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On the structure of the critical set ofnon-differentiable functions with aweak compactness conditionGabriele Bonanno a & Salvatore A. Marano ba Department of Science for Engineering and Architecture(Mathematics Section), Engineering Faculty , University ofMessina , 98166 Messina, Italyb Dipartimento di Matematica e Informatica , Università degliStudi di Catania , Viale A. Doria 6, 95125 Catania, ItalyPublished online: 19 Jan 2010.

To cite this article: Gabriele Bonanno & Salvatore A. Marano (2010) On the structure of the criticalset of non-differentiable functions with a weak compactness condition, Applicable Analysis: AnInternational Journal, 89:1, 1-10, DOI: 10.1080/00036810903397438

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Applicable AnalysisVol. 89, No. 1, January 2010, 1–10

On the structure of the critical set of non-differentiable functions

with a weak compactness condition

Gabriele Bonannoa* and Salvatore A. Maranob

aDepartment of Science for Engineering and Architecture (Mathematics Section),Engineering Faculty, University of Messina, 98166 Messina, Italy;

bDipartimento di Matematica e Informatica, Universita degli Studi di Catania,Viale A. Doria 6, 95125 Catania, Italy

Communicated by S. Carl

(Received 17 March 2009; final version received 9 September 2009)

The nature of critical points generated by a Ghoussoub’s type min–maxprinciple for locally Lipschitz continuous functionals fulfilling a weakPalais–Smale assumption, which contains the so-called non-smooth Ceramicondition, is investigated. Two meaningful special cases are then pointedout; see Theorems 3.5 and 3.6.

Keywords: critical set; locally lipschitz continuous functions; weak Palais–Smale condition

AMS Subject Classifications: Primary 58E05; 49J35; 49J52

1. Introduction

Very recently, in [1], the classical min–max principle of Ghoussoub [2] has beenextended to locally Lipschitz continuous functions satisfying a weak Palais–Smalehypothesis, which includes both the usual one [3, Definition 2] and the non-smoothCerami condition [4, p. 248]. This result is achieved by adapting Ghoussoub’sapproach to the new framework and, in particular, exploiting Ekeland’s variationalprinciple with an appropriate metric of geodesic type. When the functional is C1

while the compactness assumption is that of Cerami [5, Section 13.1], such an ideabasically goes back to Ekeland [6, Section IV.1].

Besides the existence of critical points, Ghoussoub’s result provides valuableinformation about their nature; see [7, Chapter 5], [6, pp. 145–148] and [5, Chapter 12].Making use of [1, Theorem 3.1], we address here the same question forlocally Lipschitz continuous functions that fulfil a weak Palais-Smale condition.Hence, non-differentiable variants of several known structure results are established.For instance, Theorem 3.3 extends Corollary 4.14 in [7], Theorem 3.4 comes out fromthe famous Pucci and Serrin’s multiple critical point result [8, Corollary 1], whileTheorem 3.7 is patterned after Theorem 8 of [9] and includes [10, Theorem 3.3].

*Corresponding author. Email: bonanno@unime.it

ISSN 0003–6811 print/ISSN 1563–504X online

� 2010 Taylor & Francis

DOI: 10.1080/00036810903397438

http://www.informaworld.com

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Similar results were previously obtained in [11] for functionals expressed as 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 setting [12].

However, the compactness hypothesis employed there reduces to the standard one in

the locally Lipschitz continuous case and hence it is more restrictive than that taken on

below.A special mention deserve Theorems 3.5 and 3.6, achieved by combining [13,

Theorem 3.1] with Theorem 3.4, because they deal with functionals of the type

f� :¼�� ��, where �40 while � and � are locally Lipschitz continuous. In this

case, starting from the seminal paper of Ricceri [14], many multiple critical point

results have been recently established, both in the smooth and in the non-smooth

framework; see [15] for an exhaustive bibliography. Roughly speaking, here, we get a

well-determined larger interval of parameters � for which f� possesses at least three

critical points under weaker regularity and compactness conditions.

