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arXiv:1602.01193v1 [math.AP] 3 Feb 2016
Multiple solutions for a Kirchhoff-type equation
with general nonlinearity
Sheng-Sen Lu
Chern Institute of Mathematics and LPMC,Nankai University
Tianjin, 300071, PR China
e-mail: lushengsen@mail.nankai.edu.cn
Abstract
This paper is devoted to the study of the following autonomous Kirchhoff-type
equation
−MZRN
|∇u|2∆u=f(u), u ∈H1(RN),
where Mis a continuous non-degenerate function and N≥2. Under suitable
additional conditions on Mand very general assumptions on the nonlinearity f,
we establish certain existence results of multiple solutions by variational methods,
which are also naturally interpreted from a non-variational point of view.
2010 Mathematics Subject Classification:35J20, 35J60.
Key words:Kirchhoff-type equation, Multiplicity results, Variational methods.
1 Introduction and main results
In this paper, we consider the following autonomous nonlinear elliptic problem
−MZRN
|∇u|2∆u=f(u) in RN,
u∈H1(RN), u 6≡ 0 in RN,
(KT )
where N≥2, M:R+→R+and f:R→Rare continuous functions that satisfy
some assumptions which will be stated later on.
In the case where Mis not identically equal to a positive constant, the class
of Problem (KT ) is called of Kirchhoff type because it comes from an important
application in Physic and Engineering. Indeed, if we let M(t) = a+bt with a, b > 0
and replace RNand f(u) by a bounded domain Ω ⊂RNand f(x, u) respectively
in (KT ), then we get the following Kirchhoff problem:
−a+bZΩ
|∇u|2∆u=f(x, u) in Ω,
assuming the homogeneous Dirichlet boundary condition, which is related to the
1
stationary analogue of the equation
ρ∂2u
∂t2− P0
h+E
2LZL
0
∂u
∂x
2!∂2u
∂x2= 0 in (0, T )×(0, L)
presented by G. Kirchhoff in [11]. Besides, (KT ) is also called a nonlocal problem
in this case because of the appearance of the term MRRN|∇u|2∆uwhich im-
plies that (KT ) is no longer a pointwise identity. And this phenomenon provokes
some mathematical difficulties which make the study of Problem (KT ) particularly
interesting.
On the other hand, when Mis identically equal to a positive constant, for
example M(t)≡1, there has been a considerable amount of research on this kind of
problems during the past years. The interest comes, essentially, from two reasons:
the fact that such problems arise naturally in various branches of Mathematical
Physics, indeed the solutions of (KT ) in the case where M(t)≡1 can be seen
as solitary waves (stationary states) in nonlinear equations of the Klein-Gordon or
Schr¨odinger type, and the lack of compactness, challenging obstacle to the use of
the variational methods in a standard way.
In the celebrated papers [4, 5, 6], the authors studied the case where M(t)≡1,
namely the following autonomous nonlinear scalar field problem
(−∆u=f(u) in RN,
u∈H1(RN), u 6≡ 0 in RN,(SF )
under the following assumptions on the nonlinearity f:
(f0)f∈C(R,R) is continuous and odd.
(f1) For N≥3,
−∞ <lim inf
t→0
f(t)
t≤lim sup
t→0
f(t)
t<0.(1.1)
For N= 2,
lim
t→0
f(t)
t∈(−∞,0).(1.2)
(f2) When N≥3, lim
t→∞
f(t)
|t|
N+2
N−2
= 0.
When N= 2, for any α > 0
lim
t→∞
f(t)
eαt2= 0.
(f3) There exists ζ > 0 such that F(ζ)>0, where F(t) := Rt
0f(τ)dτ.
With the aid of variational methods and critical points theory, by studying
certain constraint problems, Berestycki-Lions and Berestycki-Gallouet-Kavian es-
tablished the existence results of a ground state, namely a nontrivial solution which
minimizes the action among all the nontrivial solutions, and infinitely many bound
state solutions of (SF ) in [5, 6] for N≥3 and in [4] for N= 2 respectively.
2
As we can see, there is a difference in the assumption (f1) between the cases
N≥3 and N= 2. We remark here that, in the proofs given by [4] for the case
N= 2, the existence of a limit limt→0f(t)/t ∈(−∞,0) is used essentially to show
that the Palais-Smale compactness condition for the corresponding functional under
suitable constraint. It is hard to generalize (1.2) to the general one (1.1) in that
argument.
Later on, in a recent paper [9], Hirata, Ikoma and Tanaka revisited Problem
(SF ) in the case N≥2 assuming (f0), (f2), (f3) and
(f′
1)−∞ <lim inf
t→0
f(t)
t≤lim sup
t→0
f(t)
t<0
and tried to find radial solutions through the unconstraint functional
I(u) := 1
2ZRN
|∇u|2−ZRN
F(u), u ∈H1(RN).(1.3)
In [9], following the approach introduced by Jeanjean in [10], Hirata, Ikoma and
Tanaka considered the auxiliary functional ˜
I:R×H1
r(RN)→R
˜
I(θ, u) := 1
2e(N−2)θZRN
|∇u|2−eNθ ZRN
F(u).
