Table Of ContentA multiplicity result for the problem
δdξ = f′(hξ, ξi)ξ
7
0
0 Antonio Azzollini
2
n Dipartimento di Matematica
a
J Universit`a di Bari,
5
Via Orabona 4, 70125 Bari, Italy
2
e-mail: [email protected]
]
P
A
. Abstract
h
t
a In this paper we consider the nonlinear equation involving differ-
m
ential forms on a compact Riemannian manifold δdξ = f′(hξ,ξi)ξ.
[
This equation is a generalization of the semilinear Maxwell equations
1 recently introduced in a paper by Benci and Fortunato. We obtain a
v
multiplicity result both in the positive mass case (i.e. f′(t) ≥ ε > 0
7
5 uniformly) and in the zero mass case (f′(t) ≥ 0 and f′(0) = 0) where
7 a strong convexity hypothesis on the nonlinearity is assumed.
1
0
7 Keywords Semilinear Maxwell equations; Strongly indefinite functional; Strong
0 convexity
/
h
t
a
Introduction
m
:
v
Let (M,g) be a compact Riemannian n−manifold, where n ≥ 3, and Λk(M)
i
X
be the set of regular k-forms on M. We consider the following equation
r
a
δdξ = f′(hξ,ξi)ξ,
(1)
ξ ∈ Λk(M), 1 ≤ k ≤ n−1,
where f : R → R isa C2 map, d is the exterior differential, δ is its adjoint
with respect to the inner product
(η,ξ) = hη,ξiω = ∗(η∧∗ξ)ω (2)
2
ZM ZM
1
∗ is the Hodge operator and ω is a volume n-form.
In this paper we are looking for weak solutions of (1), namely for solutions
of
ξ ∈ H1(M),(see section 1.2 for the definition)
k
(3)
hdξ,dηiω = f′(hξ,ξi)hξ,ηiω, ∀η ∈ H1(M).
M M k
If we set R R
F(ξ) := f(hξ,ξi)ω, (4)
ZM
then, assuming a suitable condition on the growth of f′, by standard argu-
ments we have that F ∈ C1(H1(M)), so in order to solve (3) we find critical
k
points of the functional
J(ξ) = hdξ,dξiω−F(ξ) (5)
ZM
defined for all ξ ∈ H1(M).
k
The strongly indefinite nature of the functional J, largely discussed in
[4], doesn’t allow us to approach this problem in a standard way. In other
words, the functional J doesn’t present the geometry of the mountain pass
in any space with finite codimension.
Assume that
f ) f(0) = 0, and ∃ε > 0 s.t. ∀t ≥ 0 : f′(t) ≥ ε,
1
f ) f is strictly convex,
2
and for p ∈]2, 2n [
n−2
f3) ∃a > 0,b > 0 s.t. |f′(t)| ≤ atp2−1 +b, ∀t ≥ 0,
f ) ∃R > 0 s.t. 0 < p f(t) ≤ f′(t)t for t > R.
4 2
We have the following result
Theorem 1. If f )-...-f ) hold, then the problem (3) has infinitely many
1 4
solutions.
Moreover the same conclusion holds if f ) and f ) are substituted respec-
1 2
tively by
f˜) f(0) = f′(0) = 0, and ∀t ≥ 0 : f′(t) ≥ 0,
1
2
f˜) ∃c > 0 s.t. ∀ξ,η ∈ Λk(M)
2
f(hξ,ξi)−f(hη,ηi)−2f′(hη,ηi)hη,ξ−ηi ≥ chξ −η,ξ −ηi2p
pointwise in M.
Remark 2. As in [5], in the sequel we shall refer to the hypotheses f and f˜
1 1
respectively as the “positive mass” and “zero mass” case.
Moreover we want to point out the fact that f˜ is just a pointwise con-
2
vexity condition. In fact for every q ∈ M we can define the scalar product
h·,·i on the vector space Λk(M) and the functional
q
I (ξ) = f(hξ,ξi ).
q q
Since I′(η) = 2f′(hη,ηi)hη,·i, f˜ implies that for all ξ,η ∈ Λk(M) s.t. η 6= ξ
q 2
I (ξ)−I (η)−hI′(η),ξ −ηi > 0,
q q q
and then I is strictly convex. In the Appendix we shall show that f˜ is
q 2
p
satisfied when f(t) = t2.