2. Preliminaries

Let (X, k � k) be a real Banach space. If U is a subset of X, we write int(U ) for the

interior of U, U for the closure of U, and @U for the boundary of U. Moreover, when

x2X and r40, we define B(x, r) :¼ {z2X : kz�xk5r}, Bðx, rÞ :¼ Bðx, rÞ. Given

x, z2X, the symbol [x, z] indicates the line segment joining x to z, i.e.

[x, z] :¼ {xþ t(z� x): t2 [0, 1]}. We denote by X � the dual space of X, while h�, �i

stands for the duality pairing between X � and X. A function f :X! IR is called

coercive provided

limkxk!þ1

f ðxÞ ¼ þ1:

If to every x2X there correspond a neighbourhood Vx of x and a constant Lx� 0

such that

j f ðzÞ � f ðwÞj � Lxkz� wk 8z,w2X,

then we say that f is locally Lipschitz continuous. In this case f 0(x; z), x, z2X,

indicates the generalized directional derivative of f at the point x along the direction

z, namely

f 0ðx; zÞ :¼ lim supw!x,t!0þ

f ðwþ tzÞ � f ðwÞ

t:

It is known [16, Proposition 2.1.1] that f 0 is upper semi-continuous on X�X. The

generalized gradient of the function f in x, denoted by @f (x), is the set

@f ðxÞ :¼ fx� 2X � : hx�, zi � f 0ðx; zÞ 8z2X g:

Proposition 2.1.2 of [16] ensures that @f (x) is nonempty, convex, in addition to weak

star compact, and that

f 0ðx; zÞ ¼ maxfhx�, zi : x� 2 @f ðxÞg:

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Hence, it makes sense to put

mf ðxÞ :¼ minfkx�kX � : x� 2 @f ðxÞg:

We say that x2X is a critical point of f when 02 @f (x), i.e. f 0(x; z)� 0 for all z2X.Finally, given a real number c, write

Kcð f Þ :¼ fx2X : f ðxÞ ¼ c, x is a critical point of f g,

besides f c :¼ {x2X : f (x)� c} and fc :¼ {x2X : f (x)� c}.

3. Structure of the critical set

Let (X, k � k) be a real Banach space and let h : [0,þ1[! [0,þ1[ be a continuousfunction such that Z þ1

0

1

1þ hð�Þd� ¼ þ1: ð2:1Þ

Given x, z2X, we denote by P(x, z) the family of all piecewise C1 paths p : [0, 1]!Xsatisfying p(0)¼ x and p(1)¼ z. Moreover, put

lhð pÞ :¼

Z 1

0

k p0ðtÞk

1þ hðk pðtÞkÞdt, p2Pðx, zÞ,

as well as

�hðx, zÞ :¼ infflhð pÞ : p2Pðx, zÞg: ð2:2Þ

For h(�) :¼ �, � 2 [0,þ1[, the function �h :X�X! IR defined by (2.2) coincides withthe geodesic distance introduced in [6, p. 138]. Exploiting (2.1) and the arguments of[6, p. 138] yields the next basic properties of �h.

(p1) �h(x, z)�kx� zk for all x, z2X.(p2) If U is a nonempty bounded subset of X then there exists a constant cU40

such that

�hðx, zÞ � cUkx� zk 8x, z2U:

(p3) �h turns out to be a distance on X and the metric topology derived from �hcoincides with the norm topology.

(p4) �h-bounded and norm-bounded sets in X are the same.

Through (p1), (p2) and (p4) one easily verifies that the metric space (X, �h) iscomplete.

Now, let B be a nonempty closed subset of X and let F be a class of nonemptycompact sets in X. According to [2, Definition 1], we say that F is a homotopy-stablefamily with extended boundary B when for every A2F , �2C0([0, 1]�X,X ) such that�(t, x)¼ x on ({0}�X )[ ([0, 1]�B) one has �({1}�A)2F . Some meaningfulsituations are special cases of this notion.

