# Coercivity Properties for Sequences of Lower Semicontinuous Functions on Metric Spaces

Authors: D. Motreanu; V. V. Motreanu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The paper presents various results studying the asymptotic behavior of a sequence of lower semicontinuous functions on a metric space. In particular, different coercivity properties are obtained extending and refining previous results. The specific features and the structure of the terms of the sequence are used to construct appropriate quantities relevant in the verification
of Palais-Smale compactness type conditions.


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## Body

## 1. Introduction and Main Results

LetX be a complete metric space endowed with the metric d. We recall from De Giorgi et al. [1] the notion of strong slope of a lower semicontinuous function f:X→ℝ∪{+∞} (which is not identically +∞) at a point u∈dom(f):={w∈X:f(w)<+∞}:
(1)|∇f|(u)≔{0,ifuisalocalminimizeroff,limsupv→uf(u)-f(v)d(u,v),otherwise.
If X is a Banach space and f∈C1(X,ℝ), then |∇f|(u)=∥f′(u)∥ for all u∈X.LetF:X→ℝ be a function satisfying the property.H
(
F
): there exist constants γ1,γ2>0 such that
(2)d(u,v)<γ1⟹|F(u)-F(v)|<γ2(foru,v∈X).
Let fn:X→ℝ∪{+∞} (n∈ℕ) be a sequence of lower semicontinuous functions. This paper develops a general approach for studying the asymptotic behavior of this sequence with respect to F. An aspect which makes our approach general and natural is that we do not require the sequence {fn} to admit a limit (in any sense); see also Remarks 7, 9, and 11 below. We introduce the notation
(3)αF({fn}):=limr→+∞liminfn→∞inf[F>r]¯fn.Here and throughout the paper, for allr>0 we denote
(4)[F>r]:={w∈X:F(w)>r},
while [F>r]¯ and int[F>r] stand for the closure and the interior of [F>r], respectively. The expression αF({fn}) always exists, generally belonging to ℝ∪{±∞}. Equivalent expressions to αF({fn}) can be given, for instance replacing [F>r]¯ by [F>r] or by [F≥r]:={w∈X:F(w)≥r} (see Lemma 16 below).In what follows, we will always assume that.H
F
(
{
f
n
}
): there holds αF({fn})>-∞.For instance,HF({fn}) is satisfied if {fn} is uniformly bounded below.In the following we state our main result, which studies the asymptotic behavior of a sequence of lower semicontinuous functions.Theorem 1.
LetX be a complete metric space, let F:X→ℝ be a function satisfying H(F), and let fn:X→ℝ∪{+∞} (n∈ℕ) be a sequence of lower semicontinuous functions satisfying αF({fn})∈ℝ. Then, for every ε>0, there exist a subsequence {fnk} (depending on ε) of {fn} and a number kε∈ℕ such that for each k≥kε one finds uk,ε∈dom(fnk) satisfying
(5)αF({fn})-ε2<fnk(uk,ε)<αF({fn})+ε2,(6)|∇fnk|(uk,ε)≤ε,(7)F(uk,ε)>1ε.
In particular, there are a subsequence {fmℓ} of {fn} and elements uℓ∈dom(fmℓ) such that
(8)fmℓ(uℓ)⟶αF({fn}),|∇fmℓ|(uℓ)⟶0,F(uℓ)⟶+∞asℓ⟶∞.The proof of Theorem1 is done in Section 2.Note that, due to the hypothesis thatαF({fn})∈ℝ, at least a subsequence of functions fn is not identically +∞ and the sets [F>r] are nonempty for all r∈ℝ.We say that a sequence{un}⊂X is F-bounded if the sequence {F(un)}⊂X is bounded. We introduce the following notions of Palais-Smale condition and coercivity relative to the function F.Definition 2.
Letfn:X→ℝ∪{+∞} (n∈ℕ) be lower semicontinuous functions which are not identically +∞. We say that the sequence {fn} satisfies the Palais-Smale condition relative to F (condition (PS)F, for short) if whenever {fnk} is a subsequence of {fn} and {uk}⊂X is a sequence such that {fnk(uk)} is bounded and |∇fnk|(uk)→0 as k→∞, then {uk} is F-bounded.Definition 3.
Assume that the functionF satisfies in addition the requirement that supXF=+∞. We say that the sequence fn:X→ℝ∪{+∞} (n∈ℕ) is  F-coercive if αF({fn})=+∞.
If(X,∥·∥) is a Banach space, F=∥·∥, and fn≡f for all n∈ℕ, then we retrieve the usual notion of coercivity.
We state the following result on theF-coercivity of a sequence of lower semicontinuous functions.Corollary 4.
LetX be a complete metric space, let F:X→ℝ be a function satisfying H(F) and supXF=+∞, and let fn:X→ℝ∪{+∞} (n∈ℕ) be a sequence of lower semicontinuous functions satisfying HF({fn}). If {fn} satisfies condition (PS)F, then the sequence {fn} is F-coercive.The proof of Corollary4 is given in Section 2.As an immediate consequence of Corollary4, in a Banach space we have the following.Corollary 5.
Let(X,∥·∥) be a Banach space, and let fn:X→ℝ∪{+∞} (n∈ℕ) be a sequence of lower semicontinuous functions satisfying H∥·∥({fn}). If {fn} satisfies condition (PS)∥·∥, then the sequence {fn} is ∥·∥-coercive.Consider now the particular case in Theorem1 when the number αF({fn}) in HF({fn}) is also a liminf of a given lower semicontinuous function f:X→ℝ∪{+∞}. Setting
(9)αF(f):=liminfF(v)→+∞f(v)=deflimr→+∞inf[F>r]¯f,
the hypothesis αF({fn})∈ℝ in Theorem 1 is obviously satisfied if we assume the conditions
(10)αF({fn})=αF(f);αF(f)∈ℝ.
It will be noted in Lemma 19 that the first condition in (10) is satisfied if the following is assumed.H
F
(
{
f
n
}
,
f
): there exists r0∈ℝ such that for every r≥r0 one has(i)
for everyu∈dom(f)∩[F>r]¯, there exists a sequence {un} such that un→u and fn(un)→f(u) as n→∞;(ii)
liminfn→∞inf[F>r]¯fn≥inf[F>r]¯f.Concerning the second condition in (10), we have the following simple characterization: given c∈ℝ, we have that c=αF(f) if and only if,
(11)∀a<c,thereexistsr∈ℝsuchthat[f≤a]⊂[F≤r];∀a>c,∀r∈ℝ,wehave[f≤a]⊄[F≤r]
(see Lemma 17 below). With the above comments, the following result is a consequence of Theorem 1.Corollary 6 (Corvellec [2, Theorem 1′]).
LetX be a complete metric space, let F:X→ℝ be a function bounded on bounded subsets of X and satisfying H(F), and let f,fn:X→ℝ∪{+∞} (n∈ℕ) be lower semicontinuous functions satisfying that,
(12)forallu∈dom(f),thereexistsun⟶uwithfn(un)⟶f(u),(13)forallclosedY⊂X,liminfn→∞(infYfn)≥infYf.
Assume (11) for some c∈ℝ. Then there exist a subsequence {fnk} of {fn} and a sequence {uk}⊂X such that fnk(uk)→c, |∇fnk|(uk)→0, and F(uk)→+∞ as k→∞.Remark 7.
The numberc∈ℝ in Corollary 6 is necessarily c=αF(f)=αF({fn}). Moreover, hypotheses (12)-(13) are particular cases of HF({fn},f) (i)-(ii) that involve only sets of the form Y=[F>r]¯. Hence the hypotheses of Corollary 6, namely, (11), (12), (13), and c∈ℝ, imply that αF({fn})∈ℝ. Therefore, [2, Theorem 1′] (i.e., Corollary 6) is retrieved as a consequence of Theorem 1 (and then the hypothesis that F is bounded on bounded subsets of X is not even needed). As seen from Example 12, Theorem 1 is actually more general than [2, Theorem 1′], and besides it does not need an auxiliary function f in its hypotheses.
Next, we note that the hypothesisHF({fn}) in Corollary 4 is satisfied if we assume the conditions
(14)αF({fn})=αF(f),αF(f)>-∞.
As noticed above, the first condition in (14) is satisfied if HF({fn},f) holds, which in turn is satisfied if (12) and (13) are assumed. The second condition in (14) is equivalent to the property [F>b]⊂[f>a] for some a,b∈ℝ; that is,
(15)[f≤a]⊂[F≤b],forsomea,b∈ℝ
(see Lemma 18). We thus have the following consequence of Corollary 4. Here, as in [2], it is said that f satisfies condition (PSB)* if whenever {fnk} is a subsequence of {fn} and {uk}⊂X is a sequence such that {fnk(uk)} is bounded and |∇fnk|(uk)→0 as k→∞, then {uk} is bounded.Corollary 8 (Corvellec [2, Theorem 1]).
LetX be a complete metric space, let F:X→ℝ be a function bounded on bounded subsets of X satisfying H(F) and supXF=+∞, and let f,fn:X→ℝ∪{+∞} (n∈ℕ) be lower semicontinuous functions satisfying (12) and (13). If f satisfies (15) and condition (PSB)*, then f is F-coercive (i.e., f(u)→+∞ as F(u)→+∞).Remark 9.
From the above discussion, Corollary8 (i.e., [2, Theorem 1]) is obtained as a consequence of Corollary 4. In fact, Corollary 4 is more general (see Example 12) and does not rely on an auxiliary function f. In fact, on the one hand to study the coercivity of a function f we do not need to look for a sequence {fn} as in Corollary 8 (in applications it seems to be more difficult to prove the existence of a sequence {fn} related to the function f as in Corollary 8 than to prove the coercivity of f itself). On the other hand, while studying the coercivity of a sequence {fn}, the interest of Corollary 4 is to give sufficient conditions for the coercivity of the sequence {fn} without using an auxiliary function f. Finally, we note that in addition to the F-coercivity of f, the hypotheses of Corollary 8 imply also the F-coercivity of {fn}, and so αF({fn})=αF(f)=+∞.We also recall the following.Corollary 10 (Corvellec [2, Corollary 1]).
