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Multiplicity results for semilinear elliptic problems with resonance
Nonlinear Analysis 65 (2006) 634–646 www.elsevier.com/locate/na
Multiplicity results for semilinear elliptic problems with resonance A.R. El Amrouss ∗ University Mohamed I, Faculty of Sciences, Department of Mathematics, Oujda, Morocco Received 7 May 2005; accepted 28 September 2005
1. Introduction Let Ω be a bounded domain in Rn with smooth boundary, and let g : Ω × R → R be a nonlinear function satisfying the Carath´eodory conditions. We consider the Dirichlet problem −∆u = λk u + f (x, u) u=0 on ∂Ω ,
in Ω
(1.1)
where 0 < λ1 < λ2 ≤ · · · λk ≤ · · · is the sequence of eigenvalues of the problem −∆u = λu in Ω , u=0 on ∂Ω . Let F(x, s) be the primitive l± (x) = lim inf s→±∞
s 0
f (x, s) , s
f (x, t) dt, and k± (x) = lim sup s→±∞
f (x, s) . s
Let us assume that 0 ≤ l± (x) ≤ k± (x) ≤ λk+1 − λk uniformly for a.e. x ∈ Ω . There have been many papers concerning problem (1.1) at resonance in the situation where l± (x) ≡ 0 or k± (x) ≡ λk+1 − λk ; see for example [13,1,5,11,10]. Some multiplicity theorems are obtained by using the topological degree technique and the variational methods, [2,14,15,17, 6,12,8,18]. Let f (x, 0) ≡ 0, then the problem (1.1) has a trivial solution u ≡ 0. We are interested in finding multiple nontrivial solutions of (1.1). In this paper, we will prove the existence of a ∗ Tel.: +212 56 50 06 02; fax: +212 56 50 06 03.
E-mail address: [email protected]. c 2005 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2005.09.033
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nontrivial solutions when f (x, s)/s stays between 0 and λk+1 − λk for large values of |s|. We replace the non-resonance conditions of Costa–Oliviera [10] by classical resonance conditions of Ahmad–Lazer–Paul. Let us denote by E(λ j ) the λ j -eigenspace, g(x, t) = λk t + f (x, t) and let G(x, t) be the primitive of g. Now, we state the assumptions and the main results in this paper. 4 (F0) | f (x, s)| ≤ C(|s| p + 1), x ∈ Ω , s ∈ R, p < n−2 if n ≥ 3 and no restriction if n = 1, 2. 2 (F1) 0 ≤ f (x, s)s ≤ (λk+1 − λk )s for |s| ≥ r > 0 and a.e. x ∈ Ω . (F2) lim u →∞,u∈E(λk ) Ω F(x, u(x)) dx = +∞. (F3) lim u →+∞,u∈E(λk+1 ) Ω 12 (λk+1 − λk )u 2 (x) − F(x, u(x)) dx = +∞.
(F4) g (x, 0) λ1 for x ∈ Ω . Here and then, we write a(x) b(x) to indicate that a(x) ≤ b(x) on Ω , with strict inequality holding on subset of positive measure. Our main results are the following: Theorem 1.1. Suppose that f ∈ C 1 (Ω × R, R) such that f (x, 0) = 0 for a.e. x ∈ Ω , and the assumptions (F0)–(F4) k ≥ 2 are satisfied. Then problem (1.1) has at least three nontrivial solutions in which one is positive and one is negative. As an immediate consequence we obtain the corollary below, under the assumption that (F5) 0 < f (s) < λk+1 − λk for |s| ≥ r > 0. Corollary 1.1. Assume that f (x, s) = f (s) ∈ C 1 (R), f (0) = 0, k ≥ 2, g (0) < λ1 and (F5) is satisfied. Then, (1.1) has at least three nontrivial solutions in which one is positive and one is negative. Theorem 1.2. Suppose that f ∈ C 1 (Ω × R, R) such that f (x, 0) = 0 for a.e. x ∈ Ω and there is t1 = 0 such that g(x, t1 ) = 0 for x ∈ Ω . Assume that (F0)–(F3), λ2 g (x, 0) and k = 1 are satisfied. Then (1.1) has at least two nontrivial solutions. Corollary 1.2. Assume that f (x, s) = f (s) ∈ C 1 (R), f (0) = 0, k = 1, λ2 < g (0), (F5) is satisfied and there is t1 = 0 such that g(t1 ) = 0. Then, (1.1) has at least two nontrivial solutions. Next, we will prove the following result under the following assumption: (F6) There is some α > 0 such that λm t 2 ≤ g(x, t)t ≤ λm+1 t 2 , |t| ≤ α,
a.e. x ∈ Ω , m ≥ 2.
Theorem 1.3. Suppose that f ∈ C 1 (Ω × R, R) such that f (x, 0) = 0 for a.e. x ∈ Ω . Assume that f satisfies (F0)–(F3), (F6), k ≥ 2, and there is t1 = 0 such that g(x, t1 ) = 0 for x ∈ Ω . Then (1.1) has at least four nontrivial solutions. Corollary 1.3. Assume that f (x, s) = f (s) ∈ C 1 (R), f (0) = 0, k ≥ 2, (F5), (F6) are satisfied and there is t1 = 0 such that g(t1 ) = 0. Then, (1.1) has at least four nontrivial solutions. Remarks. 1. It was shown in [14] that the Ahmad–Lazer–Paul condition (F2) is weaker than of the generalized Landesman–Lazer conditions used in [14,20], when k = 1.
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2. A particular example of a nonlinear term that does not satisfy the conditions in [14] or [20] would be f (x, t) = 1t with |t| ≥ r > 0. Since lim|t |→∞ f (x, t) = 0 it follows that neither the standard Landesman–Lazer condition used in [2] nor the generalization used in [14,20] can be not satisfied. However it is obvious to see that this nonlinear term does satisfy our conditions near infinity. 3. Note that our multiplicity results are not covered by the results mentioned in [15,6,17,12]. Since the condition supt ∈R | f (t)| < λk+1 − λk used [15,6,12] implies (F3). 4. Theorem 1.3 extends the result cited in [16]. 5. The condition (F6) means the problem (1.1) is resonant near the zero at the consecutive eigenvalues. It is clear that the (F6) condition contains the case g (x, 0) ∈]λm , λm+1 [ for every x ∈ Ω . The proof of our results are based on combining the Morse theory and the minimax methods. Complete computations of the critical groups of the functional Φ at its critical point are concerned. This paper is organized as follow: in Section 2, several technical lemmas are presented and proved. In Section 3, we give the proofs of our results. 2. Preliminaries By a solution of (1.1) we mean a function u ∈ H01(Ω ) satisfying ∇u∇v − λk uv − f (x, u)v = 0, for all v ∈ H01(Ω ) Ω
Ω
Ω
is the usual Sobolev space obtained through completion of Cc∞ (Ω ) with respect where to the norm induced by the inner product ∇u∇v, u, v ∈ H01(Ω ). u, v = H01(Ω )
Ω
For u ∈ H01(Ω ) define the functional 1 1 2 2 Φ(u) = |∇u| − λk u − F(x, u). 2 Ω 2 Ω Ω It is well known that under a sub-critical growth condition on f , Φ is well defined on H01(Ω ), weakly lower semi-continuous and it is well known that, under the (F0)–(F1) conditions of f , Φ is a C 2 functional with derivatives given by Φ (u)v = ∇u∇v − λk uv − f (x, u)v, for u, v ∈ H01(Ω ), Ω Ω Ω Φ (u).v.w = ∇w∇v − λk wv − f (x, u)wv, for u, w, v ∈ H01(Ω ). Ω
Ω
Ω
Thus, finding solutions of (1.1) is equivalent to finding critical points of the functional Φ. Since we are going to apply the variational characterization of the eigenvalues, we will decompose the space H01(Ω ) as E = E − ⊕ E k ⊕ E k+1 ⊕ E + , where E − is the subspace spanned by the λ j -eigenfunctions with j < k and E j is the eigenspace generated by the λ j -eigenfunctions and E + is the orthogonal complement of E − ⊕ E k ⊕ E k+1 in H01(Ω ) and we will decompose for
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any u ∈ H01(Ω ) as following u = u − + u k + u k+1 + u + where u − ∈ E − , u k ∈ E k , u k+1 ∈ E k+1 and u + ∈ E + . We can verify easily that 2 |∇u| dx − λi |u|2 dx ≥ δi u 2 ∀u ∈ ⊕ j ≥i+1 E j (2.1) Ω Ω |∇u|2 dx − λi |u|2 dx ≤ −δi u 2 ∀u ∈ ⊕ j ≤i−1 E j . (2.2) Ω
where δi = min{1 −
Ω
λi λi λi+1 , λi−1
− 1}.
