R E S E A R C H
Open Access
Nonlinear biharmonic boundary value
problem
Tacksun Jung
1and Q-Heung Choi
2* *Correspondence:[email protected] 2Department of Mathematics Education, Inha University, Incheon, 402-751, Korea
Full list of author information is available at the end of the article
Abstract
We consider the nonlinear biharmonic equation with variable coefficient and polynomial growth nonlinearity and Dirichlet boundary condition. We get two theorems. One theorem says that there exists at least one bounded solution under some condition. The other one says that there exist at least two solutions, one of which is a bounded solution and the other of which has a large norm under some condition. We obtain this result by the variational method, generalized mountain pass geometry and the critical point theory of the associated functional.
MSC: 35J20; 35J25; 35Q72
Keywords: biharmonic boundary value problem; polynomial growth; variational method; generalized mountain pass geometry; critical point theory; (PS) condition
1 Introduction
Letbe a bounded domain inRnwith smooth boundary∂andL() be a square in-tegrable function space defined on. Letbe the elliptic operator and be the bi-harmonic operator. Letc∈R. In this paper we study the following nonlinear biharmonic equation with Dirichlet boundary condition:
u+cu=a(x)g(u) in,
u= , u= on∂,
(.)
wherea:¯ →Ris a continuous function which changes sign in. We assume thatgsatisfies the following conditions:
(g) g∈C(R,R),
(g) there are constantsa,a≥such that
g(u)≤a+a|u|μ–,
where <μ<n–n ifn≥,
(g) there exists a constantr≥such that
<μG(ξ) =μ
ξ
g(t)dt≤ξg(ξ) for|ξ| ≥r,
(g) g(u) =o(|u|)asu→.
We note that (g) implies the existence of the positive constantsa,a,asuch that
μ
ξg(ξ) +a
≥G(ξ) +a≥a|ξ|μ forξ∈R. (.)
Remark . The real numberξ in the definition (g) is not automatically nonnegative. The reason is as follows.
Since <μG(ξ) <ξg(ξ) andμ> ,G(ξ) > andξg(ξ) > . Byξg(ξ) > , we have two cases: one case is thatξ> andg(ξ) > . The other case is thatξ< andg(ξ) < . Thusξ is not nonnegative.
Remark . We obtain the boundedness ofg(u)u–G(u) as follows.
By the condition (g), g(u)u≥ μG(u) for|u| ≥r. Sinceμ> , –μ > , andG(u) + a≥a|u|μin (.),
g(u)u–G(u)≥
μG(u) –G(u)
=μ
–
μ
G(u)
≥μ
–
μ
a|u|μ–a
.
Thus we obtain the boundedness ofg(u)u–G(u).
Remark . (i) Assumption (g) implies that (.) has a trivial solution. (ii) Ifn= , (g) can be dropped. Ifn= , it suffices that
g(u)≤aexpp(ξ),
wherep(ξ)ξ–→ as|ξ| → ∞.
(iii) Ifn≥ andg(ξ) =ξ++, whereξ+=max{ξ, }and> is a small number, then (g)-(g) are satisfied.
The eigenvalue problem
u+λu= in,
u= on∂
has infinitely many eigenvalues λk, k≥, and corresponding eigenfunctionsφk,k≥,
suitably normalized with respect to theL() inner product, where each eigenvalueλ
kis
repeated as often as its multiplicity. The eigenvalue problem
u+cu= u in,
u= , u= on∂
has also infinitely many eigenvaluesλk(λk–c),k≥ and corresponding eigenfunctions
Khanfir and Lassoued [] showed the existence of at least one solution for the nonlinear elliptic boundary problem whengis locally Hölder continuous onR+. Choi and Jung [] showed that the problem
u+cu=bu++s in,
u= , u= on∂
(.)
has at least two nontrivial solutions when (c<λ, λ(λ–c) <b<λ(λ–c) ands< ) or (λ<c<λ,b<λ(λ–c) ands> ). The authors obtained these results by using the variational reduction method. The authors [] also proved that whenc<λ,λ(λ–c) < b<λ(λ–c) ands< , (.) has at least three nontrivial solutions by using degree theory. Tarantello [] also studied
u+cu=b(u+ )+– , u= , u= on∂.
(.)
She showed that ifc<λandb≥λ(λ–c), then (.) has a negative solution. She obtained this result by degree theory. Micheletti and Pistoia [] also proved that ifc<λ andb≥ λ(λ–c) then (.) has at least four solutions by the variational linking theorem and Leray-Schauder degree theory.
In this paper we are trying to find the weak solutions of (.), that is,
u+cu–a(x)g(u)v dx= for anyv∈H,
where the spaceHis introduced in Section . Let us set
+=x∈|a(x) > , –=x∈|a(x) < and let
a+=a·χ+, a–= –a·χ–.
