Boundary Value Problems
Volume 2009, Article ID 670675,20pages doi:10.1155/2009/670675
Research Article
Infinitely Many Solutions for a Boundary Value
Problem with Discontinuous Nonlinearities
Gabriele Bonanno
1and Giovanni Molica Bisci
21Mathematics Section, Department of Science for Engineering and Architecture, Engineering Faculty,
University of Messina, 98166 Messina, Italy
2PAU Department, Architecture Faculty, University of Reggio, Calabria, 89100 Reggio Calabria, Italy
Correspondence should be addressed to Gabriele Bonanno,[email protected]
Received 16 October 2008; Accepted 11 February 2009
Recommended by Ivan T. Kiguradze
The existence of infinitely many solutions for a Sturm-Liouville boundary value problem, under an appropriate oscillating behavior of the possibly discontinuous nonlinear term, is obtained. Several special cases and consequences are pointed out and some examples are presented. The technical approach is mainly based on a result of infinitely many critical points for locally Lipschitz functions.
Copyrightq2009 G. Bonanno and G. M. Bisci. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
The aim of this paper is to establish infinitely many solutions for two-point boundary value problems with the nonlinear term possibly discontinuous. We immediately emphasize the following theorem which is a particular case of our main resultTheorem 3.1.
Theorem 1.1. Letf :R → Rbe a locally bounded, and almost everywhere continuous function such thatinfRf >0. PutFξ:ξ0ftdtfor everyξ∈Rand assume that
lim inf
ξ→∞
Fξ ξ2 <
1
4lim supξ→∞
Fξ
ξ2 . 1.1
Then, for eachλ∈8/lim supξ→∞Fξ/ξ2,2/lim inf
ξ→∞Fξ/ξ2, the problem
−uλfu in0,1
u0 u1 0 G
1,0
f,λ
Clearly, whenf is continuous inR, the solutions inTheorem 1.1are classicalin this case, it is enough to assume infRf ≥ 0; seeCorollary 3.5. Moreover, substitutingξ → ∞
withξ → 0, the same results hold and, in addition, the sequence of pairwise distinct positive solutions uniformly converges to zeroseeTheorem 3.9andCorollary 3.10.
Whenfis a continuous function, results of the existence of infinitely many solutions for problemG1f,λ,0are obtained, for example, in1–7. We observe that in the very interesting paper 6, the authors assume lim infξ→∞Fξ/ξ2 0 and lim supξ→∞Fξ/ξ2 ∞,
which are conditions that imply our key assumption. Very recently, in4, a more general condition than the previous assumption has been assumed, requiring in addition, however, that limξ→∞fξ ∞. Moreover, we also observe that the results in1,2are obtained by
using the important Variational Principle of Ricceri8, which is, basically, the same as our tool. We emphasize that, also whenfis a continuos function, our theorems in this paper and the results in1–7are mutually independentseeRemark 3.13and Examples3.11and3.12. When the nonlinear term f is discontinuous, there have been many approaches to studying a nonlinear eigenvalue differential equation as it arises in physics problems, such as nonlinear elasticity theory, and mechanics, and engineering topics. Chang in9established the critical point theory for nondifferentiable functionals and presented some applications to partial differential equations with discontinuous nonlinearities. Next, Motreanu and Panagiotopoulos see 10, Chapter 3 studied the critical point theory for non-smooth functionals and in this framework, very recently, Marano and Motreanu, in 11, obtained an infinitely many critical points theorem, which extends the Variational Principle of Ricceri to non-smooth functionals, and applies this result to variational-hemivariational inequalities and semilinear elliptic eigenvalue problems with discontinuous nonlinearities.
In this paper, we present a more precise version of the infinitely many critical points theorem of Marano and MotreanuTheorem 2.1, obtained by a completely different proof seeRemark 2.2and, by using the previous theorem, we establish our main result
Theorem 3.1on the existence of infinitely many solutions for a two-point boundary value problem with the Sturm-Liouville equation having discontinuous nonlinear term.
We explicitly observe that methods and techniques used in the proof ofTheorem 3.1
can be applied to a wide class of nonlinear differential problems to investigate infinitely many solutions. The note is arranged as follows. InSection 2, we recall some basic definitions and our abstract framework, whileSection 3is devoted to infinitely many solutions for the Sturm-Liouville problem.