2.1. Preliminary results on compactness condition To apply minimax methods for finding critical points of Φ, we need to verify that Φ satisfies a compactness condition of the Palais–Smale type which was introduced by Cerami in [7]. Definition. A functional Φ ∈ C 1 (E, R), with E a real Banach space, is said to satisfy condition (C)c , at the level c ∈ R, if: every sequence (u n ) ⊂ E such that Φ(u n ) → c, u n Φ (u n ) → 0 possesses a convergent subsequence. It was shown in [3] that condition (C)c actually suffices to get a deformation theorem and the linking theorem. Now, we present some technical lemmas. Lemma 2.1. Let (u n ) ⊂ H01(Ω ) and ( pn ) ⊂ L ∞ (Ω ) be sequences, and let A be a nonnegative constant such that 0 ≤ pn (x) ≤ A
a.e. in Ω and for all n ∈ N
and pn 0 in the weak* topology of L ∞ , as n → ∞. Then, there are subsequences (u n ), ( pn ) satisfying the above conditions, and there is a positive integer n 0 such that for all n ≥ n 0 , −δk + k k+1 2
u n + u k+1 pn u n ((u − + u+ (2.3) n + u n ) − (u n n )) dx ≥ n . 2 Ω Proof. Since pn ≥ 0 a.e. in Ω , we see that k k+1 pn u n ((u − + u+ n + u n ) − (u n n )) Ω k+1 2 ≥− pn (u + n + u n ) dx Ω 2 + u n + u k+1 n k+1 2 ≥− pn dx u + n + un . + k+1
u n + u n
Ω
(2.4)
Moreover, by the compact imbedding of H01(Ω ) into L 2 (Ω ) and pn 0 in the weak* topology of L ∞ , when n → ∞, then there are subsequences (u n ), ( pn ) such that 2 + u n + u k+1 n pn dx → 0. k+1
u + Ω n + un
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Therefore, there exists n 0 ∈ N such that for n ≥ n 0 we have 2 + δk u n + u k+1 n pn dx ≤ . k+1 2
u + + u
Ω n n Combining inequalities (2.4) and (2.5), we get inequality (2.3).
(2.5)
Definition. A sequence (u n ) is said to be a (C)c sequence, at the level c ∈ R, if there is a sequence n → 0, such that Φ(u n ) → c
u n Φ (u n ), v H 1 ,H −1 ≤ n v
0
∀v ∈
(2.6) (2.7)
H01.
Lemma 2.2. Let (u n ) ⊂ H01(Ω ) be a (C)c sequence. n (x)) ∞ 1. If f n (x) = f (x,u u n (x) χ[|u n (x)|≥r] 0 in the weak* topology of L , as n → ∞. Then, there − + k+1 is a subsequence (u n ) such that ( u n + (u n + u n ) )n is uniformly bounded in n. f (x,u n (x)) χ[|u n (x)|≥r] 0 in the weak* topology of L ∞ , as 2. If f n (x) = (λk+1 −λk )uun n(x)− (x) + k n → ∞. Then, there is a subsequence (u n ) such that ( u − n + (u n + u n ) )n is uniformly bounded in n.
Proof. Since (u n )n ⊂ H01 is a (C)c sequence, (2.6) and (2.7) are satisfied. Now, we prove that + k+1 − k + k+1 the sequence ( u − n +u n +u n )n is uniformly bounded in n. Take v = (u n +u n )−(u n +u n ) in (2.7), pn (x) = f n (x), and 2 − 2 k+1 2 Λ = − |∇u − | + λ |u | dx + |∇(u + k n n n + u n )| Ω Ω Ω
+ k+1 2 k k+1 + − λk |u n + u n | dx + pn u n ((u − + u ) − (u + u )) dx n n n n Ω Ω
k+1 − k Γ = n + | f (x, u n (x))||(u + n + u n ) − (u n + u n )| dx . |u n (x)|≤r
Then Λ ≤ Γ . By the Poincar´e inequality, from (2.1)–(2.3), and Λ ≤ Γ , it follows that there exist constants A and B such that δk − k+1 2 − + k+1
u + (u + n + u n ) ≤ n + A u n + (u n + u n ) + B. 2 n + k+1 This gives that ( u − n + (u n + u n ) )n is uniformly bounded in n. The same proof of eventuality + − k 2 are given by taking v = (u n + u k+1 n ) − (u n + u n ) and
pn (x) = f n (x) =
(λk+1 − λk )u n (x) − f (x, u n (x)) χ[|u n (x)|≥r] . u n (x)
Lemma 2.3. Assume f satisfies (F1). + k+1 1. Let (u n ) ⊂ H01(Ω ) such that u − n + (u n + u n ) is uniformly bounded in n and there exists A such that if A ≤ Φ(u n ), then u kn F x, dx ≤ M. 2 Ω
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+ k 2. Let (u n ) ⊂ H01(Ω ) such that u − n + u n + u n is uniformly bounded in n and there exists A such that if Φ(u n ) ≤ A, then 2 λk+1 − λk u k+1 u k+1 n n − F x, dx ≤ M. 2 2 2 Ω
Proof. From (2.6), and Poincar´e inequality, we have uk 1 uk 2 F x, n dx ≤ A + + u k+1 + u− F x, n − F(x, u n ) dx + u + n n
2 2 2 n Ω Ω (2.8) Since, f ∈ C 1 (Ω¯ × R, R) satisfy (F1), there exist two functions γ , h : Ω × R → R such that f (x, t) = tγ (x, t) + h(x, t) ) with 0 ≤ γ (x, t) = f (x,t t χ[|t |≥r] ≤ λk+1 − λk and h(x, t) = f (x, t)χ[|t |
Set t1 = min{t ∈ [0, 1] | h(x, t uk h(x, t 2n
u kn 2
+ (1 − t)u n ) = 0} and t2 = max{t ∈ [0, 1] |
+ (1 − t)u n ) = 0}. We verify easily that
k
u (t2 − t1 )
n − u n
≤ 2r. 2
So that using (2.9), (2.10), the Poincar´e inequality, and an elementary inequality 2 a a −b + − b b ≤ (a − b)2 , 2 2 we have u kn − F(x, u n ) dx F x, 2 Ω k t2 un u kn ≤ + (1 − t)u n dt − u n dx h x, t 2 2 Ω t1 λk+1 − λk − k+1 2 +
u n + u + n + un
4λ1 λk+1 − λk − k+1 2 ≤ 2r sup | f (x, s)|meas(Ω ) +
u n + u + n + un . 4λ 1 |s|≤r
(2.10)
(2.11)
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From (2.8) and (2.11), there exists M > 0 such that u kn dx ≤ M. F x, 2 Ω Lemma 2.4. Let p satisfy p(x, t) = 0 for t < 0, x ∈ Ω and λk t 2 ≤ p(x, t)t ≤ λk+1 t 2 , t ≥ r > 0, k ≥ 2. Then the C 2 functional Φ : H01(Ω ) → R defined by 1 |∇u|2 − P(x, u) Φ(u) = 2 Ω Ω t satisfies the (PS) condition, where P(x, t) = 0 p(x, s) ds. Proof. Let (u n )n ⊂ H01 be a (P S) sequence, i.e. |Φ(u n )| ≤ A
(2.12)
Φ (u n ), v H 1 ,H −1 ≤ n v
0
∀v ∈
H01,
(2.13)
where A is a constant and n → 0. It clearly suffices to show that (u n )n remains bounded in H01. Assume by contradiction. Defining z n = uu nn , we have z n = 1 and, passing if necessary to a subsequence, we may assume that z n z weakly in H01, z n → z strongly in L 2 (Ω ) and z n (x) → z(x) a.e. in Ω . By (2.12), there is an m ∈ L 2 (Ω ) with λk ≤ m ≤ λk+1 such that Φ (u n ), u n →1− m(x)z(x) dx = 0. (2.14)
u n 2 Ω So that z ≡ 0. In other words, we verify easily that z satisfies −∆z = m(x)z + z=0 on ∂Ω
in Ω
where z + = max{z, 0}. By the maximum principle and the unique continuation property, z = z + ≥ 0 and m ≡ λk or m ≡ λk+1 . Since k ≥ 2, z ≡ 0, which contradicts (2.14). Hence u n is bounded. The proof is completed. 2.2. Preliminary results on critical groups Let H be a Hilbert space and Φ ∈ C 1 (H, R) be satisfying the compactness condition (P S) or (C) Cerami condition. Set Φ c = {u ∈ H01(Ω ) | Φ(u) ≤ c}. Denote by Hq (X, Y ) the q-th relative singular homology group with integer coefficients. The critical groups of Φ at an isolated critical point u with Φ(u) = 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. Let K = {u ∈ H | Φ (u) = 0} be the set of critical points of Φ and a < infK Φ. The critical groups of Φ at infinity are formally defined as [4] Cq (Φ, ∞) = Hq (H, Φ a );
q ∈ Z.