Let(X,∥·∥) be a Banach space, let f,fn:X→ℝ∪{+∞} (n∈ℕ) be bounded below, lower semicontinuous functions satisfying (12) and (13). Then f is ∥·∥-coercive if and only if f satisfies condition (PSB)*.The sufficiency in Corollary10 follows from Corollary 5 (or from Corollary 8). The necessity can be proved arguing by contradiction in the following way. If there are a subsequence {fnk} of {fn} and a sequence {uk}⊂X such that {fnk(uk)} is bounded, |∇fnk|(uk)→0 and ∥uk∥→+∞ as k→∞, then f(uk)→+∞ as k→∞ (by the ∥·∥-coercivity of f). Then, for every n∈ℕ, using (13) with the closed set Yn:={un,un+1,…}, we have
(16)liminfk→∞fnk(uk)≥liminfk→∞(infYnfnk)≥liminfk→∞(infYnfk)≥infYnf⟶+∞liminfxxxxk→∞(infYnfk)≥11asn⟶∞,
which contradicts the boundedness of the sequence {fnk(uk)}.Remark 11.
Hypotheses (12) and (13) imply the first part of (10); that is, α∥·∥({fn})=α∥·∥(f). Hence, in view of Corollary 5, in place of assuming that f and fn are bounded below in [2, Corollary 1] it would have been enough to assume that α∥·∥(f)>-∞, which in fact is implied just by the boundedness below of f. Corollary 5 is more general than Corollary 10 and its advantage is that it studies the coercivity of a sequence {fn} without dealing with an auxiliary function f. For the study of the coercivity of a function f we do not need to involve a sequence of functions {fn} (see Corollary 31 below).Example 12.
(a) Letf:ℝ→ℝ∪{+∞} be a lower semicontinuous, even (i.e., f(x)=f(-x) for all x∈ℝ) function which is not identically +∞ on ℝ∖{0}, let the lower semicontinuous functions fn:ℝ→ℝ∪{+∞} (n∈ℕ) be given by
(17)fn(x)={f(x),ifx≥0,+∞,ifx<0,
and let F(x)=|x| for all x∈ℝ. Then we have αF({fn})=αF(f). Condition HF({fn},f) (i) is not satisfied (thus, (12) is not satisfied) since, if x∈dom(f), x<0, then for every sequence xn→x we have fn(xn)=+∞>f(x) for n∈ℕ large enough. So Corollaries 6, 8, and 10 cannot be applied, while Theorem 1 can be applied whenever αF(f)∈ℝ. Corollaries 4 and 5 can also be applied.
(b) Letf:ℝ→ℝ be defined by f(x)=-e-2x2 for all x∈ℝ, let fn:ℝ→ℝ (n∈ℕ) be defined by fn(x)=-e-x2-(1/n) for all x∈ℝ, and let F(x)=|x| for all x∈ℝ. Then we have αF({fn})=αF(f)=0. Condition HF({fn},f) (ii) is not satisfied (so neither (13)) since for every r>0 we have
(18)liminfn→∞inf|x|≥rfn(x)=-e-r2<-e-2r2=inf|x|≥rf(x).
Hence we may apply Theorem 1 to the sequence {fn}, but not Corollary 6 with {fn} and the chosen f. Besides being more general, the advantage of Theorem 1 is to study the asymptotic behavior of a sequence {fn} without an auxiliary function f (if it exists) as in Corollary 6.
(c) Letf:ℝ→ℝ be defined by f(x)=e|x| for all x∈ℝ, let fn:ℝ→ℝ (n∈ℕ) be defined by fn(x)=(x2/2)-(1/n) for all x∈ℝ, and let F(x)=|x| for all x∈ℝ. Then αF({fn})=αF(f)=+∞. Condition HF({fn},f) (ii) is not satisfied (so neither (13)) since for every r>0 we have
(19)liminfn→∞inf|x|≥rfn(x)=r22<er=inf|x|≥rf(x).
Corollary 4 (or Corollary 5) can be applied to {fn} (note that {fn} satisfies condition (PS)F), while Corollaries 8 and 10 cannot be applied to {fn} and f (however, f is coercive).
(d) Let(X,∥·∥) be a Banach space and F:X→ℝ be a continuous function satisfying H(F). Let g:ℝ→ℝ∪{+∞} and h:ℝ→ℝ be nondecreasing, lower semicontinuous functions with the property that there exists r0>0 such that h(r)<g(r) for all r≥r0 and limr→+∞h(r)=limr→+∞g(r)>-∞. Define f=g∘F and let fn:ℝ→ℝ (n∈ℕ) be given by
(20)fn(x)={h(F(x))+sn,ifF(x)≤n,+∞,ifF(x)>n,
where {sn}⊂ℝ. Suppose liminfn→∞sn=0. Then αF({fn})=αF(f), but condition HF({fn},f) (ii) is not satisfied (so neither (13)) since for every r>r0 we have
(21)liminfn→∞infF(x)≥rfn(x)=h(r)<g(r)=infF(x)≥rf(x).
So, neither Corollary 6 nor Corollary 8 can be applied. Theorem 1 can be applied whenever αF({fn})=limr→+∞h(r)+liminfn→∞sn∈ℝ (e.g., we do not need that liminfn→∞sn=0, and then αF({fn})=αF(f)+liminfn→∞sn) and its application is not related to some auxiliary function f. For Corollary 4, we must necessarily have αF({fn})=+∞.Example 13.
(a) As examples of functionsF:X→ℝ satisfying H(F), we can consider any Lipschitz continuous function on a metric space X, or any uniformly continuous function. For example, if X is a metric space endowed with the metric d, then the function
(22)F(u)=d(u,A),∀u∈X,
for some nonempty subset A of X, satisfies hypothesis H(F) with any γ1=γ2, and if X∖A is unbounded, then F satisfies also that supXF=+∞ (so both Theorem 1 and Corollary 4 can be applied in this case). In particular, in the case where (X,∥·∥) is a Banach space, the function F=∥·∥ satisfies H(F) and that supXF=+∞.
(b) Note that ifA is a bounded subset of the metric space X, then the expression αd(·,A)({fn}) (i.e., αF({fn}) for F(·)=d(·,A)) and the notion of d(·,A)-coercivity do not depend on the choice of the set A (for this reason, we refer to d-coercivity in place of d(·,A)-coercivity). If A is unbounded, then it is not anymore the case: for example, if X=ℝ2, A={(x,y)∈ℝ2:y=0}, and f:ℝ2→ℝ given by f(x,y)=|y|, then f is d(·,A)-coercive, but it is not norm coercive (taking the Euclidean norm and denoting by d the induced distance).
(c) As another example ofF (which is not even continuous), let X=ℝ, F:ℝ→ℝ given by
(23)F(x)={0,if|x|<1,x,if|x|≥1.
The function F satisfies H(F) with γ1=1, γ2=2. Note that for every r∈ℝ, we have F(r+1)>r (so the sets [F>r] are nonempty).Remark 14.
(a) IfF:X→ℝ and F~:X→ℝ are two functions satisfying H(F) and that supXF=supXF~=+∞, and if F-F~ is bounded, then the F-coercivity and the F~-coercivity of a sequence {fn} as in Corollary 4 are equivalent.
(b) IfX is a metric space endowed with two metrics d and d~ which induce the same topology, then a sequence {fn} as in Corollary 4 may be d-coercive and non-d~-coercive.
(c) LetX be a metric space endowed with the metric d, let F:X→ℝ be a function satisfying H(F) and supXF=+∞, and let us define a new metric:
(24)d~(x,y)=d(x,y)+|F(x)-F(y)|.
The topology induced by d~ is finer than the topology induced by d (if F is Lipschitz continuous with respect to d, then they are equivalent). Applying Corollary 4 to the metric d~, if d is bounded, then the F-coercivity is equivalent to the d~-coercivity. In this case, Corollary 4 yields that if {fn} is a sequence of lower semicontinuous functions with respect to d (thus with respect to d~) satisfying HF({fn}) and such that condition (PS)F holds with respect to the metric d~, then {fn} is d~-coercive.The rest of the paper is organized as follows.Section2 contains the proofs of the results stated in Section 1, based on the Ekeland variational principle. Our approach in showing  Theorem 1 relies on the ideas in the proof of Motreanu-Motreanu [3, Theorem 3.1], which is a different approach from the one of Corvellec [2].Section3 contains further applications of Theorem 1 and Corollary 4 to special classes of sequences of lower semicontinuous functions. Section 3.1 is concerned with the coercivity of a sequence of lower semicontinuous functions fulfilling HF({fn}) in the case where liminf in the definition of αF({fn}) is actually a limit. Section 3.2 studies the coercivity of a sequence of lower semicontinuous functionals which can be written as a sum of a locally Lipschitz function and a convex, lower semicontinuous function which is not identically +∞. Section 3.3 deals with the coercivity of a continuously differentiable functional f on a Banach space under a Palais-Smale type condition relative to a sequence of Galerkin approximations of f. Section 3.4 deals with the case of a constant sequence of lower semicontinuous functions. Section 3.5 discusses what happens if in place of hypothesis HF({fn}) we consider the case where the limits in αF({fn}) are interchanged.


