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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 781746, 6 pages
http://dx.doi.org/10.1155/2013/781746
Research Article

Minimax Results with Respect to Different Altitudes in the Situation of Linking

Department of Mathematics, Ben Gurion University of the Negev, 84105 Be'er Sheva, Israel

Received 27 June 2013; Accepted 20 August 2013

Academic Editor: Kanishka Perera

Copyright © 2013 V. V. Motreanu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Consider a continuous function on a metric space. In the presence of linking between a compact pair and a closed set, depending on the different behaviors of the function on the linking sets, we establish minimax results guaranteeing existence of Palais-Smale sequences or providing gradient estimates. Our approach relies on deformation techniques.

1. Introduction and Statement of Main Results

Minimax theorems play a central role in critical point theory: in this respect, we refer to celebrated minimax results as the mountain pass theorem (see Ambrosetti and Rabinowitz [1]) and the saddle point theorem (see Rabinowitz [2]). A practical way to define minimax values is by means of a linking condition in a topological space adapted to the problem. In the above-cited results, there is a specific linking based on the geometry of the involved problem. In this paper, we provide linking results for continuous functions on metric spaces, which extend the results in [3, 4] in this setting. Beyond this general framework, the main novelty that we emphasize with our approach is that we provide a systematic study related to the involved geometry, being concerned with situations which are not covered by the “classical geometry.” This approach is original in the case of Banach spaces and smooth functions too.

Recall that when is a Banach space and is continuously differentiable, an element is called a critical point of if . In this paper, more generally, we assume that is a metric space endowed with the metric and is a continuous function. In this context, the following notion of critical point has been introduced by Degiovanni and Marzocchi (see [4]).

Definition 1. Let be a continuous function defined on a metric space . An element is called a critical point of if , where by we denote the supremum of the real numbers such that there exist and a continuous map satisfying

Remark 2. (a) Note that (take , any , and in (1)) and the map , is lower semicontinuous. The quantity is usually called the weak slope of at .
(b) If is a real Banach space, is the distance induced by the norm , and , then coincides with the norm of the differential . Thus, Definition 1 reduces to the usual notion of critical point in this context.

If , we denote by the set of critical points of at level ; that is,

The essential tool in our approach to critical point theory is the linking condition presented in the next definition. We say that is a compact pair in if are compact subsets of with and .

Definition 3. A compact pair in and a closed subset of are linking if the following property holds: where .

Given a continuous function , a compact pair in , and a nonempty, closed subset of , we set Clearly, . Moreover, if (3) holds, then we have . Note that . If , then .

The “classical” minimax principles are usually based on the assumption that (i.e., the “altitude” is positive; actually, refined results involve the weaker condition ), establishing in this situation the existence of a critical point for . Our approach covers the situation where (“negative altitude”). We will distinguish the cases: and . In the situation , we will also distinguish the cases and .

Now we formulate our main results. In all these results, unless otherwise stated, is a metric space, is a continuous function, is a compact pair in and is a closed subset of satisfying (3).

The first statement is a preliminary result providing existence and location of critical points.

Proposition 4. Assume that (see (4)) and that there exist and a number such that Then .

Hereafter, (the open -neighborhood of ). The proof of this proposition will be done in Section 2.

In the following results, the metric space is supposed to be complete.

Here are our main results in the cases and .

Theorem 5. Assume that . Then In particular, there exists a sequence such that

Theorem 6. Assume that . Then(a)for all and , one has (b)for all and , one has

Hereafter, . The notation stands for .

Remark 7. The estimate in part (b) of Theorem 6 is better than the one in part (a) when . Nevertheless, Theorem 6(a) provides an estimate for all .

The next result distinguishes the situations and in (4). Since Theorem 5 treats the case , the next result is meaningful when .

Theorem 8. Let be a compact pair in . Assume that . Then In particular, there exists a sequence such that
Let be a compact pair in and let be a closed subset of satisfying (3). Assume that . Then

As a consequence of Theorems 5, 6(b), and 8, we have the next result, which studies the situations and in (4).

