Another four critical points theorem

0.09 MB·English

by  Biagio Ricceri

Another four critical points theorem
BIAGIO RICCERI
Dedicated to the memory of Ky Fan, with my immense esteem and admiration
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1 Abstract: In this paper, making use of Theorem 2 of [5], we establish a new four
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critical points theorem which can be regarded as a companion to Theorem 1 of [4]. We also
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present an application to the Dirichlet problem for a class of quasilinear elliptic equations.
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a
J

8 Key words: Criticalpoint, globalminimum, multiplicity,minimaxinequality,Dirich-
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let problem.

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2010 Mathematics Subject Classification: 47J10, 47J30, 58E05, 49J35, 35J92.
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t
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2 The aim of this paper is to establish a new four critical points theorem (Theorem 1
v
below) that can be regarded as a companion to Theorem 1 of [4].
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Asin[4], ourkeytoolisthemultiplicityresultonglobalminimaestablishedin[3]. The
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1 use of such a result requires the validity of a strict minimax inequality which is explicitly
1. assumed in [4].
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A novelty of Theorem 1 is that no minimax inequality appears among the hypotheses.
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1 This is possible thanks to the use of Theorem 2 of the very recent [5] which just highlights
:
v a rather general situation where the strict minimax inequality occurs.
i
X In other words, Theorem 1 should be regarded as the fruit that one obtains by com-
r bining the underlying ideas of [4] with Theorem 2 of [5].
a
For the reader convenience, we start just recalling Theorem 2 of [5].
First, we introduce the following notations.
If X is a non-empty set and Γ,Ψ,Φ : X → R are three given functions, for each µ > 0
and r ∈]inf Φ,sup Φ[, we put
X X
µΓ(x)+Ψ(x)−infΦ−1(]−∞,r])(µΓ+Ψ)
α(µΓ+Ψ,Φ,r) = inf
x∈Φ−1(]−∞,r[) r −Φ(x)
and
µΓ(x)+Ψ(x)−infΦ−1(]−∞,r])(µΓ+Ψ)
β(µΓ+Ψ,Φ,r) = sup .
r −Φ(x)
x∈Φ−1(]r,+∞[)
1
When Ψ+Φ is bounded below, for each r ∈]inf Φ,sup Φ[ such that
X X
inf Γ(x) < inf Γ(x) ,
x∈Φ−1(]−∞,r]) x∈Φ−1(r)
we put
Ψ(x)−γ +r
µ∗(Γ,Ψ,Φ,r) = inf : x ∈ X,Φ(x) < r,Γ(x) < η ,
r
η −Γ(x)
r
(cid:26) (cid:27)
where
γ = inf (Ψ(x)+Φ(x))
x∈X
and
η = inf Γ(x) .
r
x∈Φ−1(r)
THEOREM A ([5], Theorem 2). - Let X be a topological space and Γ,Ψ,Φ : X → R
three sequentially lower semicontinuous functions, with Γ also sequentially inf-compact,
satisfying the following conditions:
(a) inf (µΓ(x)+Ψ(x)) = −∞ for all µ > 0 ;
x∈X
(b) inf (Ψ(x)+Φ(x)) > −∞ ;
x∈X
(c) there exists r ∈]inf Φ,sup Φ[ such that
X X
inf Γ(x) < inf Γ(x) .
x∈Φ−1(]−∞,r]) x∈Φ−1(r)
Under such hypotheses, for each µ > max{0,µ∗(Γ,Ψ,Φ,r)}, one has
α(µΓ+Ψ,Φ,r) = 0
and
β(µΓ+Ψ,Φ,r) > 0 .
As we said, the key tool in the proof of Theorem 1 is provided by the following
THEOREM B ([3], Theorem 1). - Let X be a topological space, A ⊆ R an open
interval and P : X ×A → R a function satisfying the following conditions:
(a ) for each x ∈ X, the function P(x,·) is quasi-concave and continuous ;
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(a ) for each λ ∈ A, the function P(·,λ) is lower semicontinuous and inf-compact ;
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(a ) one has
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sup inf P(x,λ) < inf supP(x,λ) .
λ∈Ax∈X x∈Xλ∈A
Under such hypotheses, there exists λ∗ ∈ A such that the function P(·,λ∗) has at least
two global minima.
