ∩ Φ 0 ) \ {0}) = 0,

∀q ∈ Z.

Hence Cq (Φ, 0) = Hq (Bρ ∩ Φ 0 , (Bρ ∩ Φ 0 ) \ {0}) = 0, The proof of Lemma 2.4 is completed.

∀q ∈ Z.



3. Proof of Theorem 1.1 Our approach is based on Morse theory by combining the result of Lemma 2.4 and the condition (f1 ). 1, p

Lemma 3.1. Φ+ and Φ− are coercive on W0 (Ω ). Proof. Put G(x, t) := F(x, t) −

λ1 p |t| . p

Then, from (f1 ) we conclude that, for every M > 0, there is R M > 0 such that G(x, t) ≤ −M,

∀|t| ≥ R M , a.e. x ∈ Ω .

(3.1)

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By contradiction, let K ∈ R and (u n ) ⊂ W0 (Ω ) be such that Φ+ (u n ) ≤ K .

ku n k → ∞ and Putting vn :=

un ku n k ,

1, p

one has kvn k = 1. For a subsequence, we may assume that for some v0 ∈ W0 (Ω ), we have 1, p

vn * v0 weakly in W0 (Ω ), vn → v0 strongly in L p (Ω ) and vn (x) → v0 (x) a.e. x ∈ Ω . Now, using (3.1) it follows that Z Z F(x, u + K Φ+ (u n ) 1 n) p |∇v | dx − ≥ = dx n p p p ku n k ku n k p Ω ku k n Ω Z Z + p G(x, u + 1 n ) + λ1 |u n | dx |∇vn | p dx − ≥ p p Ω ku n k Ω Z M1 1 (|∇vn | p − λ1 |vn+ | p ) dx + , ≥ p Ω ku n k p where M1 > 0. Letting n → ∞, we get Z Z + p lim sup |∇vn | dx ≤ 1 = lim sup |∇vn | p dx n

n

Ω

≤ λ1

Z Ω

Ω

|v0+ | p dx.

(3.2) 1, p

+ On the other hand, since ku + n k ≤ ku n k the sequence (vn ) is bounded in W0 (Ω ). Thus, for a subsequence we 1, p + + p may assume that vn * z in W0 (Ω ) and vn → z in L (Ω ). But, it is easy to see that if vn → v0 in L p (Ω ) we have vn+ → v0+ in L p (Ω ). Hence, z = v0+ and vn+ * v0+ for the whole sequence. Therefore, using Poincar´e’s inequality and the weak lower semi-continuity of the norm, it holds that Z Z Z + p + p λ1 |v0 | dx ≤ |∇v0 | dx ≤ lim inf |∇vn+ | p dx. (3.3)

Ω

n

Ω

Ω

Combining (3.2) and (3.3), one deduces Z Z |∇v0+ | p dx = λ1 |v0+ | p dx, Ω

Ω

and kvn+ k → kv0+ k. 1, p

Since W0 (Ω ) is uniformly convex, we get vn+ → v0+

1, p

in W0 (Ω ).

+ + Moreover, u + n = ku n kvn and u n → +∞ for almost every x ∈ Ω . Thus, we deduce from (f1 ) that Z Z 1 p (|∇u n | p − λ1 |u + | ) dx − G(x, u + K ≥ Φ+ (u n ) = n n ) dx p Ω Ω Z Z 1 (|∇u n | p − λ1 |u n | p ) dx − G(x, u + ≥ n ) dx p Ω Ω Z ≥− G(x, u + n ) dx → +∞.

Ω

1, p

This is a contradiction. Hence Φ+ is coercive on W0 (Ω ). 1, p Similarly, Φ− is coercive on W0 (Ω ).  Proof of Theorem 1.1. Since Φ+ is weakly lower semi-continuous and coercive, Φ+ is bounded from below and has 1, p a global minimizer u + on W0 (Ω ).