Finally, we point out that the existence of multiple solutions for nonlinear differential problems has been studied in several papers by using different techniquessee, e.g.,12,13
and references therein.
2. Infinitely Many Critical Points
LetX, · be a real Banach space. We denote byX∗the dual space ofX, while ·,·stands for the duality pairing betweenX∗andX. A functionΦ:X → Ris called locally Lipschitz continuous when, to everyx∈X, there corresponds a neighbourhoodVxofxand a constant
Lx≥0 such that
Ifx, z ∈ X, we writeΦ◦x;zfor the generalized directional derivative ofΦat the pointx
along the directionz, that is,
Φ◦x;z: lim sup
w→x, t→0
Φwtz−Φw
t . 2.2
The generalized gradient of the functionΦinx, denoted by∂Φx, is the set
∂Φx:x∗∈X∗:x∗, z≤Φ◦x;z∀z∈X. 2.3
We say thatx∈Xis ageneralizedcritical point ofΦwhen
Φ◦x;z≥0 ∀z∈X, 2.4
that clearly signifies 0 ∈ ∂Φx. When a non-smooth functional,Ψ : X →− ∞,∞, is expressed as a sum of a locally Lipschitz function,Φ : X → R, and a convex, proper, and lower semicontinuous function,j:X →− ∞,∞, that isΨ: Φ j, ageneralizedcritical point ofΨis everyu∈Xsuch that
Φ◦u;v−u jv−ju≥0 2.5
for allv∈Xsee10, Chapter 3.
Here, and in the sequel,Xis a reflexive real Banach space,Φ:X → Ris a sequentially weakly lower semicontinuous functional, Υ : X → R is a sequentially weakly upper semicontinuous functional,λ is a positive real parameter,j : X →− ∞,∞is a convex, proper and lower semicontinuous functional andDjis the effective dominion ofj.
Write
Ψ: Υ−j, Iλ: Φ−λΨ Φ−λΥ λj. 2.6
We also assume thatΦis coercive and
Dj∩Φ−1− ∞, r /∅ 2.7
for allr >infXΦ. Moreover, owing to2.7and providedr >infXΦ, we can define
ϕr inf
u∈Φ−1−∞,r
supu∈Φ−1−∞,rΨu −Ψu
r−Φu , γ :lim inf
r→∞ ϕr, δ:r→lim infinfXΦ
ϕr.
2.8
Theorem 2.1. Under the above assumptions onX,ΦandΨ, one has
afor everyr >infXΦand everyλ∈0,1/ϕr, the restriction of the functionalIλ Φ−λΨ toΦ−1− ∞, radmits a global minimum, which is a critical point (local minimum) ofI
λ inX.
bifγ <∞then, for eachλ∈0,1/γ, the following alternative holds: either
b1Iλpossesses a global minimum, or
b2there is a sequence {un} of critical points (local minima) of Iλ such that
limn→∞Φun ∞.
cifδ <∞then, for eachλ∈0,1/δ, the following alternative holds: either
c1there is a global minimum ofΦwhich is a local minimum ofIλ, or
c2there is a sequence{un}of pairwise distinct critical points (local minima) ofIλ, with
limn→∞Φun infXΦ, which weakly converges to a global minimum ofΦ.
Proof. Arguing as in the proof of14, Theorem 3.1we havea. More precisely, let 1/λ > ϕr, then there isu∈Djsuch thatΦu< r andΦu−λΨu< r−λsupΦx<rΨx. Moreover, put
M r−Φu
λ Ψu. 2.9
Clearly,
sup
Φx<rΨx< M.
2.10
Finally, put
ΨMu
Ψu, ifΨu≤M
M, ifΨu> M. 2.11
Since, owing to15, Corollary III.8jis sequentially weakly lower semicontinuous, a simple computation shows thatΨMis sequentially weakly upper semicontinuous. PutJ Φ−λΨM.
ClearlyJis a sequentially weakly lower semicontinuous functional and, as it is easy to see, it is also a coercive functional. Therefore see, e.g., 16, Theorem 1.2, it admits a global minimumu0. IfJu0 Ju, thenusatisfies the conclusion.
