A critical point theorem via the Ekeland variational principle

Nonlinear Analysis 75 (2012) 2992–3007

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A critical point theorem via the Ekeland variational principle Gabriele Bonanno Department of Science for Engineering and Architecture (Mathematics Section), Engineering Faculty, University of Messina, 98166 - Messina, Italy

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Article history: Received 15 February 2011 Accepted 7 December 2011 Communicated by S. Carl MSC: 58E30 49J52 49J50

abstract The aim of this paper is to establish the existence of a local minimum for a continuously Gâteaux differentiable function, possibly unbounded from below, without requiring any weak continuity assumption. Several special cases are also emphasized. Moreover, a novel definition of Palais–Smale condition, which is more general than the usual one, is presented and a mountain pass theorem is pointed out. As a consequence, multiple critical points theorems are then established. Finally, as an example of applications, an elliptic Dirichlet problem with critical exponent is investigated. © 2011 Elsevier Ltd. All rights reserved.

Keywords: Critical point Variational methods Palais–Smale condition Local minimum Multiple critical points

1. Introduction Critical point theorems are often used to ensure solutions to both ordinary and partial differential problems. The variational formulation which occurs in these cases is I = Φ − Ψ,

(1.1)

where Φ , Ψ are continuously Gâteaux differentiable functions defined on an infinite dimensional real Banach space. The aim of this paper is to establish the existence of a local minimum (that is, of a critical point) for functions of the type (1.1), which are, possibly, unbounded from below. It is also worth noticing that, in our main result (Theorem 3.1), no assumption of weak continuity on Φ or Ψ is required. We also observe that in order to establish Theorem 3.1 we use a novel Palais–Smale condition (see Section 2). It is more general than the usual one and, further, it holds, under additional regularity assumptions on Φ and Ψ , when Φ is coercive without requiring the coercivity of Φ − Ψ (see Proposition 2.1). The proof of Theorem 3.1 is based on a consequence of the classical Ekeland variational principle built within a non-smooth framework (see Lemma 3.1). In fact, we apply the Ekeland variational principle in a non-smooth setting since we cut off the functions Φ and Ψ so that the modified functions are only Lipschitz continuous (see the proof of Theorem 3.1). Moreover, two variants of Theorem 3.1 are pointed out (Theorems 4.1 and 4.2) and, as a consequence, several critical point theorems for functions depending on a real parameter are presented (see Section 5). In Section 6, we present a mountain pass theorem (Theorem 6.2) where a localization of the critical point is established by using the new Palais–Smale condition (see Proposition 2.3). Section 7 is devoted to multiple critical points theorems. In particular, three critical points results and infinitely many critical points theorems are presented. As a sample, the following result, which is a consequence of Theorem 7.1, is pointed out here.

E-mail addresses: [email protected], [email protected]. 0362-546X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2011.12.003

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Theorem 1.1. Let X be a real Banach space and Φ , Ψ : X → R two continuously Gâteaux differentiable functions with Φ bounded from below such that I = Φ − Ψ is bounded from below and satisfies the Palais–Smale condition. Assume that Φ (0) = Ψ (0) = 0 and there are r ∈ R and u ∈ X , with 0 < r < Φ (u), such that sup u∈Φ −1 (]−∞,r ])

Ψ (u) < r and Φ (u) < Ψ (u).

Then, function I admits at least three distinct critical points. We explicitly observe that, in Theorem 1.1, contrary to three critical points theorems in literature (see, for instance, [1–6]), no assumption of weak continuity on Φ or Ψ is assumed. The same remark also holds for Theorems 7.1–7.4. Finally, we also observe that results in Sections 5 and 7 can be applied to wide classes of nonlinear differential problems as has been already done by using multiple critical points theorems cited above (see, for instance, references in [6]). In particular, since the results in the present paper are more general than the previously cited ones, also in order to estimate of parameters, several applications to nonlinear differential problems can be improved. In Section 8, as an example of application, an elliptic Dirichlet problem with critical exponent is investigated. 2. On the Palais–Smale condition Let (X , ∥·∥) be a real Banach space. We denote the dual space of X by X ∗ , while ⟨·, ·⟩ stands for the duality pairing between X and X . A function I : X → R is called locally Lipschitz when, to every u ∈ X , there corresponds a neighbourhood U of u and a constant L ≥ 0 such that ∗

|I (v) − I (w)| ≤ L ∥v − w∥

for all v, w ∈ U .

If u, v ∈ X , the symbol I (u; v) indicates the generalized directional derivative of I at point u along direction v , namely ◦

I ◦ (u; v) := lim sup

I (w + t v) − I (w) t

w→u,t →0+

.

The generalized gradient of the function I at u, denoted by ∂ I (u), is the set

  ◦ ∂ I (u) := u∗ ∈ X ∗ : ⟨u∗ , v⟩ ≤ I (u; v) for all v ∈ X . A function I : X → R is called Gâteaux differentiable at u ∈ X if there is ϕ ∈ X ∗ (denoted by I ′ (u)) such that lim

I (u + t v) − I (u)

t →0+

t

= I ′ (u)(v) ∀v ∈ X .

It is called continuously Gâteaux differentiable if it is Gâteaux differentiable for any u ∈ X and the function u → I ′ (u) is a continuous map from X to its dual X ∗ . We recall that if I is continuously Gâteaux differentiable then it is locally Lipschitz and one has I ◦ (u; v) = I ′ (u)(v) for all u, v ∈ X . Moreover, we say that a Gâteaux differentiable function I verifies the Palais–Smale condition (in short (PS)-condition) if any sequence {un } such that

(α) {I (un )} is bounded, (β) limn→+∞ ∥I ′ (un )∥X ∗ = 0, has a convergent subsequence. For a thorough treatment of these topics we refer to [7–9] and the references therein. Now, let Φ , Ψ : X → R be two continuously Gâteaux differentiable functions; put I =Φ −Ψ and fix r1 , r2 ∈ [−∞, +∞], with r1 < r2 ; we say that function I verifies the Palais–Smale condition cut off lower at r1 and upper at r2 (in short [r1 ] (PS)[r2 ] -condition) if any sequence {un } such that (α), (β) and

(γ ) r1 < Φ (un ) < r2 ∀n ∈ N, has a convergent subsequence. Clearly, if r1 = −∞ and r2 = +∞ it coincides with the classical (PS)-condition. Moreover, if r1 = −∞ and r2 ∈ R it is denoted by (PS)[r2 ] , while if r1 ∈ R and r2 = +∞ it is denoted by [r1 ] (PS). Clearly, if I = Φ − Ψ satisfies [r1 ] (PS)[r2 ] -condition, then it satisfies [ρ1 ] (PS)[ρ2 ] -condition for all ρ1 , ρ2 ∈ [−∞, +∞] such that r1 ≤ ρ1 < ρ2 ≤ r2 . So, in particular, if I = Φ − Ψ satisfies the classical (PS)-condition, then it satisfies [ρ1 ] (PS)[ρ2 ] -condition for all ρ1 , ρ2 ∈ [−∞, +∞] with ρ1 < ρ2 . Now, we point out the following propositions which show that, under suitable regularity assumptions on Φ and Ψ , if either Φ is coercive or I = Φ − Ψ is coercive, then function I satisfies the [r1 ] (PS)[r2 ] -condition.

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Proposition 2.1. Let X be a reflexive real Banach space; Φ : X → R be a sequentially weakly lower semicontinuous, coercive and continuously Gâteaux differentiable function whose Gâteaux derivative admits a continuous inverse on X ∗ , Ψ : X → R be a continuously Gâteaux differentiable function whose Gâteaux derivative is compact. Then, for all r1 , r2 ∈ [−∞, +∞[, with r1 < r2 , the function Φ − Ψ satisfies the [r1 ] (PS)[r2 ] -condition. Proof. Since X is reflexive, Φ : X → R is sequentially weakly lower semicontinuous and coercive, then Φ is bounded from below. Put m = minX Φ . Now, fix {un } such that {(Φ − Ψ )(un )} is bounded, {(Φ − Ψ )′ (un )} converges to 0 in X ∗ and r1 < Φ (un ) < r2 for all n ∈ N. Therefore, since m ≤ Φ (un ) < r2 for all n ∈ N and Φ is coercive, then {un } is bounded. From the compactness of Ψ ′ , {Ψ ′ (un )} admits a subsequence converging that we call {Ψ ′ (uk )}. Hence, since {(Φ − Ψ )′ (uk )} converges to 0 in X ∗ , then {Φ ′ (uk )} converges in X ∗ . Now, denoted by T : X ∗ → X a continuous operator such that T [Φ ′ (u)] = u for all u ∈ X , one has that {T [Φ ′ (uk )]} converges in X , that is, {uk } converges in X .  Remark 2.1. In Proposition 2.1 it is not necessary to assume X reflexive and Φ sequentially weakly lower semicontinuous. In fact, let {un } be a sequence as in the proof, therefore, owing to the coercivity of Φ , the condition r1 < Φ (un ) < r2 for all n ∈ N implies the boundedness of {un }. Proposition 2.2. Let X be a real Banach space; Φ : X → R be a continuously Gâteaux differentiable function whose Gâteaux derivative admits a continuous inverse on X ∗ , Ψ : X → R be a continuously Gâteaux differentiable function whose Gâteaux derivative is compact. Assume that the function Φ − Ψ is coercive. Then, for all r1 , r2 ∈ [−∞, +∞], with r1 < r2 , the function Φ − Ψ satisfies the [r1 ] (PS)[r2 ] -condition. Proof. From [10, Example 38.25] Φ − Ψ satisfies the (PS)-condition, hence the conclusion follows.