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If K < ∞ then the Morse-type numbers of the pair are defined by Mq = Mq (H, Φ a ) = dim Cq (Φ, u). u∈K
By the Morse theory [8,18] if βq = dim Cq (Φ, ∞), then q j =0 ∞
(−1)q− j M j ≥
q
(−1)q− j β j ,
(2.15)
j =0
(−1)q Mq =
q=0
∞ (−1)q βq .
(2.16)
q=0
The formal expression of the Morse inequality reads as ∞ q=0
Mq t q =
∞
βq t q + (1 + t)
q=0
∞
aq t q .
(2.17)
q=0
From (2.15), we easily deduce the inequalities Mq ≥ βq for all q ∈ Z. Thus, if βq = 0 for some q then Φ must have a critical point, u, with Cq (Φ, u) 0. If equality (2.16) does not hold then Φ must have another critical point differing from the known ones and, if u, v are two critical points of Φ and Cq (Φ, u) Cq (Φ, v) for some q then u = v. Moreover, we have the following Proposition 2.1 ([4]). Assume that H = H + ⊕ H − , Φ is bounded from below on H + and Φ(u) → −∞ as u → ∞ with u ∈ H − . Then if k = dim H − < ∞.
Ck (Φ, ∞) 0,
In [19], the following result of the existence of a third critical point was proved. Theorem 2.1. Suppose Φ ∈ C 1 (H, R), H a real Hilbert space, is not bounded below and satisfies the (C) condition. Assume the origin is an isolated critical point of Φ in H satisfying C1 (Φ, 0) = 0, and Φ possesses a local minimum u 0 = 0. Then, Φ possesses at least three critical points in H . 3. Proofs of main results First, we prove that Φ satisfies the Cerami condition. Lemma 3.1. Under the assumptions (F1)–(F3), Φ satisfies the (C)c condition on H01(Ω ), for all c ∈ R. Proof. Let (u n )n ⊂ H01 be a (C)c sequence, i.e. Φ(u n ) → c
u n Φ (u n ), v H 1 ,H −1 ≤ n v
0
∀v ∈
H01,
(3.1) (3.2)
where n → 0. It clearly suffices to show that (u n )n remains bounded in H01. Assume by contradiction. Defining z n = uu nn , we have z n = 1 and, passing if necessary to a subsequence, we may assume that z n z weakly in H01, z n → z strongly in L 2 (Ω ) and z n (x) → z(x) a.e. in Ω . Set, f (x, u n (x)) χ[|u n (x)|≥r] f n (x) = u n (x)
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and l ∈ L ∞ such that f n → l in the weak* topology of L ∞ , as n → ∞. where the L ∞ -function l satisfies 0 ≤ l(x) ≤ λk+1 − λk .
(3.3)
Set m(x) = l(x) + λk , by (3.2), we have Φ (u n ), u n →1− m(x)z(x) dx = 0.
u n 2 Ω So that z ≡ 0. In other words, we verify easily that z satisfies −∆z = m(x)z z=0
in Ω
(3.4)
on ∂Ω .
Hence 1 is an eigenvalue of problem (3.4). We now distinguish three cases: (i) λk < m(x) and m(x) < λk+1 on subset of positive measure; (ii) m(x) ≡ λk ; (iii) m(x) ≡ λk+1 . In case (i), by strict monotonicity we have λk (m) < λk (λk ) = 1
and
λk+1 (λk+1 ) = 1 < λk+1 (m).
This contradicts that 1 is an eigenvalue of problem (3.4). On the other hand, by (F2), (F3), Lemmas 2.2 and 2.3 the cases (ii) and (iii) are not possible. The proof is complete. Lemma 3.2. Under hypothesis (F1)–(F3), the functional Φ has the following properties: (i) Φ(v) → −∞, as v → ∞, v ∈ E k ⊕ E − = V (ii) Φ(w) → ∞, as w → ∞, w ∈ E k+1 ⊕ E + = W . Proof. (i) Assume by contradiction, that there exist a constant B and a sequence (vn ) ⊂ V with
vn → ∞ such that B ≤ Φ(vn ) ≤ −δ vn− 2 . Therefore, vn− is bounded and by Lemma 2.3, we obtain vnk F x, lim inf dx ≤ constant. n→∞ Ω 2 This is a contradiction with assumption (F2). (ii) Suppose by contradiction that Φ is not coercive in W . Thus, there is some constant B and some sequence (wn ) ⊂ W , with wn → ∞, such that 1 1 (λk+1 − λk )wn2 − F(x, wn ) dx ≤ B. [|∇wn |2 dx − λk+1 wn2 ] + Φ(wn ) = 2 Ω Ω 2 This implies that wn+ is bounded, and so Lemma 2.3 gives us a contradiction from (F3).
Lemma 3.3. Under the assumptions of Theorem 1.1, there exists at least two nontrivial critical point u ± of Φ such that Cq (Φ, u ± ) = δq1 Z. Proof. Define the following functionals Φ± : H01(Ω ) → R as 1 |∇u|2 dx − G(x, u ± ) dx Φ± (u) = 2 Ω Ω
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where u + = max{u(x), 0}, u − = min{u(x), 0}, then Φ± ∈ C 2 . It follows from Lemma 2.4 that Φ± satisfies the (PS) condition. The (F1) condition, with k ≥ 2, shows that Φ± (tϕ1 ) → −∞,
as t → ±∞,
where ϕ1 is the first eigenfunction of −∆. Then, we can find a t0 such that t0 > R and Φ± (±t0 ϕ1 ) ≤ 0. Since g (x, 0) λ1 , u = 0 is a strictly local minimum of Φ± . Hence, there is an R > 0 and an α > 0 such that Φ± ≥ α on ∂ B R (0). The compactness condition and geometry of Φ± assure that c = infh∈Γ max0≤t ≤1 Φ± (h(t)) are critical values for Φ± , where Γ = {h ∈ C([0, 1], H01) | h(0) = 0, h(1) = ±t0 ϕ1 }, and c ≥ α. Then, by a mountain pass lemma and the maximum principle we can obtain a positive critical point u + of Φ+ and a critical point u − of Φ− such that C1 (Φ± , u ± ) = 0. Moreover, by Corollary 8.5 in [18] the critical group of Φ± at u ± is Cq (Φ± , u ± ) = δq1 Z. Using the results in [9], we have Cq (Φ, u ± ) ∼ = Cq (Φ /C 1 (Ω ) , u ± ) ∼ = Cq (Φ± /C 1 (Ω ) , u ± ) ∼ = Cq (Φ± , u ± ) ∼ = δq1 Z. 0
This completes the proof.