Sincea(x) changes sign, the open subsets+and–are nonempty. Now we can write a=a+–a–. Our main results are as follows.
Theorem . Assume thatλk<c<λk+,g satisfies(g)-(g)and g(u)u–μG(u)is bounded. Then(.)has at least one bounded nontrivial solution.
Theorem . Assume thatλk<c<λk+,g satisfies(g)-(g),g(u)u–μG(u)is not bounded and there exists a small> such that–a–(x)dx<.Then(.)has at least two
solu-tions, (i)one of which is nontrivial and bounded,and(ii)the other of which has a large norm such that
max
x∈
u(x)>M for some M.
Fréchet differentiable and satisfies the (PS) condition. In Section , we prove Theorem .. In Section , we prove Theorem . by the variational method, the generalized mountain pass geometry and the critical point theory.
2 Palais-Smale condition
Any elementuinL() can be written as
u=hkφk with
hk<∞.
We define a subspaceHofL() as follows:
H=
u∈L() λk(λk–c)hk<∞
.
Then this is a Banach space with a norm
u=λk(λk–c)hk
.
Sinceλk→+∞andcis fixed, we have (i) u+cu∈Himpliesu∈H,
(ii) u ≥CuL(), for someC> , (iii) uL()= if and only ifu= ,
which are proved in []. Let
H+=
u∈H|hk= ifλk(λk–c) < ,
H–=
u∈H|hk= ifλk(λk–c) > .
ThenH=H–⊕H+. LetP+be the orthogonal projection onH+andP–be the orthogonal projection onH–.
We are looking for the weak solutions of (.). The weak solutions of (.) coincide with the critical points of the associated functional
I(u)∈C(H,R),
I(u) =
|u|
–c |∇u|
–
a(x)G(u)
dx (.)
=
P+u–P–u
–
a(x)G(u)dx.
By (g) and (g),Iis well defined. By Proposition .,I∈C(H,R) andIis Fréchet differ-entiable inH.
Proposition . Assume thatλk<c<λk+,k≥,and that g satisfies(g)-(g).Then I(u)
is continuous and Fréchet differentiable in H with Fréchet derivative
I(u)h=
If we set
K(u) =
a(x)G(u)dx,
then K(u)is continuous with respect to weak convergence,K(u)is compact,and
K(u)h=
a(x)g(u)h dx for all h∈H.
This implies that I∈C(H,R)and K(u)is weakly continuous.
The proof of Proposition . is the same as that of Appendix B in [].
Proposition .(Palais-Smale condition) Assume thatλk<c<λk+,k≥,and g satisfies (g)-(g).We also assume that g(u)u–μG(u)is bounded or that there exists an> such that–a–(x)dx<.Then I(u)satisfies the Palais-Smale condition.
Proof Suppose that (um) is a sequence withI(um)≤MandI(um)→ asm→ ∞. Then
by (g), (g), and the Hölder inequality and the Sobolev Embedding Theorem, for largem andμ> withu=um, we have
M+
u ≥I(u) – I
(u)u=
a(x)g(u)u–a(x)G(u)
dx
=
a+(x)
g(u)u–G(u)
dx–
a–(x)
g(u)u–G(u)
dx
≥
–
μ
μ
a+(x)·G(u)dx–max
g(u)u–G(u)
–a
–(x)dx
≥
–
μ
μ
a+(x)·a|u|μ–a
dx
–max
g(u)u–G(u)
–
a–(x)dx.
Sinceg(u)u–G(u) is bounded or there exists an> such that–a–(x) <; we have
+u ≥M
|u|μdx≥M
|u|dx
·μ
. (.)
Moreover since
I(um)ϕ≤ ϕ (.)
for largemand allϕ∈H, choosingϕ=P+um∈H+gives
P+um =
um+cum
·P+umdx
=
a(x)g(um)P+umdx
≤
≤ a∞
a|um|+a|um|μ
dx
≤C
|um|μdx+CumL()
≤C
|um|μdx+Cum.
Takingϕ= –P–umin (.) yields
P–um =
um+cum
·(–P–um)dx
=
a(x)g(um)·(–P–um)dx
≤
a(x)g(um)|um|dx
≤C
|um|μdx+Cum.
Thus, by (.), we have
um =P+um+P–um≤M
|um|μdx+Mum
≤M
+um
+Mum ≤M
+um
,
from which the boundedness of (um) follows. Thus (um) converges weakly in H. Since
P±I(um) =±P±um+P±P˜(um) withP˜compact and the weak convergence ofP±umimply
the strong convergence ofP±umand hence (PS) condition holds.
3 Proof of Theorem 1.1
We shall show thatI(u) satisfies the generalized mountain pass geometrical assumptions. We recall the generalized mountain pass geometry.
