CONNEXIONS AND APPLICATIONS
IRINA MEGHEA
We prove a modified version of a theorem of Ghoussoub and Maurrey, which is a general smooth perturbation variational principle, by operating some changes on it. Two other theorems, which are also variational principles, are discussed, one of them being a generalization of a theorem due to Deville, Godefroy and Zizler, while the other is a linear perturbation variational principle in reflexive spaces.
Several connexions with other forms of variational principles are pointed out and two applications of these results are given.
AMS 2000 Subject Classification: 35A15, 47J30, 58E30, 70G75.
Key words: variational principle, linear perturbation, smooth perturbation varia- tional principle.
1. INTRODUCTION
The trends opened by the perturbed variational principle of Ekeland, which stays at the foundation of the modern Variational Calculus, have been followed by many researchers. In this paper we discuss some generalizations and variants of them due to Deville, Godefroy and Zizler and, especially, to Ghoussoub [2]. An application due to Br´ezis and Nirenberg [1] for which Ghoussoub used a perturbed linear principle is generalized. Some results on the pseudo-Laplacian which are used here have been obtained by us in [4]. The approach that is used here belongs to the author and develops ideas from the monograph [3].
This work presents several variants of perturbation variational principles and for two of them we have improved the statements and proofs. Among these variants some connexions are established.
An example of an admissible family is discussed in order to illustrate that one of the recovered theorems is a generalization of the Deville-Godefroy-Zizler principle.
The other recovered theorem is used in order to obtain a characterization of the solutions of a limit problem for the pseudo-Laplacian and represents a generalization of a minimization problem.
REV. ROUMAINE MATH. PURES APPL.,54(2009),5–6, 493–511
2. A GENERALIZATION OF
THE DEVILLE-GODEFROY-ZIZLER THEOREM
Theorem 1 below due to Deville-Godefroy-Zizler interferes in a decisive manner in the demonstration of several results, e.g., in a variant of the Ekeland principle that is obtained (see [6], [3]). In order to state this result, we fix some notation and recall some definitions.
In this paper all the Banach spaces are real.
A function f : X → R, X real normed space, is called a bump func- tion, provided it is bounded and its support suppf = {x∈X :f(x)6= 0} is nonempty and bounded.
Letf :X →(−∞,+∞] be given. Thedomaindomf off is domf ={x∈X :f(x)<+∞}.
f isproperif domf 6=∅. Letkfk∞= sup{|f(x)|:x∈X} and letk · kdenote the norm of F.
Theorem 1. Let Xbe a Banach space andF a Banach space of bounded continuous real functions defined on X such that
1◦ if f ∈ F thenkfk∞≤ kfk;
2◦ if f ∈ F and x∈X then the function y →fx(y) :=f(x+y) belongs to F and kfxk=kfk;
3◦ if f ∈ F and λ∈R then the function y→f(λy) belongs to F;
4◦ there exists a bump function in F.
If ϕ : X → (−∞,+∞] is lower bounded, lower semicontinuous and proper, then the set M of functions f of F such that ϕ+f has a point of strong minimum on X, is an everywhere dense Gδ-set.
Let (X, d) be a metric space and (M, δ) a metric space of real functions defined on X. For each nonempty subset A ofX consider
(1) MA:={f ∈ M:f upper bounded onA}.
For each f from MA and t >0 the slice
(2) S(A, f, t)def={x∈A:f(x)>supf(A)−t}, of A is obviously a nonempty set whenMA6=∅.
Definition. A metric space (X, d) is uniformly M-dentable if for each nonempty subset A of X, each f from MA, and each ε > 0 there exist g in MA and t >0 such that
(3) δ(f, g)≤ε
and
(4) diamS(A, g, t)≤ε.
The definition is extended to the case where X is a pseudo-metric space.
We associate with the metric space (X, d) the setXe :=X×Requipped with a pseudo-metric dedefined as
(5) d((x, λ),e (y, µ)) =d(x, y).
Moreover, with the metric space (M, δ) we associate the set Mf of all functions fedefined onXe by
(6) f(x, λ) =e f(x)−λ,
where f ∈ M and λ is a real number. Clearly, fe= eg ⇔ f = g. On Mf we consider the distance eδ defined as
(7) eδ(f ,eeg) =δ(f, g).
Definition. Let (X, d) be a metric space and (M, δ) the metric space of real bounded continuous functions on X. Mis anadmissible familyon X if
(I)Mis a complete metric space;
(II)∃K >0 such that f, g∈ M ⇒δ(f, g)≥Ksup{|f(x)−g(x)|:x∈X};
(III) the metric space (X,e d) is uniformlye M-dentable.f We can now state and prove our main result.
Theorem 2. Let X be a Banach space, (M, δ) an admissible family of functions on X andϕ:X→(−∞,+∞]a function bounded from below, lower semicontinuous and proper. Then the set
M1 :={f ∈ M:ϕ−f attains a strong minimum in X}
contains an everywhere dense Gδ-set.
Proof. For eachn inNconsider the set Un:=
f ∈ M:∃xn∈Xwith (ϕ−f)(xn)<
(8)
<infn
(ϕ−f)(x) :kx−xnk ≥ 1 n
o .