0
(3.5)
Proof of Theorem 1.1. By Lemma 3.1 the functional Φ satisfies the (C) condition. Since Φ is weakly lower semi-continuous, and is coercive on W by Lemma 3.2, infw∈W Φ(w) > −∞, i.e. Φ is bounded from below on W . By Lemma 3.2, Φ is anti-coercive on V . Thus Proposition 2.1 tells us that Cµ (Φ, ∞) 0, where µ = dim V ≥ k. It follows from the Morse inequality that Φ has a critical point u 0 with Cµ (Φ, u 0 ) 0,
(3.6)
Since g (x, 0) λ1 , u = 0 is a local minimum of Φ, hence Cq (Φ, 0) = δq,0 Z.
(3.7) 2 imply that u + , u −
By (3.6), (3.7), Lemma 3.3 and k ≥ of Φ. This completes the proof.
and u 0 are three nontrivial critical points
Proof of Corollary 1.1. From (F5), there are s1 , s2 such that 0 < f (s1 ), f (s2 ) < λk+1 − λk = µ. Thus, by continuity, there exist s3 ∈ R, ε > 0, δ > 0 such that ε ≤ f (s) ≤ µ − ε for every s ∈ [s3 − δ, s3 + δ]. Let s3 ≥ 0 and take s > s3 + δ, we have s3 s3 +δ s εδ ≤ f (s) = f (t) dt + f (t) dt + f (t) dt ≤ µs − εδ. 0
s3
s3 +δ
(3.8)
However, there are constants A1 , A2 such that for s > s3 + δ we have s2 − εδs + A2 . (3.9) 2 Similarly, if s3 < 0 and s < s3 −δ we have (3.8) and (3.9). Hence, the conditions of Theorem 1.1 are satisfied and the corollary follows. εδs + A1 ≤ F(s) ≤ µ
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Proof of Theorem 1.2. Without loss of generality, we suppose that t1 > 0. We define if t < 0 0 g (x, t) = λ1 t + f (x, t) if t ∈ [0, t1 ] 0 if t > t1 . : H 1(Ω ) → R as Define the cut-off functional Φ 0 1 u) dx, Φ(u) = |∇u|2 dx − G(x, 2 Ω Ω is coercive and satisfies (PS). Hence Φ t) is the primitive of where G(x, g . It is easy to see that Φ possesses a minimum u 0 . By the maximum principle we known that either u 0 ≡ 0 or 0 < u 0 < t0 for all x ∈ Ω and u 0 is a local minimum of Φ. On the other hand, since λ2 g (x, 0) we verify easily that 2 2 Φ (0).v.v = |∇v| − g (x, 0)v dx ≤ [λ2 − g (x, 0)]v 2 dx < 0, Ω
Ω
Ω
for every v ∈ ⊕ j ≤2 It follows that the Morse index d of Φ at 0 is larger than 2. The well known Gromoll–Meyer Theorem in [18] says that Cq (Φ, 0) = 0 for q ∈ [d, d + ν], with ν the nullity of Φ at 0. Hence, we have C1 (Φ, 0) = 0 and u 0 is a nontrivial local minimizer of Φ in the C01 (Ω¯ ) topology. By a standard argument [9], we know that u 0 is a local minimizer of Φ in H01(Ω ) topology. The (C) condition was established by Lemma 3.1 and Φ is not bounded below by Lemma 3.2. Consequently, we may invoke Theorem 2.1 to conclude that Φ has at least three critical points. The proof is completed. E j.
Lemma 3.4 ([16]). If (F6) is satisfied, then Cq (Φ, 0) = δq,d Z
(3.10)
where d = dim ⊕ j ≤m E(λ j ). Proof of Theorem 1.3. It follows from (3.6), (3.10) and by m = k that there exists a nontrivial critical point u 0 of Φ such that Cµ (Φ, u 0 ) = 0,
(3.11)
where µ = dim ⊕ j ≤k E(λ j ). Without loss of generality, we assume that t0 > 0. Using the condition g(x, t0 ) = 0, the same arguments in the proof of Theorem 1.2 involving the cut-off technique and the maximum principle, Φ has a local minimizer u 1 with 0 < u 1 < t0 and Cq (Φ, u 1 ) = δq,0 Z.
(3.12)
Now, define the functionals Φ± : H01(Ω ) → R as 1 2 Φ± (v) = |∇v| dx − [G(x, u 1 + v ± ) − G(x, u 1 ) − g(x, u 1)v ± ] dx 2 Ω Ω where v + = max{v(x), 0}, v − = min{v(x), 0}, then Φ± ∈ C 2 . Therefore, we obtain 1 Φ± (v) = Φ(u 1 + v ± ) − Φ(u 1 ) + |∇v ∓ |2 dx. 2 Ω
(3.13)
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It follows from (3.13) and u 1 is a local minimum of Φ that 0 is a strictly minimum of Φ± by Lemma 2.4. Φ± satisfies the (PS) condition and the (F1) condition, with k ≥ 2, shows that Φ± (tϕ1 ) → −∞,
as t → ±∞,
where ϕ1 is the first eigenfunction of −∆. However, by a mountain pass lemma and the maximum principle we can obtain a positive critical point v + of Φ+ and a critical point v − of Φ− such that C1 (Φ± , v ± ) = 0 satisfying −∆v ± = g(x, u 1 + v± ) − g(x, u 1 ) ±
v =0
in Ω
(3.14)
on ∂Ω
− + ± Hence, u ± 2 = u 1 + v are two solutions of (1.1), and u 2 < u 1 < u 2 . According to the results given in [9] and the similar arguments used in Lemma 3.3, the critical groups of Φ at u ± 2 are ± ∼ ± ∼ ± ∼ ∼ Cq (Φ, u ± 2 ) = Cq (Φ /C 1 (Ω ) , u 2 ) = Cq (Φ± /C 1 (Ω ) , v ) = Cq (Φ± , v ) = δq,1 Z. 0
0
(3.15)
− By (3.11), (3.12), (3.15), Lemma 3.4 and m, k ≥ 2, m = k imply that u 0 , u + 2 , u 2 and u 1 are four nontrivial critical points of Φ. Hence, the proof is completed.
Remark. The proofs of Corollaries 1.2 and 1.3 are similar to that in Corollary 1.1. References [1] S. Ahmad, A.C. Lazer, J.L. Paul, Elementary critical point theory and perturbations of elliptic boundary value problems at resonance, Indiana Univ. Math. J. 25 (1976) 933–944. [2] S. Ahmad, Multiple nontrivial solutions of resonant and nonresonant asymptotically problems, Proc. Amer. Math. Soc. 96 (1986) 405–409. [3] P. Bartolo, V. Benci, D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity, Nonlinear Anal. 7 (1983) 981–1012. [4] T. Bartsch, S.J. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlinear Anal. 28 (1997) 419–441. [5] H. Berestycki, D.G. DeFigueiredo, Double resonance in semilinear elliptic problems, Comm. Partial Differential Equations 6 (1981) 91–120. [6] A. Castro, A.C. Lazer, Critical point theory and the number of solutions of a nonlinear Dirichlet problem, Ann. of Math. 18 (1977) 113–137. [7] G. Cerami, Un criterio de esistenza per i punti critici su variet´a ilimitate, Rc. Ist. Lomb. Sci. Lett. 121 (1978) 332–336. [8] K.C. Chang, Infinite Dimensional Morse Theory and Multiple Solutions Problems, Birkh¨auser, Boston, 1993. [9] K.C. Chang, H 1 versus C 1 local minimizers, C. R. Acad. Sci. Paris 319 (1994) 441–446. [10] D.G. Costa, A.S. Oliveira, Existence of solution for a class of semilinear elliptic problems at double resonance, Bol. Soc. Brasil. Mat. 19 (1988) 21–37. [11] A.R. El Amrouss, M. Moussaoui, Resonance at two consecutive eigenvalues for semilinear elliptic problem: A variational approach, Ann. Sci. Math. Qu´ebec 23 (2) (1999) 157–171. [12] N. Hirano, T. Nishimura, Multiplicity results for semilinear elliptic problems at resonance and with jumping nonlinearity, J. Math. Anal. Appl. 180 (1993) 566–586. [13] E.M. Landesman, A.C. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech. 19 (1970) 609–623. [14] E. Landesman, S. Robinson, A. Rumbos, Multiple solution of semilinear elliptic problem at resonance, Nonlinear Anal. 24 (1995) 1049–1059. [15] S.Q. Liu, C.L. Tang, X.P. Wu, Multiplicity of nontrivial solutions of semilinear elliptic equations, J. Math. Anal. Appl. 249 (2000) 289–299. [16] S. Li, K. Perera, J. Su, Computation of critical groups in elliptic boundary value problems where the asymptotic limits may not exist, Proc. Roy. Soc. Edinburgh Sect. A 131 (3) (2001) 721–732.