Here is our main result:
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THEOREM 1. - Let X be a reflexive real Banach space; I : X → R a sequentially
weakly lower semicontiunuous and coercive C1 functional whose derivative admits a con-
tinuous inverse on X∗; J,Ψ,Φ : X → R three C1 functionals with compact derivative
satisfying the following conditions:
J(x) J(x)
liminf ≥ 0 , limsup < +∞ , (1)
kxk→+∞ I(x) kxk→+∞ I(x)
Ψ(x)
liminf = −∞ , (2)
kxk→+∞ I(x)
inf (Ψ(x)+λΦ(x)) > −∞ (3)
x∈X
for all λ > 0. Moreover, assume that there exist a strict local minimum x of I, with
0
I(x ) = J(x ) = Ψ(x ) = Φ(x ) = 0, and another point x ∈X such that
0 0 0 0 1
max{J(x ),Ψ(x ),Φ(x )} < 0 , (4)
1 1 1
J(x) Φ(x)
min liminf ,liminf ≥ 0 (5)
x→x0 I(x) x→x0 I(x)
(cid:26) (cid:27)
and
Ψ(x)
liminf > −∞ .
x→x0 I(x)
Under such hypotheses, for each ν,µ satisfying
I(x )
1
ν > max 0,− (6)
J(x )
(cid:26) 1 (cid:27)
and
Ψ(x)
µ > max 0,−liminf , inf µ∗(I +νJ,Ψ,Φ,r) , (7)
(cid:26) x→x0 I(x) r>supMν Φ (cid:27)
where M is the set of all global minima of I + νJ, there exists λ∗ > 0 such that the
ν
functional µ(I +νJ) +Ψ+λ∗Φ has at least four critical points. Precisely, among them,
one is x as a strict local, not global minimum and two are global minima.
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PROOF. First of all, observe that, since X is reflexive, the functionals J,Ψ,Φ are
sequentially weakly continuous, being with compact derivative ([6], Corollary 41.9). Fix ν
as in (6). For x ∈ X \I−1(0), we have
J(x)
I(x)+νJ(x) = I(x) 1+ν
I(x)
(cid:18) (cid:19)
and so, since I is coercive, in view of (1), it follows that
lim (I(x)+νJ(x)) = +∞ . (8)
kxk→+∞
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By the reflexivity of X again, this implies that the set M is non-empty and bounded. As
ν
a consequence, Φ is bounded in M . Also, by (2) and (3), we have sup Φ = +∞. Now,
ν X
fix µ as in (7). Let r > sup Φ be such that µ > µ∗((I +νJ),Ψ,Φ,r). Since Φ−1(r) is
M
ν
non-empty and sequentially weakly closed, there exists x¯ ∈ Φ−1(r) such that
I(x¯)+νJ(x¯) = inf (I(x)+νJ(x)) .
x∈Φ−1(r)
The choice of r implies that x¯ 6∈ M . So, we infer that
ν
inf (I(x)+νJ(x)) < inf (I(x)+νJ(x)) .
x∈Φ−1(]−∞,r]) x∈Φ−1(r)
Moreover, by (2), there exists a sequence {x } in X such that
n
Ψ(x )
n
lim kx k = +∞ , lim = −∞ . (9)
n
n→∞ n→∞ I(xn)
For any ρ ∈ R and for n large enough, we have
Ψ(xn)
ρ(I(x )+νJ(x ))+Ψ(x ) = (I(x )+νJ(x )) ρ+ I(xn) . (10)
n n n n n
 1+νJ(xn)
I(xn)
 
Clearly, from (1), (8), (9) and (10), it follows that
lim (µ(I(x )+νJ(x ))+Ψ(x )) = −∞ .
n n n
n→∞
So, if we consider X endowed with the weak topology, all the assumptions of Theorem A
(with Γ = I +νJ) are satisfied, and so we have
α(µ(I +νJ)+Ψ,Φ,r) < β(µ(I +νJ)+Ψ,Φ,r) .
But, by Theorem 1 of [1], this inequality is equivalent to
sup inf ((µ(I(x)+νJ(x))+Ψ(x)+λ(Φ(x)−r)) < inf sup((µ(I(x)+νJ(x))+Ψ(x)+λ(Φ(x)−r)).
λ≥0x∈X x∈Xλ≥0
At this point, after observing that, in view of (3) and (8), one has
lim (µ(I(x)+νJ(x))+Ψ(x)+λΦ(x)) = +∞ (11)
kxk→+∞
for all λ > 0, we realize that we can apply Theorem B, with A =]0,+∞[, considering X
with the weak topology again and taking
P(x,λ) = µ(I(x)+νJ(x))+Ψ(x)+λ(Φ(x)−r)) .
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Therefore, there exists λ∗ > 0 such that the functional µ(I +νJ)+Ψ+λ∗Φ has at least
two gobal minima. Now, choose ǫ,σ > 0 so that
Ψ(x)
liminf > −µ+ǫ
x→x0 I(x)
and
ǫ
σ < .