## 2. Proofs of Theorem1, Corollary 4, and Additional Lemmas

A basic ingredient in proving our results is the following version of the Ekeland variational principle (see Ekeland [4]).Theorem 15.
Let(X,d) be a complete metric space and let f:X→ℝ∪{+∞} be a lower semicontinuous function which is bounded below and not identically +∞. Then for every ε>0 and every v∈X there exists u∈X such that
(25)f(u)≤f(v)-εd(u,v),f(w)>f(u)-εd(w,u),(u)≤f(v)-εd(u,v),f(w)>f∀w∈X∖{u}.We need the following preliminary lemma.Lemma 16.
IfF:X→ℝ satisfies H(F), then for every r∈ℝ one has,(a)
[F>r]¯⊂[F>r-γ2],(b)
[F>r]⊂int[F>r-γ2].Proof.
(a) Letu∈[F>r]¯, and let {un}⊂[F>r] be a sequence such that un→u as n→∞. Since, for n sufficiently large, d(un,u)<γ1, by hypothesis H(F) we have that F(u)>F(un)-γ2>r-γ2.
(b) Letu∈[F>r], and let v∈X be such that d(v,u)<γ1. Then hypothesis H(F) yields F(v)>F(u)-γ2>r-γ2, which completes the proof.Proof of Theorem1.
We denoteα:=αF({fn})∈ℝ. Fix ε∈(0,γ1/2). By the definition of α, it follows that there exists rε>0 such that
(26)α-ε2<liminfn→∞inf[F>r]¯fn<α+ε2,∀r≥rε.
This implies that for every r≥rε we find a number nr,ε∈ℕ such that
(27)α-ε2<inf[F>r]¯fn,∀n≥nr,ε
and a subsequence {fnk(r)} (depending on r) of {fn} and a number kr,ε∈ℕ such that
(28)inf[F>r]¯fnk(r)<α+ε2,∀k≥kr,ε.
Denote(29)r¯ε=max{rε,1ε+γ2,2γ2}.
In particular, from (27), for r=r¯ε, we find a number nr¯ε,ε∈ℕ such that
(30)α-ε2<inf[F>r¯ε]¯fn,∀n≥nr¯ε,ε,
while from (28), for r=2r¯ε we find a subsequence {fnk} (which is the subsequence {fnk(2r¯ε)}) of {fn} and a number k2r¯ε,ε such that
(31)inf[F>2r¯ε]¯fnk<α+ε2,∀k≥k2r¯ε,ε.
Fix k≥kε:=max{nr¯ε,ε,k2r¯ε,ε}. Using (31) we find a point
(32)vk,ε∈[F>2r¯ε]¯
for which one has
(33)fnk(vk,ε)<α+ε2.
Corresponding to the set [F>r¯ε]¯ (see (29)), consider the function f~nk:[F>r¯ε]¯→ℝ∪{+∞} as follows:
(34)f~nk=fnkfnk|[F>r¯ε]-.
It is clear that the set [F>r¯ε]¯ is a complete metric space with respect to the metric induced by d and that the function f~nk is lower semicontinuous. In addition, f~nk is not identically +∞ since by (32), (33), and (34) we have that vk,ε∈dom(f~nk). By (34), (30) and using that nk≥k≥kε≥nr¯ε,ε, we see that
(35)f~nk(w)=fnk(w)>α-ε2,∀w∈[F>r¯ε]¯;
hence f~nk is bounded from below. Therefore we are in a position to apply Theorem 15 to the function f~nk on the complete metric space [F>r¯ε]¯. Then there exists
(36)uk,ε∈[F>r¯ε]¯
such that
(37)fnk(uk,ε)≤fnk(vk,ε)-εd(uk,ε,vk,ε),(38)fnk(w)>fnk(uk,ε)-εd(w,uk,ε),∀w∈[F>r¯ε]¯∖{uk,ε}.
In view of (33), we see that fnk(vk,ε)<+∞, which combined with (37) ensures that uk,ε∈dom(fnk).
By (36) and Lemma 16 (a) we have that uk,ε∈[F>r¯ε]¯⊂[F>r¯ε-γ2], which, in view of (29), yields F(uk,ε)>r¯ε-γ2≥1/ε, so (7) holds true.
Using (35), (36), (37), and (33) we have
(39)α-ε2<fnk(uk,ε)≤fnk(vk,ε)-εd(uk,ε,vk,ε)<α+ε2-εd(uk,ε,vk,ε),
which proves (5).
Let us show that(40)uk,ε∈int[F>r¯ε]¯.
Letμε∈(0,γ1-2ε). To get (40) it suffices to prove that
(41)d(w,uk,ε)<με⟹w∈[F>r¯ε].
To see this, let w satisfy d(w,uk,ε)<με. By (39), we have that d(uk,ε,vk,ε)<2ε, which implies that d(w,vk,ε)≤d(w,uk,ε)+d(uk,ε,vk,ε)<με+2ε<γ1. In view of H(F), this leads to |F(w)-F(vk,ε)|<γ2. On the other hand, by (32) and Lemma 16 (a), we know that vk,ε∈[F>2r¯ε]¯⊂[F>2r¯ε-γ2]. It follows that F(w)>F(vk,ε)-γ2>2r¯ε-2γ2≥r¯ε, where we have used (29). This yields (41), which proves (40).
On the other hand, (38) yields
(42)fnk(uk,ε)-fnk(w)d(uk,ε,w)<ε,∀w∈[F>r¯ε]¯∖{uk,ε}.
Taking (40) into account, we can pass to limsup as w→uk,ε in (42) to obtain
(43)limsupw→uk,εfnk(uk,ε)-fnk(w)d(uk,ε,w)≤ε,
where the limsup is taken for w∈X (using (41)). If uk,ε is not a local minimum of fnk, then inequality (43) means that |∇fnk|(uk,ε)≤ε, while if uk,ε is a local minimum of fnk, then we know that |∇fnk|(uk,ε)=0. Hence (6) is proved. Since there is no loss of generality in taking ε∈(0,γ1/2), the proof of the first part of the conclusion is complete.
For every fixed integerℓ≥1, applying the first part of the conclusion with ε=1/ℓ, we find a subsequence {fnk} (depending on ℓ) of {fn} and a number kℓ∈ℕ such that for all k≥kℓ there exists uk,ℓ∈dom(fnk) satisfying
(44)α-1ℓ2<fnk(uk,ℓ)<α+1ℓ2,|∇fnk|(uk,ℓ)≤1ℓ,α-1ℓ2<fnk(uk,ℓ)<αx+1ℓ2,F(uk,ℓ)>ℓ.
In particular, we obtain a subsequence {fmℓ} of {fnk} (setting mℓ=nkℓ) and uℓ:=ukℓ,ℓ∈dom(fmℓ) with the properties stated in the second part of the conclusion.Proof of Corollary4.