Finally, we recall the (PS)-condition for locally Lipschitz functions as given by Chang. To be precise, we say that a locally Lipschitz function I : X → R satisfies the Palais–Smale condition at level d, d ∈ R, (in short (PS)d -condition) if any sequence {un } such that

(α ′ ) {I (un )} → d, (β ′ ) limn→+∞ minω∈∂ I (un ) ∥ω∥X ∗ = 0, has a convergent subsequence. Clearly, if I is a continuously Gâteaux differentiable function, then the (PS)d -condition as given by Chang and the classical one coincide. The following proposition shows that the (PS)[r ] -condition implies the (PS)d -condition for suitable levels of d. Proposition 2.3. Let X be a real Banach space and Φ , Ψ : X → R two continuously Gâteaux differentiable functions. Assume that there are a, b ∈ R such that sup Ψ (x) < b.

(2.1)

Φ (x)
If the function Φ − Ψ satisfies the (PS)[a] -condition, then for all d ∈ R such that d < a − b the function Φ − Ψb satisfies the (PS)d -condition as given by Chang, where

Ψb (u) =

 Ψ ( u) b

if Ψ (u) < b if Ψ (u) ≥ b,

(2.2)

Proof. Fix d < a − b and let {xn } ⊆ X be a sequence such that

(α ′ ) lim (Φ − Ψb )(xn ) = d and (β ′ ) lim

min

n→+∞ ω∈∂(Φ −Ψb )(xn )

n→+∞

∥ω∥X ∗ = 0.

From (α ′ ) there is ν ∈ N such that Φ (xn ) − Ψb (xn ) < a − b for all n > ν . Hence, one has Φ (xn ) < Ψb (xn ) + a − b ≤ b + a − b = a, Φ (xn ) < a for all n > ν . Therefore, owing to (2.1) one has Ψ (xn ) < b for all n > ν . Hence, (α ′ ) and (β ′ ) become limn→+∞ (Φ − Ψ )(xn ) = d and limn→+∞ ∥(Φ − Ψ )′ (xn )∥X ∗ = 0. So, taking into account that Φ (xn ) < a for all n > ν and Φ − Ψ satisfies the (PS)[a] -condition, then {xn }n>ν admits a subsequence strongly converging in X and our conclusion is proved.  3. A critical point theorem In this section, we present the main result of the paper, that is Theorem 3.1. First, we point out the following consequence of the Ekeland variational principle. Lemma 3.1. Let X be a real Banach space and let I : X → R be a locally Lipschitz function bounded from below. Then, for all minimizing sequence of I , {un }n∈N ⊆ X , there exists a minimizing sequence of I , {vn }n∈N ⊆ X , such that I (vn ) ≤ I (un )

∀n ∈ N, ∀h ∈ X , ∀n ∈ N, where εn → 0+ .

I ◦ (vn ; h) ≥ −εn ∥h∥

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Proof. Fix σ > 0 and u ∈ X such that I (u) ≤ infX I +σ . We claim that there is v ∈ X such that I (v) ≤ I (u), ∥v− u∥ ≤ σ

1 2

and

1

1

I ◦ (v; h) ≥ −σ 2 ∥h∥ for all h ∈ X . In fact, taking X as a complete metric space endowed with the metric d(x, y) = σ − 2 ∥x − y∥, 1

1

from the Ekeland variational principle there is v ∈ X such that I (v) ≤ I (u), ∥v − u∥ ≤ σ 2 and I (w) − I (v) ≥ −σ 2 ∥w − v∥ for all w ∈ X , with w ̸= v . Therefore, taking in particular w = v + th with t > 0 and h ∈ X , one has Hence, I ◦ (v; h) ≥ lim sup

I (v + th) − I (v) t

t →0+

I (v+th)−I (v) t

1

≥ −σ 2 ∥h∥.

1

≥ −σ 2 ∥h∥

for all h ∈ X and our claim is proved. Now, let {un }n∈N ⊆ X be such that limn→∞ I (un ) = infX I and put σn = I (un ) − infX I if I (un )− infX I > 0, σn = 1

1 n

1

if I (un )− infX I = 0. As seen before, there is {vn }n∈N ⊆ X , such that I (vn ) ≤ I (un ), ∥vn − un ∥ ≤ σn 2

and I ◦ (vn ; h) ≥ −σn 2 ∥h∥ for all h ∈ X . Hence, one has limn→∞ I (vn ) = infX I and I ◦ (vn ; h) ≥ −εn ∥h∥ for all h ∈ X , where 1 2

εn = σn , and the proof is complete.



Remark 3.1. The Ekeland variational principle in nonsmooth critical point theory has already been used in earlier papers as, for instance, in [11,12]. Now, we present the main result of this paper. Theorem 3.1. Let X be a real Banach space and let Φ , Ψ : X → R be two continuously Gâteaux differentiable functions. Put I =Φ −Ψ and assume that there are x0 ∈ X and r1 , r2 ∈ R, with r1 < Φ (x0 ) < r2 , such that sup u∈Φ −1 (]r1 ,r2 [)

Ψ (u) ≤ r2 − Φ (x0 ) + Ψ (x0 ).

sup u∈Φ −1 (]−∞,r1 ])

Ψ (u) ≤ r1 − Φ (x0 ) + Ψ (x0 ).

(3.1) (3.2)

Moreover, assume that I satisfies [r1 ] (PS)[r2 ] -condition. Then, there is u0 ∈ Φ −1 (]r1 , r2 [) such that I (u0 ) ≤ I (u) for all u ∈ Φ −1 (]r1 , r2 [) and I ′ (u0 ) = 0. Proof. Put M = r2 − Φ (x0 ) + Ψ (x0 ),

ΨM (u) = Φ r1 (u) =



Ψ (u) M

 Φ (u) r1

(3.1∗ )

if Ψ (u) < M if Ψ (u) ≥ M , if Φ (u) > r1 if Φ (u) ≤ r1 ,

J = Φ r1 − ΨM . Clearly, J is locally Lipschitz and bounded from below. Now, given a sequence {un }n∈N ⊆ X such that limn→∞ J (un ) = infX J, owing to Lemma 3.1 there is a sequence {vn }n∈N ⊆ X such that limn→∞ J (vn ) = infX J and J ◦ (vn ; h) ≥ −εn ∥h∥ for all h ∈ X , for all n ∈ N, where εn → 0+ . If J (x0 ) = infX J then x0 satisfies the conclusion. In fact, if u ∈ Φ −1 (]r1 , r2 [) from (3.1) one has Ψ (u) ≤ M and J (u) = I (u) for all u ∈ Φ −1 (]r1 , r2 [); hence I (x0 ) = J (x0 ) ≤ J (u) = I (u) for all u ∈ Φ −1 (]r1 , r2 [). So, we assume infX J < J (x0 ). Therefore, there is ν ∈ N such that J (vn ) < J (x0 ) for all n > ν . Now, we claim that r1 < Φ (vn ) < r2 for all n > ν . On the one hand, one has Φ (vn ) − ΨM (vn ) ≤ Φ r1 (vn ) − ΨM (vn ) < Φ (x0 ) − Ψ (x0 ); Φ (vn ) < ΨM (vn ) + Φ (x0 ) − Ψ (x0 ) ≤ M + Φ (x0 ) − Ψ (x0 ) = r2 , Φ (vn ) < r2 . On the other hand, arguing by a contradiction, we assume Φ (vn ) ≤ r1 . Therefore, one has r1 − Ψ (vn ) = Φ r1 (vn ) − Ψ (vn ) < Φ (x0 ) − Ψ (x0 ); Ψ (vn ) > r1 − Φ (x0 ) + Ψ (x0 ) and, from (3.2), one has Φ (vn ) > r1 , that is a contradiction. Hence, our claim is proved. Therefore, one has J (vn ) = I (vn ) and J ◦ (vn ; h) = I ′ (vn )(h) for all n > ν . Hence, limn→∞ I (vn ) = limn→∞ J (vn ) = infX J and I ′ (vn )(h) ≥ −εn ∥h∥, that is limn→∞ ∥I ′ (vn )∥X ∗ = 0. Since I satisfies [r1 ] (PS)[r2 ] -condition, then {vn } admits a subsequence strongly converging to v ∗ ∈ X . So, I (v ∗ ) = infX J ≤ J (u) = I (u) for all u ∈ Φ −1 (]r1 , r2 [), that is I (v ∗ ) ≤ I (u)