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o n 1 µ Let ψ = kϕ1 with 0 < k < min ( pc0 kϕ1 k L µ ) p−µ , maxx∈Ωδ ϕ1 (x) and c0 as given in (2.5). Since 0 < ψ < δ and Z kP − Φ+ (ψ) = F(x, ψ) dx, p Ω it follows from (2.5) that Φ+ (ψ) ≤

kP µ − c0 k µ kϕ1 k L µ < 0. p

Thus, u + 6= 0 and from Lemma 2.3, u + is a local minimum of Φ and Φ(u + ) = Φ+ (u + ) < 0. Similarly, we can obtain another local minimum u − < 0 and Φ(u − ) < 0. Hence the critical groups of Φ at its local minimizers are (see [2]) Cq (Φ, u + ) = Cq (Φ, u − ) = δq,0 Z,

for all q ∈ Z.

Now, we establish the existence of the third nontrivial critical point of Φ. First, since Φ is coercive, the (PS) 1, p condition is verified. Indeed, let (u n ) ⊂ W0 (Ω ) be such that Φ(u n ) → c ∈ R,

Φ 0 (u n ) → 0

−1, p 0

in W0

(Ω ), as n → ∞.

Thus, the sequence (u n ) is bounded and it is well known that the Palais–Smale condition is satisfied on the bounded 1, p subsets of W0 (Ω ); see [4]. Take a and b satisfying a<

inf

1, p W0 (Ω )

Φ ≤ min{Φ(u + ), Φ(u − )} ≤ max{Φ(u + ), Φ(u − )} ≤ b.

Suppose that K ab = {u − , u + }. Then, by the Morse relations we obtain Hq (Φ a , Φ b ) = δq0 (Z ⊕ Z)

∀q.

On the other hand, Hq (Φ a , Φ b ) = Hq (Φ b , ∅) = Hq (Φ b )

∀q,

and hence, Hq (Φ b ) = δq0 (Z ⊕ Z)

∀q.

Taking the pair (H, Φ b ) we get an exact sequence ···

1, p

1, p

1, p

→ H1 (W0 (Ω )) → H1 (W0 (Ω ), Φ b ) → H0 (Φ b ) → H0 (W0 (Ω )) 1, p

→ H0 (W0 (Ω ), Φ b ) → 0 1, p

1, p

= · · · → 0 → H1 (W0 (Ω ), Φ b ) → Z ⊕ Z → Z → H0 (W0 (Ω ), Φ b ) → 0. 1, p

Hence H1 (W0 (Ω ), Φ b ) 6= 0 and according to the Morse relations, the functional Φ has a critical point u 0 such that Φ(u 0 ) > b and C1 (Φ, u 0 ) 6= 0. By Lemma 2.4, we have C1 (Φ, 0) = 0 and it follows that u 0 6= 0. The proof is completed.



4. Proof of Theorem 1.2 To apply the mountain pass theorem, we will do separate studies of the “compactness” of Φ± and its “geometry”. Lemma 4.1. Under (f0 ) and (f3 ), Φ± satisfies the (PS) condition.

A. Ayoujil, A.R. El Amrouss / Nonlinear Analysis 68 (2008) 1802–1815

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1, p

Proof. Let (u n ) ⊂ W0 (Ω ) be a (PS) sequence, i.e., Φ+ (u n ) bounded, and 0 hΦ+ (u n ), vi ≤ εn kvk

1, p

∀v ∈ W0 (Ω ), 1, p

where εn → 0. It clearly suffices to show that (u n ) remains bounded in W0 (Ω ). For a contradiction, define vn = kuu nn k . Then, we have kvn k = 1 and passing if necessary to subsequence, we may 1, p

1, p

assume that there is some v ∈ W0 (Ω ) such that vn * v weakly in W0 (Ω ), vn → v strongly in L p (Ω ) and vn (x) → v(x), for a.e. x ∈ Ω . Using (f0 ), the sequence f + (x, u n ) , ku n k p−1

f n,+ (x) := 0

p is bounded in L p (Ω ) where p 0 = p−1 is the conjugate exponent. For a subsequence still denoted by ( f n,+ ), we may 0 p assume that there is a f˜+ ∈ L (Ω ) such that

f n,+ * f˜+ ,

0

weakly in L p (Ω ).