In order to prove Theorem 1, as in [4] we shall use Hodge decomposition
to split ξ in this way
ξ = dα+β (6)
where δβ = 0.
The functional (5) formally becomes
J(α,β) = hdβ,dβiω− f(hdα+β,dα+βi)ω. (7)
ZM ZM
If we set
J : β 7→ J (β) = J(α,β) (8)
α α
and
J : α 7→ J (α) = J(α,β), (9)
β β
then we can define the partial derivative of J as follows
∂J
(α,β) : = dJ (α) (10)
∂α β
∂J
(α,β) : = dJ (β). (11)
α
∂β
3
We are interested in finding critical points of (7), i.e. the couples (α,β) such
that
∂J
(α,β) = 0 (12)
∂α
and
∂J
(α,β) = 0. (13)
∂β
Set
F := α 7→ F(dα+β) (14)
β
and note that by the convex nature of (14), the problem (12) is actually a
minimizing problem.
In section 1, where we assume f )...f ), we introduce some preliminary
1 4
results and the definition of the spaces V and W which respectively α and
β belong to. These spaces are constructed in such a way we have, for any
β ∈ W, a unique solution Φ(β) ∈ V for the minimizing problem (12).
Then in Theorem 5 we’ll show that in order to solve the system (12) and
(13) we are reduced to study the critical points of the functional
J(β) = J(Φ(β),β), β ∈ W. (15)
Differently from J, J bdoesn’t exhibit strong indefinitness, so the proof of
the existence of infinitely many critical points is carried out by using a well
known multiplicity rebsult for even functionals.
In section 2 we replace f and f by f˜ and f˜. Differently from the previ-
1 2 1 2
ous situation, we can’t give any proof about the regularity of the functional
(15). To overcome this difficulty we work in an indirect way. In fact we
perturb the problem (1) by adding a linear “mass term” mξ, m > 0, to the
nonlinearity on the right hand side. For this perturbed problem we have
infinitely many solutions since it satisfies the hypothesis f .
1
Then we study the behaviour of the solutions of the perturbed problem when
the perturbation goes to zero.
4
1 Positive mass case
1.1 The functional framework
For any q > 1 and k ∈ N, let H1,q(M), H1(M) and Lq(M) be defined as
k k k
follows
H1,q(M) := Λk(M)k·k1,q ,
k
k·k
H1(M) := Λk(M) ,
k
Lq(M) := Λk(M)|·|q
k
where, for every ξ ∈ Λk(M),
kξkq := hdξ,dξiq/2ω + hδξ,δξiq/2ω + hξ,ξiq/2ω
1,q
ZM ZM ZM
kξk2 := hdξ,dξiω+ hδξ,δξiω+ hξ,ξiω (16)
ZM ZM ZM
|ξ|q := hξ,ξiq/2ω.
q
ZM
Since
2n
H1(M) ֒→ Lq(M) for 1 ≤ q ≤ , (17)
k k n−2
by f we have that J(α,β) < +∞ for α ∈ H1,p (M) and β ∈ H1(M).
3 k−1 k
Observe that F : H1,p (M) → R defined by (14) is not coercive, since it
β k−1
is constant on the space
C := η ∈ H1,p (M) | dη = 0 . (18)
k−1
However we have the followi(cid:8)ng result (cid:9)
Lemma 3. For all β ∈ Lp(M), F is coercive on
k β
V := α ∈ H1,p (M) | ∀η ∈ C : hα,ηiω = 0 . (19)
k−1
(cid:26) ZM (cid:27)
Proof. First observe that, by f ,
4
p
∃c > 0,d > 0 s.t. ct2 ≤ f(t)+d,∀t ≥ 0. (20)
Let β ∈ Lp(M). By Lemma 6 in [2], we know that the norm on H1(M)
k k
defined by
kξk2 := (hdξ,dξi+hδξ,δξi+hξ0,ξ0i)ω (21)
∼
ZM
5
where ξ0 is the orthogonal projection of ξ on ker(−∆), is equivalent to the
norm defined by (16).
In particular, in the space V the norm (21) becomes
kξk2 = hdξ,dξiω.
