A VERSION OF ZHONG ’ S COERCIVITY RESULT FOR A GENERAL CLASS OF NONSMOOTH FUNCTIONALS

A version of Zhong’s coercivity result (1997) is established for nonsmooth functionals expressed as a sum Φ + Ψ, where Φ is locally Lipschitz and Ψ is convex, lower semicontinuous, and proper. This is obtained as a consequence of a general result describing the asymptotic behavior of the functions verifying the above structure hypothesis. Our approach relies on a version of Ekeland’s variational principle. In proving our coercivity result we make use of a new general Palais-Smale condition. The relationship with other results is discussed.

For the functional I as in (1.1), it was given in [9] the following definition of Palais-Smale (PS) condition.
Definition 1.1.The functional I : X → R ∪ {+∞} in (1.1) satisfies the PS condition if every sequence (u n ) ⊂ X with I(u n ) bounded and for which there exists a sequence ( n ) ⊂ R + , n → 0 + , such that

A version of Zhong's coercivity result
The notation Φ 0 in (1.2) stands for the generalized directional derivative of the locally Lipschitz functional Φ : X → R introduced by Clarke [5] as follows: In the case where Φ ∈ C 1 (X;R) and Ψ = 0, Definition 1.1 reduces to the usual PS condition.If Φ is locally Lipschitz and Ψ = 0, Definition 1.1 expresses the PS condition in the sense of Chang [4].If Φ ∈ C 1 (X;R) and Ψ is convex, l.s.c., and proper, Definition 1.1 represents the concept of PS condition introduced by Szulkin [10].
A different extension of the usual PS condition is the following one.
If h = 0, Definition 1.2 reduces to the classical PS condition.In the case where h(t) = t, for all t ≥ 0, Definition 1.2 coincides with the PS condition in the sense of Cerami [3].
It is natural to look for a concept of PS condition for functionals of type (1.1) incorporating simultaneously the two definitions above.
Definition 1.3.The functional I : X → R ∪ {+∞} in (1.1) satisfies the PS condition if every sequence (u n ) ⊂ X with I(u n ) bounded and for which there exists a sequence ( n ) ⊂ R + , n → 0 + , such that (1.5) contains a (strongly) convergent subsequence in X.
If h = 0, Definition 1.3 reduces to Definition 1.1.If Φ ∈ C 1 (X;R) and Ψ = 0, Definition 1.3 coincides with Definition 1.2 since relation (1.5) becomes (1.4) in this case.We point out that for h(t) = t, for all t ≥ 0, Definition 1.3 expresses the extension of the PS condition in the sense of Cerami [3] to the class of nonsmooth functionals in (1.1).
A problem that has been extensively studied was the relationship between the PS condition and coercivity.We recall that a functional I : X → R ∪ {+∞} is said to be coercive if the following property holds: (1.6) D. Motreanu et al. 603 The basic assertion in this direction is that, generally, the PS condition implies the coercivity.The first such result is the one of Čaklović et al. [2] who established this property for a functional I : X → R which is l.s.c., Gâteaux differentiable, and satisfying the classical PS condition (see also Brézis and Nirenberg [1] for continuously differentiable functionals).The first result of this type, for nondifferentiable functionals, is due to Goeleven [7] who has shown the coercivity property in the case where the functional I : X → R ∪ {+∞} has the structure (1.1) with Φ l.s.c. and Gâteaux differentiable, and Ψ convex, l.s.c., and proper such that the PS condition in the sense of Szulkin [10] is satisfied.For nonsmooth functionals of the general form (1.1), an analogous result has been obtained in [8] making use of the PS condition stated in Definition 1.1.The corresponding property for nonsmooth functionals, satisfying the PS condition formulated in Definition 1.2, has been given by Zhong [11].The aim of this paper is to prove the coercivity for the nonsmooth functionals verifying (1.1) together with the PS condition given in Definition 1.3.In this paper, the coercivity assertion is obtained as a consequence of Theorem 2.3 below expressing the asymptotic behavior of a nonsmooth functional of type (1.1).Specifically, the coercivity property is derived from Theorem 2.3 by assuming the PS condition as formulated in Definition 1.3.
Theorem 2.3 cannot be deduced from Zhong's corresponding result [11, Theorem 3.7] because, generally, the functionals of type (1.1), which we consider, are not Gâteaux differentiable.However, Theorem 2.3 is not an extension of Theorem 3.7 in Zhong [11] because a Gâteaux differentiable, l.s.c., and proper functional is not necessarily of form (1.1).Our Theorem 2.3 represents the version of Zhong's corresponding result for a nonsmooth functional fulfilling the structure assumption (1.1).
The method of proof for Theorem 2.3 relies, as in the case of [11,Theorem 3.7], on Zhong's variational principle [11,Theorem 2.1] which is an extension of Ekeland's variational principle [6].The proof of Theorem 2.3 takes into account, essentially, the structure of functionals in (1.1).To the end of rigourously proving Theorem 2.3, we slightly extend Zhong's variational principle [11,Theorem 2.1] in Theorem 2.1 below.The main idea is to allow the reference point x 0 to be in a larger space.Precisely, this extension is necessary because in Theorem 2.3 we encounter the situation where x 0 = 0 does not belong to the space M 0 , on which the variational principle must be applied.Our argument corrects a small gap in the proof of [11,Theorem 3.7] concerning the mentioned difficulty.Furthermore, in comparison with Zhong's paper [11], our approach in Theorem 2.3 makes other improvements, among them, the accurate treatment of the passage from the (N − 1)th to the Nth step.Moreover, our hypotheses in Theorem 2.3 and Corollary 2.4 either are slightly weaker (see (2.8)) or give the correct requirement for making the proof (see (2.5)).
The rest of the paper is organized as follows.Section 2 is devoted to the statements of the results.Section 3 contains the proof of the main result.