We first prove that Un is open. Indeed, let f0 be an arbitrary function from Un. Assume that ∀ε >0∃f inUn such that
(9) δ(f0, f)< ε
and
(10) (ϕ−f)(xn)≥inf
(ϕ−f)(x) :kx−xnk ≥ 1 n
.
We write (ϕ−f)(xn) = (ϕ−f0)(xn) + (f0 −f)(xn), combine (10), (9) and (II) (from the definition of the admissible family), in order to obtain
(11) (ϕ−f0)(xn) + ε K ≥inf
(ϕ−f)(x) :kx−xnk ≥ 1 n
. Since ∀x withkx−xnk ≥ n1 we have
(ϕ−f)(x) = (ϕ−f0)(x) + (f0−f)(x)
(II)
≥(ϕ−f0)(x)− ε K ≥
≥inf
(ϕ−f0)(x) :kx−xnk ≥ 1 n
− ε K, we deduce that
inf
(ϕ−f)(x) :kx−xnk ≥ 1 n
≥inf
(ϕ−f0)(x) :kx−xnk ≥ 1 n
− ε K. Now, we combine with (11), let ε→ 0+ and get a contradiction, taking into account that f0 ∈ Un and, consequently, that f0 is an inner point of Un.
Let us now prove that Un is everywhere dense and, in particular, that Un6=∅. Letf fromMand ε >0. We must findg inMsuch that
(12) δ(f, g)< ε
and xn inX has the property (13) (ϕ−g)(xn)<inf
(ϕ−g)(x) :kx−xnk ≥ 1 n
.
The functional feis upper bounded on the epigraph E of ϕ in domϕ×R, a nonempty set because ϕ is proper: λ ≥ ϕ(x) ⇒ fe(x, λ) = f(x) −λ ≤ f(x)−ϕ(x) ∀x∈domϕand f is bounded, while−ϕupper bounded. On the other hand, (X,e d) being uniformlye M-dentable, forf feand ∀ε0 >0∃eg inMfE and t >0 such thatδ(f, g)< ε0 and
(14) diamS(E,eg, t)< ε0. Suppose ε0 <min
ε,1n . Take
(15) (xn, λn) in S :=S(E,eg, t).
Letx in X such that kx−xnk ≥ n1. Supposex ∈domϕ. Then, by (5), (14), and (15), we get (x, ϕ(x))∈E\S. Consequently,
(16) g(xn)−λn>supeg(E)−t≥g(x)−ϕ(x).
Note that (16) still holds whenx /∈domϕ. It only remains to change the signs in (16) and then, taking into account that λn ≥ ϕ(xn), to take the greatest lower bound in order to obtain (13).
Let U :=
∞
S
n=1
Un; U is dense in M, by Baire’s Category Theorem. It remains to show that U ⊂ M1, that is,
(17) f ∈ U ⇒ϕ−f attains a strong minimum onX.
To this end, for each nfrom N∃xnin X such that (see the definition of Un) (18) (ϕ−f)(xn)<inf
(ϕ−f)(x) :kx−xnk ≥ 1 n
,
where we took into account thatxn∈domϕ∀n≥1. Further, we remark that (19) p > n⇒ kxp−xnk< 1
n,
since, otherwise, (ϕ−f)(xp)>(ϕ−f)(xn) (see (18)) and, on the other hand, askxp−xnk ≥ n1 > 1p, we have (f ∈ Up!) (ϕ−f)(xn)>(ϕ−f)(xp), that is, a contradiction. Therefore, (19) implies that (xn)n≥1 is a Cauchy sequence. Let x0 = lim
n→∞xn. We will get (17) once we prove that x0 is a strong minimum point for ϕ−f.
To see this, note that ϕ is lower semicontinuous at x0 and, taking also into account the property of xn we have
(ϕ−f)(x0)≤ lim
n→∞(ϕ−f)(xn)≤ (20)
≤ lim
n→∞
inf
(ϕ−f)(x) :kx−xnk ≥ 1 n
=:L.
However, we note that
(21) L≤inf{(ϕ−f)(x) :x∈X\ {x0}}=:m.
Indeed, to see this, let us suppose thatL > m. Then there existx0 inX\{x0}, a inR, and a subsequence (xkn)n≥1 such that
(22) inf
(ϕ−f)(x) :kx−xknk ≥ 1 kn
≥a >(ϕ−f)(x0).
Since kx0 − xknk < k1
n ∀n ≥ 1 does not hold because, otherwise, x0 =
n→∞lim xkn =x0, that is a contradiction. Let thenkN be such thatkx0−xkNk ≥
1
kN. This implies, by (22), that (ϕ−f)(x0) ≥ a > (ϕ−f)(x0), which is a contradiction, hence (21) holds. Combining (20) and (21), we get that x0 is a minimum point of ϕ−f inX.