Example 3.1 If Q denotes a compact set in X, Q0 is a nonempty closed subset of Q,�02C

0(Q0,X ), � :¼ {� 2C0(Q,X ) : �jQ0¼ �0}, and F :¼ {�(Q): � 2�}, then F is a

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homotopy-stable family with extended boundary B :¼ �0(Q0). In particular, this

occurs when Q indicates a compact topological manifold in X having a nonempty

boundary Q0 while �0 :¼ idjQ0.

The following assumptions will be posited in the sequel.

(a1) f :X! IR is a locally Lipschitz continuous function.(a2) F denotes a homotopy-stable family with extended boundary B.(a3) There exists a nonempty closed subset F of X such that

ðA \ F Þ n B 6¼ ; 8A2F ð2:3Þ

and, moreover,

supx2B

f ðxÞ � infx2F

f ðxÞ: ð2:4Þ

(a4) h : [0,þ1[! [0,þ1[ is a continuous function fulfilling (2.1), while �h indicatesthe metric defined by (2.2).

Set, as usual,

c :¼ infA2F

maxx2A

f ðxÞ: ð2:5Þ

Thanks to (2.3) one has

infx2F

f ðxÞ � c: ð2:6Þ

The next result, patterned after the seminal min–max principle of Ghoussoub [2], has

been very recently established in [1]; see [1, Theorem 3.1].

THEOREM 3.1 Let (a1)–(a4) be satisfied. Then to every sequence {An}�F such that

limn!þ1maxx2Anf (x)¼ c there corresponds a sequence {xn}�X nB having the

following properties:

(i1) limn!þ1 f (xn)¼ c.(i2) (1þ h(kxnk))f

0(xn; z)��"nkzk for all n2 IN, z2X,where "n! 0þ.(i3) limn!þ1 �h(xn, F )¼ 0 provided infx2F f (x)¼ c.(i4) limn!þ1 �h(xn,An)¼ 0.

We say that f satisfies a weak Palais-Smale condition at the level a2 IR when for

some h as in (a4) one has

ðPSÞha Every sequence {xn}�X such that

limn!þ1

f ðxnÞ ¼ a and limn!þ1

ð1þ hðkxnkÞÞmf ðxnÞ ¼ 0 ð2:7Þ

possesses a convergent subsequence.

Remark 3.1 If h(�) 0 then ðPSÞha reduces to the usual Palais–Smale condition for

locally Lipschitz continuous functions [3]. Setting h(�) :¼ �, �2 [0,þ1[, we obtain a

non-smooth version, previously introduced in [4], of the so-called Cerami

compactness assumption [5, Section 13.1].The following critical point result is an almost direct but meaningful consequence

of Theorem 3.1.

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THEOREM 3.2 Suppose (a1)–(a4) and ðPSÞhc , with c given by (2.5), hold true. Then

Kc( f ) 6¼ ;. If, moreover, infx2F f (x)¼ c then Kc( f )\F 6¼ ;.

Proof Let {xn}�X nB fulfil (i1)–(i4) of Theorem 3.1. Conclusion (i2) actually means

limn!þ1

ð1þ hðkxnkÞÞmf ðxnÞ ¼ 0: ð2:8Þ

In fact, due to [17, Lemma 1.3], for any n2 IN there exists a z�n 2X� such that

kz�nkX � � 1 and

"�1n ð1þ hðkxnkÞÞ f0ðxn; zÞ � hz

�n, zi 8z2X:

Hence,

"nð1þ hðkxnkÞÞ�1z�n 2 @f ðxnÞ,

which gives

ð1þ hðkxnkÞÞmf ðxnÞ � "nkz�nkX � � "n, n2 IN:

Now (2.8) immediately comes out from "n! 0þ. Thanks to ðPSÞhc we may thus

assume that xn! x in X, where a subsequence is considered when necessary. At this

point, (i2) and the upper semi-continuity of f 0 yield f 0(x; z)� 0 for all z2X, namely

x2Kc( f ), because f (x)¼ c by (i1). Next, suppose that infx2F f (x)¼ c. On account of

( p3) the set F turns out to be �h-closed. So, (i3) forces x2F, i.e. Kc( f )\F 6¼ ;. g

We are in a position now to establish our first structure result, which extends

Corollary 4.14 of [7] to the framework of this article.