(4.1)

We need to state the following results. Claim 1. f˜+ = 0

a.e. in A = {x ∈ Ω ; v + (x) = 0 a.e.}.

Indeed, define ϕ(x) = sign( f˜+ (x))χ A (x), where sign : R → {−1, 1} is given by  1 if x ≥ 0, sign(x) = −1 if x < 0. Thus, the hypothesis (f0 ) and (f3 ) implies that   1 + p−1 + |vn | χ A (x), | f n,+ (x)ϕ(x)| ≤ c ku n k p−1

a.e. x ∈ Ω .

Since vn → v in L p (Ω ) and so vn+ → v + in L p (Ω ), it follows by passing to the limit that 0

f n,+ (x)ϕ(x) → 0

in L p (Ω ).

(4.2)

On the other hand, from (4.1) and | f˜+ | = sign( f˜+ ) f˜+ , we conclude that Z Z Z Z f n,+ ϕ dx → f˜+ ϕ dx = | f˜+ |χ A dx = | f˜+ | dx. Ω

Ω

Ω

A

It follows from (4.2) that Z | f˜+ | dx = 0. A

Thus Claim 1 is proved. Now, let us define  f˜+ (x)  m(x) = |v + (x)| p−2 v + (x)  λ1

a.e. x ∈ Ω \ A, a.e. x ∈ A.

(4.3)

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Claim 2. λ1 ≤ m(x)

a.e. x ∈ Ω .

In fact, we have λ1 ≤ m(x)

a.e. x ∈ A.

We show λ1 ≤

f˜+ (x) |v + (x)| p−2 v + (x)

a.e. x ∈ Ω \ A.

Put B = {x ∈ Ω \ A; λ1 |v + (x)| p > v + (x) f˜+ (x) a.e.}.

(4.4) p0

It suffices to prove that mes(B) = 0. Indeed, by (f3 ), for all ε > 0, there exist bε ∈ L (Ω ) such that bε (x) + (λ1 − ε)|t| p ≤ t f (x, t),

a.e. x ∈ Ω , ∀t ∈ R.

Therefore, it results that bε (x) + (λ1 − ε)|vn+ | p ≤ vn+ f n,+ (x), ku n k p

a.e. x ∈ Ω .

Multiplying (4.5) by χ B and integrating over Ω , we obtain Z Z Z bε (x) + p χ dx + (λ1 − ε) |vn | χ B dx ≤ vn+ f n,+ (x)χ B dx. p B Ω ku n k Ω Ω

(4.5)

(4.6)

So, letting n → ∞ in (4.6) we obtain Z Z (λ1 − ε) |v + (x)| p χ B ≤ v + f˜+ (x)χ B . Ω

Ω

Since ε is arbitrary, thus Z (v + f˜+ (x) − λ1 |v + (x)| p )χ B ≥ 0.

(4.7)

Ω

In turn for (4.4), the inequality (4.7) is verified if and only if mes(B) = 0. 1, p

Claim 3. vn → v strongly in W0 (Ω ) and  − 4 p v = m(x)|v + | p−2 v + in Ω , v=0 on ∂Ω .

(4.8)

In particular, v = v + and kvk = 1, and thus v > 0. 1, p Indeed, for all w ∈ W0 (Ω ) we have Z hΦ 0 (u n ), wi = (|∇vn | p−2 ∇vn ∇w − f n,+ w) dx. ku n k p−1 Ω 0

0 (u ) → 0 in W −1, p (Ω ) and going to infinity in (4.9), we get Taking w = vn − v, via Φ+ n

limh− 4 p vn , vn − vi = 0. Since − 4 p is of type S + , one concludes vn → v and so kvk = 1.

