CRITICAL POINT THEOREMS AND APPLICATIONS

We Consider the nonlinear Dirichlet problem: (1) { −∆u = DuF (x, u) + λku dans Ω u = 0 sur ∂Ω where Ω ∈ R is a bounded open domain, F : Ω × R → R is a carathéodory function and DuF (x, u) is the partial derivative of F. We are interested in the resolution of problem (1) when F is concave. Our tool is absolutely variational. Therefore, we state and prove a critical point theorem which generalizes many other results in the literature and leads to the resolution of problem (1). Our theorem allows us to express our assumptions on the nonlinearity in terms of F and not of ∇F . Also, we note that our theorem doesn’t necessitate the verification of the famous compactness condition introduced by Palais-Smale or any of its variants.


Introduction
We Consider the nonlinear Dirichlet problem: (1) −∆u = D u F (x, u) + λ k u dans Ω u = 0 sur ∂Ω where Ω ∈ R N is a bounded open domain, F : Ω×R → R is a carathéodory function and D u F (x, u) is the partial derivative of F. We assume: D u F (x, u) = f (x, u).The spectrum of (−∆) is denoted as The aim of this paper is to resolve the problem (P) with variational tool.
We suppose that F satisfies some growth conditions and some classical assymptotic assumptions.Therefore, it turns out that the weak solutions of (1) are precisely the critical points of the functional Φ : We restrict ourselves to the case where F is concave.In contrast with classical critical point theorems, we state and prove a critical point theorem which allows us to express our assumptions on the nonlinearity in terms of F and not of ∇F in the sense of (2) below.Also, we note that our theorem doesn't necessitate the verification of the famous compactness condition introduced by Palais-smale in [1] or any of its variants, see [2] and [3].The problem (1) is equivalent to the problem (2) −∆u = D u G(x, u) dans Ω u = 0 sur ∂Ω where G(x, u) = F (x, u) + λ k 2 u 2 and D u G(x, u) = g(x, u).To resolve problem (2), the classical results considered the case when the following quotients 2G(x, s) s 2 and g(x, s) s are situated between two successive eigenvalues λ k and λ k+1 .This kind of problems has been studied a long time ago.On 1930 Hammerstein in [4] proved a result of existence for (2) if f is continuous and satisfies a linear growth condition and On 1949, in [5], Dolph proved the first result of existence when where λ k and λ k+1 are two consecutive eigenvalues of −(∆) in H 1 0 (Ω).Then, Dolph considered the last condition with G instead of g: The first variational attempt to solve problem (2) under condition (2) was carried over by Dolph, see [6].He assumed in addition the following condition: Denoting by V = ⊕ i≤k E(λ i ) where E(λ i ) is the eigenspace associated to the eigenvalue λ i and by W its orthogonal so that: The functional associated to problem (2): Φ(u) = 1  2 Ω ∇u 2 − Ω G(x, u)dx admits at most one maximum in w + V for all w ∈ W.
Our absract theorem generalizes a Mini-Max theorem due to Lazer et al. [9] to the case where X and Y are not necessarily finite dimensional.Moreover, we consider Φ of class C 1 instead of C 2 .Tersian [10] studied the case where X and Y are not necessarily finite dimensional and ∇Φ : H → H is everywhere defined and hemicontinuous on H, which means that Instead of conditions on Hessian of Φ, they supposed (1)(∇Φ(h and m 2 are strictly positives.Their result rests heavily upon two theorems on α-convex functionals and an existence theorem for a class of monotone operators due to Browder.Our theorem partially extend many other results in the literature (see e.g.[11] and [12]).On 1991, the second author of this paper proved in [13] the following theorem.
Theorem 1.1.Let H be a Hilbert space such that : H = V ⊕ W where V is finite dimensional subspace of H and W its orthogonal.Let Φ : and the convergence is uniform on bounded subsets of W . (v) for all v ∈ V , Φ is weakly lower semicontinuous on W + v. Then Φ admits a critical point in H.
In our theorem, we are based specially on theorem 1.1.We note that our theorem generalizes theorem 1.1 and our previous result [14] and our convexity conditions are weaker than all used in the previous results.In [13], Moussaoui resolved problem (1) by theorem 2.1 with the following assumptions (F1) F (x, .) is convex and differentiable for almost every x ∈ Ω. F (., s) is measurable for all s ∈ R.
The result obtained by Moussaoui is a particular case of Mawhin and Willem result, see [15].Or the proof used by Moussaoui is absolutely different from the proof used by Mawhin and Willem.In fact, Mawhin and Willem used the dual least action principle of Clarke-Ekeland which is essentially a convex analysis method to solve a perturbed problem.They combined this process to an approximation dealing to attain their aim.
For us, we will be interested te resolve problem (1) with "dual" conditions of those supposed by Moussaoui and we note that our resolution of problem (1) doesn't require the condition of Ahmed,Lazer and Paul (see [16]).