Otherwise, assumeJu0 < Ju. In this case, we have thatΨu0 < M. In fact, from
Ju0< Juone hasΦu0−λΨMu0<Φu−λΨMu. Hence,Φu0< λΨMu0 Φu−
λΨu ≤ λM Φu−λΨu r and, from2.10one hasΨu0 < M. Therefore,Φu0− λΨu0 Φu0−λΨMu0 ≤ Φu−λΨMu for allu ∈ X and, taking again 2.10 into
account,Φu0−λΨu0 ≤ Φu−λΨu for allu ∈ Φ−1− ∞, r. Hence,u0 satisfies the
conclusion.
Let us proveb. Pickλ ∈0,1/γand assume thatb1is not true. We will show that
critical points. Leta ∈ Rsuch thatλ < a < 1/γ. From lim infr→∞ϕr < 1/athere exists
a sequence {rn} such that limn→∞rn ∞ and ϕrn < 1/afor all n ∈ N. Putrn1 r1.
Owing toa, one can findu1∈Φ−1− ∞, r
n1such thatu1is a local minimum forIλ. From
our assumption,u1is not a global minimum forIλ. Therefore, there existsu1 ∈X such that
Iλu1< Iλu1. Hence,u1/∈Φ−1− ∞, rn1. Letrn2 ∈ {rn}such thatrn2 >Φu1. Again from
a, there isu2∈Φ−1− ∞, r
n2such thatu2is a local minimum for the functionalIλ. Taking
into account thatu2 is a global minimum inΦ−1− ∞, r
n2, we have that Iλu2 ≤ Iλu1
andIλu1< Iλu1. Hence,Φu2> rn1. Reasoning inductively we obtain a sequence{uk}of
distinct critical points such thatΦuk1> rnk for allk∈N. Hence,b2holds.
Finally, we provec. Fixλ ∈0,1/δand letu∈Xsuch thatΦu infXΦ. Assume
thatc1does not hold, that isuis not a local minimum forIλ. Considera∈Rsuch thatλ <
a <1/δ. By our assumption, lim infr→infXΦϕr δ <1/a, hence there exists a decreasing
sequence{rn} such that limn→∞rn infXΦand ϕrn < 1/afor alln ∈ N. Putrn1 r1.
Owing toa, there existsu1 ∈Φ−1− ∞, r
n1, which is a local minimum forIλ. Therefore,
u1/u. Then,Φu < Φu1. Letrn2 ∈ {rn}such that Φu < rn2 < Φu1. Again from a,
there isu2 ∈ Φ−1− ∞, r
n2which is a local minimum for the functionalIλ, withΦu <
Φu2<Φu1. Reasoning inductively we obtain a sequence{uk}of distinct local minima for
Iλsuch thatΦu<Φuk< rnk for allk∈N. Hence, limk→∞Φuk infXΦ. Moreover, since
{uk} ⊆ Φ−1− ∞, rn1andΦis coercive, then it is bounded. SinceX is reflexive, taking a
subsequence if necessary,{uk}weakly converges tou∗ ∈X. From the weak sequential lower
semicontinuity, one hasΦu∗≤limk→∞Φuk infXΦ, that isΦu∗ infXΦ. Hence, the
conclusion is obtained.
Remark 2.2. We explicitly observe that the proof here outlined is different from that proposed by Marano and Motreanu in11. Further we do not use the weak closure of the sub-levels
Φ−1− ∞, r, forr >inf
XΦ.
3. Sturm-Liouville Boundary Value Problem
Consider the Sturm-Liouville boundary value problem
−puquλfu in0,1
u0 u1 0, G
p,q f,λ
wherep, q ∈L∞0,1, f :R → Ris an almost everywhere continuous function andλis a positive parameter.
DenotingDf the set
Df :{z∈R:f is discontinuous atz}, 3.1
we recall thatf is said to be continuous almost everywhere ifDf isLebesguemeasurable
andmDf 0. Moreover, iffis locally essentially bounded, we write
f−t: lim
δ→0ess inf|t−z|<δ fz, f
t: lim
for eacht∈R. We observe thatf−andfare, respectively, lower semi-continuous and upper semi-continuous.
Assume that
p0 :ess inf
x∈0,1 px>0, q0:ess infx∈0,1 qx≥0. 3.3
LetW1,20,1be the Sobolev space endowed with the usual norm
u∗:
1
0
ux2 dx
1
0
ux2 dx
1/2
. 3.4
As is customary, we denote byW01,20,1the closure ofC∞0 0,1inW1,20,1. Moreover,
a functionu:0,1 → Ris said to be a weak solution ofGp,qf,λifu∈W01,20,1and
1
0
pxuxvxdx
1
0
qxuxvxdx
λ
1
0
fux vxdx ∀v∈W01,20,1 .
