A min-max principle for non-differentiable functions with a weak compactness condition

A general critical point result established by Ghoussoub is extended 
to the case of locally Lipschitz continuous functions satisfying a weak 
Palais-Smale hypothesis, which includes the so-called non-smooth Cerami condition. 
Some special cases are then pointed out.


Introduction.
A trend in today's literature on critical point theory is the attempt to weaken, in a fruitful way, the key assumptions of the famous Ambrosetti-Rabinowitz's Mountain Pass Theorem, namely (a) the Mountain Pass geometry, (b) the Palais-Smale compactness condition, and (c) the regularity of the involved functional. These questions have by now been widely investigated, and excellent monographs are already available. For instance, [10,20] contain meaningful generalizations of (a)-(b), while [17,18,8] mainly deal with (c). The book [12] represents a general reference on the subject.
In this paper we first extend the min-max principle of Ghoussoub [9] to locally Lipschitz continuous functions satisfying a weak Palais-Smale hypothesis, which includes both the usual one [2, Definition 2] and the non-smooth Cerami condition [13, p. 248]; see Theorem 3.1 below. Thus, since (a) is a very special case of the situation treated in [9], assumptions (a)-(c) are weakened here. From a technical point of view, Theorem 3.1 is achieved by adapting Ghoussoub's approach to our non-smooth framework and exploiting Ekeland's Variational Principle with a suitable metric of geodesic type. When the functional is C 1 while the compactness condition is that of Cerami, this idea basically goes back to Ekeland [6, p. 138]. Two simple but meaningful consequences of Theorem 3.1 are then pointed out, i.e., Theorems 3.2 and 3.3. The first of them provides a more refined version of several recent critical point results, as, for instance, Theorem 1.bis and Corollary 2 in [9], Corollary 1 and Theorem 2 of [19] (vide also [18,Section 2.1]), Theorem 6 in [13], besides Corollaries 9 and 10 at p. 145 of [6]. A special mention deserves the paper [14], where a general deformation lemma and a subsequent min-max principle for locally Lipschitz continuous functions are obtained under (a) and a weak Palais-Smale hypothesis similar to but different from the one adopted here. It is also worth noting that non-differentiable variants of structure or multiplicity results, as well as applications to elliptic hemivariational inequalities, might be drawn from Theorem 3.3; see for instance [1].
Let us finally observe that Ghoussoub's result has already been extended by the authors in [15] to the case of functionals which are the sum of a locally Lipschitz continuous term and of a convex, proper, lower semi-continuous function, namely the so-called Motreanu-Panagiotopoulos' non-smooth framework [17]. However, the compactness condition employed there reduces to the standard one [2,Definition 2] in the locally Lipschitz continuous setting and hence it is more restrictive than that taken on below.
2. Basic definitions and auxiliary results. Let (X, · ) be a real Banach space. If U is a nonempty subset of X, x ∈ X, and r > 0, we define B(x, r) := {z ∈ X : z − x < r} as well as We denote by X * the dual space of X, while ·, · stands for the duality pairing between X and X * . A function f : X → IR is called locally Lipschitz continuous when to every x ∈ X there correspond a neighborhood V x of x, besides a constant L x ≥ 0, such that If x, z ∈ X, we write f 0 (x; z) for the generalized directional derivative of f at the point x along the direction z, namely One evidently has f 0 (x; tz) = tf 0 (x; z) for every t ≥ 0. It is known [3, Proposition 2.1.1] that f 0 turns out to be upper semi-continuous on X × X. The generalized gradient of the function f in x, denoted by ∂f (x), is the set Proposition 2.1.2 of [3] ensures that ∂f (x) is nonempty, convex, in addition to weak* compact, and that Hence, it makes sense to write m f (x) := min{ x * X * : x * ∈ ∂f (x)} . We say that x ∈ X is a critical point of f when 0 ∈ ∂f (x), namely f 0 (x; z) ≥ 0 for all z ∈ X. Given a real number c, write In this setting, the classical Palais-Smale compactness hypothesis at the level c ∈ IR for C 1 functions becomes (cf.
We say that f satisfies a weak Palais-Smale condition at the level c ∈ IR when for some h as above one has: possesses a convergent subsequence.
, we obtain a non-smooth version, previously introduced in [13], of the so-called Cerami compactness assumption.