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[17] S. Li, M. Willem, Multiple solution for asymptotically linear boundary value problems in which the nonlinearity crosses at least one eigenvalue, Nonlinear Differential Equations Appl. 5 (1998) 479–490. [18] J. Mawhin, M. Willem, Critical Point Theory and Hamiltonian Systems, Springer-Verlag, New York, 1989. [19] E.A.B. Silva, Critical point theorems and applications to semilinear elliptic problem, NoDEA 3 (1996) 245–261. [20] J. Su, Semilinear elliptic boundary value problems with double resonance between two consecutives eigenvalues, Nonlinear Anal. 48 (2002) 881–895.
Multiplicity results for semilinear elliptic problems with resonance A.R. El Amrouss ∗ University Mohamed I, Faculty of Sciences, Department of Mathematics, Oujda, Morocco Received 7 May 2005; accepted 28 September 2005
1. Introduction Let Ω be a bounded domain in Rn with smooth boundary, and let g : Ω × R → R be a nonlinear function satisfying the Carath´eodory conditions. We consider the Dirichlet problem −∆u = λk u + f (x, u) u=0 on ∂Ω ,
in Ω
(1.1)
where 0 < λ1 < λ2 ≤ · · · λk ≤ · · · is the sequence of eigenvalues of the problem −∆u = λu in Ω , u=0 on ∂Ω . Let F(x, s) be the primitive l± (x) = lim inf s→±∞
s 0
f (x, s) , s
f (x, t) dt, and k± (x) = lim sup s→±∞
f (x, s) . s
Let us assume that 0 ≤ l± (x) ≤ k± (x) ≤ λk+1 − λk uniformly for a.e. x ∈ Ω . There have been many papers concerning problem (1.1) at resonance in the situation where l± (x) ≡ 0 or k± (x) ≡ λk+1 − λk ; see for example [13,1,5,11,10]. Some multiplicity theorems are obtained by using the topological degree technique and the variational methods, [2,14,15,17, 6,12,8,18]. Let f (x, 0) ≡ 0, then the problem (1.1) has a trivial solution u ≡ 0. We are interested in finding multiple nontrivial solutions of (1.1). In this paper, we will prove the existence of a ∗ Tel.: +212 56 50 06 02; fax: +212 56 50 06 03.
E-mail address: [email protected]. c 2005 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2005.09.033
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nontrivial solutions when f (x, s)/s stays between 0 and λk+1 − λk for large values of |s|. We replace the non-resonance conditions of Costa–Oliviera [10] by classical resonance conditions of Ahmad–Lazer–Paul. Let us denote by E(λ j ) the λ j -eigenspace, g(x, t) = λk t + f (x, t) and let G(x, t) be the primitive of g. Now, we state the assumptions and the main results in this paper. 4 (F0) | f (x, s)| ≤ C(|s| p + 1), x ∈ Ω , s ∈ R, p < n−2 if n ≥ 3 and no restriction if n = 1, 2. 2 (F1) 0 ≤ f (x, s)s ≤ (λk+1 − λk )s for |s| ≥ r > 0 and a.e. x ∈ Ω . (F2) lim u →∞,u∈E(λk ) Ω F(x, u(x)) dx = +∞. (F3) lim u →+∞,u∈E(λk+1 ) Ω 12 (λk+1 − λk )u 2 (x) − F(x, u(x)) dx = +∞.
(F4) g (x, 0) λ1 for x ∈ Ω . Here and then, we write a(x) b(x) to indicate that a(x) ≤ b(x) on Ω , with strict inequality holding on subset of positive measure. Our main results are the following: Theorem 1.1. Suppose that f ∈ C 1 (Ω × R, R) such that f (x, 0) = 0 for a.e. x ∈ Ω , and the assumptions (F0)–(F4) k ≥ 2 are satisfied. Then problem (1.1) has at least three nontrivial solutions in which one is positive and one is negative. As an immediate consequence we obtain the corollary below, under the assumption that (F5) 0 < f (s) < λk+1 − λk for |s| ≥ r > 0. Corollary 1.1. Assume that f (x, s) = f (s) ∈ C 1 (R), f (0) = 0, k ≥ 2, g (0) < λ1 and (F5) is satisfied. Then, (1.1) has at least three nontrivial solutions in which one is positive and one is negative. Theorem 1.2. Suppose that f ∈ C 1 (Ω × R, R) such that f (x, 0) = 0 for a.e. x ∈ Ω and there is t1 = 0 such that g(x, t1 ) = 0 for x ∈ Ω . Assume that (F0)–(F3), λ2 g (x, 0) and k = 1 are satisfied. Then (1.1) has at least two nontrivial solutions. Corollary 1.2. Assume that f (x, s) = f (s) ∈ C 1 (R), f (0) = 0, k = 1, λ2 < g (0), (F5) is satisfied and there is t1 = 0 such that g(t1 ) = 0. Then, (1.1) has at least two nontrivial solutions. Next, we will prove the following result under the following assumption: (F6) There is some α > 0 such that λm t 2 ≤ g(x, t)t ≤ λm+1 t 2 , |t| ≤ α,
a.e. x ∈ Ω , m ≥ 2.
Theorem 1.3. Suppose that f ∈ C 1 (Ω × R, R) such that f (x, 0) = 0 for a.e. x ∈ Ω . Assume that f satisfies (F0)–(F3), (F6), k ≥ 2, and there is t1 = 0 such that g(x, t1 ) = 0 for x ∈ Ω . Then (1.1) has at least four nontrivial solutions. Corollary 1.3. Assume that f (x, s) = f (s) ∈ C 1 (R), f (0) = 0, k ≥ 2, (F5), (F6) are satisfied and there is t1 = 0 such that g(t1 ) = 0. Then, (1.1) has at least four nontrivial solutions. Remarks. 1. It was shown in [14] that the Ahmad–Lazer–Paul condition (F2) is weaker than of the generalized Landesman–Lazer conditions used in [14,20], when k = 1.