In this way, they were able to find a Palais-Smale sequence (θj, uj)+∞
j=1 in the aug-
mented space R×H1
r(RN) such that θj→0 and uj“almost” satisfies the Poho˘zaev
identity associated to (SF ). With the aid of this extra information, it was proved
that Problem (SF) possesses a positive least energy solution and infinitely many
(possibly sign changing) radially symmetric solutions.
Our main aim of the present paper is to try to provide some multiplicity results
for Problem (KT ) under the very general assumptions (f0), (f2), (f3) and (f′
1) on
fand some suitable conditions on Mby variational methods.
In terms of (f0), (f′
1) and (f2), we conclude that the corresponding functional
Jof (KT ) given by
J(u) := 1
2c
MZRN
|∇u|2−ZRN
F(u)
is well-defined on H1(RN) and of class C1. It is easy to see that Jis invariant under
rotation. Then,
H1
r(RN) := u∈H1(RN)|u(x) = u(|x|)
is a natural constraint to look for critical points, namely critical points of the
functional restricted to H1
r(RN) are true critical points in H1(RN). Therefore,
from now on, we will directly define Jon H1
r(RN).
Before stating our assumptions on Mand the main results of this paper, it is
needed and necessary to mention the close related works of Azzollini, d’Avenia and
Pomponio [3] and Lu [12]. To the best of our knowledge, it seems that only articles
[3] and [12] consider the multiplicity of solutions for such problem under the very
general assumptions on f.
In the paper [3], under the same very general assumptions on fas above, Az-
zollini, d’Avenia and Pomponio considered a suitable perturbation of I, namely
Iq(u) := I(u) + qR(u), u ∈H1(RN),
3
where Iis given by (1.3), q > 0 is a positive parameter, R:H1(RN)→Rand
N≥3. The authors supposed that R= Σk
i=1Riand, for each i= 1,··· , k , the
functional Risatisfies:
(R1) Riis a nonnegative even C1functional on H1(RN).
(R2) There exists δi>0 such that
R′
i(u)[u]≤Ckukδi
H1(RN)for any u∈H1(RN).
(R3) If {uj}+∞
j=1 ⊂H1
r(RN) is weakly convergent to u∈H1
r(RN), then
lim sup
j→+∞
R′
i(uj)[u−uj]≤0.
(R4) There exist αi, βi≥0 such that if u∈H1(RN), t > 0 and ut(·) := u(t−1·),
then
Ri(ut(·)) = tαiRi(tβiu(·)).
(R5) Riis invariant under the action of N-dimensional orthogonal group, i.e.
Ri(u(g·)) = Ri(u(·)) for every g∈O(N).
By a suitable combination of the method described in [9] and a certain truncation
argument, they established an abstract theorem which claims the existence of (at
least) ndistinct critical points of Iqfor every n∈Nand q∈(0, qn), where qn>0 is
a suitable positive constant depending on n. As an application, in the case where
N≥3 and M(t) = a+bt with a, b > 0, they treated Problem (KT ) and obtained
finitely many distinct radial solutions for sufficiently small b > 0. For another
application to the nonlinear Schrodinger-Maxwell, we refer reader to [3].
A similar approach has also been used in [7] to the study of the following Chern–
Simons–Schr¨odinger equation
−∆u+qu h2
u(|x|)
|x|2+ 2qu Z+∞
|x|
u2(t)
thu(t)dt =f(u) in R2,(CSS )
where u∈H1
r(R2), hu(t) := Rt
0τu2(τ)dτ and q > 0 is a positive parameter. In that
paper, a multiplicity result of radial solutions for (CSS) was established under the
very general assumptions (f0), (f2), (f3) and (f′
1) on f. We refer reader to [7] for
the details.
We note that the truncation argument explored in [3] is important to the proof
of the abstract existence result. Actually, the truncation argument not only is
used to construct a suitable modified functional of Iq, which satisfies the symmetric
mountain pass geometry, but also, together with the method described in [9], plays a
vital role in obtaining (at least) ndistinct particular Palais-Smale sequences which
are bounded for every n∈Nand q∈(0, qn). Thus, it is interesting to ask the
question whether, at least for Problem (KT ) in the case where M(t) = a+bt with
a, b > 0 and N≥3, it is possible to prove the multiple result by some suitable
arguments, e.g. variational methods, but without a truncation technique similar as
that in [3].
4
In the more recent paper [12], by means of a rescaling argument based on an
idea of Azzollini [1, 2] and a new description of the critical values, we investigated
the following Kirchhoff Problem
−a+bZRN
|∇u|2∆u=f(u) in RN,
u∈H1(RN), u 6≡ 0 in RN,
(K)
where a≥0, b > 0 and N≥1. When N≥2, under some suitable conditions on
the values of the nonnegative parameters aand bif necessary and the assumptions
(f0),(f2),(f3) and (f′
1) on f, certain multiplicity results for (K) were obtained as
partial results in that paper. In particular, it is infinitely many distinct radial
solutions that we obtained in [12] for any a≥0 and b > 0 fixed when N= 2,3. We
note here that [12] not only answers the question we raise above in the affirmative
from the non-variational point of view, but also extends the result of Azzollini,
d’Avenia and Pomponio in [3] concerning the existence of multiple solutions to (K).