Main results
We start with a slight extension of Zhong's variational principle in [11].
Let M be a closed subset of a complete metric space M endowed with the metric d, let a point x 0 ∈ M, and let f : M → R ∪ {+∞} be an l.s.c., proper function which is bounded from below.Then, for all > 0, v ∈ M with and λ > 0, there exists a point w ,λ ∈ M such that where r 0 = d(x 0 ,v) and r verifies Proof.We endow M with the metric induced by the one on M, so M becomes a complete metric space.A careful examination of the proof of [11,Theorem 2.1] shows that the argument therein can be carried out with any point x 0 in M. Following the same lines as in the proof of [11, Theorem 2.1] (which goes back to Ekeland [6]), we achieve the stated conclusion.
Our main result is the following theorem.
Theorem 2.3.Let X be a Banach space and let a functional I : X → R ∪ {+∞} be of type (1.1), that is, The proof will be done in Section 3. Now we apply Theorem 2.3 for studying the coercivity of nonsmooth functionals in (1.1).
Corollary 2.4.Assume that the functional I : X → R ∪ {+∞} satisfies the structure hypothesis (1.1) with Φ : X → R locally Lipschitz and Ψ : X → R ∪ {+∞} convex, l.s.c., and proper.If I verifies the PS condition in Definition 1.3 and then I is coercive, that is property (1.6) holds.
Proof.Suppose, by contradiction, that I is not coercive.In view of (2.8), this is equivalent to (2.5) which enables us to apply Theorem 2.3.Corresponding to a sequence n → 0 + , we find a sequence (u n ) ⊂ X fulfilling (1.5), (2.6), and (2.7).
On the basis of (1.5), (2.7), and PS condition in the sense of Definition 1.3, it follows that there exists a subsequence of (u n ) which is strongly convergent in X.Thus, we arrived at a contradiction with (2.6).The proof is complete.

Proof of Theorem 2.3
Note that the condition α ∈ R, imposed in (2.5), implies that the functional I is proper outside every ball in X.We fix a positive number

A version of Zhong's coercivity result
Corresponding to > 0, the definition of the integral yields a partition r = r N < r N−1 < ••• < r 1 < r 0 = r * for which one has We consider the following sets: The requirements in Theorem 2.1 are fulfilled with M = M 0 , M = X, f = I, and x 0 = 0. To justify this, we notice that M 0 is a closed subset of the Banach space X.
We may, thus, suppose that and there exists some Using the construction leading to [11, relation (2.7)] with M 1 , 2 and in place of M, and λ, respectively, and choosing u 1 1 = w 0 , there exists a sequence (u In addition, by (3.19) and the lower semicontinuity of I in conjunction with (3.18) and (3.9), we obtain that  3.20) show that we arrived at a situation which is similar to the one described in the previous step, that is, (3.9) and (3.10).In this respect, if (an assertion analogous to (3.11)), we complete the proof as above.
It remains to consider the case where (3.22) does not hold, that is, and there exists some  Now we prove that for at least one 0 ≤ k ≤ N we are in the situation described in (3.28).Clearly, this will accomplish the proof by means of a reasoning similar to the one below relation (3.11).
In order to check (3.32), we fix k with 0 (3.39) The proof of (3.39) is done by recurrence.Considering first the case j = 2, we point out that the set (3.43) The achieved contradiction implies that u k+1 j < r k .The inductive process is accomplished, thus (3.39) holds true.The proof is complete.