1, p

strongly in W0 (Ω ),

(4.9)

A. Ayoujil, A.R. El Amrouss / Nonlinear Analysis 68 (2008) 1802–1815

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By continuity of 4 p and passing to the limit in (4.9), we obtain Z 1, p (|∇v| p−2 ∇v∇w − f˜+ w) dx = 0 ∀w ∈ W0 (Ω ), Ω

and 

− 4 p v = f˜+ v=0

in Ω , on ∂Ω .

Combining the Claim 1 and (4.3), it follows that v is a solution of (4.8). However, the maximum principle implies that v = v + ≥ 0. Since kvk = 1, the proof of Claim 3 is completed. Claim 4. λ1 . m(x)

a.e. x ∈ Ω .

In fact, let us suppose that λ1 = m(x)

a.e. x ∈ Ω .

(4.10)

It follows from (f3 ) that for every ε > 0, there exists K ε ∈ L 1 (Ω ) such that K ε (x) + (L(x) − ε)

|u n | p ≤ F(x, u n (x)) p

where L(x) = lim inf|t|→∞ Therefore, it results that

a.e. x ∈ Ω ,

p F(x,t) |t| p .

Z 1 Φ+ (u n ) 1 p kvn k = + F+ (x, u n ) p ku n k p ku n k p Z Z 1 1 Φ+ (u n ) + p + (L(x) − ε)|v | + K ε (x). ≥ n ku n k p ku n k p ku n k p

(4.11)

Since Φ+ (u n ) is bounded, and going to the limit in (4.11), we get Z + p kv k ≥ (L(x) − ε)|v + | p . Thus, since ε > 0 is arbitrary, we have Z kv + k p ≥ L(x)|v + | p .

(4.12)

But, Claim 3 implies that Z Z |∇v + | p = λ1 |v + | p .

(4.13)

Therefore, by the relations (4.12) and (4.13) we write Z (λ1 − L(x))|v + | p ≥ 0.

(4.14)

Ω

Ω

Since λ1 ≤ L(x) a.e. x ∈ Ω , it follows that Z (λ1 − L(x))|v + | p = 0. Ω

Then, (λ1 − L(x))|v + | p = 0

a.e. x ∈ Ω .

However, m(x) = λ1 and by Claim 3, v > 0 is a λ1 -eigenfunction. Thus, L(x) = λ1 a.e. x ∈ Ω . But, this contradicts (f3 ).

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We are now ready to prove Lemma 4.1. From Claim 3, we have 1 ∈ σ (−1 p , m(.)). In view of the strict monotonicity of λ1 , we get λ1 (m) < λ1 (λ1 (1)) = 1. So v = 0, which contradicts kvk = 1. The proof of Lemma 4.1 is completed.



Lemma 4.2. Under (f0 ), (f3 ) and (f4 ), one has: (i) 0 is a strict local minimum of Φ+ . (ii) Φ+ (tψ1 ) → −∞ as t → +∞, where ψ1 > 0 is a λ1 (L)-eigenfunction such that kψ1 k = 1 and L(x) = lim inf|t|→∞

p F(x,t) |t| p .

Proof. (i) For any β1 ∈ ]β, λ1 [ and by (f4 ), there exists ξ > 0 such that F(x, t) ≤

β1 p |t| p

for |t| < ξ.

(4.15)

But by (f0 ) and (f3 ), we have for all |t| ≥ ξ F(x, t) ≤ b|t| p+2 ,

b > 0.