As pointed out in [12], it is natural to know whether, at least for the non-
degenerate case a > 0, one can still obtain the multiplicity results for (K) via
variational methods. So far, this question has only been solved partially by Azzollini,
d’Avenia and Pomponio in the early work [3] and still remains open in dimensions
N= 2,3, where, in fact, it is infinitely many distinct radial solutions that Problem
(K) possesses.
Motivated by the articles [3, 9, 12] and the questions we raise above, by intro-
ducing some suitable assumptions on M, we shall show the existence of infinitely
many distinct radial solutions for Problem (KT ) as our first result of this paper.
For this purpose, we make the hypotheses on the function Mas follow:
(M1) There exists m0>0 such that M(t)≥m0for any t≥0.
(M2) Let c
M(t) := Rt
0M(τ)dτ. Then there holds
lim inf
t→+∞c
M(t)−1−2
NM(t)t= +∞.
(M3) lim
t→+∞
M(t)
t
2
N−2
= 0.
Now, our first result of the present paper can be stated as follows:
Theorem 1.1 Assume N≥2and that fsatisfies (f0),(f2),(f3)and (f′
1). In ad-
dition, suppose (M1)when N= 2 and (M1)−(M3)when N≥3. Then Problem
(KT )has infinitely many distinct (possibly sign-changing) radially symmetric solu-
tions, which are characterized by the symmetric mountain pass minimax argument
in H1
r(RN).
Remark 1.1 In our later proof of Theorem 1.1, a truncation argument similar as
that in [3] would and should be avoided; since, if not, in general it seems to be
difficult or even impossible to get infinitely many distinct solutions. This can be
seen as another reason why we try to find solutions of Problem (KT )through the
non-modified functional Jdirectly.
5
Remark 1.2 When N= 2, under the same assumptions of Theorem 1.1, Figueiredo,
Ikoma and J´unior have obtained a least energy solution of (KT )in the early work
[8]. In that paper, under certain suitable conditions on Mwhich are more than
sufficient that (M1)−(M3)hold, the existence result of a least energy solutions to
(KT )was also established for N≥3. Our Theorem 1.1 here can be viewed as a
natural extension of [8].
Next, when N≥3, for a suitable class of non-degenerate functions Mwhich
may not satisfy the hypothesis (M3), we establish the following weaker multiplicity
result, which claims the existence of finitely many radial solutions to (KT ).
Theorem 1.2 Assume that M(t) = m0+qλ(t)with q > 0and λ∈C(R+,R+),
N≥3and fsatisfies (f0),(f2),(f3)and (f′
1). Besides, suppose that either (M2)or
(M′
2)as follows:
(M′
2)There holds
lim sup
t→+∞c
M(t)−1−2
NM(t)t≤0,
is satisfied. Then there exists a positive sequence {qn}+∞
n=1 such that Problem (KT )
has at least ndistinct (possibly sign-changing) radial solutions for any q∈(0, qn).
All the solutions are characterized by the symmetric mountain pass minimax argu-
ment in H1
r(RN).
Remark 1.3 The fact that, under the assumptions of Theorem 1.2, we obtain only
finitely many nontrivial solutions for sufficiently small q > 0is not surprising and
we can hardly expect more. Actually, the function M(t) = m0+qt 2
N−2with m0, q > 0
satisfies (M2). However, in this case, Theorem A.1 in the paper [8] by Figueiredo,
Ikoma and J´unior showed the nonexistence of nontrivial solution for large enough
q > 0. In addition, by repeating certain arguments explored in [12] for the proof of
Theorem 1.2, Item (ii)in that paper, we can only show that more and more distinct
solutions of (KT )exist as q→0+. It seems to be difficult or even impossible to
get infinitely many distinct solutions of (KT )for sufficiently small but fixed q > 0.
Thus, the conclusion of Theorem 1.2 seems to be the best possible result we could
hope for when Mdoes indeed not satisfy the hypothesis (M3).
Remark 1.4 It is not difficult to see that the hypothesis (R2) is not always hold
for functions Mwhich verify the assumptions of Theorem 1.2. For example, let
m0= 1 and λ(t) = 1
2(et−1), that is M(t) = 1 + q
2(et−1), then a straightforward
computation shows that such Msatisfies assumption (M′
2). However, in this case,
(R2) is not satisfied due to the fact that, for any δ > 0and u∈H1(RN)\ {0}, there
holds
lim
t→+∞
R′(tu)[tu]
ktukδ=k∇uk2
2
2kukδlim
t→+∞et2k∇uk2
2−1t2−δ= +∞.
Thus, our Theorem 1.2 can not be obtained by applying the abstract result given by
[3] directly and the arguments there are also not valid here.
As a consequence of Theorems 1.1 and 1.2, we have the following result:
Corollary 1.1 Assume a > 0fixed, b > 0,N≥2and that fsatisfies (f0),(f2),(f3)
and (f′
1). Then the following statements hold.
6
(i)If N= 2,3, Problem (K)has infinitely many distinct radially symmetric solu-
tions for any b > 0, which are characterized by the symmetric mountain pass
minimax argument in H1
r(RN).
(ii)If N≥4, there exists a positive sequence {bn}+∞
n=1 such that Problem (K)has
at least ndistinct radially symmetric solutions for any b∈(0, bn). Moreover,
all the solutions are characterized by the symmetric mountain pass minimax
argument in H1
r(RN).
