Infinitely Many Solutions for a Boundary Value Problem with Discontinuous Nonlinearities

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

and Giovanni Molica Bisci

1Mathematics 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,bonanno@unime.it 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 that inf_Rf >0. PutFξ:_ξ

0ftdtfor everyξ∈Rand assume that lim inf

ξ→∞

Fξ ξ² < 1

4lim sup

ξ→∞

Fξ

ξ² . 1.1

Then, for eachλ∈8/lim sup_ξ→∞Fξ/ξ²,2/lim infξ→∞Fξ/ξ², the problem

−uλfu in0,1

u0 u1 0 G^1,0_f,λ

admits a sequence of pairwise distinct positive weak solutions.

Clearly, whenf is continuous inR, the solutions inTheorem 1.1are classicalin this case, it is enough to assume inf_Rf ≥ 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 problemG^1,0_f,λare obtained, for example, in1–7. We observe that in the very interesting paper 6, the authors assume lim infξ→∞Fξ/ξ² 0 and lim sup_ξ→∞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 lim_ξ→∞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

Φz−Φw≤Lxz−w ∀z, w∈Vx. 2.1

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

sup_u∈Φ−1−∞,rΨu −Ψu

r−Φu ,

γ :lim inf

r→∞ ϕr, δ: lim inf

r→infXΦϕr.

2.8

Assuming also thatΦandΥare locally Lipschitz functionals, we have the following result, which is a more precise version of11, Theorem 1.1 seeRemark 2.2.

Theorem 2.1. Under the above assumptions onX,ΦandΨ, one has

afor everyr >inf_XΦand everyλ∈0,1/ϕr, the restriction of the functionalIλ Φ−λΨ toΦ⁻¹− ∞, radmits a global minimum, which is a critical point (local minimum) ofIλ

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−λsup_Φx<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 whenγ < ∞andIλdoes not posses a global minimum inX, thenIλadmits a sequence of

critical points. Leta ∈ Rsuch thatλ < a < 1/γ. From lim infr→∞ϕr < 1/athere exists a sequence {rn} such that lim_n→∞rn ∞ and ϕrn < 1/afor all n ∈ N. Putrn1 r1. Owing toa, one can findu1∈Φ⁻¹− ∞, rn1such 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− ∞, rn2such thatu2is a local minimum for the functionalIλ. Taking into account thatu2 is a global minimum inΦ⁻¹− ∞, rn2, we have that Iλu2 ≤ Iλu1 andIλu1< Iλu1. Hence,Φu2> rn1. Reasoning inductively we obtain a sequence{uk}of distinct critical points such thatΦu_k1> 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 inf_r→infXΦϕr δ <1/a, hence there exists a decreasing sequence{rn} such that lim_n→∞rn inf_XΦand ϕrn < 1/afor alln ∈ N. Putrn1 r1. Owing toa, there existsu1 ∈Φ⁻¹− ∞, rn1, 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− ∞, rn2which 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 inf_XΦ. 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^∗≤lim_k→∞Φuk inf_XΦ, that isΦu^∗ inf_XΦ. 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, ft: lim

δ→0ess sup

|t−z|<δ fz 3.2

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 inf

x∈0,1 qx≥0. 3.3

LetW^1,20,1be the Sobolev space endowed with the usual norm

u∗: 1

0

ux²dx ₁

0

ux²dx 1/2

. 3.4

As is customary, we denote byW₀^1,20,1the closure ofC^∞₀ 0,1inW^1,20,1. Moreover, a functionu:0,1 → Ris said to be a weak solution ofG^p,q_f,λifu∈W₀^1,20,1and

₁

0

pxuxvxdx ₁

0

qxuxvxdx

λ ₁

0

f

ux vxdx ∀v∈W₀^1,2 0,1 .

3.5

We recall thatu∈AC0,1is a generalized solution ofG^p,q_f,λifpu∈AC0,1,u0 u1 and

−

pxux qxux λf

ux , 3.6

for almost everyx∈0,1.