Let H=V⊕X, whereV ={} and is finite dimensional. Suppose thatI∈C(H,R), satisfies the Palais-Smale condition, and
(i) there are constantsρ,α> and a bounded neighborhoodBρof such that
I|∂Bρ∩X≥α, and
(ii) there is ane∈∂B∩XandR>ρsuch that ifQ= (B¯R∩V)⊕ {re| <r<R}, then
I|∂Q≤.
ThenIpossesses a critical valueb≥α. Moreoverbcan be characterized as
b=inf
γ∈maxu∈QI
γ(u),
where
=γ ∈C(Q¯,H)|γ =idon∂Q .
LetHk=span{φ, . . . ,φk}. ThenHkis a subspace ofHsuch that
H=
k∈N
Let
Br=
u∈H| u ≤r .
We have the following generalized mountain pass geometrical assumptions.
Lemma . Assume thatλk<c<λk+and g satisfies(g)-(g).Then
(i) there are constantsρ> ,α> and a bounded neighborhoodBρofsuch that
I|∂Bρ∩H⊥
k ≥α,and
(ii) there is ane∈∂B∩Hk⊥andR>ρsuch that ifQ= (B¯R∩Hk)⊕ {re| <r<R},then
I|∂Q≤,and
(iii) there existsu∈Hsuch thatu>ρandI(u)≤.
Proof (i) Letu∈Hk⊥. Then
u+cuu dx≥λk+(λk+–c)uL()> .
Thus by (g), (g), and the Hölder inequality, we have
I(u) = P+u
– P–u
–
a(x)G(u)dx
≥
P+u –a
∞
C|u|μdx
≥
P+u –a
∞Cu
μ
forC,C> . Sinceμ> , there existρ> andα> such that ifu∈∂Bρ, thenI(u)≥α.
(ii) LetBrbe a ball with radiusr> ,ebe a fixed element in∂B∩Hk⊥andu∈(B¯r∩Hk)⊕ {re| <r}. Thenu=v+w,v∈Br∩Hk,w=re. We note that
sincev∈Hk,
v+cvv dx≤λk(λk–c)vL()< .
Thus we have
I(u) = r
– P–v
–
a(x)G(v+re)dx
≤
r +
λk(λk–c)
vL()–
+
a(x)a|v+re|μ–a
dx.
Sinceμ> , there existsR> such that ifu∈Q= (B¯R∩Hk)⊕ {re| <r<R}, thenI(u) < .
(iii) If we chooseψ∈Hsuch thatψ= ,ψ≥ inandsupp(ψ)⊂+, then we have
I(tψ)≤
P+(tψ)
–
P–(tψ)
–
+
a(x)atμψμ–a
dx
≤
tψ –
+a(x)
atμψμ–a
dx
= t
–
+
a(x)atμψμ–a
for allt> . Sinceμ> , fortgreat enough,u=tψis such thatu>ρandI(u)≤.
Proof of Theorem. By Proposition . and Proposition .,I(u)∈C(H,R) and satisfies the Palais-Smale condition. By Lemma ., there are constantsρ> ,α> and a bounded neighborhoodBρ of such thatI|∂Bρ∩Hm⊥≥α, and there is ane∈∂B∩H
⊥
k andR>ρ
such that ifQ= (B¯R∩Hk)⊕ {re| <r<R}, thenI|∂Q≤, and there existsu∈H such thatu>ρandI(u)≤. By the generalized mountain pass theorem,I(u) has a critical valueb≥α. Moreover,bcan be characterized as
b=inf
γ∈maxu∈QI
γ(u),
where
=γ ∈C(Q¯,H)|γ =idon∂Q .
We denote byu˜a critical point ofIsuch thatI(u˜) =b. We claim that there exists a constant C> such that
a+(x)μu˜
L()≤C
+L
–a
–(x)dx
μ
,
whereL=max|g(u˜)u˜–G(u˜)|.
In fact, we have
b≤max
≤t≤I(tu)
and
I(tu) =t
P+u
– P–u
–
a(x)G(tu)dx
≤tu–
a+(x)G(tu)dx+
a–(x)G(tu)dx
≤tu–atμ
a+(x)uμdx+a
a+(x)dx+atμ
a–(x)uμdx
=Ct–Ctμ+C+Ctμ.
Since ≤t≤,bis bounded:b<C˜. We can write
b =I(u˜) – I
(u˜)u˜
=
a(x)
g(u˜)u˜–G(u˜)
dx
=
a+(x)
g(u˜)u˜–G(u˜)
dx–
a–(x)
g(u˜)u˜–G(u˜)
≥
–
μ
a+(x)g(u˜)u dx˜ –max
g(u˜)u˜–G(u˜)
–
a–(x)dx
≥ – μ μ
a+(x)a|˜u|μ–a
dx–L
–a
–(x)dx,
whereL=max|g(u˜)u˜–G(u˜)|. Thus we have
C
+L
–
a–(x)dx
≥
a+(x)|˜u|μdx≥C
a+(x)μ|˜u|dx
μ
(.)
for some constantsC,C> , from which we conclude thatu˜ is bounded and the proof
of Theorem . is complete.