∼
ZM
Indeed, if α ∈ V, then
hδα,δαiω = hdδα,αiω = 0 (22)
ZM ZM
because dδα ∈ C.
Moreover, since α0 ∈ ker(−∆), then α0 ∈ C. But α0 ∈ V and then
hα0,α0iω = 0. (23)
ZM
By (22) and (23) we can conclude that |dα| is a norm on the space V
2
equivalent to kαk, i.e.
∃c˜> 0 s.t. ∀α ∈ V : kαk ≤ c˜|dα| . (24)
2
By (24) and since H1 (M) ֒→ Lp (M),
k−1 k−1
kαkp = |dα|p +|α|p ≤ |dα|p +c kαkp ≤ |dα|p +c |dα|p
1,p p p p 1 p 2 2
and then
kαk ≤ c (|dα| +|dα| ) ≤ c |dα| (25)
1,p 3 p 2 4 p
since Lp(M) ֒→ L2(M).
k k
Now, consider (α ) in V s.t. kα k → +∞. By (25)
n n≥1 n 1,p
p
hdαn +β,dαn +βi2 ω → +∞, (26)
ZM
so by (20) we have
p
c hdαn+β,dαn+βi2 ω ≤ dmeas(M)+ f(hdαn+β,dαn+βi)ω. (27)
ZM ZM
The coerciveness of F is a consequence of (26) and (27).
β
Theorem 4. For every β ∈ Lp(M) there exists a unique minimizer of F .
k β|V
Proof. Let β ∈ Lp(M). By f and f the functional F : Lp(M) → R defined
k 2 3 k
by (4) is strictly convex and continuous. Obviously, also F has the same
β
properties, so it is weakly lower semicontinuos. Since F is also coercive in
β
V by Lemma 3, certainly it possesses a minimizer Φ(β) ∈ V. The uniqueness
is a consequence of the strict convexity.
6
1.2 Regularity, symmetry and compactness
Assume the following definitions:
Φ : Lp(M) → V s.t. Φ(β) is the minimizer of F
k β|V
(28)
J : W → R s.t. ∀β ∈ W : J(β) = J(Φ(β),β)
where
b b
W := β ∈ H1(M) | δβ = 0 .
k
Theorem 5. If Φ ∈ C1(W,V(cid:8)), then J ∈ C1(W) a(cid:9)nd its critical points are
solutions of (12), (13).
b
Proof. Suppose Φ ∈ C1(W,V), then certainly J ∈ C1(W) since it is the
composition of C1 maps.
Now let β ∈ W be a critical point of J. We havebthat for any β ∈ W
0
∂J ∂J
0 = hJ ′(β ),βi = (Φ(β ),β ),Φ′b(β )(β) + (Φ(β ),β ),β , (29)
0 ∂α 0 0 0 ∂β 0 0
(cid:10) (cid:11) (cid:10) (cid:11)
that is b
∂J ∂J
(Φ(β ),β ) = − (Φ(β ),β )◦Φ′(β ). (30)
∂β 0 0 ∂α 0 0 0
But
∂J
(Φ(β ),β ) = 0 (31)
∂α 0 0
because Φ(β ) is a minimizer of F , so also
0 β|V
∂J
(Φ(β ),β ) = 0. (32)
∂β 0 0
Inordertostudy thefunctionalJ, we need toinvestigate theproperties of
the map Φ. Then, in the next theorem we are going to prove some regularity,
symmetry and compactness propertbies of the map Φ. To get regularity, in
particular, we will use the implicit function theorem on
∂J
: V ×W → V′. (33)
∂α
where V′ is the dual of V.
Observe that J ∈ C2(V ×W). Moreover we have the following
7
Lemma 6. Set
F˜ := (α,β) ∈ V ×W 7→ F(dα+β). (34)
Then for all β ∈ Lp(M), F˜(·,β) ∈ C2(V) is uniformly convex on V with
k
respect to the norm k·k, i.e. there exists C > 0 such that for all α, α ∈ V
∂2F˜
(α,β)[α,α] := d2F˜(·,β)(α)[α,α] ≥ Ckαk2. (35)
∂α2
Proof. Let β ∈ Lp(M) and α ∈ V. If α ∈ V, then
k
∂2F˜
(α,β)[α,α] = 4 f′′(hdα+β,dα+βi)(hdα+β,dαi)2ω
∂α2
ZM
+2 f′(hdα+β,dα+βi)hdα,dαiω
ZM
≥ ε hdα,dαiω,
ZM
where the last inequality follows from the convexity of f and assumption
f .