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2. A particular example of a nonlinear term that does not satisfy the conditions in [14] or [20] would be f (x, t) = 1t with |t| ≥ r > 0. Since lim|t |→∞ f (x, t) = 0 it follows that neither the standard Landesman–Lazer condition used in [2] nor the generalization used in [14,20] can be not satisfied. However it is obvious to see that this nonlinear term does satisfy our conditions near infinity. 3. Note that our multiplicity results are not covered by the results mentioned in [15,6,17,12]. Since the condition supt ∈R | f (t)| < λk+1 − λk used [15,6,12] implies (F3). 4. Theorem 1.3 extends the result cited in [16]. 5. The condition (F6) means the problem (1.1) is resonant near the zero at the consecutive eigenvalues. It is clear that the (F6) condition contains the case g (x, 0) ∈]λm , λm+1 [ for every x ∈ Ω . The proof of our results are based on combining the Morse theory and the minimax methods. Complete computations of the critical groups of the functional Φ at its critical point are concerned. This paper is organized as follow: in Section 2, several technical lemmas are presented and proved. In Section 3, we give the proofs of our results. 2. Preliminaries By a solution of (1.1) we mean a function u ∈ H01(Ω ) satisfying ∇u∇v − λk uv − f (x, u)v = 0, for all v ∈ H01(Ω ) Ω
Ω
Ω
is the usual Sobolev space obtained through completion of Cc∞ (Ω ) with respect where to the norm induced by the inner product ∇u∇v, u, v ∈ H01(Ω ). u, v = H01(Ω )
Ω
For u ∈ H01(Ω ) define the functional 1 1 2 2 Φ(u) = |∇u| − λk u − F(x, u). 2 Ω 2 Ω Ω It is well known that under a sub-critical growth condition on f , Φ is well defined on H01(Ω ), weakly lower semi-continuous and it is well known that, under the (F0)–(F1) conditions of f , Φ is a C 2 functional with derivatives given by Φ (u)v = ∇u∇v − λk uv − f (x, u)v, for u, v ∈ H01(Ω ), Ω Ω Ω Φ (u).v.w = ∇w∇v − λk wv − f (x, u)wv, for u, w, v ∈ H01(Ω ). Ω
Ω
Ω
Thus, finding solutions of (1.1) is equivalent to finding critical points of the functional Φ. Since we are going to apply the variational characterization of the eigenvalues, we will decompose the space H01(Ω ) as E = E − ⊕ E k ⊕ E k+1 ⊕ E + , where E − is the subspace spanned by the λ j -eigenfunctions with j < k and E j is the eigenspace generated by the λ j -eigenfunctions and E + is the orthogonal complement of E − ⊕ E k ⊕ E k+1 in H01(Ω ) and we will decompose for
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any u ∈ H01(Ω ) as following u = u − + u k + u k+1 + u + where u − ∈ E − , u k ∈ E k , u k+1 ∈ E k+1 and u + ∈ E + . We can verify easily that 2 |∇u| dx − λi |u|2 dx ≥ δi u 2 ∀u ∈ ⊕ j ≥i+1 E j (2.1) Ω Ω |∇u|2 dx − λi |u|2 dx ≤ −δi u 2 ∀u ∈ ⊕ j ≤i−1 E j . (2.2) Ω
where δi = min{1 −
Ω
λi λi λi+1 , λi−1
− 1}.
2.1. Preliminary results on compactness condition To apply minimax methods for finding critical points of Φ, we need to verify that Φ satisfies a compactness condition of the Palais–Smale type which was introduced by Cerami in [7]. Definition. A functional Φ ∈ C 1 (E, R), with E a real Banach space, is said to satisfy condition (C)c , at the level c ∈ R, if: every sequence (u n ) ⊂ E such that Φ(u n ) → c, u n Φ (u n ) → 0 possesses a convergent subsequence. It was shown in [3] that condition (C)c actually suffices to get a deformation theorem and the linking theorem. Now, we present some technical lemmas. Lemma 2.1. Let (u n ) ⊂ H01(Ω ) and ( pn ) ⊂ L ∞ (Ω ) be sequences, and let A be a nonnegative constant such that 0 ≤ pn (x) ≤ A
a.e. in Ω and for all n ∈ N
and pn 0 in the weak* topology of L ∞ , as n → ∞. Then, there are subsequences (u n ), ( pn ) satisfying the above conditions, and there is a positive integer n 0 such that for all n ≥ n 0 , −δk + k k+1 2
u n + u k+1 pn u n ((u − + u+ (2.3) n + u n ) − (u n n )) dx ≥ n . 2 Ω Proof. Since pn ≥ 0 a.e. in Ω , we see that k k+1 pn u n ((u − + u+ n + u n ) − (u n n )) Ω k+1 2 ≥− pn (u + n + u n ) dx Ω 2 + u n + u k+1 n k+1 2 ≥− pn dx u + n + un . + k+1
u n + u n
Ω
(2.4)
Moreover, by the compact imbedding of H01(Ω ) into L 2 (Ω ) and pn 0 in the weak* topology of L ∞ , when n → ∞, then there are subsequences (u n ), ( pn ) such that 2 + u n + u k+1 n pn dx → 0. k+1
u + Ω n + un
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Therefore, there exists n 0 ∈ N such that for n ≥ n 0 we have 2 + δk u n + u k+1 n pn dx ≤ . k+1 2
u + + u
Ω n n Combining inequalities (2.4) and (2.5), we get inequality (2.3).
(2.5)
Definition. A sequence (u n ) is said to be a (C)c sequence, at the level c ∈ R, if there is a sequence n → 0, such that Φ(u n ) → c
u n Φ (u n ), v H 1 ,H −1 ≤ n v
0
∀v ∈
(2.6) (2.7)
H01.
Lemma 2.2. Let (u n ) ⊂ H01(Ω ) be a (C)c sequence. n (x)) ∞ 1. If f n (x) = f (x,u u n (x) χ[|u n (x)|≥r] 0 in the weak* topology of L , as n → ∞. Then, there − + k+1 is a subsequence (u n ) such that ( u n + (u n + u n ) )n is uniformly bounded in n. f (x,u n (x)) χ[|u n (x)|≥r] 0 in the weak* topology of L ∞ , as 2. If f n (x) = (λk+1 −λk )uun n(x)− (x) + k n → ∞. Then, there is a subsequence (u n ) such that ( u − n + (u n + u n ) )n is uniformly bounded in n.
Proof. Since (u n )n ⊂ H01 is a (C)c sequence, (2.6) and (2.7) are satisfied. Now, we prove that + k+1 − k + k+1 the sequence ( u − n +u n +u n )n is uniformly bounded in n. Take v = (u n +u n )−(u n +u n ) in (2.7), pn (x) = f n (x), and 2 − 2 k+1 2 Λ = − |∇u − | + λ |u | dx + |∇(u + k n n n + u n )| Ω Ω Ω
+ k+1 2 k k+1 + − λk |u n + u n | dx + pn u n ((u − + u ) − (u + u )) dx n n n n Ω Ω
k+1 − k Γ = n + | f (x, u n (x))||(u + n + u n ) − (u n + u n )| dx . |u n (x)|≤r
Then Λ ≤ Γ . By the Poincar´e inequality, from (2.1)–(2.3), and Λ ≤ Γ , it follows that there exist constants A and B such that δk − k+1 2 − + k+1
u + (u + n + u n ) ≤ n + A u n + (u n + u n ) + B. 2 n + k+1 This gives that ( u − n + (u n + u n ) )n is uniformly bounded in n. The same proof of eventuality + − k 2 are given by taking v = (u n + u k+1 n ) − (u n + u n ) and
pn (x) = f n (x) =
(λk+1 − λk )u n (x) − f (x, u n (x)) χ[|u n (x)|≥r] . u n (x)
Lemma 2.3. Assume f satisfies (F1). + k+1 1. Let (u n ) ⊂ H01(Ω ) such that u − n + (u n + u n ) is uniformly bounded in n and there exists A such that if A ≤ Φ(u n ), then u kn F x, dx ≤ M. 2 Ω