THEOREM 3.3 Let (a1), (a2), (a4) and ðPSÞhc , with c as in (2.5), be satisfied. Assume

also that

(a5) the members of F are path-wise connected and contain B,(a6) supx2B f (x)5c.

Then Kc( f ) possesses a nonlocal minimum point.

Proof Choose F :¼ @f c. For every A2F one has A\ f c 6¼ ;, besides

ðA \ F Þ n B 6¼ ;: ð2:9Þ

Indeed, (2.5) implies maxx2A f (x)� c. Thus, the first claim is obvious. To verify (2.9)

we first observe that (A\F ) nB¼A\F, because B�X n f c on account of (a6).

If A\F¼; for some A2F then A\ int( f c) 6¼ ; and A� int( f c)[ (X n f c). Since A is

path-wise connected we get A� f c, which implies A\B¼;. However, this

contradicts (a5). We next have

c � infx2f c

f ðxÞ � infx2F

f ðxÞ � infx2A\F

f ðxÞ � maxx2A

f ðxÞ 8A2F ,

namely, c¼ infx2F f (x). Hence, by (a6), inequality (2.4) holds, and hypothesis (a3) is

satisfied. Consequently, thanks to Theorem 3.2, Kc( f )\F 6¼ ;. Pick x2Kc( f )\F.

Bearing in mind the choice of F, for every �40 there exists an x�2B(x, �) such that

f (x�)5c¼ f (x), i.e. x is not a local minimum for f. g

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Theorem 3.2 produces also the following non-smooth version of the famous

Corollary 1 in [8].

THEOREM 3.4 Suppose (a1), (a4) and ðPSÞha, a2 IR, hold true. If x0, x12X are two

local minima of f then it has at least three critical points.

Proof Obviously, x0 and x1 turn out to be critical points for f. It is not restrictive to

assume that f (x1)� f (x0)¼ 0. Let r40 satisfy

r5 kx1 � x0k, infx2Bðx0, rÞ

f ðxÞ � f ðx0Þ ¼ 0: ð2:10Þ

Define Q :¼ [x0, x1], Q0 :¼ {x0,x1}, �0 :¼ idjQ0, � :¼ {� 2C0(Q,X ) : �jQ0

¼ �0},F :¼ {�(Q) : � 2�} and B :¼ {x0, x1}. Hypothesis (a2) holds, because F is a

homotopy-stable family with extended boundary B; see Example 3.1. Setting F :¼

@B(x0, r) one evidently has (A\F ) nB 6¼ ; for all A2F as well as

supx2B

f ðxÞ ¼ maxf f ðx0Þ, f ðx1Þg ¼ 0 � infx2F

f ðxÞ, ð2:11Þ

which yields (a3). Since ðPSÞhc , with

c :¼ inf�2�

maxx2�ð½x0,x1Þ

f ðxÞ, ð2:12Þ

is true too, all the assumptions of Theorem 3.2 are satisfied. So, Kc( f ) 6¼ ; and

Kc( f )\F 6¼ ; provided infx2F f (x)¼ c. On account of (2.11) the conclusion is obvious

when infx2F f (x)5c. Otherwise, it easily follows from Kc( f )\F 6¼ ;, the choice of F,

and (2.10). g

Combining [13, Theorem 3.1] with Theorem 3.4 we obtain the two multiplicity

results below, where X is supposed to be reflexive, �, � :X! IR are locally Lipschitz