Arguing by contradiction, suppose that the sequence{fn} is not F-coercive; that is, α:=αF({fn})<+∞. Combining this with hypothesis HF({fn}), we infer that α∈ℝ. With all the hypotheses of Theorem 1 being satisfied, we then obtain a subsequence {fmℓ} of {fnk} and uℓ∈dom(fmℓ) satisfying
(45)fmℓ(uℓ)⟶α,|∇fmℓ|(uℓ)⟶0,F(uℓ)⟶+∞hhasℓ⟶∞.
The first two convergences in (45) in conjunction with condition (PS)F yield that the sequence {uℓ} is F-bounded, which contradicts the last convergence in (45).We conclude this section with the proof of some assertions stated in Section1. First, recall that
(46)αF(f)=liminfF(v)→+∞f(v)=supr∈ℝinf[F>r]¯f=supr∈ℝinf[F>r]f,
where the last equality is true in view of Lemma 16(a).Lemma 17.
LetX be a metric space and let F:X→ℝ satisfy H(F). Given c∈ℝ, for every f:X→ℝ∪{+∞} one has that c=αF(f) if and only if (11) holds.Proof.
Suppose thatc=αF(f). Then, we see from (46) that if a<c then there exists r∈ℝ such that inf[F>r]f>a (i.e., [f≤a]⊂[F≤r]), and if a>c then inf[F>r]f<a for all r∈ℝ (i.e., for every r∈ℝ there exists vr∈[f≤a]∖[F≤r]).
Conversely, suppose thatc satisfies (11). From the first part of (11), we have that if a<c, then there exists r∈ℝ such that [F>r]⊂[f>a]. It follows that αF(f)≥inf[F>r]f≥a (see (46)), whence αF(f)≥c. From the second part of (11), we have that if a>c, then for every r∈ℝ, there exists vr∈[F>r] such that f(vr)≤a; hence inf[F>r]f≤a. We conclude that αF(f)≤a for all a>c, whence αF(f)≤c.Lemma 18.
LetX be a metric space and let F:X→ℝ satisfy H(F). Then for every f:X→ℝ∪{+∞} one has that αF(f)>-∞ if and only if [F>b]⊂[f>a] for some a,b∈ℝ.Proof.
Suppose thatαF(f)>-∞. Then, by (46), there exist a,b∈ℝ with a<αF(f) such that inf[F>b]f>a; that is, [F>b]⊂[f>a]. Conversely, if [F>b]⊂[f>a] for some a,b∈ℝ, then αF(f)≥inf[F>b]f≥a>-∞ (see (46)).Lemma 19.
LetX be a metric space, let F:X→ℝ satisfy H(F), and let f,fn:X→ℝ∪{+∞} (n∈ℕ) satisfy HF({fn},f). Then
(47)αF({fn})(=limr→+∞liminfn→∞inf[F>r]¯fn)=αF(f).Proof.
Fixr≥r0, with r0 given in HF({fn},f), and let u∈dom(f)∩[F>r]¯. By HF({fn},f)  (i), there exists a sequence {un} such that un→u and fn(un)→f(u) as n→∞. By Lemma 16, we have u∈[F>r]¯⊂int[F>r-2γ2]. Then un∈[F>r-2γ2] for all n∈ℕ sufficiently large, say n≥n0. It follows that
(48)inf[F>r-2γ2]¯fn≤fn(un),∀n≥n0.
Passing to liminf as n→∞ and using HF({fn},f) (ii), we infer that
(49)inf[F>r]¯f≤liminfn→∞inf[F>r-2γ2]¯fn≤limn→∞fn(un)=f(u).
Since this inequality holds for every u∈[F>r]¯, we obtain
(50)liminfn→∞inf[F>r-2γ2]¯fn=inf[F>r]¯f.
Since r≥r0 is arbitrary, letting r→+∞ establishes the lemma.


## 3. Special Cases and Further Remarks

### 3.1. Case of the Existence of Limit inαF({fn})

LetX be a Banach space, let F:X→ℝ be a function satisfying H(F), and let fn:X→ℝ∪{+∞} (n∈ℕ) be a sequence of lower semicontinuous functions which are not identically +∞. We assume the following.H
~
F
(
{
f
n
}
): there exists r0∈ℝ such that for all r>r0, limn→∞inf[F>r]¯fn exists.Note thatH~F({fn}) is satisfied by all the functions in Example 12(a), (b), and (c). We consider another notion of Palais-Smale condition relative to F.Definition 20.
Letfn:X→ℝ∪{+∞} (n∈ℕ) be lower semicontinuous functions which are not identically +∞. We say that the sequence {fn} satisfies the generalized Palais-Smale condition relative to F (condition (gPS)F, for short) if there exists a subsequence {fnk} of {fn} such that whenever {uk}⊂X is a sequence such that {fnk(uk)} is bounded and |∇fnk|(uk)→0 as k→∞, then {uk} is F-bounded.Remark 21.
Condition(gPS)F is more general than condition (PS)F of Definition 2.Corollary 22.
Assume thatHF({fn}) and H~F({fn}) hold.(i)
AssumeαF({fn})<+∞. Then for every ε>0, there exists nε∈ℕ such that for each n≥nε one finds un,ε∈dom(fn) satisfying
(51)αF({fn})-ε2<fn(un,ε)<αF({fn})+ε2,|∇fn|(un,ε)≤ε,F(un,ε)>1ε.
In particular, there exists un∈dom(fn) (n∈ℕ) satisfying
(52)fn(un)⟶αF({fn}),|∇fn|(un)⟶0,F(un)⟶+∞asn⟶∞.(ii)
Assume that{fn} satisfies condition (gPS)F and that supXF=+∞. Then the sequence {fn} is F-coercive.Proof.
(i) We argue as in the proof of Theorem1 noting that, due to the assumption, in place of (27) and (28) we have
(53)α-ε2<inf[F>r]¯fn<α+ε2,∀n≥nr,ε
and choosing nε=max{nr¯ε,ε,n2r¯ε,ε}.
(ii) Arguing by contradiction, suppose that the sequence{fn} is not F-coercive; that is, α:=αF({fn})<+∞; thus α∈ℝ (see HF({fn})). Then, by part (i), we can find un∈dom(fn) satisfying (52). Let {fnk} be the subsequence of {fn} that satisfies condition (gPS)F. Then the convergences fnk(unk)→α and |∇fnk|(unk)→0 yield that {unk} is F-bounded, which contradicts the third convergence in (52).


