R E S E A R C H
Open Access
A nonlinear boundary value problem for
fourth-order elastic beam equations
Yuanping Song
**Correspondence: [email protected] School of Science, Shandong University of Technology, Zibo, 255049, P.R. China
Abstract
By using an infinitely many critical points theorem, we study the existence of infinitely many solutions for a fourth-order nonlinear boundary value problem, depending on two real parameters. No symmetric condition on the nonlinear term is assumed. Some recent results are improved and extended.
1 Introduction
In this paper, we consider a beam equation with nonlinear boundary conditions of the type:
⎧ ⎪ ⎨ ⎪ ⎩
u()=λf(x,u) +μh(x,u), <x< , u() =u() = ,
u() = , u() =g(u()),
(.)
whereλ,μ are two positive parameters, f, h are twoL-Carathéodory functions, and g∈C(R) is real function. This kind of problem arises in the study of deflections of elastic beams on nonlinear elastic foundations. The problem has the following physical descrip-tion: a thin flexible elastic beam of length is clamped at its left endx= and resting on an elastic device at its right endx= , which is given byg. Then the problem models the static equilibrium of the beam under a load, along its length, characterized byf andh. The derivation of the model can be found in [, ].
Fourth-order boundary value problems modeling bending equilibria of elastic beams have been considered in several papers. Most of them are concerned with nonlinear equations with null boundary conditions; see [–]. When the boundary conditions are nonzero or nonlinear, fourth-order equations can model beams resting on elastic bear-ings located in their extremities; see for instance [, , –] and the references therein. More precisely, in [], using variant fountain theorems, the author obtains the existence of infinitely many solutions for problem (.) withλ= andμ= under the symmetric condition and some other suitable assumptions of the nonlinear termf.
Motivated by the above works, in the present paper we establish some multiplicity re-sults for problem (.) under rather different assumptions on the functionsf,handg. It is worth noticing that in our results neither the symmetric nor the monotonic condition on the nonlinear term is assumed. We require thatf has a suitable oscillating behavior ei-ther at infinity or at zero. In the first case, we obtain an unbounded sequence of solutions
(Theorem .); in the second case, we obtain a sequence of nonzero solutions strongly converging at zero (Theorem .), which improve and extend the results in [].
The remainder of this paper is organized as follows. In Section , some preliminary results are presented. In Section , we give the proofs of our main results.
2 Variational setting and preliminaries
We prove our results applying the following smooth version of Theorem . of Bonanno and Bisci [], which is a more precise version of Ricceri’s variational principle [, Lem-ma .].
Theorem . Let E be a reflexive real Banach space,let,:E→Rbe two Gâteaux differentiable functionals such thatis sequentially weakly lower semicontinuous,strongly continuous and coercive, and is sequentially weakly upper semicontinuous.For every r>infE,let
ϕ(r) := inf
u∈–(–∞,r)
(supu∈–(–∞,r)(v)) –(u) r–(u) ,
γ :=lim inf
r→+∞ ϕ(r) and δ:=r→(inflim infE)+ϕ(r).
Then the following properties hold:
(a) For everyr>infEand everyλ∈(, /ϕ(r));the restriction of the functional
Iλ:=–λ
to–(–∞,r)admits a global minimum,which is a critical point(local minimum)of IλinE.
(b) Ifγ < +∞;then for eachλ∈(, /γ),the following alternative holds:either (b) Iλpossesses a global minimum,or
(b) there is a sequence{un}of critical points(local minima)ofIλsuch that
lim
n→+∞(un) = +∞.
(c) Ifδ< +∞;then for eachλ∈(, /δ),the following alternative holds:either (c) there is a global minimum ofwhich is a local minimum ofIλ,or
(c) there is a sequence{un}of pairwise distinct critical points(local minima)ofIλ
that converges weakly to a global minimum of.
LetEbe the Hilbert space
E=u∈H(, );u() =u() =
with the inner product and norm
u,v=
whereH(, ) is the Sobolev space of all functionsu: [, ]→Rsuch thatuand its
distri-butional derivativeuare absolutely continuous andubelongs toL([, ]), and ·
p de-notes the standardLpnorm. In addition,Eis compactly embedded in the spacesL([, ])
andC([, ]), and therefore, there exist immersion constantsS,S¯> , such that
u≤Su and u∞≤ ¯Su. (.)
We recall thatf : [, ]×R→Ris anL-Carathéodory function if (a) the mappingx→f(x,u)is measurable for everyu∈R;
(b) the mappingu→f(x,u)is continuous for almost everyx∈[, ]; (c) for everyρ> there exists a functionlρ∈L([, ])such that
sup |u|≤ρ
f(x,u)≤lρ(x),
for almost everyx∈[, ].