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+ k 2. Let (u n ) ⊂ H01(Ω ) such that u − n + u n + u n is uniformly bounded in n and there exists A such that if Φ(u n ) ≤ A, then 2 λk+1 − λk u k+1 u k+1 n n − F x, dx ≤ M. 2 2 2 Ω
Proof. From (2.6), and Poincar´e inequality, we have uk 1 uk 2 F x, n dx ≤ A + + u k+1 + u− F x, n − F(x, u n ) dx + u + n n
2 2 2 n Ω Ω (2.8) Since, f ∈ C 1 (Ω¯ × R, R) satisfy (F1), there exist two functions γ , h : Ω × R → R such that f (x, t) = tγ (x, t) + h(x, t) ) with 0 ≤ γ (x, t) = f (x,t t χ[|t |≥r] ≤ λk+1 − λk and h(x, t) = f (x, t)χ[|t |
Set t1 = min{t ∈ [0, 1] | h(x, t uk h(x, t 2n
u kn 2
+ (1 − t)u n ) = 0} and t2 = max{t ∈ [0, 1] |
+ (1 − t)u n ) = 0}. We verify easily that
k
u (t2 − t1 )
n − u n
≤ 2r. 2
So that using (2.9), (2.10), the Poincar´e inequality, and an elementary inequality 2 a a −b + − b b ≤ (a − b)2 , 2 2 we have u kn − F(x, u n ) dx F x, 2 Ω k t2 un u kn ≤ + (1 − t)u n dt − u n dx h x, t 2 2 Ω t1 λk+1 − λk − k+1 2 +
u n + u + n + un
4λ1 λk+1 − λk − k+1 2 ≤ 2r sup | f (x, s)|meas(Ω ) +
u n + u + n + un . 4λ 1 |s|≤r
(2.10)
(2.11)
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From (2.8) and (2.11), there exists M > 0 such that u kn dx ≤ M. F x, 2 Ω Lemma 2.4. Let p satisfy p(x, t) = 0 for t < 0, x ∈ Ω and λk t 2 ≤ p(x, t)t ≤ λk+1 t 2 , t ≥ r > 0, k ≥ 2. Then the C 2 functional Φ : H01(Ω ) → R defined by 1 |∇u|2 − P(x, u) Φ(u) = 2 Ω Ω t satisfies the (PS) condition, where P(x, t) = 0 p(x, s) ds. Proof. Let (u n )n ⊂ H01 be a (P S) sequence, i.e. |Φ(u n )| ≤ A
(2.12)
Φ (u n ), v H 1 ,H −1 ≤ n v
0
∀v ∈
H01,
(2.13)
where A is a constant and n → 0. It clearly suffices to show that (u n )n remains bounded in H01. Assume by contradiction. Defining z n = uu nn , we have z n = 1 and, passing if necessary to a subsequence, we may assume that z n z weakly in H01, z n → z strongly in L 2 (Ω ) and z n (x) → z(x) a.e. in Ω . By (2.12), there is an m ∈ L 2 (Ω ) with λk ≤ m ≤ λk+1 such that Φ (u n ), u n →1− m(x)z(x) dx = 0. (2.14)
u n 2 Ω So that z ≡ 0. In other words, we verify easily that z satisfies −∆z = m(x)z + z=0 on ∂Ω
in Ω
where z + = max{z, 0}. By the maximum principle and the unique continuation property, z = z + ≥ 0 and m ≡ λk or m ≡ λk+1 . Since k ≥ 2, z ≡ 0, which contradicts (2.14). Hence u n is bounded. The proof is completed. 2.2. Preliminary results on critical groups Let H be a Hilbert space and Φ ∈ C 1 (H, R) be satisfying the compactness condition (P S) or (C) Cerami condition. Set Φ c = {u ∈ H01(Ω ) | Φ(u) ≤ c}. Denote by Hq (X, Y ) the q-th relative singular homology group with integer coefficients. The critical groups of Φ at an isolated critical point u with Φ(u) = 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. Let K = {u ∈ H | Φ (u) = 0} be the set of critical points of Φ and a < infK Φ. The critical groups of Φ at infinity are formally defined as [4] Cq (Φ, ∞) = Hq (H, Φ a );
q ∈ Z.
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If K < ∞ then the Morse-type numbers of the pair are defined by Mq = Mq (H, Φ a ) = dim Cq (Φ, u). u∈K
By the Morse theory [8,18] if βq = dim Cq (Φ, ∞), then q j =0 ∞
(−1)q− j M j ≥
q
(−1)q− j β j ,
(2.15)
j =0
(−1)q Mq =
q=0
∞ (−1)q βq .
(2.16)
q=0
The formal expression of the Morse inequality reads as ∞ q=0
Mq t q =
∞
βq t q + (1 + t)
q=0
∞
aq t q .
(2.17)
q=0
From (2.15), we easily deduce the inequalities Mq ≥ βq for all q ∈ Z. Thus, if βq = 0 for some q then Φ must have a critical point, u, with Cq (Φ, u) 0. If equality (2.16) does not hold then Φ must have another critical point differing from the known ones and, if u, v are two critical points of Φ and Cq (Φ, u) Cq (Φ, v) for some q then u = v. Moreover, we have the following Proposition 2.1 ([4]). Assume that H = H + ⊕ H − , Φ is bounded from below on H + and Φ(u) → −∞ as u → ∞ with u ∈ H − . Then if k = dim H − < ∞.
Ck (Φ, ∞) 0,
In [19], the following result of the existence of a third critical point was proved. Theorem 2.1. Suppose Φ ∈ C 1 (H, R), H a real Hilbert space, is not bounded below and satisfies the (C) condition. Assume the origin is an isolated critical point of Φ in H satisfying C1 (Φ, 0) = 0, and Φ possesses a local minimum u 0 = 0. Then, Φ possesses at least three critical points in H . 3. Proofs of main results First, we prove that Φ satisfies the Cerami condition. Lemma 3.1. Under the assumptions (F1)–(F3), Φ satisfies the (C)c condition on H01(Ω ), for all c ∈ R. Proof. Let (u n )n ⊂ H01 be a (C)c sequence, i.e. Φ(u n ) → c
u n Φ (u n ), v H 1 ,H −1 ≤ n v
0
∀v ∈
H01,
(3.1) (3.2)
where n → 0. It clearly suffices to show that (u n )n remains bounded in H01. Assume by contradiction. Defining z n = uu nn , we have z n = 1 and, passing if necessary to a subsequence, we may assume that z n z weakly in H01, z n → z strongly in L 2 (Ω ) and z n (x) → z(x) a.e. in Ω . Set, f (x, u n (x)) χ[|u n (x)|≥r] f n (x) = u n (x)
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and l ∈ L ∞ such that f n → l in the weak* topology of L ∞ , as n → ∞. where the L ∞ -function l satisfies 0 ≤ l(x) ≤ λk+1 − λk .
(3.3)
Set m(x) = l(x) + λk , by (3.2), we have Φ (u n ), u n →1− m(x)z(x) dx = 0.
u n 2 Ω So that z ≡ 0. In other words, we verify easily that z satisfies −∆z = m(x)z z=0
in Ω
(3.4)
on ∂Ω .
Hence 1 is an eigenvalue of problem (3.4). We now distinguish three cases: (i) λk < m(x) and m(x) < λk+1 on subset of positive measure; (ii) m(x) ≡ λk ; (iii) m(x) ≡ λk+1 . In case (i), by strict monotonicity we have λk (m) < λk (λk ) = 1
and
λk+1 (λk+1 ) = 1 < λk+1 (m).
This contradicts that 1 is an eigenvalue of problem (3.4). On the other hand, by (F2), (F3), Lemmas 2.2 and 2.3 the cases (ii) and (iii) are not possible. The proof is complete. Lemma 3.2. Under hypothesis (F1)–(F3), the functional Φ has the following properties: (i) Φ(v) → −∞, as v → ∞, v ∈ E k ⊕ E − = V (ii) Φ(w) → ∞, as w → ∞, w ∈ E k+1 ⊕ E + = W . Proof. (i) Assume by contradiction, that there exist a constant B and a sequence (vn ) ⊂ V with
vn → ∞ such that B ≤ Φ(vn ) ≤ −δ vn− 2 . Therefore, vn− is bounded and by Lemma 2.3, we obtain vnk F x, lim inf dx ≤ constant. n→∞ Ω 2 This is a contradiction with assumption (F2). (ii) Suppose by contradiction that Φ is not coercive in W . Thus, there is some constant B and some sequence (wn ) ⊂ W , with wn → ∞, such that 1 1 (λk+1 − λk )wn2 − F(x, wn ) dx ≤ B. [|∇wn |2 dx − λk+1 wn2 ] + Φ(wn ) = 2 Ω Ω 2 This implies that wn+ is bounded, and so Lemma 2.3 gives us a contradiction from (F3).