It remains to show that for any sequence (yn)n≥1 we have
n→∞lim(ϕ−f)(yn) = (ϕ−f)(x0)⇒ (23)
⇒ lim
n→∞yn=x0. (24)
To this end, assume that the contrary holds. Then there exist ε > 0 and a subsequence (ykn)n≥1 of (yn)n≥1 such that
(25) kykn−x0k ≥ε ∀n≥1.
From (25) we get
kykn−xpk+kxp−x0k ≥ε ∀n, p≥1.
As xp →x0, there existsN from Nsuch that (26) kykn−xpk+ ε
2 ≥ε ∀n≥1, ∀p≥N.
Take p1 sufficiently large such that p1
1 ≤ ε2 andp1≥N. Then, from (26),
(27) kykn−xp1k ≥ 1
p1 ∀n≥1.
So, as x0 is a minimum point, we have (ϕ−f)(x0)≤(ϕ−f)(xp1)<inf
(ϕ−f)(x) :kx−xp1k ≥ 1 p1
(27) (28) ≤
(27)
≤(ϕ−f)(ykn) ∀n≥1.
Now, remark that (28) implies (ϕ−f)(ykn)9(ϕ−f)(x0), and this contradicts (23). This proves that (24) holds.
Theorem 2 is, essentially, Theorem 1.27 (general variational principle) from [2, Chapter 1, 1.7]. In that theorem, a flaw occurs: X is a complete metric space and then, the definition Un :=
g ∈A : ∃xn ∈ X with (ϕ−g) (xn)<inf
(ϕ−g)(x) : d(x, xn) ≥ n1
is problematical, since the inequality d(x, xn)≥ n1 may have no solution inX. We changed Xinto a Banach space, where the inequality kx−x0k ≥ R has always solutions. Two other errors in Theorem 1.27 in [2] have been corrected and we got our Theorem 2. We consider that, in this way, the importance and power of Ekeland Principle, as a general variational principle in metric spaces, so far remain unattainable.
3. AN EXAMPLE OF ADMISSIBLE FAMILY The next statement is an example of admissible family.
Proposition 1. LetX be a Banach space and F a Banach space of real functions that are bounded and continuous on X such that
1◦f ∈ F ⇒ kfk∞≤ kfk;
2◦f ∈ F and x∈X ⇒ the function y →fx(y) :=f(x+y) is in F and kfxk=kfk;
3◦ f ∈ F and λ∈R⇒ the functiony →f(λy) is in F;
4◦ there exists a bump function in F.
Then F is an admissible family on X ([2], [3], I, §1, 1.2).
Proof. We verify properties (I), (II), (III) from the definition. (I) and (II) follow by the hypotheses (K = 1). In order to verify (III), let febe any function from MfA andε >0. We should find eg inMfA and t >0 such that
(29) eδ(f ,eeg)≤2ε
and
(30) diamS(A,eg, t)≤2ε.
By 4◦and 2◦there exists a bump functionbwithb(0)6= 0. Then by 3◦we can replace b(x) byλ1b(λ2x), with λ1 and λ2 conveniently chosen inR such that
b(0)>0, (31)
kbkF ≤ε, (32)
kxk ≥ε⇒b(x) = 0.
(33)
Let (x0, λ0) fromA such that (see (31))
(34) fe(x0, λ0) =f(x0)−λ0>supfe(A)−b(0).
Consider the function
(35) h:X→R, h(x) =b(x−x0) and take eg fromMfdefined as
(36) eg:=f]+g.
We have eg∈MfAbecause on the one handf+h∈ Mand on the other hand (x, λ)∈A⇒eg(x, λ) = [f(x)−λ] +h(x), andfeandh are upper bounded on A, respectivelyX. We have khkF(2)=kbkF, hence
δ(ef ,eeg)(7)=δ(f, g) =kf −gkF =khkF (32)
≤ ε, i.e., (29). Transcribe (34) as
(37) eg(x0, λ0)>supf(A).e
Obviously, eg(x0, λ0)≤supeg(A). Take now t >0 such that (38) eg(x0, λ0)−t >supfe(A)
(see (37)). It remains to prove (30). Let (x, λ) arbitrary fromS :=S(A,eg, t).
Then
(39) kx−x0k< ε.
Indeed, suppose kx−x0k ≥ ε. This gives h(x)(33),=(35)0, but eg(x, λ) >
supeg(A)−t, consequently f(x)−λ > eg(x0, λ0) −t(38)> supfe(A), that is, a contradiction, since (x, λ)∈A.
Thus, if (x1, λ1),(x2, λ2)∈S then
ed((x1, λ1),(x2, λ2))(5)= d(x1, x2) =kx1−x2k(39)= 2ε, i.e., (30) holds.
Remark. Proposition 1 shows that Theorem 2 generalizes Theorem 1 (Deville-Godefroy-Zizler theorem; [3], I, §1, 1.2; F being a Banach space, F ={−f :f ∈ F }).