(4.16)

Combining (4.15) and (4.16), we get F(x, t) ≤

β1 p |t| + b|t| p+2 , p

(4.17)

∀t ∈ R. 1, p

Using (4.17) and Poincar´e’s inequality, for all u ∈ W0 (Ω ), we get β1 1 p p+2 kuk p − ku + k p − bku + k p+2 , p p β1 1 p p+2 p ≥ kuk − kuk p − bkuk p+2 , p p 1 β1 ≥ kuk p − kuk p − kkuk p+2 , k > 0, p pλ1   λ1 − β1 − kkuk2 kuk p . ≥ pλ1 q 1 −β1 So, for r = 12 λkpλ we have 1 Φ+ (u) ≥

ρ := inf Φ+ (u) > 0 = Φ+ (0). kuk=r

(ii) From (f3 ), we have Z Z Z L(x) − ε |t∇ψ1 | p dx − |tψ1 | p dx + K ε (x) dx. Φ+ (tψ1 ) ≤ p p Ω Ω Ω R R But, Ω L(x)|ψ1 | p dx = λ11(L) and Ω |ψ1 | p dx ≤ λ11 . Therefore, Z γ Φ+ (tψ1 ) ≤ |t| p + K ε (x) dx, p Ω where γ = 1 − λ11(L) + λε1 , Since 1 − λ11(L) < 0, then for ε small enough such that γ < 0, the affirmation is satisfied.



Proof of Theorem 1.2. From the second assertion of Lemma 4.2, we can find t0 > 0 such that Φ+ (t0 ψ1 ) < 0. In view of Lemma 4.1 and (i) of Lemma 4.2, we may apply the classical mountain pass theorem to get that c = inf max Φ(h(t)) ≥ ρ h∈Γ 0≤t≤1

A. Ayoujil, A.R. El Amrouss / Nonlinear Analysis 68 (2008) 1802–1815

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is a critical value for Φ+ , where Γ = {h ∈ C([0, 1], W0 (Ω )); h(0) = 0, h(1) = t0 ψ1 }. Then, the functional Φ+ has a critical point u + with Φ+ (u + ) ≥ ρ. But, Φ+ (0) = 0, that is, u + 6= 0. Therefore, the problem (P+ ) has a nontrivial solution which, by Lemma 2.2, is a positive nontrivial solution of the problem (P). Similarly, using Φ− , we show that there exists furthermore a negative nontrivial solution.  References [1] A. Anane, Etude des valeurs propres et de la r´esonance pour l’op´erateur p-Laplacien, Ph.D. Thesis, Universit´e Libre de Bruxelles, 1987; C.R. Acad. Sci. Paris S´er. I Math. 305(1987) 725–728. [2] K.C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkh¨auser, Boston, 1993. [3] D.G. Costa, C.A. Magalh˜aes, Existence results for perturbations of the p-Laplacian, Nonlinear Anal. TMA 24 (1995) 409–418. [4] A.R. El Amrouss, M. Moussaoui, Minimax principle for critical point theory in applications to quasilinear boundary value problems, Electron. J. Differential Equations 18 (2000) 1–9. [5] A.R. El Amrouss, Critical point theorems and applications to differential equations, Acta Math. Sin. (Engl. Ser.) 21 (1) (2005) 129–142. [6] J. Liu, J. Su, Remarks on multiple nontrivial solutions for quasi-linear resonant problems, J. Math. Anal. Appl. 258 (2001) 209–222. [7] J. Mawhin, M. Willem, Critical Point Theory and Hamiltonian systems, Springer-Verlag, New York, 1989. [8] V. Moroz, Solutions of superlinear at zero elliptic equations via Morse theory, Topol. Methods Nonlinear Anal. 10 (1997) 387–397. [9] Y. Guo, J. Liu, Solutions of p-sublinear p-Laplacian equation via Morse theory, J. London Math. Soc. 72 (2) (2005) 632–644. [10] Z. Zhang, S. Li, S. Liu, W. Feng, On an asymptotically linear elliptic Dirichlet problem, Abstr. Appl. Anal. 10 (2002) 509–516.