3.5
We recall thatu∈AC0,1is a generalized solution ofGp,qf,λifpu∈AC0,1,u0 u1 and
−pxux qxux λfux , 3.6
for almost everyx∈0,1.
Clearly, the weak solutions ofGp,qf,λare also generalized solutions.
Iff andqare continuous functions we recall thatu∈C10,1is a classical solution
ofGp,qf,λifpu∈C10,1,u0 u1and
−pxux qxux λfux , 3.7
for everyx∈0,1. We recall that
u, v:
1
0
qxuxvxdx
1
0
is an inner product that induces inW01,20,1the norm
u:
1
0
pxux2dx
1
0
qxux2dx
1/2
, 3.9
which is equivalent to the usual one.
It is well known thatW01,20,1is compactly embedded inC00,1and in particular
one has
u∞≤ 2√1p0u, 3.10
for everyu∈W01,20,1.
ConsiderΦ:W01,20,1 → RandΥ:W01,20,1 → Rdefined as follows
Φu: u
2
2 , Υu:
1
0
Fux dx, 3.11
where
Fξ:
ξ
0
ftdt, 3.12
for everyξ∈R.
By standard arguments, one has that Φ is Gˆateaux differentiable and sequentially weakly lower semicontinuous. Moreover, the Gˆateaux derivative is the functional Φu ∈
W01,20,1∗given by
Φuv 1
0
pxuxvxdx
1
0
qxuxvxdx, 3.13
for everyv∈W01,20,1.
Now, put
κ:3 p0
q∞12p∞, 3.14
A:lim inf
ξ→∞
max|t|≤ξFt
ξ2 ,
B:lim sup
ξ→∞
Fξ ξ2 ,
3.15
λ1: 2 3
q∞12p∞
B ,
λ2:2p0
A.
3.16
Our main result is the following theorem.
Theorem 3.1. Letf : R → Rbe a locally essentially bounded and almost everywhere continuous function. PutFξ:ξ0ftdtfor everyξ∈Rand assume that
iξ0Ftdt≥0, for everyξ≥0;
ii
lim inf
ξ→∞
max|t|≤ξFt
ξ2 < κlim sup
ξ→∞
Fξ
ξ2 , 3.17
whereκis given by3.14;
iiifor almost everyx∈0,1, for eachz∈Dfand for eachλ∈λ1, λ2(whereλ1,λ2are given by3.16) the condition
λf−z−qxz≤0≤λfz−qxz 3.18
impliesλfz qxz.
Then, for each λ ∈λ1, λ2, the problem Gp,qf,λ possesses a sequence of weak solutions which is unbounded inW01,20,1.
Proof. Our aim is to applyTheorem 2.1b. For this end, fixλ ∈λ1, λ2and denote byX the Banach spaceW01,20,1endowed with the norm
u:
1
0
pxux2dx
1
0
qxux2dx
1/2
For eachu∈X, put
Φu: u
2
2 , Υu:
1
0
Fux dx, ju:0
Ψu: Υu−ju Υu,
Iλu: Φu−λΨu Φu−λΥu.
3.20
Clearly, Φ is sequentially weakly lower semicontinuous and coercive, Υ is, in particular, sequentially weakly upper semicontinuous; moreover, they are locally Lipschitz functions and one hasIλ◦u;v Φuv λ−Υ◦u;vfor allu, v∈X.
Now, let{cn}be a real sequence such that limn→∞cn ∞and
lim
n→∞
max|t|≤cnFt
c2
n
A. 3.21
Putrn 2p0c2nfor alln ∈ N. Taking3.10into account, one has maxt∈0,1|vt| ≤ cn for all
v∈Xsuch thatv2<2rn. Hence,taking also into account that the functionu0t 0 for all
t∈0,1is such thatu020<2r
nfor alln∈None has
ϕrn inf
u2<2r
n
supv2<2rn
1
0F
vx dx−10Fux dx rn− u2/2
≤ supv2<2rn 1
0F
vx dx−10Fu0x dx rn−u02/2
supv2<2rn
1
0F
vx dx rn
≤ max|t|≤cnFt
rn
1
2p0
max|t|≤cnFt
c2n
.