Critical point theorems
1 The main result.Theorem 2.1.Let E be a Hilbert space such that: E = V ⊕ W where V and W are two closed subspaces of E. Let Φ : There exists an increasing function γ : (0, +∞) → (0, +∞) such that: Then Φ admits at least a critical point u ∈ H.Moreover, this critical point is characterized by the equality Remark 2.2.Condition (ii) in theorem 2.1 doesn't contain the " uniform convergence on bounded subsets of W" which is essential in theorem 1.1.This elimination will permit us to improve the resolution of problème (1).In fact we will solve problem (1) (paragraph 4) without Ahmed,lazer and Paul condition (A.L.P) which was used in many works (see e.g.[15]and [13] ).
The proof of theorem 2.1 will depend on three lemmas.Lemmas 3.4 and 2.4 were used by Moussaoui to prove the existence of critical points of Φ when V is of finite dimension (theorem 1.1, introduction).Lemma 2.3.For all w ∈ W , there exists a unique v ∈ V such that : Proof: From (iii), for w fixed in W , v → Φ(v+w) is continuous and strictly convex on V.Then, it is weakly lower semicontinuous on V .Moreover, from (ii), it is coercive on V.So that, from a lemma in [17] and [18] for axample, it admits a minimum on V .We affirm that this minimum is unique, otherwise we suppose that there exist two minimums v 1 and v In the proof of this theorem, we will adopt the notations: and Lemma 2.4.There exists u ∈ S such that

1)
Proof: There exists a sequence ( From (iv), Φ(w n ) → −∞, hence Φ(u n ) → −∞.A contradiction.From (5), there exists a subsequence denoted also w n such that w n w.Take v in V , by (v), we have: This is true for all v ∈ V , in particular for v ∈ V (w).Then u = v + w satisfies (4).
Lemma 2.5.The application f : W → V such that is continuous.
Proof: We suppose that f isn't continuous, thus there exist δ > 0 and a sequence (w n ) converging to w ∈ W such that for enough big n, we have: Let P be the projection of H onto V defined by P (v + w) = v, and let P * be the adjoint of P. Then we obtain for each n: Hence, from ( 4) and (iii), we conclude that for enough big n, there exists a > 0 such that In the other hand,
Proof of theorem 2.1.
Let w ∈ W and u ∈ S w .We will prove that if u satisfies (4), then u is a critical point of Φ. From lemma 3.4, we have (∇Φ(u), v) = 0 for all v in V, so it suffices to prove that (∇Φ(u), h) = 0 ∀h ∈ W.
Recall that u ∈ S w is written: u = v + w where w ∈ W and v ∈ V (w).Take h ∈ W and w t = w + th for |t| ≤ 1.For each t such that 0 < |t| ≤ 1, there exists a unique v tn ∈ V (w tn ).Since w tn → w when n → +∞, we deduce by lemma 2.4 that v t n converges to a certain v 0 and v 0 ∈ V (w).Then, by lemma 3.4, we conclude that v 0 = v.For t > 0, since v 0 + w ∈ S w we have Then, (∇Φ(v t + w + λ t th), h) ≥ 0 0 < λ t < 1.
At the limit, we obtain Then, u is a critical point of Φ.

Variants of theorem 2.1
The first theorem that we will present is the "dual" version of theorem 1.1.
Theorem 3.1.Let H be a Hilbert space such that: H = V ⊕ W where V and W are two closed subspaces of H. Let Φ : (ii) For all w ∈ W , Φ is anticoercive on V + w. i.e, Φ(v + w) → −∞ when v → +∞.(iii) There exists an increasing function γ : (0, +∞) → (0, +∞) such that: Then Φ admits at least a critical point u ∈ H.Moreover, this critical point is characterized by the equality The proof of theorem 3.1 is made by a dual manner of the proof of theorem 2.1.Now, before announcing the second variant of theorem 3.1, We will give some definitions.
• Let A a convex set.The function f : A → R is quasiconcave if for all x 1 , x 2 in A, and for all λ in ]0, 1[, we have • The function f is quasiconvex if (-f) is quasiconcave, and it is strictly quasiconcave if the inequality above is strict.
• It is clear that any strictly concave function is strictly quasiconcave.Proposition 3.2.Let E be a reflexive Banach space.If Φ : E → R is quasiconvex and lower semicontinuous, then Φ is weakly lower semicontinuous.Theorem 3.3.Let E be a reflexive Banach space such that: Then Φ admits at least a critical point u ∈ H.Moreover, this critical point is characterized by the equality:

Proof of theorem 3.3
The structure of the proof of theorem 3.3 is the same as theorem 2.1.So, we will report only the changes wich concern lemmas 3.4 and 2.5.Lemma 3.4.For all w ∈ W , there exists a unique v ∈ V such that : Proof.For each w in W , v → Φ(v+w) is continuous and quasiconvex on V. Thus, by propsition 3.2, it's weakly lower semi-continuous on V .Moreover, from (iii), it's coercive, then it admits a minimum on the reflexive Banach space V. Since v → Φ(v+w) is strictly quasiconvex, this minimum is unique.Otherwise, there exist two minimums v 1 and v 2 .Let Absurde.
Lemma 3.5.The application f : W → V such that Proof.First, we prove that the application (f) defined in lemma 3.5 is bounded on bounded sets of W. Let M > 0, from (v) and (vi) of theorem 4.1 we conclude that Φ admits a maximum on W, so there exists a constant N > 0 such that Φ(w) ≤ N for w ∈ W and w ≤ M .By (iii), there exists a constante δ > 0 such that for v ∈ V, w ∈ W, w ≤ M and v ≥ δ, we have Φ(v + w) ≥ 2N .Since Φ(w + f (w)) = min g∈V Φ(w + g) ≤ φ(w) < N , we conclude that f (w) ≤ δ.Next, we prove that (f) is continuous.Suppose that w n converges to w, thus f (w n ) is bounded, so there exists a subsequence of f (w n ) denoted also f (w n ) which converges weakly to v 0 .Since Φ is weakly continuous, we obtain Remark 3.6.Theorem 3.3 is a complete generalization of theorem (3.7) in [14] because it deals with a more general class of functionals than the second theorem.

Application
We Consider the nonlinear Dirichlet problem: (1) where Ω ∈ R N is a bounded open domain, F : Ω×R → R is a carathéodory function and D u F (x, u) is the partial derivative of F par rapport à u.We assume: (F1) F (x, s) is concave and of class C 1 for almost every x ∈ Ω. F(.,s) is measurable for all s ∈ R N (F2) There exist a > 0 and β(x) ∈ L p (Ω) such that

Proof of theorem
Denote by Φ the functional associated to problem (1) We note that The critical point of Φ is the weak solution of problem (1).So, for the proof of therem 4.1, we will verifie successively hypotheses (i) to (v) of theorem 3.1.
Then there exists c 1 in R such that Since F is concave, we conclude Since the roles of V and W are symetriques, we obtain Thus, we obtain We choise ε such that ε < λ k+1 − λ k .Hence In the other hand, we have (L(w We conclude that there exists a (iv) We show that Φ is anticoercive on V. Claim: From (F4), there exists δ < 0 such that , so that v is an eigenvector associated to λ k .Or 0 = ρ(w) = Ω −α(x)v 2 = 0 implies that v(x) = 0 when α(x) > 0, so v = 0 We suppose that the claim is false, then there exist a sequence v n such that v n 2 = 1 and v n v (v n converges weakly in V), v n → v strongly in L 2 (Ω) and 0 ≥ ρ(v n ) → 0, n → ∞.We have ρ(v) ≥ lim sup n ρ(v n ) = 0.
Hence v = 0, so v n → 0 strongly in L 2 (Ω) and which contradicts the fact that ρ(w n ) → 0, and so the claim is proved.On the other hand, ∀ε > 0, ∃M > 0 such that for all s in R and for almost every x in Ω.Then Since V is of finite dimension, there exists l > 0 such that v 2 ≤ v 2 2 .So Consequently, it suffices to take ε < −2δl to conclude that Φ is coercive on V .
(v) Φ is weakely upper semi-continuous on V. Indeed, q is a negative quadratic form on V, then q is concave on V, and since it's continuous, q is weakly upper semi-continuous on V.Moreover, F (x, u) dx is weakely continuous on H.So the result.