Corollary 9. Assume that . Then (10) holds. If , then (6) holds.
Assume that . Then (12) holds. If , then (10) holds.

Theorems 5 and 8(a) and Corollary 9(a) lead to the construction of a Palais-Smale sequence at level (i.e., a sequence such that and as ). Moreover, in the case of Theorem 5 (referring to the limiting case ), the Palais-Smale sequence is located near (i.e., as ). Recall from [3] the following Palais-Smale condition.

Definition 10. We say that a continuous function satisfies the Palais-Smale condition at level (condition for short), with , if every sequence such that and as has a convergent subsequence in .

If we assume condition (see (4)), the previous results guarantee the existence of critical points.

Corollary 11. If conditions (3) and hold, then one has the following.(a)In the case , then .(b)In the case , if one assumes that (for instance, if ), then .

Remark 12. (a) Corollary 11 actually holds under a weaker condition than , namely, : every sequence such that and as has a convergent subsequence in .
(b) Corollary 11(a) holds under a weaker condition than , namely, : every sequence such that , , and as has a convergent subsequence in . The conclusion in Corollary 11(a) is also more precise since it establishes the location of the critical point in the set .

Theorems 6 and 8(b) and Corollary 9(b) go beyond the “classical geometry” in the situation of linking, allowing the case and providing an estimate of the infimum of at the points with in terms of and of or of .

We may still obtain a Palais-Smale sequence from Theorem 8(b) (the case of “nonclassical geometry”) under appropriate hypotheses for a sequence of sets (in the spirit of [5]) which are linking.

Corollary 13. Let be a continuous function on the metric space , and let , , and be sequences of subsets of such that for all , is closed, is a compact pair in , and satisfy (3) with replaced by . Assume that Then, denoting and , there exists a sequence satisfying In particular, if satisfies condition for all , then there exists such that and .

Remark 14 (assume that ). Considering particular situations of linking as in (3) in a Banach space , we obtain through Corollary 11 generalizations of classical minimax results in smooth critical point theory.(a)If , , and (with and such that ), then we get the mountain pass theorem (see [1]).(b)If , with , , , and , then we get the saddle point theorem (see [2]).(c)If , with ,   +  , and (with and such that ), we get the generalized mountain pass theorem (see [2]).
It is worth noting that generalizations of the minimax results cited in (a)–(c), relative to critical point theories for certain classes of nondifferentiable functionals, can already be found in the literature and have numerous applications to partial differential equations, differential inclusions, variational inequalities, hemivariational inequalities, and variational-hemivariational inequalities. In this respect, motivated by the study of existence of solutions of the so-called variational-hemivariational inequalities, introduced by Motreanu-Panagiotopoulos [6], we mention the critical point theory developed by these authors in [6] for functionals , with locally Lipschitz and lower semicontinuous, convex, and not identically (see also Chang [7] for the case when , and see Szulkin [8] for the case when is of class and is lower semicontinuous, convex, and not identically ).

The rest of the paper is organized as follows. Section 2 contains the proof of Proposition 4. Section 3 deals with the proofs of Theorems 5, 6, and 8 and Corollaries 9, 11, and 13. The method of proofs is based on suitable deformation results given in Sections 2, 3.

2. Proof of Proposition 4

We start with stating the following deformation result.

Theorem 15. Let be a metric space endowed with the metric , and let be a continuous function. For all , there exists a deformation (i.e., is continuous and satisfies for all ) such that for all and , one has(a); (b); (c), .

Proof. First, assume that . We consider the continuous function given by We have for all with . Using that is lower semicontinuous (see Remark 2(a)), we see that Applying [3, Theorem ()], we find a deformation and a continuous map with the properties (for all and ): (i) ; (ii) ; (iii) . Then the deformation defined by has properties (a)–(c) of the statement. If , we make the same reasoning taking an arbitrary positive constant in place of .

Proof of Proposition 4. Let , and let be a deformation satisfying properties (a)–(c) of Theorem 15 (with in place of ). Note that . Let be a continuous function such that Defining by with as in the statement of Proposition 4, we have . Fix with (see (3)). Then, using property (a) in Theorem 15, we infer that Hence, using properties (b), (c) of Theorem 15, relation (5), and the assumption that , we obtain that and . This combined with property (b) of Theorem 15 yields , which completes the proof.