4. A GHOUSSOUB-MAUREY PRINCIPLE We shall prove
Theorem 3 (Ghoussoub-Maurey principle). Let X be a Banach space and Ma Banach space of real bounded continuous functions on X such that 1◦, 2◦, 3◦ and 4◦ from Proposition 1 all hold. Then there exists ρ > 0 such that for any function φ: X → (−∞,+∞] that is bounded from below, lower semicontinuous, and proper, and for any ∈ (0,1) and any x0 ∈ X that verifies
(I)ϕ(x0)<infϕ(X) +ρε2. There exist f in Mand ν in X, subject to the following properties:
(II) νε is a strongly minimum point for ϕ+f, (III) kfkM< ε,
(IV)kνε−x0k< ε.
Proof. We can suppose that there exists a bump functionb inMwith b(0) = 1,
(40)
b(X)⊂[0,1], (41)
suppb⊂S(0,1) (42)
(as indicated in the proof of Proposition 1). SetM :=kbkM. We haveM ≥1 as
(43) M1
◦
≥ kbk∞
(40),(41)
= 1.
Take
(44) ρ:= 1
4M and consider the function
(45) g:X →(−∞,+∞], g(x) =ϕ(x)−2ρε2b
x−x0 ε
.
g is bounded from below, l.s.c. and proper. Apply Proposition 1 and Theo- rem 2 to it: there exists h∈ Mwith
(46) khkM< ρε2
2
and g+h has in X a strong minimum point νε (g+h = g−(−h), M is a Banach space).
VerifyIV. Suppose thatkν0−x0k ≥ε. Then νε−xε 0 ∈/ S(0,1) (open hall), hence b νε−xε 0
= 0 (see (42)), so that
(47) g(νε)(45)= ϕ(νε)≥infϕ(X).
Moreover,
(48) g(x0)(45),=(40)ϕ(x0)−2ρε2(I)<infϕ(X)−ρε2. But
(g+h)(νε)≤(g+h)(x0), hence
g(νε)≤g(x0) +h(x0)−h(νε)≤g(x0) + 2khk∞(46),1
◦
< g(x0) +ρε2, consequently infϕ(X)(47)< g(x0) + ρε2, which contradicts (48), and (IV) is verified.
Take nowf :X→Rwith
(49) f(x) =−2ρε2b
x−x0 ε
+h(x).
We have f ∈ M and ϕ+f =g+h. Consequently, νε is a strongly minimum point for ϕ+f, and (II) is verified.
Verify (III). kfkM (49)
≤ 2ρε2kbkM + khkM (46)
< 2ρε2M +ρε22, but 2ρ(44)=
1 2M
(43)
≤ 12 and then 2ρε2M +ρε22 ≤ 12ε2 + 18ε2 < ε2 < ε. Consequently, kfkM< ε.
Remark. Compare the hypotheses of the G-M principle (Theorem 3) with those of the D-G-Z principle, Theorem 1 ([3], II, 2, 2.3).
5. THE SECOND VARIATIONAL PRINCIPLE
This is a linear perturbed variational principle in reflexive spaces. We first introduce the following
Definition. Let X be a real normed space, f : X → (−∞,+∞], C nonempty subset of X and x0 ∈ C. f is said to strongly exposes C from below at x0 if
1◦ f(x0) = inff(C)<+∞
and
2◦ xn∈C ∀n≥1, f(xn)→f(x0)⇒xn→x0.
When C =X we get the definition of a strongly minimum point ([3], I, 1, above 1.2).
In the same manner we can definef strongly exposesC from above atx0. Theorem 4 (Ghoussoub-Maurey linear principle). Let X be a reflex- ive separable space and ϕ : X → (−∞,+∞] a lower semicontinuous and proper function.
(I)Ifϕis bounded from below on the closed bounded nonempty subsetC, the set {ξ ∈ X∗ : ϕ+ξ strongly exposes C f rom below} is an everywhere dense Gδ-set.
(II)If, for anyξ fromX∗,ϕ+ξ is bounded from below, the set {ξ ∈X∗ : ϕ+ξ strongly exposes X f rom below} is an everywhere dense Gδ-set.
Theorem 4 follows from the more general Theorem 5 and we pass to the preparation of this with definitions and some auxiliary results.
Definition.LetXbe a real normed space andC, DwithC⊂Dnonempty subsets of X∗.C is astrict w-Hδ set in Dresp. strictw∗-Hδ set inD if
(50) D\C =
∞
[
n=1
Kn,
dist(Kn, C)>0,Kn convex and weakly compact resp. ∗-weakly compact.
We exemplify this definition by
Proposition 2.Any nonempty closed setCof a separable reflexive space X, C 6=X, is a strict w-Hδ set inX. In particular, if ϕ:X→(−∞,+∞] is lower semi-continuous and proper, then the epigraph of ϕin X×Ris a strict w-Hδ set inX×R.
Proof. Let (xn)n≥1be a sequence such that the set of its terms is dense in X\C (open set). For eachnfromN, take a closed ballKncentred atxnwith radius rn:= 14dist(xn, C). Kn is convex, weakly compact (Kakutani-˘Smulian theorem, [5], vol. III, p. 750) and dist(Kn, C)> rn: letxfromKn, dist(x, C)≥ dist(xn, C)−dist(x, xn) ≥ 4rn−rn = 3rn. Take the greatest lower bound.