Remark 1.5 As we can see in Sections 3 and 4, the proofs of Theorems 1.1 and
1.2 are all based on a certain variational method described in [9] but without a
truncation argument similar as that in [3]. Thus, noting the fact that Corollary 1.1
follows from Theorems 1.1 and 1.2 directly, we answer the first question we raise
above in the affirmative again from the variational point of view and address the
second problem we raise above in the remaining case N= 2,3. As a by-product,
in the case a > 0and N≥4, we provide another variational proof of the multiple
result of (K)through the non-modified functional J, which is different from that in
[3].
The rest of this paper is organized as follows. In Section 2, an auxiliary problem
is constructed in the spirit of [9] and the corresponding conclusions are shown at
the same time. With the aid of the method described in [9] and the conclusions
in Section 2, the proofs of Theorems 1.1 and 1.2 are completed in Sections 3 and
4 respectively. Lastly, in Section 5, the non-variational proofs are presented which
actually provide us a better understanding of the multiplicity results.
2 The auxiliary problem and its result
In this section, we shall construct an auxiliary problem in the spirit of [9], which
will play a important role in the proofs of the main results of this paper. To be
more precise, it will be proved that there is a sequence of positive critical values
{en}+∞
n=1 corresponding to the auxiliary problem which is divergent to infinity. This
fact allows us to prove the multiplicity results for our original problem (KT ) based
on the level sets argument.
Following [9], we set
ω:= −1
2lim sup
t→0
f(t)
t∈(0,+∞)
and equip H1
r(RN) with the norm k · k := m0k∇ · k2
2+ωk · k2
21
2.
Consider p0∈1,N+2
N−2if N≥3, p0∈(1,+∞) if N= 2 and set
h(t) := (max{ωt +f(t),0},for t≥0,
−h(−t),for t < 0,H(t) := Zt
0
h(τ)dτ,
h(t) :=
tp0max
0<τ ≤t
h(τ)
τp0,for t > 0,
0,for t= 0,
−h(−t),for t < 0,
H(t) := Zt
0
h(τ)dτ.
7
Then, the functions h, h, H and Hsatisfy the properties stated in Lemmas 2.1-2.3
below.
Lemma 2.1 The following hold:
(i)For all t≥0,ωt +f(t)≤h(t)≤h(t).
(ii)For all t≥0,h(t), h(t)≥0.
(iii)There exists δ > 0such that h(t) = h(t) = 0 for all t∈[0, δ]
(iv)There exists ξ > 0such that 0< h(ξ)≤h(ξ).
(v)The map t7→ h(t)
tp0is non-decreasing in t∈(0,+∞).
(vi)The functions h, hsatisfy (f2).
Lemma 2.2 The following hold:
(i)For all t≥0,1
2ωt2+F(t)≤H(t)≤H(t).
(ii)For all t≥0,H(t), H(t)≥0.
(iii)There exists δ > 0such that H(t) = H(t) = 0 for all t∈[0, δ]
(iv)It holds that H(ζ)−1
2ωζ2>0.
(v)For all t∈R,0≤(p0+ 1)H(t)≤th(t).
(vi)The functions H, Hsatisfy
lim
|t|→+∞
H(t)
t2N
N−2
= lim
|t|→+∞
H(t)
t2N
N−2
= 0,when N≥3,
lim
|t|→+∞
H(t)
eαt2= lim
|t|→+∞
H(t)
eαt2= 0,for any α > 0when N= 2.
Lemma 2.3 Let N≥2and suppose that {uj}+∞
j=1 ⊂H1
r(RN)converges to u∈
H1
r(RN)weakly in H1
r(RN). Then
(i)RRNH(uj)→RRNH(u)and RRNH(uj)→RRNH(u).
(ii)h(uj)→h(u)and h(uj)→h(u)strongly in (H1
r(RN))−1.
Now, we can construct the auxiliary problem as follows:
(−m0∆u+ωu =h(u),in RN,
u∈H1
r(RN), u 6≡ 0 in RN,(A)
where N≥2, m0>0 given by (M1), ω > 0 and h∈C(R,R) defined as above. It
is not difficult to see that the corresponding functional of (A) given by
K(u) := 1
2kuk2−ZRN
H(u)
8
is well-defined on H1
r(RN) and of class C1. Moreover, as stated in the next lemma,
Khas the geometry of the Symmetric Mountain Pass theorem and satisfies the
Palais-Smale compactness condition. In what follows, we set
Dn:= {σ= (σ1,··· , σn)∈Rn| |σ| ≤ 1}and Sn−1:= ∂Dn.
Lemma 2.4 The functional Ksatisfies the following properties.
(i)There exist r > 0and ρ > 0such that
K(u)≥0for any u∈H1
r(RN)with kuk ≤ r,
K(u)≥ρfor any u∈H1
r(RN)with kuk=r.
(ii)For every n∈N, there exists an odd continuous mapping γ0n:Sn−1→
H1
r(RN)such that
K(γ0n(σ)) <0for all σ∈Sn−1.
(iii)The Palais-Smale compactness condition holds.