continuous,

f� :¼ �� ��, �2 IR, ð2:13Þ

and, for any r, r1, r24infx2X �(x) such that r15r2,

’ð1ÞðrÞ :¼ infx2Xn�r

supz2Xn�r�ðzÞ ��ðxÞ

r��ðxÞ,

’ð2ÞðrÞ :¼ supx2Xn�r

�ðxÞ � supz2�r�ðzÞ

�ðxÞ � ras soon as r5 sup

x2X�ðxÞ,

’1ðr1, r2Þ :¼ maxf’ð1Þðr1Þ, ’ð1Þðr2Þg,

’2ðr1, r2Þ :¼ infx2Xn�r1

supz2�r1 n�r2

�ðzÞ ��ðxÞ

�ðzÞ ��ðxÞ:

THEOREM 3.5 Let (a4) be fulfilled. If, moreover,

(a7) � is sequentially weakly lower semi-continuous and coercive,(a8) � is sequentially weakly upper semi-continuous,(a9) there exist r1, r24infx2X �(x) such that r15r2 and ’1(r1, r2)5’2(r1, r2),

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(a10) the function f� given by (2.13) satisfies ðPSÞha for all �40, a2 IR,

then f� possesses at least three critical points (two local minima and a nonlocal

minimum point) provided 1/’2(r1, r2)5�51/’1(r1, r2).

Remark 3.2 When

infx2X

�ðxÞ ¼ �ð0Þ ¼ �ð0Þ ¼ 0,

the same reasoningmade in the proof of [13, Corollary 3.1] (see also [14, Remark 3.11])

shows that the condition

(a9)0 for some r1, r240, x12X one has r15�(x1)5r2 as well as

supz2�r1

�ðzÞ5r1�ðx1Þ

2�ðx1Þ, sup

z2�r2

�ðzÞ5r2�ðx1Þ

2�ðx1Þ

forces (a9). Since

’1ðr1, r2Þ � max1

r1supz2�r1

�ðzÞ,1

r2supz2�r2

�ðzÞ

( ), ’2ðr1, r2Þ �

�ðx1Þ

2�ðx1Þ,

by Theorem 3.5 the function �� �� possesses at least three critical points provided

2�ðx1Þ

�ðx1Þ5 �5 min

r1supz2�r1

�ðzÞ,

r2supz2�r2

�ðzÞ

( ): ð2:14Þ

THEOREM 3.6 Assume that (a4), (a7), (a8), (a10) hold, and that

(a11) there exists r2 ]infx2X �(x), supx2X �(x)[ fulfilling ’(1)(r)5’(2)(r),(a12) f� turns out to be coercive for every �2 ]1/’(2)(r), 1/’(1)(r)[.

Then the conclusion of Theorem 3.5 is true whenever 1/’(2)(r)5�51/’(1)(r).

Proof Theorem 3.1 and Remark 3.3 in [13] yield a global minimum x0 of f�jXn�r,

because �51/’(1)(r). We next claim that f�jXn�rhas a global minimum. In fact, from

�41/’(2)(r) it follows

�ð �xÞ � ��ð �xÞ5 r� � supz2�r

�ðzÞ ð2:15Þ

for some �x2X n�r. Let �� : X! IR defined by setting, for every x2X,

��ðxÞ :¼r if �ðxÞ � r ,�ðxÞ otherwise.

�

Through (a7) we easily see that �� is sequentially weakly lower semi-continuous and

coercive. Thanks to (a8) and (a12), the function ��� �� enjoys the same properties.

Since X is reflexive, there exists an x12X satisfying

��ðx1Þ � ��ðx1Þ � ��ðxÞ � ��ðxÞ 8x2X: ð2:16Þ

Thus, the assertion is verified once we show that x12X n�r. If, on the contrary,

x12�r then, in view of (2.16),

r� ��ðx1Þ � �ð �xÞ � ��ð �xÞ: ð2:17Þ

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Due to (2.15) one has

�ð �xÞ � ��ð �xÞ5 r� ��ðx1Þ: ð2:18Þ

Gathering (2.17) and (2.18) together gives

r� ��ðx1Þ5 r� ��ðx1Þ,

which is evidently impossible. Hence, the function f� possesses two local minima, x0and x1. Since it is locally Lipschitz continuous, the conclusion directly follows from