3.22
Therefore, since from assumptioniione hasA <∞, we obtain
γ≤lim inf
n→∞ ϕ
rn ≤ A
2p0 <∞. 3.23
Now we claim that the functional Iλ is unbounded from below. Let {dn} be a real
sequence such that limn→∞dn ∞and
lim
n→∞
Fdn
dn2
For alln∈Ndefine
wnx:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
4dnx ifx∈
0,1
4
dn ifx∈
1 4, 3 4
−4dnx−1 ifx∈
3 4,1
.
3.25
Clearly,wn∈Xand
wn2≤
8d2n
q∞
12 p∞
. 3.26
Therefore,
Φwn −λΥ
wn
wn2
2 −λ
1
0
Fwnx dx
≤4dn2
q∞
12 p∞
−λ
1
0
Fwnx dx.
3.27
Takingiinto account, we have
1
0
Fwnx dx≥
3/4
1/4
Fdn dt 1
2F
dn . 3.28
Then,
Φwn −λΥ
wn ≤4d2n
q∞
12 p∞
−λ
2F
dn , 3.29
for alln∈N.
Now, ifB <∞, we fixε∈8/λBq∞/12p∞,1. From3.24there existsνε∈N
such that
for alln > νε. Therefore,
Φwn −λΥ
wn ≤4d2n
q∞
12 p∞
−λ
2εBd
2
n
d2n
4
q∞
12 p∞
−λ
2εB
3.31
for alln > νε.
From the choice ofε, one has
lim
n→∞
Φwn −λΥ
wn −∞. 3.32
On the other hand, ifB ∞, we fixM >8/λq∞/12p∞and, again from3.24there
existsνM∈Nsuch that
Fdn > Md2n, 3.33
for alln > νM. Therefore,
Φwn −λΥ
wn ≤4d2n
q∞
12 p∞
− λ
2Md
2
n
dn2
4
q∞
12 p∞
−λ
2M
3.34
for alln > νM.
From the choice ofM, also in this case, one has
lim
n→∞
Φwn −λΥ
wn −∞. 3.35
Hence, our claim is proved.
Since all assumptions of Theorem 2.1b are verified, the functional Iλ admits a
sequence{un} of generalized critical points such that limn→∞un ∞, that is{un} is
unbounded inX.
Now, we claim that the generalized critical points of Iλ are weak solutions for the
problemGp,qf,λ. To this end, letu0 ∈X a generalized critical point ofIλ, that isIλ◦u0, v≥0,
for allv∈X. From which, we obtain
for allv∈X. Hence,Φu0v≥ −λ−Υ◦u0;vfor allv∈X, that is
−
1
0
pxu0xvx qxu0xvxdx≤λ−Υ◦u0;v , 3.37
for allv∈X. Clearly, setting
Lv:−
1
0
pxu0xvx qxu0xvxdx ∀v∈X, 3.38
Lis a continuous and linear functional onX, for which3.37signifiesL∈λ∂−Υu0. Now, since X is dense inL20,1, from 9, Theorem 2.2one has Lv ≤ λ−Υ◦u0;v for all v∈L20,1, so thatLis continuous and linear onL20,1. Therefore, there ish∈L20,1
such thatLv 10hxvxdxfor allv ∈ L20,1. From a standard resultsee, e.g.,15,
Example 2, page 219 there is a uniqueu ∈ W2,20,1∩X such thatpu−qu h. In
particular, one has10pxux−qxuxvxdx−10pxuxvxqxuxvxdx
for all v ∈ X. Hence, −10pxu0xvx qxu0xvxdx Lv 10hxvxdx
1
0pxu
x−qxuxvxdx −1 0pxu
xvx qxuxvxdx for all v ∈ X,
and since a continuous and linear functional onXis uniquely determined by a function inX
see17, Theorem 5.9.3, page 295, we haveuu0; so that,u0∈W2,20,1and
1
0
pxu0x −qxu0x
vxdx−
1
0
pxu0xvx qxu0xvx
dx 3.39
for allv∈X. From3.37and3.39one has
1
0
pxu0x −qxu0xvxdx≤λ−Υ◦u0;v 3.40
for all v ∈ X. Hence, 9, Corollary page 111 ensures that pxu0x − qxu0x ∈
−λf−u0x,−λfu0x for almost every x ∈ 0,1, that is pxu0x ∈
−λf−u0x qxu0x,−λfu0x qxu0x, for almost every x ∈ 0,1. From which
−pxu0x ∈λf−u0x −qxu0x, λfu0x −qxu0x, 3.41
for almost everyx∈0,1.