µν +λ∗
In view of (5) and recalling that x is a strict local minimum of I (with I(x ) = 0), we can
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find a neighbourhood V of x such that, for each x ∈ V \{x }, one has
0 0
I(x) > 0 ,
Ψ(x)
> −µ+ǫ ,
I(x)
J(x)
> −σ ,
I(x)
and
Φ(x)
> −σ .
I(x)
Consequently, for each x ∈ V \{x }, we have
0
J(x) Ψ(x) Φ(x)
µ(I(x)+νJ(x))+Ψ(x)+λ∗Φ(x) = I(x) µ+µν + +λ∗ >
I(x) I(x) I(x)
(cid:18) (cid:19)
> I(x)(ǫ−σ(µν +λ∗)) > 0 .
Hence, since µ(I(x )+νJ(x ))+Ψ(x )+λ∗Φ(x ) = 0, it follows that x is a strict local
0 0 0 0 0
minimum of the functional µ(I +νJ)+Ψ+λ∗Φ. On the other hand, in view of (4) and
(6), we have
µ(I(x )+νJ(x ))+Ψ(x )+λ∗Φ(x ) < 0 ,
1 1 1 1
and hence x is not a global minimum of the functional µ(I +νJ) +Ψ+ λ∗Φ. Now, we
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remark that this functional, due to (11) and to our assumptions on I,J,Φ,Ψ turns out to
satisfythePalais-Smalecondition([6],Example38.25). Summarizing: thefunctional µ(I+
νJ)+Ψ+λ∗Φ is C1, satisfies the Palais-Smale condition, has at least two global minima
and admits x as a local, not global minimum. At this point, we can invoke Theorem
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(1.ter) of [2] to ensure the existence of a fourth critical point for the same functional, and
the proof is complete. △
Now, we are going to present an application of Theorem 1 to quasilinear elliptic
equations.
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So, let Ω ⊂ Rn be a bounded domain with smooth boundary and let p > 1. On the
Sobolev space W1,p(Ω), we consider the norm
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1
p
kuk = |∇u(x)|pdx .
(cid:18)ZΩ (cid:19)
If n ≥ p, we denote by A the class of all Carath´eodory functions f : Ω×R → R such that
|f(x,ξ)|
sup < +∞ ,
1+|ξ|q
(x,ξ)∈Ω×R
where 0 < q < pn−n+p if p < n and 0 < q < +∞ if p = n. While, when n < p, we denote
n−p
by A the class of all Carath´eodory functions f : Ω×R → R such that, for each r > 0, the
function x → sup |f(x,ξ)| belongs to L1(Ω).
|ξ|≤r
Given f ∈ A, consider the following Dirichlet problem
−div(|∇u|p−2∇u) = f(x,u) in Ω
(P )
f
(
u = 0 on ∂Ω .
Let us recall that a weak solution of (P ) is any u ∈ W1,p(Ω) such that
f 0
|∇u(x)|p−2∇u(x)∇v(x)dx− f(x,u(x))v(x)dx = 0
ZΩ ZΩ
for all v ∈ W1,p(Ω).
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The functionals T,J : W1,p(Ω) → R defined by
f 0
1
T(u) = kukp
p
J (u) = F(x,u(x))dx ,
f
ZΩ
where
ξ
F(x,ξ) = f(x,t)dt ,
Z0
are C1 with derivatives given by
T′(u)(v) = |∇u(x)|p−2∇u(x)∇v(x)dx
ZΩ
J′(u)(v) = f(x,u(x))v(x)dx
f
ZΩ
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for all u,v ∈ W1,p(Ω). Consequently, the weak solutions of problem (P ) are exactly the
0 f
critical points in W1,p(Ω) of the functional T −J which is called the energy functional of
0 f
problem (P ). Moreover, J′ is compact, while T′ is a homeomorphism between W1,p(Ω)
f f 0
and its dual.