Theorem 3.4. g

Remark 3.3 Simple calculations guarantee that the condition(a11)

0 There exist r2 IR, x0,x12X fulfilling �(x0)5r5�(x1) and, moreover,

supz2�r

�ðzÞ5ð�ðx1Þ � rÞ�ðx0Þ þ ðr��ðx0ÞÞ�ðx1Þ

�ðx1Þ ��ðx0Þ

implies (a11). In particular, if �(0)¼�(0)¼ 0 then (a11)0 obviously reduces to

(a11)00 For some r40, x12X one has r5�(x1) as well as

supz2�r

�ðzÞ5 r�ðx1Þ

�ðx1Þ: ð2:19Þ

In this case, by Theorem 3.6, the function �� �� possesses at least three critical

points provided

�ðx1Þ

�ðx1Þ5 �5

r

supz2�r�ðzÞ

: ð2:20Þ

Indeed, since ’ð1ÞðrÞ � 1r supz2�r

�ðzÞ while, due to (2.19),

’ð2ÞðrÞ ��ðx1Þ � supz2�r

�ðzÞ

�ðx1Þ � r�

�ðx1Þ � r �ðx1Þ�ðx1Þ

�ðx1Þ � r¼

�ðx1Þ

�ðx1Þ,

we obtain

1

’ð2ÞðrÞ�

�ðx1Þ

�ðx1Þ5

r

supz2�r�ðzÞ�

1

’ð1ÞðrÞ:

Remark 3.4 Suppose

infx2X

�ðxÞ ¼ �ð0Þ ¼ �ð0Þ ¼ 0:

Then inequalities (2.20) written for r¼ r1 evidently are less restrictive than (2.14), i.e.

Theorem 3.6 allows a wider range of parameters �. However, in Theorem 3.5 the

coercivity of f� is not assumed.

Remark 3.5 By [18, Corollary 2.4], hypothesis (a12) holds true whenever the

function f� turns out to be bounded from below and (a10) is fulfilled for some non-

decreasing h.

Remark 3.6 Theorem 3.6 generalizes [13, Theorem 3.2] and, in the smooth case, [19,

Theorem 2.1]; see also [20, Theorem 2.1]. In fact, besides requiring weaker regularity

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and compactness conditions on the involved functionals, it exhibits a well-determined interval of parameters �, which contains those of the above-mentionedresults. Accordingly, their applications to nonlinear differential problems (as forinstance Dirichlet, Neumann, mixed, Hamiltonian, p-Laplacian problems) might beimproved through Theorem 3.6.

Remark 3.7 Let us explicitly point out a useful sufficient condition in order that �

and � be weakly continuous on X:

There is a Banach space ~X such that X compactly embeds in ~X and �,� : ~X! IRare locally Lipschitz continuous on the whole ~X.

Finally, a critical point x2X is called a saddle point for f when to every �40there correspond x0, x00 2B(x, �) such that f (x0)5f (x)5f (x00). Arguing as in theproofs of Theorem 2 and Corollary 3 of [21], but with Theorem 3.2 in place ofGhoussoub–Preiss min–max principle (see also [11, Theorem 4.3]), yields the nextresult.

THEOREM 3.7 Suppose X is infinite dimensional and x0, x12X. If (a1), (a4) and ðPSÞhc ,

with c as in (2.12), hold true, while max{ f (x0), f (x1)}5c, then Kc( f ) contains asaddle point.

References

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partial differential equations, J. Math. Anal. Appl. 80 (1981), pp. 102–129.

[4] N.C. Kourogenis and N.S. Papageorgiou, Nonsmooth critical point theory and nonlinear

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for nondifferentiable functions, Differ. Integral Equ. 16 (2003), pp. 1001–1012.[11] R. Livrea and S.A. Marano, Existence and classification of critical points for

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