Due to Item (ii) of Lemma 2.4, for every n∈N, we can define a family of
mapping Γnby
Γn:= γ∈C(Dn, H1
r(RN)) |γis odd and γ(σ) = γ0n(σ) on σ∈Sn−1,(2.1)
which is nonempty since
γn(σ) :=
|σ|γ0nσ
|σ|,for σ∈Dn\ {0},
0,for σ= 0,
belongs to Γn. Thus, the symmetric mountain pass values of Kdefined by
en:= inf
γ∈Γn
max
σ∈Dn
K(γ(σ))
for any n∈N, are all meaningful. Moreover, we have
Theorem 2.1 The following statements hold.
(i)For every n∈N,enis a critical value of Kand en≥ρ > 0.
(ii)en→+∞as n→+∞.
Remark 2.1 All of the conclusions stated in this section and their proofs can be
found in [9]. For ease of exposition and completeness of this paper, it is better to
outline the necessary conclusions that we need.
3 Proof of Theorem 1.1
In this section, we shall give the detailed proof of Theorem 1.1. Before going
further, we would like to point out that assumption (M3) is almost necessary when
it comes to obtaining infinitely many distinct solutions to (KT ) in the case N≥3,
see Remark 1.3 in Section 1. On the other hand, as we can see below, actually
assumption (M3) is important to verifying the symmetric mountain pass geometry
of Jfor every n∈Nand, together with assumptions (M1)−(M2), is also sufficient
to establish the existence result of infinite many distinct solutions to (KT ).
9
3.1 Symmetric mountain pass geometry of J
Lemma 3.1 Items (i)and (ii)of Lemma 2.4 in Section 2 are also applied to J.
Proof. In terms of Item (i) of Lemma 2.2 and (M1), we have
J(u)≥K(u) for all u∈H1
r(RN),(3.1)
which implies that Item (i) of Lemma 2.4 is applied to J.
For every n∈N, arguing as in Theorem 10 of [6], an odd and continuous map
πn:Sn−1→H1
r(RN) is defined such that
0/∈πnSn−1and ZRN
F(πn(σ)) ≥1,for all σ∈Sn−1.
It is easy to see that, for every n∈N, there exists αn>0 such that
k∇πn(σ)k2
2≤αn,for all σ∈Sn−1.
For every n∈Nand any σ∈Sn−1, setting βt
n(σ)(x) := πn(σ)(t−1x), we have
Jβt
n(σ)=1
2c
MtN−2k∇πn(σ)k2
2−tNZRN
F(πn(σ))
≤1
2c
MtN−2αn−tN=: gn(t).
When N= 2, it is clear that gn(tn)<0 for sufficiently large tn>0. When N≥3,
in terms of (M3), there also exists sufficiently large tn>0 such that
gn(tn) = tN
n 1
2c
M(sn)
s
N
N−2
n
α
N
N−2
n−1!<0,
where sn:= tN−2
nαn>0. Thus the proof is completed by redefining γ0n:= βtn
n.
Now, for every n∈N, we can defined the symmetric mountain pass value dnof
J:
dn:= inf
γ∈Γn
max
σ∈Dn
J(γ(σ)),
where Γnis given by (2.1). In view of (3.1) and Theorem 2.1, we have that
dn≥en≥ρ > 0 and dn→+∞as n→+∞.
It is easy to see that the proof of Theorem 1.1 is completed if we can prove that,
for every n∈N,dndefined above is a critical value of J.
For every n∈N, by Ekeland’s principle, we can find a Palais-Smale sequence
{uj}+∞
j=1 at level dn, that is, {uj}+∞
j=1 satisfies
J(uj)→dnand J′(uj)→0 in (H1
r(RN))−1,as j→+∞.(3.2)
However, merely under the condition (3.2), it seems difficult to show the existence of
strongly convergent subsequence and even the boundedness of {uj}+∞
j=1 in H1
r(RN).
Inspired by [9], by introducing an auxiliary functional, we find a Palais-Smale se-
quence that “almost” satisfies the Pohoˇzaev identity associated to (KT ), which
makes it possible for us to overcome these difficulties.
In the following subsection, based on the key idea above, we will show that dn
is indeed a critical value of Jfor every n∈N.
10
3.2 Auxiliary functional Φ(θ, u)and conclusion
Analogously to [9], we equip a standard product norm k(θ, u)kR×H1
r:= θ2+kuk21
2
to the augmented space R×H1
r(RN) and define the auxiliary functional
Φ(θ, u) := 1
2c
Me(N−2)θZRN
|∇u|2−eNθ ZRN
F(u).
It is easy to conclude that Φ is of class C1and
Φ(θ, u(x)) = Jue−θx for all θ∈Rand u∈H1
r(RN).(3.3)
In particular, Φ(0, u) = J(u) for all u∈H1
r(RN). We denote its derivative as
Φ′:= (∂θΦ, ∂uΦ) with
∂θΦ(θ, u) = N−2
2Me(N−2)θZRN
|∇u|2e(N−2)θZRN
|∇u|2−NeN θ ZRN
F(u)
and
∂uΦ(θ, u)[v] = Me(N−2)θZRN
|∇u|2e(N−2)θZRN
∇u· ∇v−eNθ ZRN
f(u)v,
for all v∈H1
r(RN).