(3.3)

for all u ∈ Φ (]r1 , r2 [). Since r1 < Φ (vn ) < r2 for all n > ν , from the continuity of Φ we obtain v ∗ ∈ Φ −1 ([r1 , r2 ]). If v ∗ ∈ Φ −1 (]r1 , r2 [), (3.3) immediately ensures the conclusion. If Φ (v ∗ ) = r1 , from (3.2) we obtain I (v ∗ ) = r1 − Ψ (v ∗ ) ≥ r1 − supΦ (u)≤r1 Ψ (u) ≥ Φ (x0 ) − Ψ (x0 ) = I (x0 ) and, hence, from (3.3), I (x0 ) ≤ I (u) for all u ∈ Φ −1 (]r1 , r2 [) and the conclusion is achieved. If Φ (v ∗ ) = r2 , first we observe that Ψ (v ∗ ) ≤ M; in fact, taking into account that I (v ∗ ) = J (v ∗ ), one −1

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has r2 − Ψ (v ∗ ) = r2 − ΨM (v ∗ ), Ψ (v ∗ ) = ΨM (v ∗ ) ≤ M. Next, we prove that I (v ∗ ) = I (x0 ). In fact, arguing by contradiction and assuming I (v ∗ ) < I (x0 ), from (3.1∗ ) one has I (v ∗ ) = r2 − Ψ (v ∗ ) ≥ r2 − M = Φ (x0 )− Ψ (x0 ) = I (x0 ), that is I (v ∗ ) ≥ I (x0 ) and this is absurd. Hence, from (3.3) one has I (x0 ) ≤ I (u) for all u ∈ Φ −1 (]r1 , r2 [) and also in this case the conclusion is achieved.  Remark 3.2. We again recall that if I satisfies (PS)-condition, then it satisfies [r1 ] (PS)[r2 ] -condition for all r1 , r2 ∈ R, with r1 < r2 . Hence, when I satisfies (PS)-condition and the algebraic inequalities (3.1) and (3.2) hold, then the conclusion of Theorem 3.1 is clearly true. 4. Two variants of the critical point theorem In this section, we present two relevant variants of Theorem 3.1. The first one is the following. Theorem 4.1. Let X be a real Banach space and let Φ , Ψ : X → R be two continuously Gâteaux differentiable functions with Φ bounded from below. Put I =Φ −Ψ and assume that there are x0 ∈ X and r ∈ R, with r > Φ (x0 ), such that sup u∈Φ −1 (]−∞,r [)

Ψ (u) ≤ r − Φ (x0 ) + Ψ (x0 ).

(4.1)

Moreover, assume that I satisfies (PS)[r ] -condition. Then, there is u0 ∈ Φ −1 (] − ∞, r [) such that I (u0 ) ≤ I (u) for all u ∈ Φ −1 (] − ∞, r [) and I ′ (u0 ) = 0. Proof. Put M = r − Φ (x0 ) + Ψ (x0 ),

Ψ M ( u) =

 Ψ (u) M

if Ψ (u) < M if Ψ (u) ≥ M ,

J = Φ − ΨM . Clearly, J is locally Lipschitz and bounded from below. Hence, arguing as in the proof of Theorem 3.1, the conclusion is obtained.  Remark 4.1. We explicitly observe that if I satisfies (PS)-condition and (4.1) holds then the conclusion of Theorem 4.1 is true. On the other hand, when Φ and Ψ are regular enough and Φ is coercive (see Proposition 2.1), inequality (4.1) ensures the conclusion of Theorem 4.1, without requiring the classical (PS)-condition on I. The second result is the following. Theorem 4.2. Let X be a real Banach space and let Φ , Ψ : X → R be two continuously Gâteaux differentiable functions. Put I =Φ −Ψ and assume that I is bounded from below and there are x1 ∈ X and r ∈ R, with r < Φ (x1 ), such that sup u∈Φ −1 (]−∞,r ])

Ψ (u) ≤ r − Φ (x1 ) + Ψ (x1 ).

(4.2)

Moreover, assume that I satisfies [r ] (PS)-condition. Then, there is u1 ∈ Φ −1 (]r , +∞[) such that I (u1 ) ≤ I (u) for all u ∈ Φ −1 (]r , +∞[) and I ′ (u1 ) = 0. Proof. Put

Φ r ( u) =



Φ ( u) r

if Φ (u) > r if Φ (u) ≤ r ,

J = Φr − Ψ . Clearly, J is locally Lipschitz and bounded from below. Hence, arguing as in the proof of Theorem 3.1, the conclusion is obtained.  Remark 4.2. We explicitly observe that if I satisfies (PS)-condition and (4.2) holds then the conclusion of Theorem 4.2 is true. We explicitly observe that when I is bounded from below and satisfies (PS)-condition the existence of a critical point (global minimum) is obtained owing to a classical result (see, for instance, [7, Corollary 1.5]). In Theorem 4.2, in addition, a localization of the critical point (local minimum) is established.

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5. Functions depending on a real parameter In this section, we point out some corollaries of theorems in Sections 2 and 3 when the function I depends on a real parameter, that is, it is of the type Φ − λΨ , with λ > 0. To this end, given Φ , Ψ : X → R, put sup

β(r1 , r2 ) =

u∈Φ −1 (]r1 ,r2 [)

inf

Ψ (u) − Ψ (v) (5.1)

r2 − Φ (v)

v∈Φ −1 (]r1 ,r2 [)

for all r1 , r2 ∈ R, with r1 < r2 , sup

ϕ(r ) =

u∈Φ −1 (]−∞,r [)

inf

Ψ (u) − Ψ (v) (5.2)

r − Φ (v)

v∈Φ −1 (]−∞,r [)

for all r ∈ R,

Ψ (v) −

ρ(r ) =

sup v∈Φ −1 (]r ,∞[)

sup u∈Φ −1 (]−∞,r ])

Ψ (u) (5.3)

Φ (v) − r

for all r ∈ R,

Ψ (v) −

ρ2 (r1 , r2 ) =

sup

sup u∈Φ −1 (]−∞,r1 ])

Ψ (u) (5.4)

Φ (v) − r1

v∈Φ −1 (]r1 ,r2 [)

for all r1 , r2 ∈ R, with r1 < r2 ,

ϕ2 (r1 , r2 ) =

inf

sup

v∈Φ −1 (]−∞,r1 ]) u∈Φ −1 (]r1 ,r2 [)

Ψ (u) − Ψ (v) Φ (u) − Φ (v)

(5.5)

for all r1 , r2 ∈ R, with r1 < r2 . Clearly, if r1 < infX Φ then β(r1 , r2 ) = ϕ(r2 ), and ρ(r1 ) ≥ ρ2 (r1 , r2 ) for all r2 > r1 . The next result is a consequence of Theorem 3.1. Theorem 5.1. Let X be a real Banach space and let Φ , Ψ : X → R be two continuously Gâteaux differentiable functions. Assume that there are r1 , r2 ∈ R, with r1 < r2 , such that

β(r1 , r2 ) < ρ2 (r1 , r2 ), where β and ρ2 are given by (5.1) and (5.4), and for each λ ∈



1 ρ2 (r1 ,r2 )

 , β(r11,r2 ) the function Iλ = Φ − λΨ satisfies [r1 ] (PS)[r2 ] -

condition.   Then, for each λ ∈ ρ (r1 ,r ) , β(r 1,r ) there is u0,λ ∈ Φ −1 (]r1 , r2 [) such that Iλ (u0,λ ) ≤ Iλ (u) for all u ∈ Φ −1 (]r1 , r2 [) and 2 1 2 1 2 Iλ′ (u0,λ ) = 0. Proof. Fix λ as in the conclusion. One has β(r1 , r2 ) < λ1 < ρ2 (r1 , r2 ), that is there is v0 ∈ Φ −1 (]r1 , r2 [) such that sup

u∈Φ −1 (]r1 ,r2 [)

Ψ (u)−Ψ (v0 )

Ψ (v 0 )−sup

−1

Ψ (u)

u∈Φ (]−∞,r1 ]) < λ1 and there is v 0 ∈ Φ −1 (]r1 , r2 [) such that λ1 < . Therefore, calling Φ (v 0 )−r1 x0 the point of Φ −1 (]r1 , r2 [) such that Φ (x0 ) − λΨ (x0 ) = min{Φ (v0 ) − λΨ (v0 ), Φ (v 0 ) − λΨ (v 0 )}, one has

r2 −Φ (v0 )

sup u∈Φ −1 (]r1 ,r2 [)