Now, sincemDf 0, from18, Lemma 1we obtain −pxu0x 0 for almost
everyx ∈u−1
0 Df. Hence, fromiiiwe obtainλfu0x−qxu0x 0 for almost every
x∈u−01Df. From which
for almost every x ∈ u−01Df. On the other hand, for almost every x ∈ 0,1\u−01Df,
condition3.41reduces to
−pxu0x qxu0x λfu0x . 3.43
Hence, our claim is proved and the assertion follows.
Remark 3.2. Ifqx 0 for allx∈0,1assumptioniiibecomes
iiifor eachz∈Df, the condition
f−z≤0≤fz 3.44
impliesfz 0.
Ifz∈Df is such that 0∈f−z, fzandfz 0, theniiiis verified. Otherwise,
ifz∈Df and there is a neighborhoodV ofzand a positive constantm >0 such that either
ft> mfor almost everyt∈V, orft<−mfor almost everyt∈V, theniiiis verified. In particular, if infRf >0 theniiiis verified for allz∈Df.
Ifqis a nonzero function,z ∈ Df andλ ∈λ1, λ2,iiiis verified, e.g., when there is
a neighborhoodV ofzand a positive constantm > 0 such that eitherft>q∞/λzm
for almost everyt∈V, orft<q0/λz−mfor almost everyt∈V. In particular, whenever
λ1>0 and one hasfz>q∞/λ1zmfor somem >0, theniiiis verified.
Finally, since a functionu∈W01,20,1such that
−pxux qxux∈λf−ux ,fux 3.45
for almost everyx ∈ 0,1is called multi-valued solution forGp,qf,λ see9, we explicitly observe that, without assuming condition iii, the same proof of Theorem 3.1 ensures a sequence of pairwise distinct multi-valued solutions to problemGp,qf,λ.
Remark 3.3. The following condition
iithere exist two real sequences {bn},{cn}, withbn <
p0/2q∞/12p∞cn
for alln∈Nand limn→∞cn ∞, such that
A1: lim
n→∞
max|t|≤cnFt−1/2F
bn
2p0c2
n−4
q∞/12p∞ b2n
< κ
2p0lim supξ→∞ Fξ
ξ2 3.46
is more general than condition ii of Theorem 3.1. In fact, from ii we obtain ii, by choosingbn0 for alln∈N.
Assuming in Theorem 3.1 condition ii instead of condition ii, for each λ ∈
Theorem 3.1, one hasϕrn infu2<2r
nsupv2<2rn
1
0Fvxdx−
1
0Fuxdx/rn−u2/2≤
supv2<2rn
1
0Fvxdx−
1
0Funxdx/rn−un2/2≤max|t|≤cnFt−1/2Fbn/2p0c
2
n−
4q∞/12p∞b2n, by choosing
unx:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
4bnx ifx∈
0,1
4
bn ifx∈
1 4, 3 4
−4bnx−1 ifx∈
3 4,1
.
3.47
Among the consequences ofTheorem 3.1we point out the following results.
Corollary 3.4. Letf :R → Rbe a continuous function andp∈C10,1,q∈C00,1. Assume
that (i) and (ii) ofTheorem 3.1hold. Then, for eachλ∈λ1, λ2, problemGp,qf,λpossesses a sequence of pairwise distinct classical solutions.
Iffis nonnegative, using the Strong Maximum Principlesee, e.g.,19, Theorem 8.19, page 198we can get the following two results.
Corollary 3.5. Let f : R → Rbe a continuous, nonnegative function and p ∈ C10,1,q ∈ C00,1. Assume that
iia
lim inf
ξ→∞
Fξ
ξ2 < κlim sup
ξ→∞
Fξ
ξ2 . 3.48
Then, for eachλ ∈λ1, λ2, problemGp,qf,λpossesses a sequence of pairwise distinct positive classical solutions.