4 Proof of Theorem 1.2
Assume that
g(u)u–G(u) is not bounded and there exists an> such that
–a–(x,
t)dx<. By Proposition . and Proposition .,I∈C(H,R) and satisfies the Palais-Smale condition. By Lemma . and the generalized mountain pass theorem,I(u) has a critical valuebwith critical pointu˜such thatI(u˜) =b. If–a–(x)dxis sufficiently small, by (.),
we have
a+(x)|˜u|μdx≤C
forC> , from which we can conclude thatu˜is bounded and the proof of Theorem .(i) is complete.
Next we shall prove Theorem .(ii). We may assume thatRn<Rn+for alln∈N. Let us setDn=BRn∩Hn,∂Dn=∂BRn∩Hn.
Lemma . Assume that g satisfies(g)-(g),g(u)u–G(u)is not bounded and there exists an> such that–a–(x,t) <.Then there exists an Rn> such that
I(u)≤ for u∈Hn\BRn. (.)
Proof Let us chooseψ∈Hsuch thatψ= ,ψ≥ inandsupp(ψ)⊂+. Then, by (g), (g), and the Hölder inequality, we have
I(tψ) = P+tψ
– P–tψ
–
a(x)G(tψ)dx
= P+tψ
– P–tψ
–
+a
+(x)G(tψ)dx+
–a
–(x)G(tψ)dx
≤
P+tψ –
P–tψ –
+
a+(x)a
tμψμ–a
dx
+G(tψ)∞
–a
–(x)dx
≤
t –
+a
+(x)a
tμψμ–a
dx+
for small> . Sinceμ> , there existtngreat enough for eachnand anRn> such that
Let us set
n=
γ∈C[, ],H|γ() = andγ() =un
and
bn= inf γ∈n
max
[,]I
γ(u), n∈N.
Proof of Theorem.(ii) We assume thatg(u)u–μG(u) is not bounded and there exists an > such that–a–(x)dx<. By Proposition . and Proposition .,I∈C(H,R) and
satisfies the Palais-Smale condition. By Lemma ., there exists anRn> such thatI(un)≤
forun∈Hn\BRn. We note thatI() = . By Lemma . and the generalized mountain pass theorem, fornlarge enoughbn> is a critical value ofIandlimn→∞bn= +∞. Letu˜nbe
a critical point ofIsuch thatI(u˜n) =bn. Then for each real numberM,max|˜un(x)| ≥M.
In fact, by contradiction,u+cu=a(x)g(u) andmax|˜un(x)| ≤Kimply that
I(u˜n)≤ max
|˜un|≤K
g(u˜n)u˜n–G(u˜n)
a(x)dx,
which means thatbnis bounded. This is absurd because of the fact thatlimn→∞bn= +∞.
Thus we complete the proof.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
TJ and Q-HC participated in the sequence alignment and drafted the manuscripted. Both authors read and approved the final manuscript.
Author details
1Department of Mathematics, Kunsan National University, Kunsan, 573-701, Korea.2Department of Mathematics
Education, Inha University, Incheon, 402-751, Korea.
Acknowledgements
This work (Choi) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (KRF-2013010343).
Received: 28 June 2013 Accepted: 8 January 2014 Published:04 Feb 2014
References
1. Khanfir, S, Lassoued, L: On the existence of positive solutions of a semilinear elliptic equation with change of sign. Nonlinear Anal., Theory Methods Appl.22(11), 1309-1314 (1994)
2. Choi, QH, Jung, T: Multiplicity results on nonlinear biharmonic operator. Rocky Mt. J. Math.29(1), 141-164 (1999) 3. Jung, TS, Choi, QH: Multiplicity results on a nonlinear biharmonic equation. Nonlinear Anal., Theory Methods Appl.
30(8), 5083-5092 (1997)
4. Tarantello, G: A note on a semilinear elliptic problem. Differ. Integral Equ.5(3), 561-565 (1992)
5. Micheletti, AM, Pistoia, A: Multiplicity results for a fourth-order semilinear elliptic problem. Nonlinear Anal. TMA31(7), 895-908 (1998)
6. Choi, QH, Jung, T: Multiplicity of solutions and source terms in a fourth order nonlinear elliptic equation. Acta Math. Sci.19(4), 361-374 (1999)
7. Rabinowitz, PH: Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Regional Conf. Ser. Math., vol. 65. Am. Math. Soc., Providence (1986)
10.1186/1687-2770-2014-30
Cite this article as:Jung and Choi:Nonlinear biharmonic boundary value problem.Boundary Value Problems