Moreover, X\C =
∞
S
n=1
Kn; let x be arbitrary from X\C, ∃ a subsequence (xpn)n≥1 of (xn)n≥1 such that xpn → x, which also implies dist(xpn, C) → dist(x, C), and ifu, v are taken such that 0< u < v <dist(x, C), from some rank on, we have 4dist(xpn, x)< u, but on the other hand 4rn= dist(xpn, C)>
v, hencex∈Kpn.
LetX be a reflexive space and C, D subsets ofX∗,C⊂D. Set M(C, D) :={x∈X: ∃ξ∈C such thatJ x(ξ)≥J x(η)∀η∈D}.
In other words,M(C, D) is the set ofxfromXfor whichJ xis upper bounded on D and the least upper bound is attained at a point of C, J the Hahn imbedding ofX intoX∗∗. So,X being reflexive,J is an isomorphism of vector spaces which preserves the norms.
In the following, sometimes x designates J x, when no possibility of con- fusion may occur.
Note that ifC is ∗-weakly compact thenM(C, D) is closed.
Notation. SX(x0, r)≡closed ball centred atx0 of radiusrin the normed space X.
SX ≡SX(0,1).
E∗ ≡the closure of the subset E from X∗ for the∗-weak topology.
We give now some auxiliary results.
Proposition 3. Let X be a reflexive space, D ⊂ X∗ and K ⊂ D, K is convex ∗-weakly compact. If SX(x, α) ⊂ M(K, D), then, for any ε > 0, S(D, J x, ε) ⊂ K + αεSX∗. In particular, when C ⊂ D ⊂ conv∗C, we have dist(K, C) = 0.
Proof. First assertion. This is equivalent to
(51) ξ /∈K+ ε
αSX∗ ⇒ξ /∈S(D, x, ε).
Suppose for a contradiction that ξ ∈S(D, x, ε), i.e. (see (2))
(52) x(ξ)>supx(D)−ε (x∈M(K, D)⇒J x(D) upper bounded).
The left hand side of (51) yields
(53) kξ−ηkX∗ > ε
α ∀η∈K.
Let z be from X such that
(54) kJ zkX∗∗(=kzk) = 1 and J z(ξ−η) =kξ−ηkX∗
(Hahn lemma, [5], Vol. III, p. 697). Combining with (53) and taking the least upper bound, we obtain
(55) supz(K)≤z(ξ)− ε
2.
But x+αz(54)∈ SX(x, α)⊂M(K, D), hence∃η0 inK such that (56) (x+αz)(η0)≥(x+αz)(ξ).
But
(x+αz)(ξ) =x(ξ) +αz(ξ)(52),(55)> [supx(D)−ε] +α h
supz(K) + ε α i
≥
≥x(η0) +z(η0) and we contradict (56), so that (51) is proved.
Second assertion. This follows from
(57) \
ε>0
S(D, x, ε)⊂K and
(58) \
ε>0
S(D, x, ε)∩C 6=∅.
For (57): ξ ∈ T
ε>0
S(D, x, ε)(51)⇒ ξ = ηε + αεuε, ηε ∈ K, kuεk ≤ 1, hence kξ−ηεk ≤ αε, ξ is a strong adherent point ofK and the strong closure of K is included in the ∗-weak closure of this, that is, in K.
Proposition 4. LetX be a reflexive space, C⊂X∗ a nonempty set and U ⊂X a nonempty open set such thatsupJ x(C)<+∞ ∀x∈U.Then for any x fromU, J x is upper bounded on conv∗C and attains its least upper bound.
Proof. SetD:= conv∗C. We have
(59) supJ x(C) = supJ x(D).
[ξ∈convC⇒ξ=λ1ξ1+λ2ξ2,ξ1, ξ2∈C,λ1+λ2 = 1,λ1, λ2≥0⇒J x(ξ) = λ1J x(ξ1) +λ2J x(ξ2) ≤ supJ x(C); ξ ∈ D ⇒ ∃ξn ∈ convC, ξn∗-weak
−→ ξ ⇒ ξn(x)→ξ(x), ξn(x)≤supJ x(C)∀n≥1, hence ξ(x)≤supJ x(C)] and so (60) supJ x(D)<+∞ ∀x∈U,
and the first assertion is proved.
We pass to the proof of the second assertion. Fixx fromU,∃ε >0 with x+εz∈U ∀z inSX. Then (see (60)) we have
supJ(x+εz)(D) = sup(J x+εJ z)(D)<+∞ ∀z∈SX. Consequently, using once again (60) we get
(61) supJ z(D)<+∞ ∀z∈SX, which implies
(62) supJ z(D)<+∞ ∀z∈X
[for any fixed z 6= 0, replace z by kzkz in (61), J y(ξ) =ξ(y)]. Replacing z by
−z in (62) we get
(63) infJ z(D)>−∞ ∀z∈X.
ButX is reflexive, hence (62) and (63) show that Dis weakly bounded, consequently D is even bounded.
D being also ∗-weak closed, it is ∗-weak compact ([5], Vol. III, p. 744), hence the conclusion follows by applying the Weierstrass theorem.