Define the functionsF,H: [, ]×R→Ras follows:
F(x,u) = u
f(x,s)ds and H(x,u) = u
h(x,s)ds,
for all (x,u)∈[, ]×R, andG(t) =tg(x)dx. Thus we define the functionalIλ,μ∈C(E,R)
by
Iλ,μ(u) :=
u
–λ
F(x,u)dx–μ
H(x,u)dx+Gu(), for allu∈E.
Definition . We say that a functionu∈Eis a weak solution of problem (.) if
u(x)v(x)dx–λ
f(x,u)v dx–μ
h(x,u)v dx+gu()v() =
holds for anyv∈E.
3 Main results
In this section we establish the main abstract results of this paper. Let
A:=lim inf
ξ→+∞
max|u|≤ξF(x,u)dx
ξ ,
B:=lim sup
ξ→+∞
b
a max|u|≤ξF(x,u)dx
ξ ,
c:= α
c +
β σ+ c
σ–
and
λ:=
a
|d|dx+
b|e|dx
B , λ:=
SA¯ ,
Theorem . Let f : [, ]×R→Rbe an L-Carathéodory function and <a<b< . Assume that
(A) there exist constantsα,β> andσ∈[, )such that g(u)≤α+β|u|σ, for allu∈R;
(A) F(x,u)≥for all(x,u)∈([,a]∪[b, ])×R;
(A) there exist two functionsd∈C([,a])ande∈C([b, ])satisfying d() =d() = , d(a) =e(b) = , d(a) =e(b) =
and a
ddx+
b
edx= ,
such that
¯
SA
a
ddx+
b
edx
<B. (.)
Then,for everyλ∈(λ,λ)and for any L-Carathéodory function h: [, ]×R→R, whose potential H(x,u) =uh(x,s)ds for all(x,u)∈[, ]×R,is a nonnegative function satisfying the condition
H∞:=lim sup
ξ→+∞
max|u|≤ξH(x,u)dx
ξ < +∞, (.)
if we put
μH,λ:=
SH∞¯ ( – S¯λA),
whereμH,λ= +∞when H∞= ,for everyμ∈[,μH,λ)problem(.)has an unbounded
sequence of weak solutions in E.
Proof Obviously, it follows from (A) thatλ<λ. Fixλ¯∈(λ,λ). Sinceλ¯<λ, we have
μH,¯λ=
SH¯ ∞( –
¯
Sλ¯A) > .
Now fixμ¯∈(,μH,λ¯) and set
J(x,u) :=F(x,u) +μ¯¯
λH(x,u), for all (x,u)∈[, ]×R. Let the functionals,:E→Rbe defined by
(u) = u
,
(u) =
J(x,u)dx–¯ λG
whereG(t) =tg(x)dx. Put
I¯λ,μ¯(u) :=(u) –λ(¯ u), for allu∈E.
Using the property off,hand the continuity ofg, we obtain,∈C(E,R) and for any
v∈E, we have
(u),v=
u(x)v(x)dx
and
(u),v=
fx,u(x)v(x)dx+μ¯¯ λ
hx,u(x)v(x)dx– ¯ λg
u()v().
So, with standard arguments, we deduce that the critical points of the functionalI¯λ,μ¯ are
the weak solutions of problem (.) and so they are classical. We first observe that the functionalsandsatisfy the regularity assumptions of Theorem ..
First of all, we show thatλ¯ <γ. Let{ξn}be a sequence of positive numbers such that
limn→+∞ξn= +∞and
lim
n→+∞
max|u|≤ξnF(x,u)dx
ξ
n
=A.
Setrn:=¯Sξnfor alln∈N. Then, for allv∈Ewith(v) <rn, taking (.) into account, one hasv∞<ξn. Note that() =() = . Then, for alln∈N,
ϕ(rn) = inf
u∈–(–∞,rn)
(supv∈–(–∞,rn)(v)) –(u) rn–(u)
≤ supu∈–(–∞,rn)(u) rn
≤
max|u|≤ξnJ(x,u)dx+ ¯
λ(αξn+ β σ+ξ
σ+
n )
S¯ξn
≤S¯
max|u|≤ξnF(x,u)dx
ξ
n
+μ¯¯ λ
max|u|≤ξnH(x,u)dx
ξ
n
+¯ λξn
.
Sincelimn→+∞ξn= , from the assumption (A) and the condition (.), we have
γ <lim inf
n→+∞ϕ(rn)≤S¯
A+μ¯¯ λH∞
< +∞,
and combining the assumptionμ¯∈(,μG,λ¯), we obtain
γ <lim inf
n→+∞ϕ(rn)≤S¯
A+μ¯¯ λH∞
< SA¯ + – ¯S¯λ¯A λ .