λΨ (u) < r2 − Φ (x0 ) + λΨ (x0 )

and supu∈Φ −1 (]−∞,r1 ]) λΨ (u) ≤ r1 − Φ (x0 ) + λΨ (x0 ). Hence, applying Theorem 3.1 to the function Φ − λΨ the conclusion is obtained.  Remark 5.1. If we assume there exist r1 , r2 ∈ R, x0 ∈ X and l ∈ [0, 1], with r1 < Φ (x0 ) < lr2 + (1 − l)r1 , such that 1. supu∈Φ −1 (]r1 ,r2 [) Ψ (u) < lΨ (x0 ) Φ (2x )−1 r 0 1 r −r 2. supu∈Φ −1 (]−∞,r1 ]) Ψ (u) < (1 − l)Ψ (x0 ) r −2 Φ (1x ) 2 0 3. for each λ > 0 the function Φ − λΨ satisfies [r1 ] (PS)[r2 ] -condition r −r

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G. Bonanno / Nonlinear Analysis 75 (2012) 2992–3007

then there is Λ ⊆]0, +∞[ such that

Φ (x0 ) − r1 r2 − Φ (x0 ) ∈ Λ, Ψ (x0 ) [lr2 + (1 − l)r1 ] − Φ (x0 ) and for all λ ∈ Λ the conclusion of Theorem 5.1 holds. In fact, from 1. and 2. one has



  Ψ (u) (Φ (x0 ) − r1 ) +

sup u∈Φ −1 (]r1 ,r2 [)

< Ψ (x0 )(r2 − r1 ); −





u∈Φ −1 (]r1 ,r2 [)

sup u∈Φ −1 (]−∞,r1 ])



sup u∈Φ −1 (]r1 ,r2 [)

u∈Φ −1 (]−∞,r1 ])

 Ψ (u) (r2 − Φ (x0 ))

  Ψ (u) Φ (x0 ) −

sup u∈Φ −1 (]r1 ,r2 [)

  Ψ (u) r1 +

sup u∈Φ −1 (]−∞,r1 ])

 Ψ (u) r2

 Ψ (u) Φ (x0 ) < Ψ (x0 )r2 − Ψ (x0 )r1 + Ψ (x0 )Φ (x0 ) − Ψ (x0 )Φ (x0 );

 Ψ (u) − Ψ (x0 ) (Φ (x0 ) − r1 )



< Ψ (x0 ) −

sup

sup

sup u∈Φ −1 (]−∞,r1 ])

 Ψ (u) (r2 − Φ (x0 )); β(r1 , r2 ) < ρ2 (r1 , r2 ).

Moreover, again from 2. one has 1

ρ2 (r1 , r2 )

≤

Φ (x0 ) − r1 Ψ (x0 ) − sup

u∈Φ −1 (]−∞,r1 ])

=

Ψ (u)

Φ (x0 ) − r1

<





Ψ (x0 ) − (1 − l)Ψ ( )

r −r x0 r −2 Φ (1x ) 2 0

Φ (x0 ) − r1 r2 − Φ (x0 ) Ψ (x0 ) [lr2 + (1 − l)r1 ] − Φ (x0 )

and, again from 1. one has 1

β(r1 , r2 )

≥

r2 − Φ (x0 ) sup u∈Φ −1 (]r1 ,r2 [)

=

Ψ (u) − Ψ (x0 )

r2 − Φ (x0 )

> 

 lΨ ( )

r −r x0 Φ (2x )−1 r 0 1

− Ψ (x0 )

Φ (x0 ) − r1 r 2 − Φ ( x0 ) . Ψ (x0 ) [lr2 + (1 − l)r1 ] − Φ (x0 )

Hence, in particular, if we assume there exist r1 , r2 ∈ R, with 0 < r1 < r2 , and x0 ∈ X , with r1 < Φ (x0 ) < (r2 + r1 )/2, such that 1′ . supu∈Φ −1 (]−∞,r2 [) Ψ (u) <

Ψ (x0 )

r2 −r1 2 Φ (x0 )−r1 Ψ (x0 ) r2 −r1 2 r2 −Φ (x0 )

2′ . supu∈Φ −1 (]−∞,r1 ]) Ψ (u) < 3′ . for each λ > 0 the function Φ − λΨ satisfies (PS)[r2 ] -condition then, setting

λ∗ := 2

(Φ (x0 ) − r1 )(r2 − Φ (x0 )) , Ψ (x0 )[(r2 + r1 ) − 2Φ (x0 )]

Theorem 5.1 ensures that there is ε > 0 such that for each λ ∈]λ∗ − ε, λ∗ + ε[ the function Φ − λΨ admits a critical point which lies in Φ −1 (]r1 , r2 [). Now, we point out the following consequence of Theorem 5.1. Corollary 5.1. Let X be a real Banach space and let Φ , Ψ : X → R be two continuously Gâteaux differentiable functions. Put sup

β ∗ := lim inf r →+∞

u∈Φ −1 (]−∞,r [)

Ψ ( u)

r

and assume that there is r ∈ R such that

β ∗ < ρ(r ),

(5.6)

G. Bonanno / Nonlinear Analysis 75 (2012) 2992–3007

2999

where ρ is given be (5.3). Moreover, assume that for each λ ∈] ρ(1r ) , β1∗ [ the function Φ − λΨ satisfies [r ] (PS)[r ] -condition for all r > r. Then, there is r2 > r such that for each λ ∈] ρ(1r ) , β1∗ [ there is u0,λ ∈ Φ −1 (]r , r2 [) such that Iλ (u0,λ ) ≤ Iλ (u) for all u ∈ Φ −1 (]r , r2 [) and Iλ′ (u0,λ ) = 0.

Proof. Fix λ ∈] ρ(1r ) , β1∗ [. One has lim infr →+∞ has sup lim inf

u∈Φ −1 (]r ,r [)

r →+∞

Ψ (u)

u∈Φ −1 (]−∞,r [)

u∈Φ −1 (]−∞,r [)

r →+∞

Ψ (u)

r

sup

≤ lim inf

r − Φ (x0 )

sup

<

Ψ ( u)

r − Φ (x0 )

= β∗ <

1

λ

1

λ

and ρ(r ) > λ1 . Fixed x0 ∈ X such that Φ (x0 ) > r, one

.

Hence, there is r ∗ such that for all r > r ∗ one has sup u∈Φ −1 (]r ,r [)

Ψ ( u)

<

r − Φ (x0 )

1

λ

.

(5.7)

On the other hand, since limr →+∞ ρ(r , r ) = ρ(r ) > λ1 there is r ∗∗ ∈ R such that

ρ(r , r ) >

1

(5.8)

λ

for all r > r ∗∗ . So picking r2 > max{r ∗ , r ∗∗ , Φ (x0 )}, from (5.7) and (5.8) one has

β(r , r2 ) <

1

λ

< ρ(r , r2 ).

Hence, from Theorem 5.1 the conclusion follows.



Remark 5.2. If we assume that β ∗ := lim infr →+∞

sup

sup u∈Φ −1 (]−∞,r ])

u∈Φ −1 (]−∞,r [)

Ψ (u)

r

= 0 and there are r > 0 and v ∈ X such that

Ψ (u) < r − Φ (v) + Ψ (v)

then one has ρ(r ) > 0 = β ∗ . Therefore, in this case, for each λ ∈] ρ(1r ) , +∞[ Corollary 5.1 ensures the existence of a critical point for Iλ which lies in Φ −1 (]r , r2 [) for some r2 > r. The next result is a consequence of Theorem 4.1. Theorem 5.2. Let X be a real Banach space and let Φ , Ψ : X → R be twocontinuously Gâteaux differentiable functions with Φ bounded from below. Fix r > infX Φ and assume that for each λ ∈ 0, ϕ(1r ) , where ϕ is given by (5.2), the function Iλ = Φ − λΨ satisfies (PS)[r ] -condition.  Then, for each λ ∈

Iλ′ (u0,λ ) = 0.



0, ϕ(1r ) there is u0,λ ∈ Φ −1 (] − ∞, r [) such that Iλ (u0,λ ) ≤ Iλ (u) for all u ∈ Φ −1 (] − ∞, r [) and

Proof. Fix λ as in the conclusion. One has ϕ(r ) < λ1 , so there is x0 ∈ Φ −1 (] − ∞, r [) such that It follows sup u∈Φ −1 (]−∞,r [)

sup

u∈Φ −1 (]−∞,r [)

Ψ (u)−Ψ (x0 )

r −Φ (x0 )

< λ1 .

λΨ (u) < r − Φ (x0 ) + λΨ (x0 ).

Hence, Theorem 4.1 ensures the conclusion.