Remark. Proposition 4 is Lemma 2.7 from [2, Ch. 2]. The proof of this in [2] is not correct.
Proposition 5. Let X be a reflexive space, C ⊂ X∗ nonempty and U a nonempty open subset of X such that supJ x(C) <+∞ ∀x ∈U. If C is a strict w∗-Hδ set inD:= conv∗C, then the set
V :={x∈U :J x attains supJ x(D)in C}
contains a Gδ-set dense in U.
Proof. According to the definition, (64) D\C=
∞
[
n=1
Kn, Kn convex∗-weakly compact and dist(Kn, C)>0.
Every J x, x ∈ U, is upper bounded on D and attains its supremum there (Proposition 4). As dist(Kn, C) > 0, Proposition 3 prevents M(Kn, D) to include any nonempty ball, in other words ∀n≥1 intM(Kn, D) = ∅, and so M(Kn, D), being also closed, is thin and Un := X\M(Kn, D) is open and
dense in X. Then,
∞
T
n=1
Un is dense in X (Baire theorem), hence
∞
T
n=1
(U ∩Un), a Gδ-set, is dense in U. We see that if x ∈
∞
T
n=1
(U ∩Un) , then J x, which is upper bounded on D, attains a fortiori its least upper bound on C, as x /∈M(Kn, D)∀n≥1 and taking into account (64).
Proposition 6. Let X be a reflexive space, C a subset of X∗ separable, D:= conv∗C andU a nonempty open subset ofX such thatsupJ x(C)<+∞
∀x ∈ U. Suppose that M(C, D) includes a dense Gδ-subset of U. Then, for any K⊂D ∗-weakly compact with K∩C=∅ and for any ε >0, the set
G(K, ε) :={x∈U :∃r >0 such that S∗(D, J x, r)∩K=∅ anddiamS∗(D, J x, r)< ε}
is open and dense in U.
Proof. G(K, ε) is open. Use the fact that, D being bounded (proof of Proposition 4), the subsetS(D, J x, r) also is bounded.
G(K, ε)is dense inU. LetV ⊂U be a nonempty open set. Cbeing sepa- rable, we can find a sequence (Cn)n≥1,Cn⊂D, Cn convex∗-weakly compact, such that
C⊂
∞
[
n=1
Cn, (65)
dist(Cn, K)>0 ∀n, (66)
diam(Cn)≤ ε 2 ∀n.
(67)
Obviously, we have V ⊃
∞
S
n=1
M(Cn, D)∩V (65)⊃ M(C, D) ∩V. But the last member includes, according to the hypothesis, a Gδ-set dense in V, which forces, via the Baire theorem, the second member to have at least one term.
Let this be M(Cn0, D)∩V, with nonempty interior. From this interior take a point x. It remains to show that
(68) x∈G(K, ε).
By Theorem 3, for every ρ >0 there exists r >0 such that (69) S(D, J x, r)⊂Cn0 +ρSX∗
(∃α >0 such thatSX(x, α)⊂M(Cn0, D), take r=αρ). Take
(70) ρ <minnε
4,dist(Cn0, K)o .
Then (69) yields
(71) S∗(D, J x, r)⊂Cn0+ρSX∗∗ =Cn0 +ρSX∗,
because the last term being ∗-weakly compact, it is∗-weakly closed, too. But (72) (Cn0 +ρSX∗)∩K =∅:
for a contradiction, let ξ +ρu = ζ, ξ ∈ Cn0, u ∈ SX∗, ζ ∈ K; then ρ = kξ −ζk ≥ dist(Cn0, K) and this contradicts (70). So, (71) and (72) yield S∗(D, J x, r)∩K =∅. Moreover,
diamS∗(D, J x, r)
(71)
≤ diamCn0 + 2ρ(67),<(70)ε 2 +ε
2 =ε, which completes the proof.
And now we can state another result of this section.
Theorem 5.LetX be a reflexive space,Ca separable subset ofX∗ which is a strict w∗-Hδ set in D := conv∗C and U an open subset of X such that supJ x(C)<+∞ ∀x∈U. Then
(I) the set {x ∈U : J x strongly exposes D from above at a point of C}
is a Gδ-set dense in U;
(II)ifϕ:C →(−∞,+∞]is proper lower semicontinuous and ϕ+J xis,
∀x from X, bounded from below on C, then the set {x ∈X : ϕ+J x strongly exposes C from below} is a Gδ-set dense inX.
Proof. (I). According to the hypothesis, D\C =
∞
S
n=1
Kn, Kn convex
∗-weakly compact, dist(Kn, D) > 0, M(C, D) includes a Gδ-subset dense in U (Proposition 5); but then, for eachn from N, the set Vn := G(K1∪K2∪ . . .∪Kn,n1) is open and dense in U (Proposition 6), consequently
∞
T
n=1
Vn is dense in U (relativized Baire theorem) and it only remains to observe that
∞
T
n=1
Vn={x∈X :J xstrongly exposes Dfrom above at a point of C}.