Corollary 3.6. Letf :R → Rbe a locally essentially bounded, nonnegative and almost everywhere continuous function. Assume thatiiaofCorollary 3.5and (iii) ofTheorem 3.1hold. Then, for each
λ ∈λ1, λ2, problemGp,qf,λpossesses a sequence of positive weak solutions which is unbounded in
W01,20,1.
Finally, we present the following result.
Corollary 3.7. Letf :R → Rbe a locally essentially bounded and almost everywhere continuous function. Assume that (i) and (iii) ofTheorem 3.1hold. Further, assume that
ii1
lim sup
ξ→∞
Fξ ξ2 >
2 3
ii2
lim inf
ξ→∞
max|t|≤ξFt
ξ2 <2p0. 3.50
Then, problem
−pu qufu in0,1
u0 u1 0
Gp,qf,1
possesses a sequence of weak solutions which is unbounded inW01,20,1.
Remark 3.8. Clearly, also Theorem 1.1 in Introduction is a particular case of Theorem 3.1, takingRemark 3.2and the Strong Maximum Principle into account.
Now, we present the other main result. First, put
A∗:lim inf
ξ→0
max|t|≤ξFt
ξ2 ,
B∗:lim sup
ξ→0 Fξ
ξ2 ,
3.51
λ∗1: 2 3
q∞12p∞
B∗ ,
λ∗2:2p0
A∗.
3.52
Theorem 3.9. Letf : R → Rbe a locally essentially bounded and almost everywhere continuous function. Assume that
jξ0Ftdt≥0, for everyξ≥0;
jj
lim inf
ξ→0
max|t|≤ξFt
ξ2 < κlim sup
ξ→0 Fξ
ξ2 , 3.53
whereκis given by3.14;
jjjfor almost everyx∈0,1, eachz ∈Df and eachλ ∈λ∗1, λ∗2(whereλ∗1, λ∗2are given by
3.52), the condition
λf−z−qxz≤0≤λfz−qxz 3.54
Then, for eachλ ∈λ∗1, λ2∗, problemGp,qf,λpossesses a sequence of pairwise distinct weak solutions, which strongly converges to zero inW01,20,1.
Proof. The proof is the same ofTheorem 3.1applying partcofTheorem 2.1instead of part
b.
Clearly, fromTheorem 3.9 we obtain similar consequences to those ofTheorem 3.1. Here, we present only one of them.
Corollary 3.10. Let f : R → Rbe a continuous, nonnegative function andp ∈ C10,1,q ∈ C00,1. Assume that
jja
lim inf
ξ→0 Fξ
ξ2 < κlim sup
ξ→0 Fξ
ξ2 . 3.55
Then, for eachλ ∈λ∗1, λ2∗, problemGp,qf,λpossesses a sequence of pairwise distinct positive classical solutions, which strongly converges to zero inC00,1.
Now, we present some examples of application ofTheorem 3.1for which the results in
1–4,6,7cannot be appliedseeRemark 3.13.
Example 3.11. Letq0be a nonnegative real constant, put
an: 2n!n2!−1
4n1! , bn:
2n!n2!1
4n1! , 3.56
for everyn∈Nand define the nonnegative, continuous functionf:R → Ras follows
fξ:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
32n1!2n1!2−n!2
π
1 16n1!2 −
ξ−n!n2
2
2
ifξ∈
n∈N
an, bn
0 otherwise.
3.57
One has nn! 1!ftdt bn
anftdt n 1!
2 − n!2 for every n ∈ N. Then, one has
limn→∞Fbn/b2n 4 and limn→∞Fan/a2n 0. Therefore, by a simple computation, we
have lim infξ→∞Fξ/ξ20 and lim supξ→∞Fξ/ξ24. Hence,
0lim inf
ξ→∞
Fξ ξ2 <
3
q0124 3
q012lim supξ→∞
Fξ
Owing toCorollary 3.5, for eachλ >q012/6 the problem
−uq0uλfu in0,1
u0 u1 0
G1,q0
f,λ
possesses a sequence of pairwise distinct positive classical solutions.
Now, let f∗ : R → R be the positive, continuous function defined as f∗ξ fξ 1 for all ξ ∈ R, wheref is given by 3.57. Clearly, lim infξ→∞F∗ξ/ξ2 0 and
lim supξ→∞F∗ξ/ξ2 4. Hence, again owing to Corollary 3.5, for each λ > q0 12/6
the problem
−uq0uλf∗u in0,1
u0 u1 0
G1f,λ,q0
possesses a sequence of pairwise distinct positive classical solutions.