3. Proofs of Theorems 5, 6, and 8 and Corollaries 9, 11, and 13

The following deformation result was shown in [9].

Theorem 16. Let be a metric space endowed with the metric , let be a continuous function, let be a subset of , and let , and be real numbers. Assume that the metric space (endowed with the induced metric) is complete for all and that Then there exist a continuous function and a continuous map          such that for all and , one has(a); (b); (c); (d).

Hereafter, we denote and .

Proof of Theorem 5. For a fixed , it suffices to prove that To this end, let be a number such that Let such that In particular, these choices combined with (24) guarantee that Hereafter, we denote . Thus, applying Theorem 16 with the numbers and , , and , we find a deformation such that
Using that , we see that . Let be a continuous function such that if and if . Define the deformation     []  →   by To prove the continuity of , we note that the restriction of to the sets and , which are closed in , is continuous. Indeed, the restriction of to coincides with the projection , whereas the restriction of to coincides with the map (since on ) which is continuous (by the continuity of ).
Let with . Then the map defined by is well defined and . Finally, let such that . Since (by the first part of (27)), we have that , so ; thus (by the second part of (27)). Consequently, we can apply Proposition 4. Therefore, . This concludes the proof.

Proof of Theorem 6. (a) By (3), we see that . For a fixed and , it suffices to prove that To this end, let be a number such that Let be positive numbers such that Setting , since , we have This allows us to apply Theorem 16 with the numbers and , , and . Hence we find a deformation     []  →   such that
The fact that in conjunction with the inequality yields . Let be a continuous function such that if and if . Define the deformation     []   by The continuity of can be shown as in the proof of Theorem 5. Let with . Then the map given by is well defined, and we have . Let such that . Since it follows that , which yields Consequently, Proposition 4 (with replaced by ) can be applied, and we conclude that .
(b) We see that . For a fixed and , it suffices to show that To this end, let be a number such that Let be positive numbers such that Setting , we have Thus, we can apply Theorem 16 with the numbers and , , and . So we find a deformation such that
Let be a continuous function such that if and if . Define the deformation     []  →   by The continuity of can be shown as in the proof of Theorem 5.
Let with . Define by We show that . To prove that is continuous, it suffices to note that the restrictions of to the closed sets and are continuous. This is immediate for the latter closed set. Concerning the former, this follows from the fact that the equality holds whenever (which is a consequence of the definition of and relation (45)). Whence is continuous. Using that , we infer that .
Let an arbitrary point with . If , then (by the first part of (46)), so , and thus (by the second part of (46)). If , then it is clear that . Consequently, in both cases, we have . This allows us to apply Proposition 4 (with replaced by ) and thus to obtain that , which completes the proof.

Proof of Theorem 8. (a) The hypotheses allow us to apply Theorem 5 with . Indeed, since , we have that . Moreover, since is compact, for each , there exists with . It follows that condition (3) is satisfied and so . The conclusion follows now from Theorem 5.
(b) This follows from Theorem 6(b).

Proof of Corollary 9. (a) If , then we have . Thus, we can apply Theorem 8(a). In the case , we apply Theorem 5.
(b) If , then we obtain from Theorem 8(a) that (10) holds (so that a fortiori (12) holds since ). If , then we apply Theorem 8(b).

Proof of Corollary 11. Parts (a) and (b) follow from Theorems 5 and 8(a), respectively, taking into account the lower semicontinuity of the map (see Remark 2(a)).

Proof of Corollary 13. Fix an arbitrary . If , then applying Corollary 9(a), we have If , then using Corollary 9(b), we have Let Then we have as and This completes the proof.

Acknowledgments

The author is grateful to Jean-Noël Corvellec for his suggestion to study this problem and his valuable advices. This work was funded by a Marie Curie Intra-European Fellowship for Career Development within the European Community’s 7th Framework Program (Grant Agreement no. PIEF-GA-2010-274519).

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