For every n∈N, we define the class
Γn:=
γ∈C(Dn,R×H1
r(RN))
γ(σ) = (θ(σ), η(σ)) satisfies
(θ(−σ), η(−σ)) = (θ(σ),−η(σ)) ,∀σ∈Dn,
(θ(σ), η(σ)) = (0, γ0n(σ)) ,∀σ∈Sn−1.
,
where γ0nis given in Item (ii) of Lemma 2.4. In terms of the nonemptyness of
Γnand the fact that {(0, γ )|γ∈Γn} ⊂ Γn, we conclude that Γnis nonempty, the
minimax value dnof Φ given by
dn:= inf
γ∈Γn
max
σ∈Dn
Φ (γ(σ))
is well-defined and dn≤dn. On the other hand, for any given γ(σ) = (θ(σ), η(σ)) ∈
Γn, setting γ(σ)(x) = η(σ)e−θ(σ)x, we can verify that γ(σ)∈Γnand, by (3.3),
I(γ(σ)) = Φ(γ(σ)) for any σ∈Dn, which imply that dn≥dn. Thus we have
Lemma 3.2 For all n∈N,dn=dn.
Based on Lemma 3.2, arguing as the proof of Proposition 4.2 in [9], we have the
following lemma:
Lemma 3.3 For every n∈N, there exists a sequence {(θj, uj)}+∞
j=1 ⊂R×H1
r(RN)
such that
(i)θj→0,
(ii) Φ(θj, uj)→dn,
11
(iii)∂uΦ (θj, uj)→0strongly in (H1
r(RN))−1,
(iv)∂θΦ (θj, uj)→0.
Lemma 3.4 Let {(θj, uj)}+∞
j=1 be the sequence given by Lemma 3.3. Then {uj}+∞
j=1
is bounded and has a strongly convergent subsequence in H1
r(RN).
Proof. We shall prove the boundedness of {k∇ujk2
2}+∞
j=1 in Step 1, complete the
boundedness of {uj}+∞
j=1 in H1
r(RN) by showing, in Step 2, that {kujk2
2}+∞
j=1 is
bounded and conclude the existence of a strongly convergent subsequence in Step
3.
Claim 1. {k∇ujk2
2}+∞
j=1 is bounded.
In view of Items (ii) and (iv) of Lemma 3.3, setting µj:= e(N−2)θjRRN|∇uj|2,
we have
c
M(µj)−1−2
NM(µj)µj= 2Φ(θj, uj)−2
N∂θΦ(θj, uj) = 2dn+oj(1).
Thus, in association with the Item (i) of Lemma 3.3, we conclude the boundedness
of {k∇ujk2
2}+∞
j=1 from (M1) when N= 2 and (M2) when N≥3 respectively.
Claim 2. {kujk2
2}+∞
j=1 is bounded and then, by Claim 1, {uj}+∞
j=1 is bounded in
H1
r(RN).
Arguing by contradiction, let us assume that, up to a subsequence, kujk2→+∞.
For every j∈N, set tj:= kujk−2
N
2and vj(·) := uj(t−1
j·). Then,
tj→0,as j→+∞.(3.4)
By some simple calculations, we have
∇vj(·) = t−1
j∇uj(t−1
j·),k∇vjk2
2=tN−2
jk∇ujk2
2,kvjk2
2= 1,(3.5)
which imply the boundedness of {vj}in H1
r(RN) with the aid of Claim 1 and (3.4).
Without loss of generality, up to a subsequence, we may assume that vj⇀ v0in
H1
r(RN). Set εj:= k∂uΦ(θj, uj)k(H1
r(RN))−1, with the help of (3.5) and Item (i) of
Lemma 2.1, some calculations show that
ωeN θj≤Me(N−2)θjZRN
|∇uj|2e(N−2)θjt2
jZRN
|∇vj|2+ωeN θjZRN
v2
j
=∂uΦ(θj, uj)tN
juj+eNθjZRN
(f(vj) + ωvj)vj
≤εjm0tN
jk∇ujk2
2+ω1
2+eNθjZRN
h(vj)vj.
Then, by (3.4), Claim 1, Item (ii) of Lemma 2.3 and Items (i) and (iii) of Lemma
3.3, we have
0< ω ≤ZRN
h(v0)v0,
12
which implies v06≡ 0.
On the other hand, let ϕ∈H1
r(RN) be a function with compact support and, for
every j∈N, set ψj(·) := ϕ(tj·). With the aid of the fact that vj⇀ v0in H1
r(RN),
Items (i) and (iii) of Lemma 3.3, Claim 1 and (3.4), we have
ZRN
f(v0)ϕ=eNθjZRN
f(vj)ϕ+oj(1)
≤∂uΦ(θj, uj)tN
jψj
+Me(N−2)θjZRN
|∇uj|2e(N−2)θjt2
jZRN
∇vj∇ϕ+oj(1)
≤εjm0t2
jk∇ϕk2
2+ωkϕk2
21
2+Ct2
j+oj(1) →0.
Thus, there holds
ZRN
f(v0)ϕ= 0,for any ϕ∈H1
r(RN) with compact support,
which implies f(v0)≡0. However, from (f′
1), it follows that 0 is an isolated zero
point of f. In association with the fact that H1
r(RN)⊂C(RN\ {0}) and v0(x)→0
as |x| → +∞, e.g. see [5], we have v0≡0, which is a contradiction.