Clearly, the weak solutions ofG^p,q_f,λare also generalized solutions.

Iff andqare continuous functions we recall thatu∈C¹0,1is a classical solution ofG^p,q_f,λifpu∈C¹0,1,u0 u1and

−

pxux qxux λf

ux , 3.7

for everyx∈0,1.

We recall that

u, v: ₁

0

qxuxvxdx

₁

0

pxuxvxdx u, v∈W₀^1,2

0,1 3.8

is an inner product that induces inW₀^1,20,1the norm

u: 1

0

pxux²dx ₁

0

qxux²dx 1/2

, 3.9

which is equivalent to the usual one.

It is well known thatW₀^1,20,1is compactly embedded inC⁰0,1and in particular one has

u_∞≤ 1 2√

p0u, 3.10

for everyu∈W₀^1,20,1.

ConsiderΦ:W₀^1,20,1 → RandΥ:W₀^1,20,1 → Rdefined as follows

Φu: u²

2 , Υu:

₁

0

F

ux 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 ∈ W₀^1,20,1^∗given by

Φuv ₁

0

pxuxvxdx ₁

0

qxuxvxdx, 3.13

for everyv∈W₀^1,20,1.

Moreover, Υ is locally Lipschitz continuous in W₀^1,20,1. So it makes sense to consider the generalized directional derivative Υ^◦. Finally, by a standard argument, Υ is sequentially weakly continuous.

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:_ξ

0ftdtfor everyξ∈Rand assume that i_ξ

0Ftdt≥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 G^p,q_f,λ possesses a sequence of weak solutions which is unbounded inW₀^1,20,1.

Proof. Our aim is to applyTheorem 2.1b. For this end, fixλ ∈λ1, λ2and denote byX the Banach spaceW₀^1,20,1endowed with the norm

u: ₁

0

pxux²dx ₁

0

qxux²dx _1/2

. 3.19

For eachu∈X, put

Φu: u²

2 , Υu:

₁

0

F

ux 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 hasI_λ^◦u;v Φuv λ−Υ^◦u;vfor allu, v∈X.

Now, let{cn}be a real sequence such that limn→∞cn ∞and

nlim→∞

max_|t|≤cnFt cn²

A. 3.21

Putrn 2p0c²_nfor alln ∈ N. Taking3.10into account, one has max_t∈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<2rnfor alln∈None has

ϕ

rn inf

u²<2rn

sup_v2<2rn

₁

0F

vx dx−₁

0F

ux dx rn− u²/2

≤ sup_v2<2rn

₁

0F

vx dx−₁

0F

u0x dx rn−u0²/2

sup_v2<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 lim_n→∞dn ∞and

nlim→∞

F dn

dn²

B. 3.24

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≤8d²_n q∞

12 p∞

. 3.26

Therefore,

Φ

wn −λΥ

wn wn² 2 −λ

₁

0

F

wnx dx

≤4d_n² q∞

12 p∞

−λ ₁

0

F

wnx dx.

3.27

Takingiinto account, we have ₁

0

F

wnx dx≥ _3/4

1/4

F

dn dt 1 2F

dn . 3.28

Then,

Φ

wn −λΥ

wn ≤4d²_n 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

F

dn > εBd_n², 3.30

for alln > νε. Therefore,

Φ

wn −λΥ

wn ≤4d²_n q∞

12 p∞

−λ 2εBd²_n d²_n

4

q∞

12 p∞

−λ 2εB

3.31

for alln > νε.

From the choice ofε, one has

nlim→∞

Φ

wn −λΥ

wn −∞. 3.32

On the other hand, ifB ∞, we fixM >8/λq_∞/12p_∞and, again from3.24there existsνM∈Nsuch that

F

dn > Md²_n, 3.33

for alln > νM. Therefore,

Φ

wn −λΥ

wn ≤4d²_n q∞

12 p_∞

− λ 2Md²_n d_n²

4

q∞

12 p∞

−λ 2M

3.34

for alln > νM.