(II).C×R, a separable subset ofX∗×R, is a strictw∗-Hδ set inD×R. Then,ϕbeing l.s.c., the epigraph epiϕinC×R(nonempty set,ϕis proper) is a strictw∗-Hδset inD×R, hence also in conv∗epiϕ. Next,W :={(x, α) :x∈ X, α <0}is open inX×Rand sup(J x, α)(epiϕ)<+∞ ∀(x, α)∈W [(J x, α), a continuous linear functional, acts onC×Rby the rule (J x, α)(ξ, λ) =J x(ξ)+
αλ]. Indeed,J x(ξ) +αλ≤J x(ξ) +αϕ(ξ), ∃ainRsuch that Jαx(ξ) +ϕ(ξ)≥ a ∀ξ ∈C (the hypothesis), hence J x(ξ) +αϕ(ξ) ≤αa ∀ξ ∈C. Let us show that for each ε >0∃y0 with ky0k ≤2εand ϕ+J y0 strongly exposesC from below, which is enough to validate (II). Apply (I), ∃(xε, αε) in W such that (73) k(xε, αε)−(0,−1)k ≤ε
((0,−1) ∈ W !) and (J xε, αε) strongly exposes epiϕ from above at a point (ξ0, λ0). Then∀(ξ, λ) from epiϕwithξ6=ξ0we haveJ xε(ξ0)+αελ0> J xε(ξ)+
αελ.Consequently, takingy0 := αxε
ε, in particular we have ϕ(ξ0) +J y0(ξ0)< ϕ(ξ) +J y0(ξ) ∀ξ∈C\ {ξ0},
ξ0 is a strict global minimum point forϕ+J y0. Moreover, as one can suppose ε < 12, we haveky0k= kx|αεk
ε| ≤2ε, because, by (73),kxεk ≤εand |αε+ 1| ≤ε, hence αε ∈(−32,−12).
Finally, let (ξn)n≥1 be a minimizing sequence for ϕ+J y0 on C. Then (ξn, ϕ(ξn))n≥1 is a maximizing sequence for (J xε, αε) which strongly exposes epiϕin (ξ0, λ0), which imposes ξn→ξ0.
Proof of Ghoussoub-Maurey linear principle (Theorem 4) (I). Set Y :=
X∗, a separable reflexive space [5, Vol. III, p. 761]. Then Y∗ =X (identifi- cation via the Hahn embedding, X is reflexive). C is separable and a strict w∗-Hδ set in Y∗ (Proposition 2, the weak and ∗-weak topologies coincide), hence also in D := conv∗C (X\C =
∞
S
n=1
Kn with the properties from (50), take the intersection with D). Apply (II), Theorem 5 transcribed with Y re- placed by X, this is correct asϕ+ξ, ξ∈ X∗ =Y∗∗, is bounded from below (|ξ(x)| ≤ kξk kxk and C is bounded).
(II). The epigraph epiϕ of ϕ in X×R is a strict w-Hδ set in X×R (Proposition 2). Next, we use the proof for (II), Theorem 5 begining from (73), epiϕis that considered above.
Corrolary. Let X be a reflexive space, C a subset of X∗ that is a separable bounded strict w∗-Hδ set in D:= conv∗C and ϕ:X → (−∞,+∞]
bounded from below l.s.c. proper. For any ε > 0 there exist x0 in X with kx0k ≤ε andξ0 in C such that
1◦ (ϕ+J x0)(ξ0)<(ϕ+J x0)(ξ)∀ξ∈C\ {ξ0},
2◦ any minimizing sequence from C for ϕ+J x0 converges to ξ0.
Proof. C bounded implies ϕ+J x bounded from below ∀x ∈X, conse- quently II, Theorem 5 can be used in order to obtain 1◦ and 2◦.
6. APPLICATION
We apply Theorem 4 in a generalization of a minimization problem from [1], in its generalized form
(74) Cf := min (Z
Ω
1 p
N
X
i=1
∂u
∂xi
p
−f(u)
dx:u∈W01,p(Ω), kuk2∗ = 1 )
,
where Ω is open set of C1-class in RN,N ≥3, f ∈W−1,p0(Ω)(= (W01,p(Ω))∗),
1
p+p10 = 1, 2∗ = N−22N – the critical exponent of the Sobolev embedding (for the necessary explanations, here and in the following, see [3], I,§4, last section).
Let Ω be an open bounded set of C1 class in RN, N ≥ 3. Consider the problem
(75)
(−∆spu=f(x, u) in Ω,
u= 0 on ∂Ω,
where f : Ω×R→Ris a Carath´eodory function satisfying the growth condi- tion
(76) |f(x, s)| ≤c|s|p−1+b(x) with
c >0, 2≤p≤ 2N
N−2, b∈Lp0(Ω), 1 p + 1
p0 = 1.
The functionalϕ:W01,p(Ω)→R defined by
(77) ϕ(u) =
Z
Ω
1 p
N
X
i=1
∂u
∂xi
p
−F(x, u(x))
! dx with
F(x, s) :=
Z s
0
f(x, t)dt,
is of C1-Fr´echet class and its critical points are the weak solutions of (75).