Claim 3. {uj}+∞
j=1 has a strongly convergent subsequence in H1
r(RN).
From Claim 2 and Item (i) of Lemma 3.3, up to a subsequence, we may assume
that, when jtends to infinity, uj⇀ u0weakly in H1
r(RN) and
αj:= Me(N−2)θjk∇ujk2
2→α0∈(0,+∞).
Then, by Items (i) and (iii) of Lemma 3.3, it is not difficult to see that u0satisfies
−α0u0=f(u0) in RN,
which implies
α0k∇u0k2
2=ZRN
f(u0)u0.(3.6)
On the other hand, a straightforward computation yields
αje(N−2)θjk∇ujk2
2+ωeN θjkujk2
2=∂uΦ(θj, uj)[uj] + β1
jeNθj−β2
jeNθj(3.7)
where
β1
j:= ZRN
h(uj)ujand β2
j:= ZRNh(uj)uj−f(uj)uj−ωu2
j.
Noting that, by Item (iii) of Lemma 3.3, Item (ii) of Lemma 2.3 and Item (i) of
Lemma 2.1 and Fatou’s lemma, there hold
lim
j→+∞∂uΦ(θj, uj)[uj] = 0,lim
j→+∞β1
j=ZRN
h(u0)u0,(3.8)
13
and
lim inf
j→+∞β2
j≥ZRNh(u0)u0−f(u0)u0−ωu2
0.(3.9)
Now, from Item (i) of Lemma 3.3 and (3.6)-(3.9) above, we conclude
lim sup
j→+∞α0k∇ujk2
2+ωkujk2
2= lim sup
j→+∞αje(N−2)θjk∇ujk2
2+ωeN θjkujk2
2
≤ZRNf(u0)u0+ωu2
0
=α0k∇u0k2
2+ωku0k2
2,
which implies that uj→u0in H1
r(RN). Thus the proof of Lemma 3.4 is com-
pleted.
Conclusion Let {(θj, uj)}+∞
j=1 be the sequence given by Lemma 3.3. By Lemma
3.4, we may assume that uj→u0nin H1
r(RN). Then, in association with Items (i)
and (iii) of Lemma 3.3, it follows that
Φ(0, u0n) = dnand ∂uΦ(0, u0n) = 0,
that is
J(u0n) = dnand J′(u0n) = 0,
Thus, for every n∈N, the symmetric mountain pass value dndefined in Subsection
3.1 is indeed a critical value of Jand, by (3.2), we complete the proof of Theorem
1.1.
4 Proof of Theorem 1.2
In this section, the proof of Theorem 1.2 shall be completed. It is worth pointing
out that, due to the loss of assumption (M3), finding suitable candidate critical
values of Jbecomes the major difficulty that we need to overcome in the proof of
Theorem 1.2. Fortunately, as we can see below, we are able to get through this
obstacle by Item (ii) of Theorem 2.1 and the non-negativeness of function λ. As
the core of this section, the process of finding suitable candidate critical values will
be shown in detail.
For convenience, we rewrite the corresponding functional of (KT ) as
Jq(u) := 1
2m0ZRN
|∇u|2+q
2ΛZRN
|∇u|2−ZRN
F(u)
=I0(u) + q
2ΛZRN
|∇u|2,
where Λ(t) := Rt
0λ(τ)dτ and I0∈C1(H1
r(RN),R) given by
I0:= 1
2m0ZRN
|∇u|2−ZRN
F(u).
Apparently, there holds Jq(u)≥I0(u)≥K(u) for all u∈H1
r(RN).
14
Redefining γ0nif necessary, by Lemma 3.1, we have that Items (i) and (ii) of
Lemma 2.4 in Section 2 are also applied to I0. It is easy to see that, for such γ0n,
there exist αn, βn>0 such that
I0(γ0n(σ)) ≤ −2αn<0 and Λ ZRN
|∇γ0n(σ)|2≤2βn,for all σ∈Sn−1.
Let q∈(0, αnβ−1
n], then
Jq(γ0n(σ)) = I0(γ0n(σ)) + q
2ΛZRN
|∇γ0n(σ)|2≤ −2αn+qβn≤ −αn<0
for all σ∈Sn−1. Therefor, we can define a candidate critical value cq
nof Jqby
cq
n:= inf
γ∈Γn
max
σ∈Dn
Jq(γ(σ)),
where Γnis given by (2.1). Obviously, for any 0 < q ≤q′≤αnβ−1
n,
cq′
n≥cq
n≥en≥ρ > 0.
We claim that, for every m∈N, there exist {nk}m
k=1 ⊂Nand qm>0 such that,
for q∈(0, qm], the minimax values {cq
nk}m
k=1 of Jqgiven by
cq
nk:= inf
γ∈Γnk
max
σ∈Dnk
Jq(γ(σ)), k = 1,2,··· , m
are all well-defined and satisfy
0< ρ ≤cq
n1<···< cq
nk<···< cq
nm<+∞.
Actually, let n1= 1, qn1:= αn1β−1
n1and q∈(0, qn1], it is easy to see that cq
n1is
well defined and
0< ρ ≤en1≤cq
n1≤cqn1
n1<+∞.