Example 3.12. Letq0be a nonnegative real constant, put
a1:2, an1:
an 3/2, 3.59
for everyn∈NandS:n≥2an1−1, an11. Define the continuous functionf :R → R
as follows
ft:
⎧ ⎪ ⎨ ⎪ ⎩
an1 3e1/t−an1−1t−an111
2an1−t
t−an1−1
2
t−an11
2 ift∈S
0 otherwise.
3.60
For which, one has
Fξ
ξ
0 ftdt
an1 3e1/ξ−an1−1ξ−an111 ifξ∈S
0 otherwise,
3.61
andFan1 an13 for everyn ≥ 2. Hence, one has lim supξ→∞Fξ/ξ2 ∞. On the
other hand, by settingxnan1−1 for everyn≥2, one has maxξ∈−xn,xnFξ an
3for every n ≥2. Therefore, one has limn→∞maxξ∈−xn,xnFξ/xn2 1 and, by a simple computation,
one has lim infξ→∞maxt∈−ξ,ξFt/ξ2 1. Hence,
lim inf
ξ→∞
maxt∈−ξ,ξFt
ξ2 1<
3
q012lim supξ→∞
Fξ
Owing toCorollary 3.4, for eachλ∈0,2the problemG1,q0
f,λpossesses a sequence of pairwise
distinct classical solutions.
Remark 3.13. In3the existence of infinitely many solutions for the problem
−ufu in0,1
u0 u1 0 3.63
was studied under suitable assumptions on the function f, as fξ > 0 for all ξ large enough. We explicitly observe that we cannot apply3, Theorem 2.11to problemG1,q0
f,λof
Example 3.11, even in the caseq00, since our function is not positive for allξlarge enough. The same remark for4, Corollary 3.1holds, since limξ→∞fξ ∞hence, in particular,
fξ>0 forξlarge enoughis requested.
The same problem was studied in1,7. Assumptions of1, Theorem 2.1imply that
fis negative in suitable real intervals. Hence,1, Theorem 2.1cannot be applied to problem
G1,q0
f,λofExample 3.11. Moreover, assumptions in7, Theorem 3.1, as inf{t∈R:ft>0}<
0, cannot be applied to the functionf ofExample 3.11since, in this case, one has inf{t∈R:
ft>0}11/8.
In2,6, the authors studied the existence of infinitely many weak solutions of the following autonomous Dirichlet problem
−Δpufu inΩ,
u0 on∂Ω, D
whereΩis a bounded open subset of the Euclidean spaceRN,| · |,N ≥ 1, with boundary of classC1,Δ
pu:div|∇u|p−2∇u,p >1, andfis a continuous function. In6, Remark 3.3
the key assumption to obtain infinitely many solutions to Dis: lim infξ→∞Fξ/ξp 0
and lim supξ→∞Fξ/ξp ∞. Clearly, the functionf inExample 3.11does not satisfy this
condition. Hence, we cannot apply6, Theorem 1.1to our problemG1,q0
f,λ, even in the case
q00.
On the other hand, we cannot apply 2, Theorem 1.1 to G1f,λ,q0, since one of the
key assumptions is that function f is nonpositive in suitable real intervals. Another key assumption of2, Theorem 1.1is
lim sup
ξ→∞
Fξ
ξp <∞. 3.64
Hence, we cannot apply2, Theorem 1.1toG1,q0
f,λinExample 3.12, even in the caseq00.
Clearly, we cannot apply 6, Remark 3.3,3, Theorem 2.11,4, Corollary 3.1, 7, Theorem 3.1toG1,q0
f,λinExample 3.12for the same previous reasons.
Example 3.14. Let{an}and{bn}be the sequences defined as in theExample 3.11and define
the nonnegativeand discontinuousfunctionf:R → Ras follows
fξ:
⎧ ⎨ ⎩
2n1!n1!2−n!2 if ξ∈
n∈N
an, bn
0 otherwise.
3.65
By a similar computation as inExample 3.11, we have
lim sup
ξ→∞
Fξ
ξ2 4, lim infξ→∞ Fξ
ξ2 0. 3.66
FromCorollary 3.6, for eachλ > 2 the problemG1f,λ,0possesses a sequence of positive weak solutions which is unbounded inW01,20,1.
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