Let λ1 be the first eigenvalue of −∆sp in W01,p(Ω) with homogeneous boundary conditions. We have
(78) λ1 = inf
( |u|p1,p
ki(u)kp0,p :u∈W01,p(Ω)\ {0}
)
(see [4], 7.2). We should note that |u|1,p= N
P
i=1
k∂x∂u
ikpLp(Ω)1p
and the dual of (W01,p(Ω),| · |1,p) isW−1,p0(Ω), wherep0 is the conjugate ofp(i.e., 1p+p10 = 1), i:W01,p(Ω)→Lp(Ω) is a linear compact embedding.
And now we can state
Proposition 7. Under the above assumptions and in addition the growth condition
(79) F(x, s)≤c1
s2
2 +α(x)s
with 0 < c1 < λ1 and α ∈ Lq0(Ω) for some 2 ≤ q ≤ N−22N and f(x,−s) =
−f(x, s), ∀x from Ω,∀sfrom R, the following assertions hold.
1◦ The set of functions h from W−1,p0(Ω) such that the functional ϕh : W01,p(Ω)→R, defined as
ϕh(u) = Z
Ω
1 p
N
X
i=1
∂u
∂xi
p
−F(x, u(x)) +h(u(x))
! dx
attains its minimum at only one point includes an everywhere dense Gδ-set.
2◦ The set of functions h from W−1,p0(Ω)such that the problem (−∆psu=f(x, u) +h(u) in Ω,
u= 0 on∂Ω
has solutions in W01,p(Ω) in the sense of W−1,p0(Ω) includes an everywhere dense Gδ-set.
3◦ Moreover, ifs→f(x, s)is increasing, then the set of functionshfrom W−1,p0(Ω) such that the problem
(−∆psu=f(x, u) +h(u) in Ω,
u= 0 on∂Ω
has a unique solution in W01,p(Ω) in the sense of W−1,p0(Ω) includes an everywhere dense Gδ-set.
Proof. It is sufficient to justify 1◦. For each h from W−1,p0(Ω) consider the functionalξh∈W−1,p0(Ω) defined by
ξh(u) = Z
Ω
h(u(x))dx, u∈W01,p(Ω).
It is obvious thatϕh =ϕ+ξh(see (77)). Consequently, according to (II), Theo- rem 4, if we show thatϕhis bounded from below for anyhfromW−1,p0(Ω) (this is enough by the Riesz representation theorem), then 1◦ is proved. However, taking into account the Sobolev embedding and (79), ∀u∈W01,p(Ω) we have
(ϕ+ξh)(u)≥ Z
Ω
1 p
N
X
i=1
∂u
∂xi
p
−c1|u(x)|2
!
dx− kαkq0kukq−
−khkW−1,p0kukW1,p
0 ≥ 1
2
1− c1 λ1
Z
Ω
1 p
N
X
i=1
∂u
∂xi
p
dx−rkukW1,p
0 , r ∈R,
hence the conclusion, because 1−λc1
1 >0.
7. CONCLUSIONS
A general variational principle was recovered by inserting some ma- jor changes.
An example of admissible family was reconsidered in order to show that the recovered variational principle is a generalization of the Deville-Godefroy- Zizler theorem.
Two variants of generalized (perturbed) variational principle (one of them linear) are presented (Theorems 3 and 4).
The proof of Theorem 4 was improved by giving a correct demonstration for Proposition 4.
For Proposition 4, an original proof was given in order to strengthen the construction of the proof of Theorem 4.
A novel application of Theorem 4 was proposed. We applied Theorem 4 in a minimization problem which generalizes a minimization problem due to Br´ezis and Nierenberg [1]. We characterized a set of functions which are involved in some equations involving the pseudo-Laplacian.
REFERENCES
[1] H. Brezis and L. Nirenberg,A minimization problem with critical exponent and non-zero data. Symmetry in Nature, Scuola Norm. Sup. Pisa (1989), 129–140.
[2] N. Ghoussoub,Duality and Perturbation Methods in Critical Point Theory. Cambridge Univ. Press, Cambridge, 1993.
[3] I. Meghea,Ekeland Variational Principles with Generalizations and Variants. Old City Publishing, Philadelphia & Editions des Archives Contemporaines, Paris, 2009.
[4] I. Meghea,Surjectivity and Fredholm alternative type theorems for operators of the form λJϕ−S, Jϕ duality map andS Nemytskii operator. PhD Thesis, 1999.
[5] C. Meghea and I. Meghea, Differential Calculus and Integral Calculus, Vol. I, Ed. Teh- nic˘a, Bucharest, 1997; Vol. II, Ed. Tehnic˘a, Bucharest, 2000; Vol. III, Ed. Printech, Bucharest, 2002.
[6] R.R. Phelps,Convex Functions, Monotone Operators and Differentiability, 2nd Edition.
Lecture Notes in Math.1364. Springer-Verlag, Berlin, 1993.
Received 14 September 2008 “Politehnica” University of Bucharest Faculty of Applied Sciences Department of Mathematics II
Splaiul Independent¸ei 313 060042 Bucharest, Romania irina.meghea@upb.ro; i−meghea@yahoo.com