From the choice ofM, also in this case, one has

nlim→∞

Φ

wn −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 lim_n→∞un ∞, that is{un} is unbounded inX.

Now, we claim that the generalized critical points of Iλ are weak solutions for the problemG^p,q_f,λ. To this end, letu0 ∈X a generalized critical point ofIλ, that isI_λ^◦u0, v≥0, for allv∈X. From which, we obtain

Φ

u0 v λ−Υ^◦

u0;v ≥0, 3.36

for allv∈X. Hence,Φu0v≥ −λ−Υ^◦u0;vfor allv∈X, that is

− ₁

0

pxu₀xvx qxu0xvx

dx≤λ−Υ^◦

u0;v , 3.37

for allv∈X. Clearly, setting

Lv:− ₁

0

pxu₀xvx qxu0xvx

dx ∀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 ₁

0hxvxdxfor allv ∈ L²0,1. From a standard resultsee, e.g.,15, Example 2, page 219 there is a uniqueu ∈ W^2,20,1∩X such thatpu−qu h. In particular, one has₁

0pxux−qxuxvxdx−₁

0pxuxvxqxuxvxdx for all v ∈ X. Hence, −₁

0pxu₀xvx qxu0xvxdx Lv ₁

0hxvxdx

₁

0pxux−qxuxvxdx −₁

0pxuxvx 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∈W^2,20,1and

₁

0

pxu₀x −qxu0x

vxdx− ₁

0

pxu₀xvx qxu0xvx

dx 3.39

for allv∈X. From3.37and3.39one has ₁

0

pxu₀x −qxu0x

vxdx≤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, λf

u0x −qxu0x

, 3.41

for almost everyx∈0,1.

Now, sincemDf 0, from18, Lemma 1we obtain −pxu₀x 0 for almost everyx ∈u⁻¹₀ Df. Hence, fromiiiwe obtainλfu0x−qxu0x 0 for almost every x∈u⁻¹₀ Df. From which

−

pxu₀x qxu0x λf

u0x , 3.42

for almost every x ∈ u⁻¹₀ Df. On the other hand, for almost every x ∈ 0,1\u⁻¹₀ Df, condition3.41reduces to

−

pxu₀x qxu0x λf

u0x . 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 inf_Rf >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∈W₀^1,20,1such that

−

pxux qxux∈λ f⁻

ux ,f

ux 3.45

for almost everyx ∈ 0,1is called multi-valued solution forG^p,q_f,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 problemG^p,q_f,λ.

Remark 3.3. The following condition

iithere exist two real sequences {bn},{cn}, withbn <

p0/2q_∞/12p_∞cn

for alln∈Nand lim_n→∞cn ∞, such that

A1: lim

n→∞

max_|t|≤cnFt−1/2F bn

2p0cn²−4

q_∞/12p_∞ b²n

< κ 2p0

lim sup

ξ→∞

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 ∈ λ1,1/A1 the conclusion in Theorem 3.1 again holds. In fact, arguing as in the proof of

Theorem 3.1, one hasϕrn inf_u²_<2rnsup_v2<2rn

₁

0Fvxdx−₁

0Fuxdx/rn−u²/2≤ sup_v2<2rn

₁

0Fvxdx−₁

0Funxdx/rn−un²/2≤max_|t|≤cnFt−1/2Fbn/2p0c_n²− 4q_∞/12p_∞b²_n, 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, problemG^p,q_f,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, problemG^p,q_f,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, problemG^p,q_f,λpossesses a sequence of positive weak solutions which is unbounded in W₀^1,20,1.

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

q_∞12p_∞ ; 3.49

ii2

lim inf

ξ→∞

max_|t|≤ξFt

ξ² <2p0. 3.50

Then, problem

−

pu qufu in0,1 u0 u1 0

G^p,q_f,1

possesses a sequence of weak solutions which is unbounded inW₀^1,20,1.

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

j_ξ

0Ftdt≥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

impliesλfz qxz.

Then, for eachλ ∈λ^∗₁, λ^∗₂, problemG^p,q_f,λpossesses a sequence of pairwise distinct weak solutions, which strongly converges to zero inW₀^1,20,1.