In view of Item (ii) of Theorem 2.1, there exists n2∈Nsuch that cqn1
n1< en2. For
such n2∈N, let qn2:= min{αn2β−1
n2, qn1}and q∈(0, qn2], we have that {cq
nk}2
k=1
are well-defined and satisfy
0< ρ ≤en1≤cq
n1< en2≤cq
n2≤cqn2
n2<+∞.
Thus, for every fixed m∈N, the desired sequence {nk}m
k=1 ⊂Ncan be obtained by
an iterative procedure, and the desired positive number qmcan also be found by
letting qm:= min
1≤k≤m{αnkβ−1
nk}.
It is not difficult to see that, under the assumptions of Theorem 1.2, the ar-
guments explored in Subsection 3.2 are also valid here. This fact meant that the
candidate critical values of Jqwe define above are indeed critical values of Jq.
Therefor, the proof of Theorem 1.2 is finished.
15
5 Non-variational proofs of the multiplicity results
In this last section, inspired by [1, 12], we shall present another new proofs of the
multiplicity results which are non-variational, simple and fundamental. As we can
see below, this gives us natural interpretations of the results we prove in previous
sections.
Before going into details of the non-variational proofs, some preliminary results
are needed. Firstly, we have the following proposition which concerns the multiplic-
ity result for (SF ) under the very general assumptions (f0), (f2), (f3) and (f′
1) on
f, see [6, 9].
Proposition 5.1 Assume N≥2and that fsatisfies (f0),(f2),(f3)and (f′
1).
Then Problem (SF)possesses infinitely many distinct radial solutions {vn}+∞
n=1
which satisfy k∇vnk2
2→+∞as n→+∞. Without loss of generality, we may
assume that
k∇vnk2
2<k∇vn+1k2
2for every n∈N.(5.1)
On the other hand, similar as Proposition 2.1 in [12], we have
Proposition 5.2 When N≥2,u∈H1(RN)is a nontrivial solution to (KT )if
and only if there exist v∈H1(RN)a nontrivial solution to (SF )and t > 0such
that
h(v, t) := Mt2−Nk∇vk2
2t2= 1 and u(·) = v(t·).
Now, under the assumptions of Theorem 1.1, the existence result of infinitely
many distinct solutions to (KT ) can be proved in a convenient way.
Actually, when N= 2, let un(·) := vn(tn·) for every n∈N, where tn>0 is
uniquely determined by h(vn, tn) = 1. When N≥3, (M1) and (M3) show that, for
every v6≡ 0,
h(v, t)→+∞as t→+∞and h(v, t)→0+as t→0+.
Thus, there exists a positive sequence {tn}+∞
n=1 such that h(vn, tn) = 1. In terms of
(M3) and (5.1), we can also assume that t2−N
nk∇vnk2
2< t2−N
n+1 k∇vn+1k2
2for every
n∈N. For such {tn}+∞
n=1, set un(·) := vn(tn·), n= 1,2,···. From Propositions 5.1
and 5.2 and the fact that k∇unk2
2=t2−N
nk∇vnk2
2, we conclude easily that {un}+∞
n=1
defined as above are the desired solutions for N≥2.
Similarly, under the assumptions of Theorem 1.2, the existence result of finitely
many distinct solutions to (KT ) can also be proved from the non-variational point
of view. The detailed proof is provided here for reader’s convenience.
For every fixed n∈N, let q∈(0, qn), where
qn:= m0
1 + max
1≤i≤nλ(2m0)N−2
2k∇vik2
2>0.
Obviously, hvi,1
√2m0<1 for every i∈ {1,··· , n}. On the other hand, (M1)
yields that h(vi, t)→+∞as t→+∞for every i∈ {1,··· , n}. Thus, there exists a
positive sequence {ti}n
i=1 such that h(vi, ti) = 1 for every i∈ {1,··· , n}. In terms
of (5.1), we also have that t2−N
ik∇vik2
26=t2−N
jk∇vjk2
2for every i, j ∈ {1,··· , n}
and i6=j. For such {ti}n
i=1, set ui(·) := vi(ti·), i= 1,2,··· , n. Now, it is easy to
see that {ui}n
i=1 are the desired solutions.
16
Remark 5.1 In some sense, Jcan be seen as a suitable perturbation of I. Ad-
ditionally, Proposition 5.2 provides a clear and vital relation between the solutions
of (KT )and that of (SF ). Thus, in terms of Proposition 5.1, it is natural and
well-founded to ask the existence of multiple solutions to (KT ).
Remark 5.2 As we can see in this section, the assumptions on Mare mainly used
to ensure the existence of t > 0such that h(v, t) = 1. In this procedure, we observe
that, when N≥3, the behavior of function c
M(t)−(1 −2/N)M(t)tat infinity is
actually not used, which, in contrast, plays a important role in the variational proofs,
see Claim 1 and its proof in Subsection 3.2. This significant difference seems to,
at least, imply that, in the variational arguments, the boundedness of {k∇ujk2
2}+∞
j=1
could be established under some weaker assumptions on Mor in a more natural
way.
Acknowledgment
The author would like to express his sincere gratitude to his advisor Professor Zhi-
Qiang Wang for his patient guidance, constant encouragement and timely help. The
author also thanks Professor Kazunaga Tanaka for sharing the full text of [9].
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18