Now, we point out the following consequence of Theorem 5.2. Corollary 5.2. Let X be a real Banach space and let Φ , Ψ : X → R be two continuously Gâteaux differentiable functions with Φ bounded from below. Assume that are r1 , r2 ∈ R with r1 < r2 , such that

ϕ(r2 ) < ϕ2 (r1 , r2 ), where ϕ and ϕ2 are given by (5.2) and (5.5), and for each λ ∈



1

ϕ2 (r1 ,r2 )

 , ϕ(1r2 ) the function Iλ = Φ − λΨ satisfies (PS)[r2 ] -

condition.   Then, for each λ ∈ ϕ (r1 ,r ) , ϕ(1r ) there is u0,λ ∈ Φ −1 (]r1 , r2 [) such that Iλ (u0,λ ) ≤ Iλ (u) for all u ∈ Φ −1 (]r1 , r2 [) and 2 1 2 2 Iλ′ (u0,λ ) = 0.

3000

G. Bonanno / Nonlinear Analysis 75 (2012) 2992–3007



Proof. Fix λ ∈



1

ϕ2 (r1 ,r2 )

, ϕ(1r2 ) . Since λ <

1

ϕ(r2 )

owing to Theorem 5.2 there is u0 ∈ Φ −1 (] − ∞, r2 [) such that

Φ (u0 ) − λΨ (u0 ) ≤ Φ (u) − λΨ (u)

(5.9)

for all u ∈ Φ (] − ∞, r2 [). We claim that u0 ∈ Φ −1 (]r1 , r2 [). Arguing by a contradiction and assuming Φ (u0 ) ≤ r1 , from (5.9) one has −1

Ψ ( u) − Ψ ( u0 ) 1 ≤ Φ (u) − Φ (u0 ) λ for all u ∈ Φ −1 (]r1 , r2 [). Hence, ϕ2 (r1 , r2 ) ≤ λ1 and this is absurd.



Remark 5.3. Assume Φ (0) = Ψ (0) = 0 and Φ (v) ≥ 0 for all v ∈ X . If there are r1 , r2 ∈ R, u ∈ X , with 0 < nr1 < Φ (u) < r2 (for some n ∈ N), such that sup

(A) sup

(B)

u∈Φ −1 (]−∞,r1 ])

Ψ (u)

r1

Ψ (u) u∈Φ −1 (]−∞,r2 [) r2

≤

n Ψ (u) n+1 Φ (u)

<

n Ψ (u) n+1 Φ (u)

then ϕ(r2 ) < ϕ2 (r1 , r2 ). Moreover, one has





 n + 1 Φ (u) 

n

Ψ ( u)



r2

,

sup u∈Φ −1 (]−∞,r2 [)

Ψ (u)

 ⊆

1

,



1

ϕ2 (r1 , r2 ) ϕ(r2 )

.

In fact, sup

ϕ(r2 ) =

u∈Φ −1 (]−∞,r2 [)

inf

Ψ (u) − Ψ (v)

r2 − Φ (v)

v∈Φ −1 (]−∞,r2 [)

sup

≤

u∈Φ −1 (]−∞,r2 [)

Ψ (u)

r2

<

n

Ψ ( u)

n + 1 Φ ( u)

and, taking into account also that from (A) one has supu∈Φ −1 (]−∞,r1 ]) Ψ (u) ≤ Ψ (u), it follows

ϕ2 (r1 , r2 ) =

inf

sup

v∈Φ −1 (]−∞,r1 ]) u∈Φ −1 (]r1 ,r2 [)

≥

inf

v∈Φ −1 (]−∞,r1 ])

≥

Ψ (u) − Φ ( u)

Ψ (u) − Ψ (v) Ψ (u) − Ψ (v) ≥ inf Φ (u) − Φ (v) v∈Φ −1 (]−∞,r1 ]) Φ (u) − Φ (v)

Ψ (u) − Ψ (v) ≥ Φ (u) sup Ψ (v)

v∈Φ −1 (]−∞,r1 ])

≥

nr1

Ψ ( u) −

sup v∈Φ −1 (]−∞,r1 ])

Ψ (v)

Φ (u) Ψ ( u) 1 Ψ ( u) n Ψ ( u) − = . Φ (u) n + 1 Φ ( u) n + 1 Φ (u)

Ψ (u)

n Hence, ϕ(r2 ) < n+ ≤ ϕ2 (r1 , r2 ). 1 Φ (u)

The next result is a consequence of Theorem 4.2. Theorem 5.3. Let X be a real Banach space and let Φ , Ψ : X → R be two continuously Gâteaux differentiable functions. Fix infX Φ < r < supX Φ and assume that

ρ(r ) > 0, where ρ is given by (5.3), and for each λ > ρ(1r ) the function Iλ = Φ − λΨ is bounded from below and satisfies [r ] (PS)-condition. Then, for each λ > ρ(1r ) there is u0,λ ∈ Φ −1 (]r , +∞[) such that Iλ (u0,λ ) ≤ Iλ (u) for all u ∈ Φ −1 (]r , +∞[) and Iλ′ (u0,λ ) = 0. Proof. Fix λ as in the conclusion. One has ρ(r ) > λ1 , so there is x0 ∈ Φ −1 (]r , +∞[)

Ψ (x0 ) −

sup u∈Φ −1 (]−∞,r ])

Φ (x0 ) − r

Ψ ( u)

>

1

λ

.

It follows sup u∈Φ −1 (]−∞,r ])

λΨ (u) < r − Φ (x0 ) + λΨ (x0 ).

Hence, Theorem 4.2 ensures the conclusion.



G. Bonanno / Nonlinear Analysis 75 (2012) 2992–3007

3001

Remark 5.4. We explicitly observe that in Corollary 5.1 (see Remark 5.2), on the contrary to the previous theorem, lim infr →+∞

sup

u∈Φ −1 (]−∞,r [)

Ψ (u)

r

= 0 is requested, while there the function I is not assumed bounded from below.

Now, taking into account Propositions 2.1 and 2.2 we point out the following immediate consequences of Theorems 5.1– 5.3 when the functions Φ , Ψ are regular enough. From Theorem 5.2 and Proposition 2.1 one has the following result. Theorem 5.4. Let X be a reflexive real Banach space; let Φ : X → R be a sequentially weakly lower semicontinuous, coercive and continuously Gâteaux differentiable function whose Gâteaux derivative admits a continuous inverse on X ∗ ; let Ψ : X → R be a continuously Gâteaux differentiable function  whose  Gâteaux derivative is compact. Then, fixed r > infX Φ , for each λ ∈

u∈Φ

−1

(] − ∞, r [) and

Iλ′ (u0,λ )

0, ϕ(1r )

there is u0,λ ∈ Φ −1 (] − ∞, r [) such that Iλ (u0,λ ) ≤ Iλ (u) for all

= 0.

From Theorem 5.3 and Proposition 2.2 one has the following result. Theorem 5.5. Let X be a real Banach space; let Φ : X → R be a continuously Gâteaux differentiable function whose Gâteaux derivative admits a continuous inverse on X ∗ ; let Ψ : X → R be a continuously Gâteaux differentiable function whose Gâteaux derivative is compact. Assume that the function Φ − Ψ is coercive and that there is r ∈ R, with infX Φ < r < supX Φ , such that ρ(r ) > 0. Then, for each λ > ρ(1r ) there is u0,λ ∈ Φ −1 (]r , +∞[) such that Iλ (u0,λ ) ≤ Iλ (u) for all u ∈ Φ −1 (]r , +∞[) and Iλ′ (u0,λ ) = 0. From Corollary 5.1 (see Remark 5.2) and Proposition 2.1 one has the following result. Theorem 5.6. Let X be a real Banach space; let Φ : X → R be a continuously Gâteaux differentiable function whose Gâteaux derivative admits a continuous inverse on X ∗ ; let Ψ : X → R be a continuously Gâteaux differentiable function whose Gâteaux derivative is compact. Assume that sup lim inf

u∈Φ −1 (]−∞,r [)

Ψ (u)

r

r →+∞

=0

and there is r ∈ R, with infX Φ < r < supX Φ , such that ρ(r ) > 0. Then, for each λ > ρ(1r ) there is u0,λ ∈ Φ −1 (]r , +∞[) such that Iλ (u0,λ ) ≤ Iλ (u) for all u ∈ Φ −1 (]r , +∞[) and Iλ′ (u0,λ ) = 0. Remark 5.5. Theorem 5.4 is a version of [13, Theorem 2.5] (see also [6, Theorem 3.1]), while Theorem 5.5 is a version of [14, Theorem 2.1]. 6. On a mountain pass theorem In this section, we point out a version of a mountain pass theorem where a localization of the third critical point is established. First we recall a result of Marano and Livrea which is a non-smooth version of the classical Mountain Pass Theorem as given by Pucci and Serrin. Taking the proofs of [15, Theorem 4.2] and of [15, Lemma 2.2] into account, [15, Theorem 4.2] can be stated as follows. Theorem 6.1 (See [15, Theorem 4.2]). Let X be a real Banach space, I : X → R a locally Lipschitz function and x0 , x1 ∈ X two local minima of I. Put c := inf max I (γ (t )), γ ∈Γ t ∈[0,1]

where Γ = {γ ∈ C ([0, 1], X ) : γ (x0 ) = 0, γ (x1 ) = 1}, and assume that I satisfies the (PS)c -condition as given by Chang. Then, I admits at least a third critical point distinct from x0 and x1 such that I (x3 ) = c. From the preceding result and Proposition 2.3 we obtain the following version of the mountain pass theorem. Theorem 6.2. Let X be a real Banach space and let Φ , Ψ : X → R be two continuously Gâteaux differentiable functions with Φ convex. Put I = Φ − Ψ and assume that x0 , x1 ∈ X are two local minima of I. Put m := mint ∈[0,1] Ψ (tx1 + (1 − t )x0 ) and assume that there are r > max{Φ (x0 ), Φ (x1 )} and s ≥ 0 such that sup

Φ (x)
Ψ (x) < s + m

and I satisfies (PS)[r +s] -condition. Then I admits at least a third critical point x3 such that

Φ (x3 ) < r + s.