Remark 2.
Under the further request that h be nondecreasing in [0, +∞[, condition (PS) h c is already known. We refer to [22] for C 1 functions and to [16] for a much more general situation. A similar but different hypothesis has recently been exploited in the paper [14].
A weaker form of (PS) h c is the one below, where U denotes a nonempty closed subset of X.
holds true possesses a convergent subsequence.
δ h turns out to be a distance on X and the metric topology derived from δ h coincides with the norm topology. (p 4 ) δ h -bounded and norm-bounded sets in X are the same. Through (p 1 ), (p 2 ), and (p 4 ) one easily verifies that the metric space (X, δ h ) is complete.
Finally, the following version [5, pp. 444, 456] of the famous variational principle of Ekeland will be employed.
3. Existence of critical points. Let B be a nonempty closed subset of X and let F be a class of nonempty compact sets in X. According to [9, Definition 1], we say that F is a homotopy-stable family with extended boundary B when for every Some meaningful situations are special cases of this notion. For instance, if Q denotes a compact set in X, Q 0 is a non-empty closed subset of Q, γ 0 ∈ C 0 (Q 0 , X), Γ := {γ ∈ C 0 (Q, X) : γ| Q0 = γ 0 } , and F := {γ(Q) : γ ∈ Γ}, then F enjoys the above-mentioned property with B := γ 0 (Q 0 ). In particular, it holds true when Q indicates a compact topological manifold in X having a nonempty boundary Q 0 while γ 0 := id| Q0 .
The following assumptions will be posited in the sequel. (a 1 ) f : X → IR is a locally Lipschitz continuous function. (a 2 ) F denotes a homotopy-stable family with extended boundary B. (a 3 ) There exists a nonempty closed subset F of X such that and, moreover, sup Thanks to (5) one has inf Theorem 3.1. Let (a 1 )-(a 4 ) be satisfied. Then to every sequence {A n } ⊆ F such that lim n→+∞ max x∈An f (x) = c there corresponds a sequence {x n } ⊆ X \ B having the following properties: Proof. On account of (6), we first consider the case Pick an ǫ > 0 and choose A ǫ ∈ F fulfilling We shall prove the existence of a point x ǫ ∈ X \ B such that which obviously provides a sequence {x n } ⊆ X \ B enjoying properties (i 1 )-(i 4 ). To shorten notation, write We denote by L the space of all η ∈ C 0 ([0, 1] × X, X) such that A simple computation then shows that L, equipped with the uniform distance ρ(η 1 , η 2 ) := sup (t,x)∈[0,1]×X δ h (η 1 (t, x), η 2 (t, x)) , η 1 , η 2 ∈ L , is complete. Define, for every x ∈ X,

Moreover, set
The function I : L → IR is evidently lower semi-continuous. Gathering (13) and (4) together yields (η(A ′ ǫ ) ∩ F ) \ B = ∅ ∀ η ∈ L . Consequently, on account of (7), Letη(t, x) := x for all (t, x) ∈ [0, 1] × X. Since, by (8) and (14), Theorem 2.1 can be applied. Hence, there exists an η 0 ∈ L such that Define Obviously, the set C turns out nonempty and compact. We claim that for some In fact, letẑ . Ifẑ ∈ B, then the assertion is true with z 0 :=ẑ. Otherwise, exploiting (4), pick any and (19) holds. The point z 0 does not lie in (A ǫ \ F ǫ ) ∪ B. So, f 2 (z 0 ) > 0. Because of (7) this implies Through (8), (5), and (7) we then achieve Let which clearly forces We shall next prove that: There is a point If (25) were false then for each x ∈ C one could find a v x ∈ X such that Without loss of generality, suppose v x = 1. Since the function turns out upper semi-continuous in X × X and less than zero at (x, x), there exists an r x > 0 such that Let r x be small enough to have also f Lipschitz continuous on B(x, r x ), by (24), as well as Put V x := B(x, r x ). The family B := {V x : x ∈ C} represents an open covering of C.
For simplicity of notation, write v j := v xj , r j := r xj , besides V j := V xj . Moreover, define, for every Observe that v(x) ≤ 1 for all x ∈ X because v j = 1, j = 1, 2, . . . , m. Hence, the function u : X → X is continuous, and, on account of (28), u(x) = 0 in B ′ .
Proof. Theorem 3.1 provides a sequence {x n } ⊆ X \ B with properties (i 1 )-(i 4 ). Observe that (i 2 ) actually means In fact, by [21, Lemma 1.3], for any n ∈ IN there exists a z * n ∈ X * such that z * n X * ≤ 1 and n ≤ ǫ n , n ∈ IN . Now, (42) is an immediate consequence of ǫ n → 0 + . Bearing in mind (p 1 )-(p 2 ), for bounded F one has δ h (x n , F ) → 0 if and only if d(x n , F ) → 0. By the weak Palais-Smale condition we may thus assume that x n → x in X, where a subsequence is considered when necessary. At this point, the conclusion comes from (i 1 )-(i 3 ). In fact, (i 2 ) and the upper semi-continuity of f 0 yield f 0 (x; z) ≥ 0 for all z ∈ X, namely x ∈ K c (f ).
Making suitable choices of F , B, and F , more refined versions of several recent results can easily be obtained through Theorem 3.2. By way of example, we find the result below, which includes Theorem (1.bis) in [11], Corollary 1 and Theorem 2 of [19] (vide also [18, Section 2.1]), as well as Theorem 6 in [13]. Keep the same notation introduced at the beginning of this section. Theorem 3.3. Let (a 1 ) and (a 4 ) be satisfied. Suppose that: (a 5 ) There exists a closed subset F of X complying with (γ(Q) ∩ F ) \ γ 0 (Q 0 ) = ∅ for all γ ∈ Γ and, moreover, sup x∈Q0 f (γ 0 (x)) ≤ inf x∈F f (x). (a 6 ) Setting c := inf γ∈Γ max x∈Q f (γ(x)), either (PS) h c holds or F is bounded and (PS) h F,c holds, according to whether inf x∈F f (x) < c or inf x∈F f (x) = c. Then the conclusion of Theorem 3.2 is true. c , that appears in Theorems 3.2 and 3.3, can be replaced by assumption (C) α c .
Remark 5. The above result reveals useful to get non-smooth variants of structure or multiplicity results for C 1 functions where, instead of (PS) c , a weak Palais-Smale condition is requested; see [1].