This implies that
¯
Letλ¯be fixed. We claim that the functionalI¯λ,μ¯ is unbounded from below. Since
¯
λ<
B
a
|d|dx+
b|e|dx ,
there exist a sequence{ηn}of positive numbers andτ > such thatlimn→+∞ηn= +∞and
¯
λ<τ<
abF(x,ηn)dx
η
n[ a
|d|dx+
b|e|dx] ,
for eachn∈Nlarge enough. For alln∈Nwe definevnby
vn(x) := ⎧ ⎪ ⎨ ⎪ ⎩
d(x)ηn, x∈[,a], ηn, x∈(a,b],
e(x)ηn, x∈(b, ].
(.)
From the condition (A), it is easy to verify thatvn∈E. For anyn∈N, one has
(vn) = vn
=ηn
a
ddx+
b
edx
. (.)
On the other hand, by (A) and sinceHis nonnegative, from the definition of, we infer
(vn)≥ b
a
F(x,ηn)dx–¯ λη
n(¯Sηn).
By (.) and (.), we have
I¯λ,μ¯(vn)≤
ηn
a
ddx+
b
edx
–λ¯
b
a
F(x,ηn)dx+ηn(S¯ηn)
< η
n
a
ddx+
b
edx
( –λτ¯ ) +ηn(¯Sηn),
for everyn∈Nlarge enough. Sinceσ< ,λτ¯ > andlimn→+∞ηn= +∞, we have
lim
n→+∞I¯λ,μ¯(vn) = –∞.
Then the functionalI¯λ,μ¯ is unbounded from below, and it follows thatI¯λ,μ¯ has no global
minimum. Therefore, by Theorem .(b), there exists a sequence{un}of critical points of
I¯λ,μ¯ such that
lim
n→+∞un= +∞,
and the conclusion is achieved.
Remark . Indeed, it is not difficult to find such functionsd(x) ande(x) satisfying the condition (A). For example, leta=andb=. We can choose
d(x) = –x
x–
, x∈
,
and
e(x) = –x
x–
, x∈
,
.
Remark . Under the conditionsA= andB= +∞, from Theorem . we see that for everyλ> and for eachμ∈[,SH¯∞), problem (.) admits a sequence of classical solu-tions which is unbounded inE. Moreover, ifH∞= , the result holds for everyλ> and μ> .
Corollary . Let f :R→Rbe an L-Carathéodory function and <a<b< .Suppose that hypotheses(A)-(A)hold.Moreover,the condition(A)is satisfied if(.)is replaced by
a
ddx+
b
edx< B, SA¯ < .
Then, for any L-Carathéodory function h: [, ]×R→R, whose potential H(x,u) :=
u
h(x,s)ds for all(x,u)∈[, ]×R,is a nonnegative function satisfying the condition(.), if we put
μH:=
SH∞¯ ( – SA¯ ),
whereμH= +∞when H∞= ,the problem
⎧ ⎪ ⎨ ⎪ ⎩
u()=f(x,u) +μh(x,u), <x< , u() =u() = ,
u() = , u() =g(u()),
has an unbounded sequence of weak solutions for everyμ∈[,μH)in E.
Corollary . Under the assumptions of Corollary.,for any nonnegative continuous function h: [, ]→R,the problem
⎧ ⎪ ⎨ ⎪ ⎩
u()=f(x,u) +h(x), <x< , u() =u() = ,
u() = , u() =g(u()),
has infinitely many distinct weak solutions in E.
Now, let
¯
A:=lim inf
ξ→+
max|u|≤ξF(x,u)dx
ξ ,
¯
B:=lim sup
ξ→+
b
a max|u|≤ξF(x,u)dx
ξ ,
c:=min
|u|≤c u()
and
¯
λ:=
a
|d|dx+
b|e|dx
B¯ , λ¯:=
S¯A¯.
Theorem . Let f : [, ]×R→Rbe an L-Carathéodory function and <a<b< . Moreover,assume that(A)and
(A) g(u)≤for allu∈Randlimu→+ u
g(s)ds
u = ;
(A) there exist two functionsd∈C([,a])ande∈C([b, ])satisfying
d() =d() = , d(a) =e(b) = , d(a) =e(b) = , e() >
and a
ddx+
b
edx= ,
such that
¯
SA¯
a
ddx+
b
edx
<B¯,
are satisfied.Then, for everyλ∈(λ¯,λ¯)and for anyL-Carathéodory function h:
[, ]×R→R,whose potentialH(x,u) :=uh(x,s)dsfor all(x,u)∈[, ]×R,is a nonnegative function satisfying the condition
H:=lim sup
ξ→+
max|u|≤ξH(x,u)dx
ξ < +∞,
if we put
¯
μH,λ:=
SH¯ ( – S¯λA¯),
whereμ¯H,λ= +∞whenH= ,for everyμ∈[,μ¯H,λ)problem(.)has a sequence of
weak solutions,which strongly converges to zero inE.