(C)

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G. Bonanno / Nonlinear Analysis 75 (2012) 2992–3007

Proof. Since I satisfies (PS)[r +s] -condition and (C) holds, Proposition 2.3 ensures that the function Φ − Ψ(s+m) , for all d < r − m, satisfies the (PS)d -condition as given by Chang. Moreover, one has c = infγ ∈Γ maxt ∈[0,1] (Φ (γ (t )) − Ψ(s+m) (γ (t ))) ≤ maxt ∈[0,1] (Φ (tx1 + (1 − t )x0 ) − Ψ(s+m) (tx1 + (1 − t )x0 )) = Φ (t ∗ x1 + (1 − t ∗ )x0 ) − Ψ(s+m) (t ∗ x1 + (1 − t ∗ )x0 ) ≤ t ∗ Φ (x1 ) + (1 − t ∗ )Φ (x0 ) − Ψ(s+m) (t ∗ x1 + (1 − t ∗ )x0 ) < r − Ψ(s+m) (t ∗ x1 + (1 − t ∗ )x0 ). Now, either Ψ(s+m) (t ∗ x1 + (1 − t ∗ )x0 ) = s + m ≥ m, or Ψ(s+m) (t ∗ x1 + (1 − t ∗ )x0 ) = Ψ (t ∗ x1 + (1 − t ∗ )x0 ) ≥ m. Hence, in both cases, one has c < r − m.

(1)

Therefore, the function Φ − Ψ(s+m) satisfies the (PS)c -condition as given by Chang. Owing to Theorem 6.1 the function Φ − Ψ(s+m) admits a (generalized) critical point x3 such that c = Φ (x3 ) − Ψ(s+m) (x3 ). From (1) one has

Φ (x3 ) − Ψ(s+m) (x3 ) < r − m.

(2)

We claim that Ψ (x3 ) < s + m. Arguing by a contradiction, we assume that Ψ (x3 ) ≥ s + m. From (2) one has Φ (x3 )−(s + m) < r − m, Φ (x3 ) < s + r, for which, owing to (C) one has Ψ (x3 ) < s + m and this is a contradiction, so our claim is proved. Hence, x3 is a classical critical point of Φ − Ψ and from (2) one has Φ (x3 ) − Ψ (x3 ) < r − m, Φ (x3 ) < Ψ (x3 ) + r − m ≤ (s + m) + (r − m) = s + r, that is Φ (x3 ) < s + r and the proof is completed.  Remark 6.1. If m ≥ 0, condition sup

Φ (x)
Ψ (x) < s

(C1 )

implies condition (C). On the other hand, if we assume sup Ψ (x)

lim inf

Φ (x)
=p<1

q

q→+∞

(C2 )

condition (C) again holds. In fact, from (C2 ) there is a sequence qn → +∞ such that limn→+∞

supΦ (x)
sup

lim

Φ (x)
= p and, setting sn = qn − r, one has

Ψ (x)

sn + m

n→+∞

sup

= lim

Φ (x)
n→+∞

Ψ (x)

sn + r

sn + r sn + m

= p.

7. Multiple critical points theorems In this section we present some multiple critical points results that are consequences of the theorems in Section 5. We give three different versions of three critical points theorems and an infinitely many critical points theorem. Theorem 7.1. Let X be a real Banach space and let Φ , Ψ : X → R be two continuously Gâteaux differentiable functions with Φ bounded from below. Assume that there is r ∈] infX Φ , supX Φ [ such that

ϕ(r ) < ρ(r ), where ϕ and ρ are given by (5.2) and (5.3), and for each λ ∈ satisfies (PS)-condition. 



1

ρ(r )

 , ϕ(1r ) the function Iλ = Φ − λΨ is bounded from below and

 , ϕ(1r ) the function Iλ admits at least three critical points.   Proof. Fix λ ∈ ρ(1r ) , ϕ(1r ) . Since λ < ϕ(1r ) , from Theorem 5.2 there is a local minimum u1 ∈ X for Iλ such that Φ (u1 ) < r. Then, for each λ ∈

1 ρ(r )

Since λ > ρ(1r ) , from Theorem 5.3 there is a local minimum u2 ∈ X for Iλ such that Φ (u2 ) > r. Hence, Iλ admits two distinct local minima and, taking into account that it satisfies the (PS)-condition, the classical mountain pass theorem as given by Pucci and Serrin [16, Corollary 1] ensures the third critical point.  Remark 7.1. If we assume that Φ (0) = Ψ (0) = 0 and there are r > 0 and u ∈ X , with Φ (u) > r, such that sup u∈Φ −1 (]−∞,r ])

r

Ψ (u)

<

Ψ (u) Φ ( u)

then one has ϕ(r ) < ρ(r ) and, in addition,



Φ (u) r , Ψ (u) supu∈Φ −1 (]−∞,r ]) Ψ (u)

 ⊆



1 ρ(r )

 , ϕ(1r ) .

G. Bonanno / Nonlinear Analysis 75 (2012) 2992–3007

3003

In fact, sup

ϕ(r ) =

u∈Φ −1 (]−∞,r [)

inf

Ψ (u) − Ψ (v)

r − Φ (v)

v∈Φ −1 (]−∞,r [)

sup

≤

u∈Φ −1 (]−∞,r [)

Ψ (u)

sup

≤

r

u∈Φ −1 (]−∞,r ])

Ψ (u)

r

and

Ψ (v) −

ρ(r ) =

sup sup

u∈Φ −1 (]−∞,r ])

Ψ (u)

r

<

Ψ (u) Φ (u)

Ψ ( u) −

Ψ (u)

≥

Φ (v) − r

v∈Φ −1 (]r ,∞[)

Hence, ϕ(r ) ≤

sup u∈Φ −1 (]−∞,r ])

sup u∈Φ −1 (]−∞,r ])

Ψ (u)

≥

Φ ( u) − r

Ψ (u) − r ΨΦ ((uu)) Φ ( u) − r

=

Ψ (u) . Φ ( u)

≤ ρ(r ).

Remark 7.2. Taking Remark 7.1 into account, Theorem 1.1 in the introduction is an immediate consequence of Theorem 7.1. In fact, from assumptions of Theorem 1.1 one has holds.

sup

u∈Φ −1 (]−∞,r ])

r

Ψ (u)

<1<

Ψ (u) Φ (u)

and the conclusion of Theorem 7.1 for λ = 1

Theorem 7.2. Let X be a real Banach space and let Φ , Ψ : X → R be two continuously Gâteaux differentiable functions with Φ bounded from below and convex. Assume that there is r ∗ ∈] infX Φ , supX Φ [ such that

ϕ(r ∗ ) < ρ(r ∗ ) and

β ∗ < ρ(r ∗ ), ∗ where ϕ,  ρ and β are given by  (5.2), (5.3) and (5.6). Assume also that for each

λ∈

, min{ ϕ(1r ∗ ) , β1∗ } the function Iλ = Φ − λΨ satisfies (PS)[r ] -condition for all r > r ∗ .   Then, for each λ ∈ ρ(1r ∗ ) , min{ ϕ(1r ∗ ) , β1∗ } the function Iλ admits at least three critical points.   Proof. Fix λ ∈ ρ(1r ∗ ) , min{ ϕ(1r ∗ ) , β1∗ } . Since λ < ϕ(1r ∗ ) , from Theorem 5.2 there is a local minimum u1 ∈ X for Iλ such 1

ρ(r ∗ )

that Φ (u1 ) < r ∗ . Since ρ(1r ∗ ) < λ < β1∗ , from Corollary 5.1 there are r ′ > r ∗ and a local minimum u2 ∈ X for Iλ such that r ∗ < Φ (u2 ) < r ′ . Hence, Iλ admits two distinct local minima which lie in Φ −1 (] − ∞, r ′ [). Now, since u3 ∈ Φ

−1

supΦ (x)
= λβ ∗ < 1, Theorem 6.2, taking (C2 ) in Remark 6.1 into account, ensures a third critical point (] − ∞, r + s[) for some s > 0. 

lim infq→+∞

′

Remark 7.3. If we assume that Φ (0) = Ψ (0) = 0, there are r > 0 and u ∈ X , with Φ (u) > r, such that sup u∈Φ −1 (]−∞,r ])

Ψ (u)

<

r

Ψ (u) Φ (u)

and sup lim inf

u∈Φ −1 (]−∞,r ])

Ψ (u)

=0

r

r →+∞

then one has ϕ(r ) < ρ(r ) and, in addition,



Φ (u) r , Ψ (u) supu∈Φ −1 (]−∞,r ]) Ψ (u)

 ⊆



1

ρ(r )

 , min{ ϕ(1r ) , β1∗ } .