### 3.2. Case of Functionals with Special Structure

Let(X,∥·∥) be a Banach space, F:X→ℝ be a function satisfying H(F), and fn:X→ℝ∪{+∞} (n∈ℕ) be of the form
(54)fn=Φn+Ψn
with Φn:X→ℝ locally Lipschitz and Ψn:X→ℝ∪{+∞} convex, lower semicontinuous, not identically +∞. In this setting we consider an appropriate version of Palais-Smale condition (see Motreanu and Panagiotopoulos [5, Chapter 3]).Definition 23.
The sequence of functionals{fn} as in (54) satisfies the Palais-Smale condition in the sense of Motreanu and Panagiotopoulos relative to F (condition (PS+)F for short) if whenever {fnk} is a subsequence of {fn} and {uk}⊂X is a sequence such that {fnk(uk)} is bounded and for which there exists a sequence {εk}⊂ℝ+, εk↓0, such that
(55)Φnk0(uk;v-uk)+Ψnk(v)-Ψnk(uk)≥-εk∥v-uk∥,∀v∈X,∀k≥1,
then {uk} is F-bounded.Hereafter, the notationΦ0(u;v) stands for the generalized directional derivative of a locally Lipschitz functional Φ:X→ℝ at the point u∈X in the direction v∈X (see Clarke [6]) given by
(56)Φ0(u;v)=limsupw→ut↓01t(Φ(w+tv)-Φ(w)).Remark 24.
Condition(PS+)F in the above definition generalizes the Palais-Smale conditions of Chang [7] (for the case where Φn≡Φ is locally Lipschitz, Ψ=0, and F=∥·∥) and Szulkin [8] (for the case where Φn≡Φ∈C1(X,ℝ), Ψn≡Ψ is lower semicontinuous, convex, not identically +∞, and F=∥·∥).Lemma 25.
(a) LetΦ:X→ℝ be a locally Lipschitz functional, let Ψ:X→ℝ∪{+∞} be a convex, lower semicontinuous function which is not identically +∞, and let f=Φ+Ψ. Then
(57)Φ0(u;v-u)+Ψ(v)-Ψ(u)≥-|∇f|(u)∥u-v∥,∀u,v∈X.
(b) For a sequence{fn} as in Definition 23, one has
(58)(PS+)F(ofDefinition23)⟹(PS)F(ofDefinition2).Proof.
(a) Using the convexity ofΨ, for every u,v∈X, v≠u, we have
(59)Φ0(u;v-u)∥u-v∥+Ψ(v)-Ψ(u)∥u-v∥≥limsupt↓0Φ(u+t(v-u))-Φ(u)t∥v-u∥+limt↓0Ψ(u+t(v-u))-Ψ(u)t∥u-v∥≥liminfw→u(Φ(w)-Φ(u)∥w-u∥+Ψ(w)-Ψ(u)∥w-u∥)=-limsupw→uf(u)-f(w)∥u-w∥.
If u is not a local minimum of f, then the desired inequality follows. If u is a local minimum of f, then it is a critical point of f in the sense of Motreanu-Panagiotopoulos [5, Definition 3.1]; that is, Φ0(u;v-u)+Ψ(v)-Ψ(u)≥0 for all v∈X, and so again we are done.
(b) This is an immediate consequence of part (a).Corollary 26.
LetX be a Banach space and let F:X→ℝ satisfy H(F). Let fn=Φn+Ψn:X→ℝ∪{+∞} (n∈ℕ) as in (54). Assume that HF({fn}) holds.  (i)
AssumeαF({fn})<+∞. Then, for every ε>0, there exist a subsequence {fnk} (depending on ε) of {fn} and a number kε∈ℕ such that for each k≥kε one finds uk,ε∈dom(fnk) satisfying
(60)αF({fn})-ε2<fnk(uk,ε)<αF({fn})+ε2,F(uk,ε)>1ε,Φnk0(uk,ε;v-uk,ε)+Ψnk(v)-Ψnk(uk,ε)≥-ε∥v-uk,ε∥,∀v∈X,∀k≥1.(ii)
Assume that{fn} satisfies condition (PS+)F and that supXF=+∞. Then the sequence {fn} is F-coercive.Proof.
Part (i) follows from Theorem1 by using Lemma 25(a), while part (ii) follows from Corollary 4 by using Lemma 25(b).Remark 27.
When all the terms of the sequence{fn} coincide (and F=∥·∥), an extension of Corollary 26 has been obtained in Motreanu et al. [9] by means of a general Palais-Smale condition incorporating the Palais-Smale conditions in the sense of Cerami [10] and Zhong [11].


### 3.3. Case of Galerkin Approximations

Let(X,∥·∥) be a Banach space, let {Xn} be a sequence of closed vector subspaces of X (not necessarily increasing) such that ⋃n=1∞‍Xn¯=X, and let F:X→ℝ be a function satisfying H(F). Let f∈C1(X,ℝ) and let
(61)fn:=fC1|Xn∈C1(Xn;ℝ),∀n∈ℕ.
We consider the following Palais-Smale condition (see Li and Willem [12]).Definition 28.
The functionf∈C1(X,ℝ) satisfies the Palais-Smale condition in the sense of Li-Willem relative to F (condition (PS*)F for short) if every sequence {un}⊂X with un∈Xαn, αn→+∞, {f(un)} bounded and fαn′(un)→0 is F-bounded.For alln∈ℕ, define the functions f~n:X→ℝ∪{+∞} by
(62)f~n(u)={f(u),ifu∈Xn,+∞,otherwise.
It is clear that the functions f~n are lower semicontinuous and not identically +∞.Lemma 29.
(a)|∇f~n|(u)=∥fn′(u)∥Xn* for all u∈Xn and all n∈ℕ.
(b)(PS*)F (of Definition 28 for f∈C1(X,ℝ)) ⇔(PS)F (of Definition 2 for {f~n}).Proof.
(a) Using thatfn∈C1(Xn,ℝ), for every u∈Xn which is not a local minimizer of f~n (equivalently, nor of fn) we have
(63)|∇f~n|(u)=limsupw→uw∈Xnf~n(u)-f~n(w)∥u-w∥=|∇fn|(u)=∥fn′(u)∥Xn*.
The case where u∈Xn is a local minimizer of f~n is straightforward.
(b) This easily follows from part (a).DenoteαfF:=limr→+∞liminfn→∞inf[F>r]¯∩Xnf.Corollary 30.
LetX be a Banach space, let {Xn} be a sequence of closed vector subspaces of X such that ⋃n=1∞‍Xn¯=X, and let F:X→ℝ be a function satisfying H(F). Let f∈C1(X,ℝ) and fn:=f|Xn (n∈ℕ). Assume that αfF>-∞.(i)
AssumeαfF<+∞. Then, for every ε>0, there exist a subsequence {fnk} (depending on ε) of {fn} and a number kε∈ℕ such that for each k≥kε one finds uk,ε∈Xnk satisfying
(64)αfF-ε2<f(uk,ε)<αfF+ε2,∥fnk′(uk,ε)∥Xnk*≤ε,F(uk,ε)>1ε.(ii)
Assume thatf satisfies condition (PS*)F and that supXF=+∞. Then αfF=+∞.Proof.
Part (i) follows by applying Theorem1 to the sequence {f~n} in (62), noting that αF({f~n})=αfF and using Lemma 29(a). To prove part (ii), apply Corollary 4 to {f~n} and use Lemma 29(b).