Proof It follows from (A)thatλ¯<λ¯. Fixλ¯∈(λ¯,λ¯). Sinceλ¯<λ¯, we have
μH,¯λ=
SH¯
( – S¯λ¯A¯) > .
Now fixμ¯∈(,μ¯H,λ¯) and set
J(x,u) :=F(x,u) +μ¯¯
λH(x,u), for all (x,u)∈[, ]×R.
We take,, andI¯λ,μ¯as in the proof of Theorem .. Now, as has been pointed out before,
first step, we will prove thatλ¯< /δ. Let{ξn}be a sequence of positive numbers such that
limn→+∞ξn= and
lim
n→+∞
max|u|≤ξnF(x,u)dx
ξ
n
=A¯.
By the fact thatinfu∈E(u) = and the definition ofδ, we haveδ=lim infr→+ϕ(r). Set rn:=¯Sξ
n for alln∈N. Then, for allv∈Ewith(v) <rn, taking (.) into account, one hasv∞<ξn. Note that() =() = . Then, for alln∈N,
ϕ(rn) = inf
u∈–(–∞,rn)
(supv∈–(–∞,rn)(v)) –(u) rn–(u)
≤ supu∈–(–∞,rn)(u)
rn ≤
max|u|≤ξnJ(x,u)dx– ¯
λξn
S¯ξn
≤S¯
max|u|≤ξnF(x,u)dx
ξ
n
+μ¯¯ λ
max|u|≤ξnH(x,u)dx
ξ n –¯ λ ξn ξ n .
It follows from (A)thatlimn→+∞
ξn
ξn = . Then we have
δ<lim inf
n→+∞ϕ(rn)≤S¯
¯
A+μ¯¯ λH
< +∞.
Fromμ¯∈(,μG,λ¯), we obtain
δ≤S¯
¯
A+μ¯¯ λH
< S¯A¯+ – ¯S¯λ¯A¯ λ ,
which implies that
¯
λ< δ.
Letλ¯ be fixed. We claim that the functionalI¯λ,μ¯ does not have a local minimum at zero.
Since
¯
λ<
B¯
a
|d|dx+
b|e|dx ,
there exist a sequence{ηn}of positive numbers andτ> such thatlimn→+∞ηn= and
¯
λ<τ<
abF(x,ηn)dx η
n[ a
|d|dx+
b|e|dx] ,
for eachn∈Nlarge enough. For alln∈N, letvnbe defined by (.) with the aboveηn. Note thatλτ¯ > . Then, sinceg(u)≤ for allu∈Rande() > , we obtain
I¯λ,μ¯(vn)≤ ηn
a
ddx+
b
edx
–λ¯
b
a
F(x,ηn)dx+ vn()
g(x)dx
<η n a
ddx+
b
edx
for everyn∈Nlarge enough. Thus, since
lim
n→+∞I¯λ,μ¯(vn) =I¯λ,μ¯() = ,
we see that zero is not a local minimum ofI¯λ,μ¯. This, together with the fact that zero is
the only global minimum of, we deduce that the energy functionalI¯λ,μ¯ does not have a
local minimum at the unique global minimum of. Therefore, by Theorem .(c), there exists a sequence{un}of critical points ofI¯λ,μ¯, which converges weakly to zero. In view
of the fact that the embeddingE→C([, ]) is compact, we know that the critical points converge strongly to zero, and the proof is complete.
Remark . Applications similar to Corollaries . and . can also be made to Theo-rem .. Now we give an example illustrating TheoTheo-rem .. Consider the problem
⎧ ⎪ ⎨ ⎪ ⎩
u()=λf(x,u), <x< , u() =u() = ,
u() = , u() =g(u()),
(.)
wheref(x,u) =|u|. Obviously,A¯ =B¯=. Leta= andb=, and choose
d(x) = – x
√S¯
x–
, x∈
,
and
e(x) = – x √S¯
x–
, x∈
,
.
By calculating, we have
|d|dx+
|e
|dx= ¯
S(
+
×). Thus,λ¯=S¯( +× )
andλ¯=S¯. Furthermore, the conditions (A) and (A)are satisfied. Letg(u) = –u. Then
(A)holds. Therefore, by Theorem ., we find that problem (.) has a sequence of weak solutions which strongly converges to zero inEfor allλ∈(λ¯,λ¯).
Competing interests
The author declares that they have no competing interests.
Received: 25 February 2014 Accepted: 25 July 2014 References
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doi:10.1186/s13661-014-0191-6