It is enough to argue as in Remark 7.1 taking also into account that ρ(r ) lim infr →+∞

Ψ (u) u∈Φ −1 (]−∞,r ]) . r

sup

>

ϕ(r )

≥

0

=

β∗

=

Theorem 7.3. Let X be a real Banach space and let Φ , Ψ : X → R be two continuously Gâteaux differentiable functions with Φ bounded from below and convex. Assume that there are r1 , r2 > infX Φ , with r1 < r2 , and s > 0 such that

ϕ(r1 ) < ϕ2 (r1 , r2 ) ϕ(r2 ) < ϕ2 (r1 , r2 ) sup Ψ (x) x∈Φ −1 (]−∞,r2 +s[)

s

< ϕ2 (r1 , r2 ),

where ϕ and ϕ2 are given by (5.2) and (5.5).

3004

G. Bonanno / Nonlinear Analysis 75 (2012) 2992–3007



Assume also that for each λ ∈ Λ =

(PS)

[r2 +s]

1 ϕ2 (r1 ,r2 )

, min{ ϕ(1r1 ) , ϕ(1r2 ) , sup

s

x∈Φ −1 (]−∞,r2 +s[)

 } the function Iλ = Φ − λΨ satisfies Ψ (x)

-condition and

min Ψ (tx1 + (1 − t )x0 ) ≥ 0

t ∈[0,1]

for all x0 , x1 ∈ X which are two local minima of Iλ and such that Ψ (x0 ) ≥ 0 and Ψ (x1 ) ≥ 0. Then, for each λ ∈ Λ the function Iλ admits at least three critical points which lie in Φ −1 (] − ∞, r2 + s[). Proof. Fix λ ∈ Λ. Since λ < ϕ(1r ) , from Theorem 5.2 the function Iλ admits a local minimum u1 such that Φ (u1 ) < r1 . 1 Further, applying Corollary 5.2, the function Iλ admits a local minimum u2 such that r1 < Φ (u2 ) < r2 . Hence, Theorem 6.2, taking also into account (C1 ) of Remark 6.1, ensures a third critical point u3 such that Φ (u3 ) < r2 + s.  Remark 7.4. If we assume that infX Φ = Φ (0) = Ψ (0) = 0, there are ρ1 , ρ2 ∈ R and u ∈ X , with 0 < 2ρ1 < Φ (u) < ρ2 /2, such that sup u∈Φ −1 (]−∞,ρ1 [)

ρ1 sup u∈Φ −1 (]−∞,ρ2 [)

Ψ ( u)

< Ψ (u)

<

ρ2

2 Ψ (u) 3 Φ (u) 1 Ψ (u) 3 Φ (u)

,

then the inequalities in Theorem 7.3 hold and, in addition

  3 Φ ( u) 

2 Ψ (u)



ρ1

, min

sup u∈Φ −1 (]−∞,ρ1 [)

Ψ (u)

ρ2 /2

,

sup u∈Φ −1 (]−∞,ρ2 [)

Ψ (u)

    ⊆ Λ.

In fact, setting r1 = ρ1 , r2 = ρ2 /2, s = ρ2 /2 and arguing as in Remark 5.3 we obtain ϕ(r1 ) ≤ 2 3 2 3

Ψ (u) ; ϕ(r2 ) ≤ Φ (u) Ψ (u) ; ϕ2 (r1 , r2 ) Φ (u)

Ψ (u) u∈Φ −1 (]−∞,ρ2 /2[) ρ2 / 2 2 Ψ (u) . 3 Φ (u)

sup

>

sup

≤

Ψ (u) u∈Φ −1 (]−∞,ρ2 [) ρ2 /2

<

2 Ψ (u) 3 Φ (u)

sup

;

Ψ (u) u∈Φ −1 (]−∞,r2 +s[) s

sup

Ψ (u)

ρ1

sup

=

u∈Φ −1 (]−∞,ρ1 [)

Ψ (u) u∈Φ −1 (]−∞,ρ2 [) ρ2 /2

< <

Now, put

γ := lim inf ϕ(r ), r →+∞

δ := lim inf ϕ(r ), r →(inf Φ )+ X

where ϕ is given by (5.2). Theorem 7.4. Let X be a real Banach space and let Φ , Ψ : X → R be two continuously Gâteaux differentiable functions with Φ bounded from below.

  (a) If γ < +∞ and for each λ ∈ 0, γ1 the function Iλ = Φ − λΨ satisfies (PS)[r ] -condition for all r ∈ R, then, for each λ ∈]0, γ1 [, the following alternative holds:

either (a1 ) Iλ possesses a global minimum, or (a2 ) there is a sequence {un } of  critical  points (local minima) of Iλ such that limn→+∞ Φ (un ) = +∞. (b) If δ < +∞ and for each λ ∈ 0, 1δ the function Iλ = Φ − λΨ satisfies (PS)[r ] -condition for some r > infX Φ , then, for each

λ ∈]0, 1δ [, the following alternative holds:

either (b1 ) there is a global minimum of Φ which is a local minimum of Iλ , or (b2 ) there is a sequence of pairwise distinct critical points (local minima) of Iλ such that limn→+∞ Φ (un ) = infX Φ . Proof. Starting from Theorem 5.2 and arguing as in the proof of [17, Theorem 2.1] the conclusion is obtained.



Remark 7.5. We explicitly observe that the previous results with respect to [18, Theorem 3.6], [6, Theorem 3.3], [13, Theorem 2.5], [17, Theorem 2.1] do not require the weak semi-continuity of Φ and Ψ . 8. Applications to nonlinear partial differential equations In this section, as an example of application of previous results, we investigate elliptic Dirichlet problems with critical exponent. In fact, in these cases the (PS)-condition as well as the weak continuity of the associated function may fail

G. Bonanno / Nonlinear Analysis 75 (2012) 2992–3007

3005

(see Remark 8.2). Precisely, consider the following problem



−1u = λf (u) u|∂ Ω = 0,

in Ω ,

(Dfλ )

where Ω is a non-empty bounded open subset of the Euclidean space (RN , | · |), N ≥ 3, with boundary of class C 1 , λ is a positive parameter and f : R → R is a continuous function such that

(h) there exist two non-negative constants a1 , a2 and q ∈]1, 2N /(N − 2)] such that |f (t )| ≤ a1 + a2 |t |q−1 for every t ∈ R. Clearly, f has a ‘‘critical growth’’ whenever q = 2∗ = N2N . −2 Now, as an example of application of theorems established in previous sections, we point out the following result. Theorem 8.1. Let f : R → R be a non-negative and non-zero continuous function satisfying (h). Put F (ξ ) = every ξ ∈ R and assume that lim sup

F (t )

t →0+

t2

ξ 0

f (t )dt for

= +∞. f

Then, there is λ∗ > 0 such that, for each λ ∈]0, λ∗ [, the problem (Dλ ) admits at least one positive weak solution. Proof. In order to apply the previous results the usual settings are made. Precisely, put X = H01 (Ω ) endowed with the norm