### 3.4. Case of a Constant Sequence of Functions

When all the terms of the sequence{fn} coincide, say fn=f, for all n, Theorem 1 and Corollary 4 yield the following.Corollary 31.
LetX be a complete metric space and let F:X→ℝ satisfy H(F). Let f:X→ℝ∪{+∞} be a lower semicontinuous function with the property
(65)αF(f):=liminfF(v)→+∞f(v)(=limr→+∞inf[F>r]¯f)>-∞.(i)
IfαF(f)<+∞, then for every ε>0 there exists uε∈dom(f) such that
(66)αF(f)-ε2<f(uε)<αF(f)+ε2,|∇f|(uε)≤ε,F(uε)>1ε.(ii)
Assume thatf satisfies the following Palais-Smale condition: every sequence {uk}⊂X such that {f(uk)} is bounded and |∇f|(uk)→0 as k→∞ is F-bounded. Assume also that supXF=+∞. Then f is F-coercive; that is, αF(f)=+∞, or, equivalently, f(v)→+∞ as F(v)→+∞.Proof.
Part (i) follows from Theorem1 applied with fn=f for all n∈ℕ (note that αF(f)=αF({f})), while part (ii) follows from Corollary 4 in the same way.Remark 32.
(a) The condition thatαF(f)∈ℝ ensures that the function f is not identically +∞ and that the sets {v∈X:F(v)>r} are nonempty for all r∈ℝ.
(b) In Motreanu et al. [13], results concerning the asymptotic behavior as in Corollary 31(i) and the F-coercivity as in Corollary 31(ii) are given in the more general setting of a metric space endowed with a quasiorder ≤ and of ≤-lower semicontinuous functions, by means of an appropriate notion of strong slope. Corollary 31 can be obtained from [13, Theorems 6.1, and 6.2] in the case where the quasiorder is the trivial one.We apply Corollary31 to the special situation F=d(·,A) (see Example 13(a)).Corollary 33.
LetX be a complete metric space, let A be a nonempty subset of X, and let f:X→ℝ∪{+∞} be a lower semicontinuous function with the property
(67)αd(·,A)(f):=liminfd(v,A)→+∞f(v)>-∞.(i)
Ifαd(·,A)(f)<+∞, then for every ε>0 there is uε∈dom(f) such that
(68)αd(·,A)(f)-ε2<f(uε)<αd(·,A)(f)+ε2,|∇f|(uε)≤ε,d(uε,A)>1ε.(ii)
Assume thatf satisfies the following Palais-Smale condition: every sequence {uk}⊂X such that {f(uk)} is bounded and |∇f|(uk)→0 as k→∞ satisfies that {d(uk,A)} is bounded. If X∖A is unbounded, then f is coercive in the sense that f(v)→+∞ as d(v,A)→+∞.Proof.
This readily follows from Corollary31 applied to F=d(·,A).Remark 34.
In the case whereA is bounded, the expression αd(·,A)(f) in Corollary 33 and the coercivity property in (ii) do not depend on the set A⊂X.In particular, we can considerA={u0}, for some u0∈X. If X is a Banach space and u0=0, then Corollary 33 becomes the following.Corollary 35.
Let(X,∥·∥) be a Banach space and let f:X→ℝ∪{+∞} be a lower semicontinuous function with the property α∥·∥(f):=liminf∥v∥→+∞f(v)>-∞.(i)
Ifα∥·∥(f)<+∞, then for every ε>0 there exists uε∈dom(f) such that
(69)α∥·∥(f)-ε2<f(uε)<α∥·∥(f)+ε2,|∇f|(uε)≤ε,∥uε∥>1ε.(ii)
Assume thatf satisfies the following Palais–Smale condition: every sequence {uk}⊂X such that {f(uk)} is bounded and |∇f|(uk)→0 as k→∞ is bounded. Then f is coercive in the sense that f(v)→+∞ as ∥v∥→+∞.Remark 36.
Corollary35 extends the corresponding result in the smooth case, that is, for f of class C1 (see Brezis and Nirenberg [14]).


### 3.5. Case Where the Limits inαF({fn}) Are Interchanged

It is natural to ask what happens when the expressionαF({fn}) in HF({fn}) is replaced with
(70)α~F({fn}):=limn→∞liminfF(v)→+∞fn(v)=deflimn→∞limr→+∞inf[F>r]¯fn=limn→∞supr∈ℝinf[F>r]¯fn,
that is, when the limits in HF({fn}) are interchanged (we work directly with a sequence {fn} for which the limit as n→∞ in (70) exists in place of considering liminf).Proposition 37.
LetX be a complete metric space, let F:X→ℝ be a function satisfying H(F), and let fn:X→ℝ∪{+∞} (n∈ℕ) be a sequence of lower semicontinuous functions with the property α~F({fn})>-∞. (i)
Assumeα~F({fn})<+∞. Then for every ε>0, there exists nε∈ℕ such that for each n≥nε one finds un,ε∈dom(fn) satisfying
(71)α~F({fn})-ε2<fn(un,ε)<α~F({fn})+ε2,|∇fn|(un,ε)≤ε,F(un,ε)>1ε.(ii)
Assume that{fn} satisfies condition (gPS)F (see Definition 20) and that supXF=+∞. Then α~F({fn})=+∞.Proof.
(i) We denoteα~:=α~F({fn})∈ℝ. Fix ε>0. By (70), there exists nε∈ℕ such that
(72)α~-ε22<limr→+∞inf[F>r]¯fn<α~+ε22,∀n≥nε.
Denote αn:=limr→+∞inf[F>r]¯fn. In view of (72) we have αn∈ℝ for all n≥nε. Then, for each n≥nε we can apply Corollary 31(i) to the function f:=fn and the number ε~:=ε/2. Thus, we find un,ε∈dom(fn) such that
(73)α~-ε2<αn-ε22<fn(un,ε)<αn+ε22<α~+ε2,|∇fn|(un,ε)≤ε2<ε,F(un,ε)>2ε>1ε.
(ii) Arguing by contradiction, suppose thatα~:=α~F({fn})∈ℝ. Let a subsequence of {fn}, denoted again by {fn}, with the property in condition (gPS)F. Applying part (i) of the proposition to the subsequence {fn} and ε=1/k (k∈ℕ), we find a number nk∈ℕ and an element uk∈dom(fnk) with the properties
(74)α~-1k2<fnk(uk)<α~+1k2,|∇fnk|(uk)≤1k,α~-1k2<fnk(uk)<α~++1k2,F(uk)>k.
Consequently, we have fnk(uk)→α~, |∇fnk|(uk)→0, and F(uk)→+∞ as k→∞. The first two convergences in conjunction with condition (gPS)F yield that the sequence {uk} is F-bounded, which contradicts the third convergence.Remark 38.
The above proof shows that Proposition37(i) is a consequence of Corollary 31(i). On the other hand, Corollary 31(i) can be obtained from Proposition 37(i) by taking fn=f, for all n∈ℕ. So, Proposition 37(i) and Corollary 31(i) are conceptually equivalent, in the sense that
(75)Corollary31(i)⟹Proposition37(i)⟹Corollary31(i).
Thus our main result, Theorem 1 (involving the expression αF({fn}) in HF({fn}) is more general than Proposition 37(i) (involving the expression α~F({fn}) in (70)).