1
Now we are ready to prove
Theorem 7. The following properties hold
1. Φ ∈ C1(W);
2. Φ is odd;
3. Φ is compact.
Proof. 1. Since Φ(β) is a minimizer of F
β
∂F˜
∀β ∈ W : (Φ(β),β) = 0, (36)
∂α
so, by Lemma 6 and the implicit function theorem, we have that Φ ∈
C1(W,V) and for any β ∈ W:
−1
∂2F˜ ∂2F˜
Φ′(β) = − (Φ(β),β) ◦ (Φ(β),β).
(∂α)2 ∂α∂β
!
8
2. If β ∈ W, then some calculations show that
∂F˜ ∂F˜
(−Φ(β),−β) = (Φ(β),β) = 0,
∂α ∂α
so, by uniqueness, −Φ(β) = Φ(−β).
3. By the same arguments as above, we can prove that Φ is also in
C1(Lp(M),V) so, if (β ) is a sequence in W and
k n n
β ⇀ β ∈ W
n
with respect to the norm k·k, then
β → β in Lp(M)
n k
and so
Φ(β ) → Φ(β) in V.
n
1.3 Main Theorem (first part)
We introduce some results on the Laplace Beltrami operator −∆.
It is well known that −∆ is a self adjoint operator on L2(M) with a
k
nonnegative, discrete and divergent spectrum σ(∆).
Set
λ ≤ λ ≤ ... (37)
1 2
thesequence oftheeigenvalues different fromzerorepeatedaccordingtotheir
finite multiplicity. The corresponding eigenvectors
η ,η ,... (38)
1 2
constitute an orthonormal basis of (ker(−∆))⊥. Take a basis h ,h ,...,h
1 2 N
of ker(−∆) so that, if β ∈ L2(M), we have
k
∞
β = β η +β0,
i i
i≥1
X
where (β ) are the Fourier components ofβ corresponding to η ,η ..., and
i i≥1 1 2
β0 is its projection on ker(∆).
For every s ∈ R define the Sobolev space
Ws,2(M) := β ∈ L2(M) | kβk < ∞ ,
k k s,2
where n o
9
∞
kβk2 := (λs +1)β2 +|β0|2. (39)
s,2 i i 2
i≥1
X
It is easy to prove that (Ws,2(M),k·k ) is a Banach space and W1,2(M) ≡
k s,2 k
H1(M). Since p < 2n , by Sobolev embedding theorem, there exist s < 1
k n−2
and c˜> 0 such that
|β|2 ≤ c˜kβk2 , ∀β ∈ Ws,2(M). (40)
p s,2 k
Now we recall the following abstract multiplicity theorem whose proof
can be found in [3] (see also [1]).
Theorem 8. Let H be an Hilbert space and I be a C1 even functional on H
such that
1. I(0)=0,
2. I satisfies (P-S) condition i.e. any sequence (x ) such that
n n
I(x ) is bounded
n
I′(x ) → 0,
n
admits a convergent subsequence,
3. there exist H−, H+ two closed subspaces of H such that
(a) codim(H+) < dim(H−) < +∞
(b) ∃c ,ρ > 0 s.t. I(x) ≥ c , ∀x ∈ ∂S (0) ∩ H+, (where ∂S (0) :=
0 0 ρ ρ
{x ∈ H | kxk = ρ})
(c) ∃c > 0 s.t. ∀x ∈ H− : I(x) < c
1 1
then I possesses at least dim(H−) − codim(H+) couples of critical points
whose corresponding critical values are in [c ,c ].
0 1
Inthenext lemmas we shallverify thehypotheses of theprevious theorem
for the functional J on the Hilbert space W.
Lemma 9. J is a C1 even functional satisfying the (P.S.) condition
b
Proof. The regularity and symmetry properties of J are an immediate con-
b
sequence of the structure of J, and Theorem 7.
b
10