 1 2 |∇ u(x)|2 dx 2 and Φ (u) = ∥u2∥ , Ψ (u) = Ω F (u(x))dx for all u ∈ X . Now, fixed λ > 0, we claim that Iλ = Φ − λΨ satisfies (PS)[r ] -condition for all r > 0. Indeed, let {un } ⊆ X be a sequence such that Φ (un ) < r ∀n ∈ N, {Φ (un ) − λΨ (un )} is bounded and limn→+∞ ∥Φ ′ (un ) − λΨ ′ (un )∥X ∗ = 0. Since Φ is coercive, from Φ (un ) < r ∀n ∈ N one has that {un } is bounded in X . Therefore, since X is reflexive, there is a subsequence, called again {un }, which is weakly converging to u0 ∈ X . Hence, by the Sobolev embedding theorem, {un } is strongly converging to u0 in L2 (Ω ). By a standard computation from limn→+∞ ∥Φ ′ (un ) − λΨ ′ (un )∥X ∗ = 0 one has Φ ′ (un )(v − un ) − λΨ ′ (un )(v − un ) ≥ −εn ∥v − un ∥ for all v ∈ X , where εn → 0+ . From the above expression, written with v = u0 , we infer Φ′ (un )(u0 − un ) − λΨ ′ (un )(u0 − u into account n ) ≥ −εn ∥u0 − un ∥ for   all n ∈ N. Now, taking that Φ ′ (un )(u0 − un ) = Ω ∇ un (x)∇(u0 (x) − un (x))dx = Ω ∇ un (x)∇ u0 (x)dx − Ω |∇ un (x)|2 dx ≤ 12 Ω |∇ un (x)|2 dx +  1 |∇ u0 (x)|2 dx − ∥un ∥2 = 12 ∥u0 ∥2 − 21 ∥un ∥2 , one has −εn ∥u0 − un ∥ + 12 ∥un ∥2 ≤ −λΨ ′ (un )(u0 − un ) + 12 ∥u0 ∥2 . From this, 2 Ω taking into account the continuity of Ψ ′ in L2 (Ω ) × L2 (Ω ) it actually results lim supn→+∞ ∥un ∥ ≤ ∥u0 ∥. Therefore, since X is uniformly convex [19, Proposition III.30] ensures that {un } is strongly converging to u0 in X and our claim is proved. Hence, in particular, Φ − λΨ satisfies [r1 ] (PS)[r2 ] -condition for all r1 and r2 such that r1 < r2 < +∞ (see Section 2). Our aim is to apply Theorem 5.1. To this end we observe that, by choosing r1 = 0, the following condition ∥ u∥ =



Ω

sup u∈Φ −1 (]−∞,r2 [)

Ψ (u)

<

r2

Ψ (v0 ) , Φ (v0 )

with 0 < Φ (v0 ) < r2 , implies that β(r1 , r2 ) < ρ2 (r1 , r2 ) and



Φ (v0 ) r2 , Ψ (v0 ) supu∈Φ −1 (]−∞,r [) Ψ (u) 2

 ⊆



1

ρ2 (r1 ,r2 )

 , β(r11,r2 ) . In fact, from

the above inequality one has sup

β(0, r2 ) ≤

u∈Φ −1 (]−∞,r2 [)

Ψ (u) − Ψ (v0 )

Ψ (v )

<

r2 − Φ (v0 )

r2 Φ (v0 ) − Ψ (v0 ) 0

r2 − Φ (v0 )

=

Ψ (v0 ) ≤ ρ2 (0, r2 ). Φ (v0 )

Now, since there exists a positive constant cq such that

∥u∥Lq (Ω ) ≤ cq ∥u∥,

u ∈ H01 (Ω ),

(we observe that the embedding H01 (Ω ) ↩→ Lq (Ω ) is not compact if q = 2∗ ), one has

 sup u∈Φ −1 (]−∞,r2 [)

r2

Ψ (u)



sup

≤

Ω

u∈Φ −1 (]−∞,r2 [)

F (u(x))dx

u∈Φ −1 (]−∞,r2 [)

sup

≤

 a1 c1 ∥u∥ + r2

a2 c q q

∥ u∥

q Lq (Ω )

r2

 u∈Φ −1 (]−∞,r2 [)

a1 ∥u∥L1 (Ω ) +

sup

≤

r2

 a2 q

∥ u∥



q

√

q

2r2 c1 a1 +

≤

2q/2 cq a2 q/2 r2 q

r2

 .

3006

G. Bonanno / Nonlinear Analysis 75 (2012) 2992–3007

Hence, by choosing r2 = 1, one has sup u∈Φ −1 (]−∞,r2 [)

Ψ ( u)

 ≤

r2 Put λ∗ = √

1 2c1 a1 +

q 2q/2 cq a2 q

2q/2 cq a2 q

√ 2c1 a1 +

q

 .

 and fix λ < λ∗ .

Next, fix R = sup d(x, ∂ Ω ) x∈Ω

and let x0 ∈ Ω such that B(x0 , R) ⊆ Ω . Moreover, put

vδ (x) :=

if x ∈ Ω \ B(x0 , R)

 0   

2δ

  R δ

(R − |x − x0 |)

if x ∈ B(x0 , R) \ B(x0 , R/2) if x ∈ B(x0 , R/2).

Clearly, one has that vδ ∈ X and Φ (vδ ) = meas(B(x0 , R/2)))



Ω

F (vδ (x)) dx ≥

π N /2 1 (2δ)2 2 R2 Γ (1+N /2)

=



B(x0 ,R/2)

1 2



Ω

|∇vδ (x)|2 dx =

1 2



B(x0 ,R)\B(x0 ,R/2)

(2δ)2 R2

dx =

1 (2δ)2 2 R2

(meas(B(x0 , R)) −

(RN − (R/2)N ), where Γ is the Euler function. Moreover, one has Ψ (vδ ) = N /2

F (δ) dx ≥ F (δ) Γ (π1+N /2) R2N and, hence, N

Ψ (vδ ) R2 F (δ) ≥ . N Φ (vδ ) 2(2 − 1) δ 2 From lim supt →0+

F (t ) t2

F (δ)

R2

2(2N − 1) δ 2

= +∞ and Φ (vδ ) =

1 (2δ)2 π N /2 2 R2 Γ (1+N /2)

(RN − (R/2)N ), there is δ > 0 such that

1

>

λ

and Φ (vδ ) < 1. Therefore,

sup

u∈Φ −1 (]−∞,r2 [)

Ψ (u)

√

2q/2 cq a2 q



<

<

F (δ) R2 2(2N −1) δ 2

Ψ (v )

≤ Φ (vδ ) with δ 0 = r1 < Φ (vδ ) < r2 = 1. Hence, taking into account the observation at the beginning, one has β(r1 , r2 ) < ρ2 (r1 , r2 ). From Theorem 5.1 the functional Φ − λΨ admits a critical point uλ such that 21 ∥uλ ∥2 > 0, which is a positive weak solution for f problem (Dλ ) and the conclusion is achieved.  r2

≤

2c1 a1 +

q

=

1

λ∗

1

λ

Remark 8.1. As it can be seen in the proof of Theorem 8.1, one has

λ∗ = 

1

√

2q/2 cq a2 q

2c1 a1 +

.

q

So, taking into account that



1

c2∗ = √ N (N − 2)π

1/N

N! 2Γ (1 + N /2)

(see, for instance, [20]) and, owing to Hölder’s inequality, meas(Ω )

2∗ −q 2∗ q

cq ≤ √ N (N − 2)π



N! 2Γ (N /2 + 1)

1/N ,

a precise numerical estimate of λ∗ can be made. f We also observe that the weak solution uλ of (Dλ ) ensured in Theorem 8.1 satisfies the further condition

∥ uλ ∥ <

√

2

for all λ ∈]0, λ∗ [.

G. Bonanno / Nonlinear Analysis 75 (2012) 2992–3007

3007

Example 8.1. Consider the problem

    −1u = λ u NN −+22 + u 21 in Ω ,   u|∂ Ω = 0.

(Pλ )

Owing to Theorem 8.1, taking also Remark 8.1 into account, for each positive parameter λ such that 1

λ< 23/2 [meas(Ω )](N +2)/2N

1/N



[N (N +2)π]1/2

N! 2Γ (1+N /2)

+

2N /(N −2) (N −2) N



1/N 2N /(N −2) ,

 1

[N (N +2)π ]1/2

N! 2Γ (1+N /2)

the problem (Pλ ) admits at least one positive weak solution uλ such that

 Ω

|∇ uλ (x)|2 dx < 2. N +2

In fact, it is enough to observe that |f (t )| ≤ 2 + 2|u| N −2 for all t ∈ R and limt →0+

f (t ) t

= +∞.

Remark 8.2. The function f in Theorem 8.1 has a critical growth when q = 2∗ (see, for instance, Example 8.1) for which, in ∗ these cases, the associated function Iλ = Φ − λΨ is not enough regular. In fact, since the embedding H01 (Ω ) ↩→ L2 (Ω ) is not compact, the (PS)-condition as well as the weak continuity of Iλ may fail. Remark 8.3. We recall that the study of critical nonlinearities has been successfully addressed in the very well known work of Brézis and Nirenberg (see [21]). In such a paper the following equation ∗ −2

− 1u = |u|2

u + λu

(8.1)

has been investigated by obtaining, for suitable values of λ, the existence of one positive weak solution to the corresponding Dirichlet problem (see also [22, Chapter 1.10]). Clearly, the Eq. (8.1) and the following



 2 ∗ −2

− 1u = λ |u|

u + | u|

1 2

,

(8.2)

applied in the Example 8.1, are of a completely different type, even for the position of λ. Moreover, it is worth noticing that the solution of (8.1) obtained in [21] is a critical point of type mountain pass, while the solution of (8.2) is a local minimum for the associated functional. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

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