Lemma 3.3. Under the assumptions of Theorem 1.1, there exists at least two nontrivial critical point u ± of Φ such that Cq (Φ, u ± ) = δq1 Z. Proof. Define the following functionals Φ± : H01(Ω ) → R as 1 |∇u|2 dx − G(x, u ± ) dx Φ± (u) = 2 Ω Ω
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where u + = max{u(x), 0}, u − = min{u(x), 0}, then Φ± ∈ C 2 . It follows from Lemma 2.4 that Φ± satisfies the (PS) condition. The (F1) condition, with k ≥ 2, shows that Φ± (tϕ1 ) → −∞,
as t → ±∞,
where ϕ1 is the first eigenfunction of −∆. Then, we can find a t0 such that t0 > R and Φ± (±t0 ϕ1 ) ≤ 0. Since g (x, 0) λ1 , u = 0 is a strictly local minimum of Φ± . Hence, there is an R > 0 and an α > 0 such that Φ± ≥ α on ∂ B R (0). The compactness condition and geometry of Φ± assure that c = infh∈Γ max0≤t ≤1 Φ± (h(t)) are critical values for Φ± , where Γ = {h ∈ C([0, 1], H01) | h(0) = 0, h(1) = ±t0 ϕ1 }, and c ≥ α. Then, by a mountain pass lemma and the maximum principle we can obtain a positive critical point u + of Φ+ and a critical point u − of Φ− such that C1 (Φ± , u ± ) = 0. Moreover, by Corollary 8.5 in [18] the critical group of Φ± at u ± is Cq (Φ± , u ± ) = δq1 Z. Using the results in [9], we have Cq (Φ, u ± ) ∼ = Cq (Φ /C 1 (Ω ) , u ± ) ∼ = Cq (Φ± /C 1 (Ω ) , u ± ) ∼ = Cq (Φ± , u ± ) ∼ = δq1 Z. 0
This completes the proof.
0
(3.5)
Proof of Theorem 1.1. By Lemma 3.1 the functional Φ satisfies the (C) condition. Since Φ is weakly lower semi-continuous, and is coercive on W by Lemma 3.2, infw∈W Φ(w) > −∞, i.e. Φ is bounded from below on W . By Lemma 3.2, Φ is anti-coercive on V . Thus Proposition 2.1 tells us that Cµ (Φ, ∞) 0, where µ = dim V ≥ k. It follows from the Morse inequality that Φ has a critical point u 0 with Cµ (Φ, u 0 ) 0,
(3.6)
Since g (x, 0) λ1 , u = 0 is a local minimum of Φ, hence Cq (Φ, 0) = δq,0 Z.
(3.7) 2 imply that u + , u −
By (3.6), (3.7), Lemma 3.3 and k ≥ of Φ. This completes the proof.
and u 0 are three nontrivial critical points
Proof of Corollary 1.1. From (F5), there are s1 , s2 such that 0 < f (s1 ), f (s2 ) < λk+1 − λk = µ. Thus, by continuity, there exist s3 ∈ R, ε > 0, δ > 0 such that ε ≤ f (s) ≤ µ − ε for every s ∈ [s3 − δ, s3 + δ]. Let s3 ≥ 0 and take s > s3 + δ, we have s3 s3 +δ s εδ ≤ f (s) = f (t) dt + f (t) dt + f (t) dt ≤ µs − εδ. 0
s3
s3 +δ
(3.8)
However, there are constants A1 , A2 such that for s > s3 + δ we have s2 − εδs + A2 . (3.9) 2 Similarly, if s3 < 0 and s < s3 −δ we have (3.8) and (3.9). Hence, the conditions of Theorem 1.1 are satisfied and the corollary follows. εδs + A1 ≤ F(s) ≤ µ
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Proof of Theorem 1.2. Without loss of generality, we suppose that t1 > 0. We define if t < 0 0 g (x, t) = λ1 t + f (x, t) if t ∈ [0, t1 ] 0 if t > t1 . : H 1(Ω ) → R as Define the cut-off functional Φ 0 1 u) dx, Φ(u) = |∇u|2 dx − G(x, 2 Ω Ω is coercive and satisfies (PS). Hence Φ t) is the primitive of where G(x, g . It is easy to see that Φ possesses a minimum u 0 . By the maximum principle we known that either u 0 ≡ 0 or 0 < u 0 < t0 for all x ∈ Ω and u 0 is a local minimum of Φ. On the other hand, since λ2 g (x, 0) we verify easily that 2 2 Φ (0).v.v = |∇v| − g (x, 0)v dx ≤ [λ2 − g (x, 0)]v 2 dx < 0, Ω
Ω
Ω
for every v ∈ ⊕ j ≤2 It follows that the Morse index d of Φ at 0 is larger than 2. The well known Gromoll–Meyer Theorem in [18] says that Cq (Φ, 0) = 0 for q ∈ [d, d + ν], with ν the nullity of Φ at 0. Hence, we have C1 (Φ, 0) = 0 and u 0 is a nontrivial local minimizer of Φ in the C01 (Ω¯ ) topology. By a standard argument [9], we know that u 0 is a local minimizer of Φ in H01(Ω ) topology. The (C) condition was established by Lemma 3.1 and Φ is not bounded below by Lemma 3.2. Consequently, we may invoke Theorem 2.1 to conclude that Φ has at least three critical points. The proof is completed. E j.
Lemma 3.4 ([16]). If (F6) is satisfied, then Cq (Φ, 0) = δq,d Z
(3.10)
where d = dim ⊕ j ≤m E(λ j ). Proof of Theorem 1.3. It follows from (3.6), (3.10) and by m = k that there exists a nontrivial critical point u 0 of Φ such that Cµ (Φ, u 0 ) = 0,
(3.11)
where µ = dim ⊕ j ≤k E(λ j ). Without loss of generality, we assume that t0 > 0. Using the condition g(x, t0 ) = 0, the same arguments in the proof of Theorem 1.2 involving the cut-off technique and the maximum principle, Φ has a local minimizer u 1 with 0 < u 1 < t0 and Cq (Φ, u 1 ) = δq,0 Z.
(3.12)
Now, define the functionals Φ± : H01(Ω ) → R as 1 2 Φ± (v) = |∇v| dx − [G(x, u 1 + v ± ) − G(x, u 1 ) − g(x, u 1)v ± ] dx 2 Ω Ω where v + = max{v(x), 0}, v − = min{v(x), 0}, then Φ± ∈ C 2 . Therefore, we obtain 1 Φ± (v) = Φ(u 1 + v ± ) − Φ(u 1 ) + |∇v ∓ |2 dx. 2 Ω
(3.13)
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It follows from (3.13) and u 1 is a local minimum of Φ that 0 is a strictly minimum of Φ± by Lemma 2.4. Φ± satisfies the (PS) condition and the (F1) condition, with k ≥ 2, shows that Φ± (tϕ1 ) → −∞,
as t → ±∞,
where ϕ1 is the first eigenfunction of −∆. However, by a mountain pass lemma and the maximum principle we can obtain a positive critical point v + of Φ+ and a critical point v − of Φ− such that C1 (Φ± , v ± ) = 0 satisfying −∆v ± = g(x, u 1 + v± ) − g(x, u 1 ) ±
v =0
in Ω
(3.14)
on ∂Ω
− + ± Hence, u ± 2 = u 1 + v are two solutions of (1.1), and u 2 < u 1 < u 2 . According to the results given in [9] and the similar arguments used in Lemma 3.3, the critical groups of Φ at u ± 2 are ± ∼ ± ∼ ± ∼ ∼ Cq (Φ, u ± 2 ) = Cq (Φ /C 1 (Ω ) , u 2 ) = Cq (Φ± /C 1 (Ω ) , v ) = Cq (Φ± , v ) = δq,1 Z. 0
0
(3.15)
− By (3.11), (3.12), (3.15), Lemma 3.4 and m, k ≥ 2, m = k imply that u 0 , u + 2 , u 2 and u 1 are four nontrivial critical points of Φ. Hence, the proof is completed.
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