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. Letf : R → Rbe a continuous, nonnegative function andp ∈ C¹0,1,q ∈ C⁰0,1. Assume that

jj_a

lim inf

ξ→0

Fξ

ξ² < κlim sup

ξ→0

Fξ

ξ² . 3.55

Then, for eachλ ∈λ^∗₁, λ^∗₂, problemG^p,q_f,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!²

n1!²−n!² π

1 16n1!² −

ξ−n!n2 2

₂

ifξ∈

n∈N

an, bn

0 otherwise.

3.57 One has _n1!

n! ftdt _bn

anftdt n 1!² − n!² for every n ∈ N. Then, one has lim_n→∞Fbn/b²_n 4 and lim_n→∞Fan/a²_n 0. Therefore, by a simple computation, we have lim inf_ξ→∞Fξ/ξ²0 and lim sup_ξ→∞Fξ/ξ²4. Hence,

0lim inf

ξ→∞

Fξ ξ² < 3

q0124 3

q012lim sup

ξ→∞

Fξ

ξ² . 3.58

Owing toCorollary 3.5, for eachλ >q012/6 the problem

−uq0uλfu in0,1 u0 u1 0

G^1,q_f,λ⁰

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 sup_ξ→∞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

G^1,q_f,λ⁰

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≥2a_n1−1, an11. Define the continuous functionf :R → R as follows

ft:

⎧⎪

⎨

⎪⎩ an1 3

e^{1/t−an1−1t−an111} 2

an1−t t−

a_n1−1 2 t−

a_n11 2 ift∈S

0 otherwise.

3.60

For which, one has

Fξ _ξ

0

ftdt

a_n1 ³e^{1/ξ−an1−1ξ−an111} ifξ∈S

0 otherwise,

3.61

andFa_n1 a_n1³ for everyn ≥ 2. Hence, one has lim sup_ξ→∞Fξ/ξ² ∞. On the other hand, by settingxnan1−1 for everyn≥2, one has max_{ξ∈−xn,xn}Fξ an³for every n ≥2. Therefore, one has limn→∞max_ξ∈−xn,xnFξ/xn2 1 and, by a simple computation, one has lim inf_ξ→∞max_t∈−ξ,ξFt/ξ² 1. Hence,

lim inf

ξ→∞

max_t∈−ξ,ξFt

ξ² 1< 3

q012lim sup

ξ→∞

Fξ

ξ² ∞. 3.62

Owing toCorollary 3.4, for eachλ∈0,2the problemG^1,q_f,λ⁰possesses 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 problemG^1,q_f,λ⁰of Example 3.11, even in the caseq00, since our function is not positive for allξlarge enough.

The same remark for4, Corollary 3.1holds, since lim_ξ→∞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 G^1,q_f,λ⁰ofExample 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 spaceR^N,| · |,N ≥ 1, with boundary of classC¹,Δpu: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 inf_ξ→∞Fξ/ξ^p 0 and lim sup_ξ→∞Fξ/ξ^p ∞. Clearly, the functionf inExample 3.11does not satisfy this condition. Hence, we cannot apply6, Theorem 1.1to our problemG^1,q_f,λ⁰, even in the case q00.

On the other hand, we cannot apply 2, Theorem 1.1 to G^1,q_f,λ⁰, 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

ξ^p <∞. 3.64

Hence, we cannot apply2, Theorem 1.1toG^1,q_f,λ⁰inExample 3.12, even in the caseq00.

Clearly, we cannot apply 6, Remark 3.3,3, Theorem 2.11,4, Corollary 3.1, 7, Theorem 3.1toG^1,q_f,λ⁰inExample 3.12for the same previous reasons.

The following example deals with a discontinuous function.

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 inf

ξ→∞

Fξ

ξ² 0. 3.66

FromCorollary 3.6, for eachλ > 2 the problemG^1,0_f,λpossesses a sequence of positive weak solutions which is unbounded inW₀^1,20,1.

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