## 3.1. Case of the Existence of Limit inαF({fn})

LetX be a Banach space, let F:X→ℝ be a function satisfying H(F), and let fn:X→ℝ∪{+∞} (n∈ℕ) be a sequence of lower semicontinuous functions which are not identically +∞. We assume the following.H
~
F
(
{
f
n
}
): there exists r0∈ℝ such that for all r>r0, limn→∞inf[F>r]¯fn exists.Note thatH~F({fn}) is satisfied by all the functions in Example 12(a), (b), and (c). We consider another notion of Palais-Smale condition relative to F.Definition 20.
Letfn:X→ℝ∪{+∞} (n∈ℕ) be lower semicontinuous functions which are not identically +∞. We say that the sequence {fn} satisfies the generalized Palais-Smale condition relative to F (condition (gPS)F, for short) if there exists a subsequence {fnk} of {fn} such that whenever {uk}⊂X is a sequence such that {fnk(uk)} is bounded and |∇fnk|(uk)→0 as k→∞, then {uk} is F-bounded.Remark 21.
Condition(gPS)F is more general than condition (PS)F of Definition 2.Corollary 22.
Assume thatHF({fn}) and H~F({fn}) hold.(i)
AssumeαF({fn})<+∞. Then for every ε>0, there exists nε∈ℕ such that for each n≥nε one finds un,ε∈dom(fn) satisfying
(51)αF({fn})-ε2<fn(un,ε)<αF({fn})+ε2,|∇fn|(un,ε)≤ε,F(un,ε)>1ε.
In particular, there exists un∈dom(fn) (n∈ℕ) satisfying
(52)fn(un)⟶αF({fn}),|∇fn|(un)⟶0,F(un)⟶+∞asn⟶∞.(ii)
Assume that{fn} satisfies condition (gPS)F and that supXF=+∞. Then the sequence {fn} is F-coercive.Proof.
(i) We argue as in the proof of Theorem1 noting that, due to the assumption, in place of (27) and (28) we have
(53)α-ε2<inf[F>r]¯fn<α+ε2,∀n≥nr,ε
and choosing nε=max{nr¯ε,ε,n2r¯ε,ε}.
(ii) Arguing by contradiction, suppose that the sequence{fn} is not F-coercive; that is, α:=αF({fn})<+∞; thus α∈ℝ (see HF({fn})). Then, by part (i), we can find un∈dom(fn) satisfying (52). Let {fnk} be the subsequence of {fn} that satisfies condition (gPS)F. Then the convergences fnk(unk)→α and |∇fnk|(unk)→0 yield that {unk} is F-bounded, which contradicts the third convergence in (52).


## 3.2. Case of Functionals with Special Structure

Let(X,∥·∥) be a Banach space, F:X→ℝ be a function satisfying H(F), and fn:X→ℝ∪{+∞} (n∈ℕ) be of the form
(54)fn=Φn+Ψn
with Φn:X→ℝ locally Lipschitz and Ψn:X→ℝ∪{+∞} convex, lower semicontinuous, not identically +∞. In this setting we consider an appropriate version of Palais-Smale condition (see Motreanu and Panagiotopoulos [5, Chapter 3]).Definition 23.
The sequence of functionals{fn} as in (54) satisfies the Palais-Smale condition in the sense of Motreanu and Panagiotopoulos relative to F (condition (PS+)F for short) if whenever {fnk} is a subsequence of {fn} and {uk}⊂X is a sequence such that {fnk(uk)} is bounded and for which there exists a sequence {εk}⊂ℝ+, εk↓0, such that
(55)Φnk0(uk;v-uk)+Ψnk(v)-Ψnk(uk)≥-εk∥v-uk∥,∀v∈X,∀k≥1,
then {uk} is F-bounded.Hereafter, the notationΦ0(u;v) stands for the generalized directional derivative of a locally Lipschitz functional Φ:X→ℝ at the point u∈X in the direction v∈X (see Clarke [6]) given by
(56)Φ0(u;v)=limsupw→ut↓01t(Φ(w+tv)-Φ(w)).Remark 24.
Condition(PS+)F in the above definition generalizes the Palais-Smale conditions of Chang [7] (for the case where Φn≡Φ is locally Lipschitz, Ψ=0, and F=∥·∥) and Szulkin [8] (for the case where Φn≡Φ∈C1(X,ℝ), Ψn≡Ψ is lower semicontinuous, convex, not identically +∞, and F=∥·∥).Lemma 25.
(a) LetΦ:X→ℝ be a locally Lipschitz functional, let Ψ:X→ℝ∪{+∞} be a convex, lower semicontinuous function which is not identically +∞, and let f=Φ+Ψ. Then
(57)Φ0(u;v-u)+Ψ(v)-Ψ(u)≥-|∇f|(u)∥u-v∥,∀u,v∈X.
(b) For a sequence{fn} as in Definition 23, one has
(58)(PS+)F(ofDefinition23)⟹(PS)F(ofDefinition2).Proof.
(a) Using the convexity ofΨ, for every u,v∈X, v≠u, we have
(59)Φ0(u;v-u)∥u-v∥+Ψ(v)-Ψ(u)∥u-v∥≥limsupt↓0Φ(u+t(v-u))-Φ(u)t∥v-u∥+limt↓0Ψ(u+t(v-u))-Ψ(u)t∥u-v∥≥liminfw→u(Φ(w)-Φ(u)∥w-u∥+Ψ(w)-Ψ(u)∥w-u∥)=-limsupw→uf(u)-f(w)∥u-w∥.
If u is not a local minimum of f, then the desired inequality follows. If u is a local minimum of f, then it is a critical point of f in the sense of Motreanu-Panagiotopoulos [5, Definition 3.1]; that is, Φ0(u;v-u)+Ψ(v)-Ψ(u)≥0 for all v∈X, and so again we are done.
(b) This is an immediate consequence of part (a).Corollary 26.
LetX be a Banach space and let F:X→ℝ satisfy H(F). Let fn=Φn+Ψn:X→ℝ∪{+∞} (n∈ℕ) as in (54). Assume that HF({fn}) holds.  (i)
AssumeαF({fn})<+∞. Then, for every ε>0, there exist a subsequence {fnk} (depending on ε) of {fn} and a number kε∈ℕ such that for each k≥kε one finds uk,ε∈dom(fnk) satisfying
(60)αF({fn})-ε2<fnk(uk,ε)<αF({fn})+ε2,F(uk,ε)>1ε,Φnk0(uk,ε;v-uk,ε)+Ψnk(v)-Ψnk(uk,ε)≥-ε∥v-uk,ε∥,∀v∈X,∀k≥1.(ii)
Assume that{fn} satisfies condition (PS+)F and that supXF=+∞. Then the sequence {fn} is F-coercive.Proof.
Part (i) follows from Theorem1 by using Lemma 25(a), while part (ii) follows from Corollary 4 by using Lemma 25(b).Remark 27.
When all the terms of the sequence{fn} coincide (and F=∥·∥), an extension of Corollary 26 has been obtained in Motreanu et al. [9] by means of a general Palais-Smale condition incorporating the Palais-Smale conditions in the sense of Cerami [10] and Zhong [11].


## 3.3. Case of Galerkin Approximations

Let(X,∥·∥) be a Banach space, let {Xn} be a sequence of closed vector subspaces of X (not necessarily increasing) such that ⋃n=1∞‍Xn¯=X, and let F:X→ℝ be a function satisfying H(F). Let f∈C1(X,ℝ) and let
(61)fn:=fC1|Xn∈C1(Xn;ℝ),∀n∈ℕ.
We consider the following Palais-Smale condition (see Li and Willem [12]).Definition 28.
The functionf∈C1(X,ℝ) satisfies the Palais-Smale condition in the sense of Li-Willem relative to F (condition (PS*)F for short) if every sequence {un}⊂X with un∈Xαn, αn→+∞, {f(un)} bounded and fαn′(un)→0 is F-bounded.For alln∈ℕ, define the functions f~n:X→ℝ∪{+∞} by
(62)f~n(u)={f(u),ifu∈Xn,+∞,otherwise.
It is clear that the functions f~n are lower semicontinuous and not identically +∞.Lemma 29.
(a)|∇f~n|(u)=∥fn′(u)∥Xn* for all u∈Xn and all n∈ℕ.
(b)(PS*)F (of Definition 28 for f∈C1(X,ℝ)) ⇔(PS)F (of Definition 2 for {f~n}).Proof.
(a) Using thatfn∈C1(Xn,ℝ), for every u∈Xn which is not a local minimizer of f~n (equivalently, nor of fn) we have
(63)|∇f~n|(u)=limsupw→uw∈Xnf~n(u)-f~n(w)∥u-w∥=|∇fn|(u)=∥fn′(u)∥Xn*.
The case where u∈Xn is a local minimizer of f~n is straightforward.
(b) This easily follows from part (a).DenoteαfF:=limr→+∞liminfn→∞inf[F>r]¯∩Xnf.Corollary 30.
LetX be a Banach space, let {Xn} be a sequence of closed vector subspaces of X such that ⋃n=1∞‍Xn¯=X, and let F:X→ℝ be a function satisfying H(F). Let f∈C1(X,ℝ) and fn:=f|Xn (n∈ℕ). Assume that αfF>-∞.(i)
AssumeαfF<+∞. Then, for every ε>0, there exist a subsequence {fnk} (depending on ε) of {fn} and a number kε∈ℕ such that for each k≥kε one finds uk,ε∈Xnk satisfying
(64)αfF-ε2<f(uk,ε)<αfF+ε2,∥fnk′(uk,ε)∥Xnk*≤ε,F(uk,ε)>1ε.(ii)
Assume thatf satisfies condition (PS*)F and that supXF=+∞. Then αfF=+∞.Proof.
Part (i) follows by applying Theorem1 to the sequence {f~n} in (62), noting that αF({f~n})=αfF and using Lemma 29(a). To prove part (ii), apply Corollary 4 to {f~n} and use Lemma 29(b).