The announced application of Theorem 1 is as follows:
THEOREM 2. - Let q > p, with q < pn when n > p, and let f,g,h: Ω×R → R be
n−p
three functions belonging to A and satisfying the following conditions:
inf F(x,ξ) sup F(x,ξ)
lim x∈Ω = +∞ , limsup x∈Ω < +∞ , (12)
ξ→+∞ ξp |ξ|→+∞ |ξ|q
inf G(x,ξ)
x∈Ω
lim = +∞ , (13)
|ξ|→+∞ |ξ|q
sup H(x,ξ) inf H(x,ξ)
limsup x∈Ω ≤ 0 , liminf x∈Ω > −∞ , (14)
|ξ|→+∞ |ξ|p |ξ|→+∞ |ξ|p
sup F(x,ξ)
limsup x∈Ω < +∞ , (15)
|ξ|p
ξ→0
inf G(x,ξ)
x∈Ω
liminf ≥ 0 , (16)
ξ→0 |ξ|p
sup H(x,ξ)
limsup x∈Ω ≤ 0 . (17)
|ξ|p
ξ→0
Finally, assume that there exist a measurable set B ⊂ Ω, with meas(B) > 0, and ξ ∈ R
1
such that
max{−F(x,ξ ),G(x,ξ ),−H(x,ξ )} < 0
1 1 1
for all x ∈ B.
Under such hypotheses, for each ν > 0 large enough, there exists ǫ > 0 with the
ν
following property: for each ǫ ∈]0,ǫ [ there exists λ∗ > 0 such that the problem
ν
−div(|∇u|p−2∇u) = ǫf(x,u)−λ∗g(x,u)+νh(x,u) in Ω
(
u = 0 on ∂Ω
has at least three non-zero weak solutions, two of which are global minima in W1,p(Ω) of
0
the corresponding energy functional.
PROOF. First, observe that from the first assumption in (12) it follows
J (u)
f
limsup = +∞ .
kukp
kuk→+∞
This is proved in the proof of Theorem 4 of [5], and so we do not repeat the argument
here. Moreover, from the second assumption in (12) and from (13), it clearly follows that,
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for each λ > 0, the function λG − F is bounded below in R (see [5] again), and so the
functional λJ −J is bounded below in W1,p(Ω). Moreover, by (14), there is c > 0 and,
g f 0
for each ǫ > 0, another c > 0, such that
ǫ
−c(|ξ|p +1) ≤ H(x,ξ) ≤ ǫ|ξ|p +c
ǫ
for all (x,ξ) ∈ Ω×R. This clearly implies that
J (u)
h
limsup ≤ 0
kukp
kuk→+∞
and
J (u)
h
liminf > −∞ .
kuk→+∞ kukp
Furthermore, by (15) and by the second assumption in (12), there is a constant c > 0
1
such that
F(x,ξ) ≤ c (|ξ|p+|ξ|q)
1
for all (x,ξ) ∈ Ω×R. Since q > p, this implies that
J (u)
f
limsup < +∞ .
kukp
u→0
Now, suppose n ≥ p. By (16) and (17), taking into account that g,h ∈ A, for some r > p
and for each ǫ > 0 there is d > 0 such that
ǫ
G(x,ξ) ≥ −ǫ|ξ|p −d |ξ|r
ǫ
and
H(x,ξ) ≤ ǫ|ξ|p +d |ξ|r
ǫ
for all (x,ξ) ∈ Ω×R. From this, we get
J (u)
g
liminf ≥ 0
u→0 kukp
and
J (u)
h
limsup ≤ 0 .
kukp
u→0
In the case n < p, we get again these two inequalities thanks to (16) and (17) and to the
continuous embedding of W1,p(Ω) into C0(Ω). Now, let ω ∈ L1(Ω) be such that
0
max{|F(x,ξ)|,|G(x,ξ)|,|H(x,ξ)|}≤ ω(x)
for all (x,ξ) ∈ Ω×[−|ξ |,|ξ |]. Next, choose a closed set C ⊂ B, an open set D ⊂ Ω, with
1 1
C ⊂ D, and θ ∈ R in such a way that
η := max − F(x,ξ )dx, G(x,ξ )dx,− H(x,ξ )dx < θ < 0
1 1 1
(cid:26) ZC ZC ZC (cid:27)
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and
ω(x)dx < θ−η .
ZD\C
Finally, let v : Ω → [−|ξ |,|ξ |] be a function belonging to W1,p(Ω) such that v (x) = ξ
1 1 1 0 1 1
for all x ∈ C and v (x) = 0 for all x ∈ Ω\D. Clearly, we have
1
max − F(x,v (x))dx, F(x,v (x))dx,− H(x,v (x))dx < η +(θ−η) < 0 .
1 1 1
(cid:26) ZΩ ZΩ ZΩ (cid:27)
At this point, the conclusion comes directly from that of Theorem 1 applied taking I(u) =
1kukp, J(u) = −J (u), Ψ(u) = −J (u), Φ(u) = J (u) for all u ∈ W1,p(Ω). △
p h f g 0
References
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Verlag, 1985.
Department of Mathematics
University of Catania
Viale A. Doria 6
95125 Catania
Italy
e-mail address: [email protected]
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