## 3.4. Case of a Constant Sequence of Functions

When all the terms of the sequence{fn} coincide, say fn=f, for all n, Theorem 1 and Corollary 4 yield the following.Corollary 31.
LetX be a complete metric space and let F:X→ℝ satisfy H(F). Let f:X→ℝ∪{+∞} be a lower semicontinuous function with the property
(65)αF(f):=liminfF(v)→+∞f(v)(=limr→+∞inf[F>r]¯f)>-∞.(i)
IfαF(f)<+∞, then for every ε>0 there exists uε∈dom(f) such that
(66)αF(f)-ε2<f(uε)<αF(f)+ε2,|∇f|(uε)≤ε,F(uε)>1ε.(ii)
Assume thatf satisfies the following Palais-Smale condition: every sequence {uk}⊂X such that {f(uk)} is bounded and |∇f|(uk)→0 as k→∞ is F-bounded. Assume also that supXF=+∞. Then f is F-coercive; that is, αF(f)=+∞, or, equivalently, f(v)→+∞ as F(v)→+∞.Proof.
Part (i) follows from Theorem1 applied with fn=f for all n∈ℕ (note that αF(f)=αF({f})), while part (ii) follows from Corollary 4 in the same way.Remark 32.
(a) The condition thatαF(f)∈ℝ ensures that the function f is not identically +∞ and that the sets {v∈X:F(v)>r} are nonempty for all r∈ℝ.
(b) In Motreanu et al. [13], results concerning the asymptotic behavior as in Corollary 31(i) and the F-coercivity as in Corollary 31(ii) are given in the more general setting of a metric space endowed with a quasiorder ≤ and of ≤-lower semicontinuous functions, by means of an appropriate notion of strong slope. Corollary 31 can be obtained from [13, Theorems 6.1, and 6.2] in the case where the quasiorder is the trivial one.We apply Corollary31 to the special situation F=d(·,A) (see Example 13(a)).Corollary 33.
LetX be a complete metric space, let A be a nonempty subset of X, and let f:X→ℝ∪{+∞} be a lower semicontinuous function with the property
(67)αd(·,A)(f):=liminfd(v,A)→+∞f(v)>-∞.(i)
Ifαd(·,A)(f)<+∞, then for every ε>0 there is uε∈dom(f) such that
(68)αd(·,A)(f)-ε2<f(uε)<αd(·,A)(f)+ε2,|∇f|(uε)≤ε,d(uε,A)>1ε.(ii)
Assume thatf satisfies the following Palais-Smale condition: every sequence {uk}⊂X such that {f(uk)} is bounded and |∇f|(uk)→0 as k→∞ satisfies that {d(uk,A)} is bounded. If X∖A is unbounded, then f is coercive in the sense that f(v)→+∞ as d(v,A)→+∞.Proof.
This readily follows from Corollary31 applied to F=d(·,A).Remark 34.
In the case whereA is bounded, the expression αd(·,A)(f) in Corollary 33 and the coercivity property in (ii) do not depend on the set A⊂X.In particular, we can considerA={u0}, for some u0∈X. If X is a Banach space and u0=0, then Corollary 33 becomes the following.Corollary 35.
Let(X,∥·∥) be a Banach space and let f:X→ℝ∪{+∞} be a lower semicontinuous function with the property α∥·∥(f):=liminf∥v∥→+∞f(v)>-∞.(i)
Ifα∥·∥(f)<+∞, then for every ε>0 there exists uε∈dom(f) such that
(69)α∥·∥(f)-ε2<f(uε)<α∥·∥(f)+ε2,|∇f|(uε)≤ε,∥uε∥>1ε.(ii)
Assume thatf satisfies the following Palais–Smale condition: every sequence {uk}⊂X such that {f(uk)} is bounded and |∇f|(uk)→0 as k→∞ is bounded. Then f is coercive in the sense that f(v)→+∞ as ∥v∥→+∞.Remark 36.
Corollary35 extends the corresponding result in the smooth case, that is, for f of class C1 (see Brezis and Nirenberg [14]).


## 3.5. Case Where the Limits inαF({fn}) Are Interchanged

It is natural to ask what happens when the expressionαF({fn}) in HF({fn}) is replaced with
(70)α~F({fn}):=limn→∞liminfF(v)→+∞fn(v)=deflimn→∞limr→+∞inf[F>r]¯fn=limn→∞supr∈ℝinf[F>r]¯fn,
that is, when the limits in HF({fn}) are interchanged (we work directly with a sequence {fn} for which the limit as n→∞ in (70) exists in place of considering liminf).Proposition 37.
LetX be a complete metric space, let F:X→ℝ be a function satisfying H(F), and let fn:X→ℝ∪{+∞} (n∈ℕ) be a sequence of lower semicontinuous functions with the property α~F({fn})>-∞. (i)
Assumeα~F({fn})<+∞. Then for every ε>0, there exists nε∈ℕ such that for each n≥nε one finds un,ε∈dom(fn) satisfying
(71)α~F({fn})-ε2<fn(un,ε)<α~F({fn})+ε2,|∇fn|(un,ε)≤ε,F(un,ε)>1ε.(ii)
Assume that{fn} satisfies condition (gPS)F (see Definition 20) and that supXF=+∞. Then α~F({fn})=+∞.Proof.
(i) We denoteα~:=α~F({fn})∈ℝ. Fix ε>0. By (70), there exists nε∈ℕ such that
(72)α~-ε22<limr→+∞inf[F>r]¯fn<α~+ε22,∀n≥nε.
Denote αn:=limr→+∞inf[F>r]¯fn. In view of (72) we have αn∈ℝ for all n≥nε. Then, for each n≥nε we can apply Corollary 31(i) to the function f:=fn and the number ε~:=ε/2. Thus, we find un,ε∈dom(fn) such that
(73)α~-ε2<αn-ε22<fn(un,ε)<αn+ε22<α~+ε2,|∇fn|(un,ε)≤ε2<ε,F(un,ε)>2ε>1ε.
(ii) Arguing by contradiction, suppose thatα~:=α~F({fn})∈ℝ. Let a subsequence of {fn}, denoted again by {fn}, with the property in condition (gPS)F. Applying part (i) of the proposition to the subsequence {fn} and ε=1/k (k∈ℕ), we find a number nk∈ℕ and an element uk∈dom(fnk) with the properties
(74)α~-1k2<fnk(uk)<α~+1k2,|∇fnk|(uk)≤1k,α~-1k2<fnk(uk)<α~++1k2,F(uk)>k.
Consequently, we have fnk(uk)→α~, |∇fnk|(uk)→0, and F(uk)→+∞ as k→∞. The first two convergences in conjunction with condition (gPS)F yield that the sequence {uk} is F-bounded, which contradicts the third convergence.Remark 38.
The above proof shows that Proposition37(i) is a consequence of Corollary 31(i). On the other hand, Corollary 31(i) can be obtained from Proposition 37(i) by taking fn=f, for all n∈ℕ. So, Proposition 37(i) and Corollary 31(i) are conceptually equivalent, in the sense that
(75)Corollary31(i)⟹Proposition37(i)⟹Corollary31(i).
Thus our main result, Theorem 1 (involving the expression αF({fn}) in HF({fn}) is more general than Proposition 37(i) (involving the expression α~F({fn}) in (70)).



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