Infinitely many solutions for a class of fourth-order partially sublinear elliptic problem

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R E S E A R C H

Open Access

Inﬁnitely many solutions for a class of

fourth-order partially sublinear elliptic

problem

Hua Gu

*

_{and Tianqing An}

*_{Correspondence:}

[email protected]

College of Science, Hohai University, Nanjing, China

Abstract

In this paper, we study the existence of inﬁnitely many solutions for a class of

fourth-order partially sublinear elliptic problem with Navier boundary value condition by using an extension of Clark’s theorem.

Keywords: multiple solution; fourth-order elliptic problem; critical point; variational method

1 Introduction

Consider the following fourth-order boundary value problem:

_u₊_c_u₌_g₍_x_,_u₎ _in_Ω_,

u=u=  on∂Ω, ()

where_{denotes the biharmonic operator,}_Ω_⊂_RN ₍_N_{> ) is a bounded domain with}

smooth boundary,g∈C(Ω×R,R), andc<λis a parameter, whereλis the ﬁrst

eigen-value of –inH (Ω).

The fourth-order elliptic equations can describe the static form change of beam or the motion of rigid body, so they are widely applied in physics and engineering. In the s, Lazer and Mckenna (see [, ]) investigated the problem () asg(x,u) =d[(u+ )+_{– ]. In [],}

they pointed out that this type of nonlinearity furnishes a model to study traveling waves in suspension bridges. In [], they got k–  solutions whenN=  andd>λk(λk–c) (λkis

the sequence of the eigenvalues of –inH

(Ω)) by the global bifurcation method. In [],

Micheletti and Pistoia used a variational linking theorem to investigate the existence of two solutions for a more general nonlinearityg(·,u). In , Micheletti and Saccon (see []) obtained two results about the existence of two nontrivial solutions and four non-trivial solutions by a similar variational approach, depending on the position of a suitable parameter with respect to the eigenvalues of the linear part.

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f ∈C(Ω×R,R)). In [], Hu and Wang obtained the existence of nontrivial solutions to problem () under suitable assumptions ofg(x,u) by a variant version of the mountain pass theorem. In [], we have studied the existence of multiple solutions for problem () by us-ing the variant fountain theorem without the conditionc<λ(λis the ﬁrst eigenvalue of

–inH

(Ω)). In [], Wang and Shen, under an improved Hardy-Rellich inequality,

stud-ied the existence of multiple and sign-changing solutions for a class of biharmonic equa-tions in an unbounded domain by the minimax method and linking theorem. In , Liu and Chen (see []) investigated the existence of ground-state solution and nonexistence of nontrivial solution for a similar biharmonic equation in [] by using variational meth-ods, also they explored the phenomenon of concentration of solutions. For other related results, see [–] and the references therein.

In critical point theory, Clark’s theorem [] asserts the existence of a sequence of neg-ative critical values tending to  for even coercive functionals. It is constantly and effec-tively applied to sublinear differential equations with symmetry. In , Wang (see []) explored a variant of the Clark theorem given by Heinz in [] to investigate a variety of nonlinear boundary value problems. Then, in , Liu and Wang improved Clark’s the-orem and gave an extension of Clark’s thethe-orem in []. Their new results gave a more detailed structure of the set of critical points near the origin and are powerful in applica-tions. In this paper, inspired by [], we will use the new result of Liu and Wang in [] to investigate the existence of infinitely many solutions for partially sublinear problems ().

Our main result is the following theorem.

Theorem  Assume that there exists a constantδ> such that g∈C(Ω×[–δ,δ],R)is odd and bounded.The primitive G(x,u) :=_ug(x,s)ds of the nonlinearity g satisﬁes

lim inf

|u|→ G(x,u)

|u| =∞ uniformly for x∈Ω. ()

Then problem()possesses inﬁnitely many solutions uksuch thatuk∞→as k→ ∞,

where · ∞is the usual norm of L∞≡L∞(Ω).

Remark The advantage of this theorem is that it not only obtained the existence of inﬁnite solutions, but it also pointed out their positions.

2 Preliminaries

LetE=H₍_Ω₎_∩_H

(Ω) be the Hilbert space equipped with the inner product

(u,v)E=

Ω

uv–c∇u,∇vdx

and the norm

u= (u,u)

  E.

A weak solution of problem () is au∈Esuch that

Ω

uv–c∇u,∇vdx–

Ω

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for any v∈E. Here and in the sequel,·,· always denotes the standard inner product inRN_.

Let:E→Rbe the functional deﬁned by

(u) =  

Ω

|u|–c|∇u|dx–

Ω

G(x,u)dx

=  u

_–

Ω

G(x,u)dx. ()

It is well known that a critical point of the functionalinEcorresponds to a weak solution of problem ().

Direct computation shows that

(u)v= (u,v)E–

Ω

g(x,u)v dx ()

for allu,v∈E. It is well known that:E→Eis compact. Denote by · pthe usual norm ofLp≡Lp(Ω) as

up=

Ω |u|pdx



p

, foru∈Lp, ≤p< +∞.

Lemma  The normuis equivalent to the normuin E.

Proof This result can be found in [] (Lemma .), so we omit the proof here. Then by Lemma  and Sobolev embedding theorem, there exists aτp>  such that

up≤τpu, ∀u∈E, ()

for all ≤p≤_NN_–.

Letλi(i= , , . . .) be the eigenvalues of –inH(Ω). Then the eigenvalue problem

_u₊_c_u₌_μ_u _in_Ω_, u=u=  on∂Ω,

has inﬁnitely many eigenvaluesμi=λi(λi–c),i= , , . . . .

Deﬁne a selfadjoint linear operatorA :L→Lby

(Au,v)=

Ω

uv–c∇u,∇vdx ()

with domainD(A) =E. Here, (·,·)denotes the inner product inL. Then the sequence of

eigenvalues ofA is just{μi}(i= , , . . .). Denote the corresponding system of

eigenfunc-tions by{en}, it forms an orthogonal basis inL.

Denote

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Here,{·}denotes the cardinality of a set andncan be . Let

L–=span{e, . . . ,en–}, L=span{e_n–_+, . . . ,e_n}, L+=L–⊕L⊥=span{e_n_+, . . .}.

DecomposeL_as L=L–⊕L⊕L+.

ThenEalso possesses the orthogonal decomposition

E=E–⊕E⊕E+ ()

with

E–=L–, E=L and E+=E∩L+=span{en+, . . .}. ()

To prove our main result, Theorem , we give the improved Clark theorem in [].

Theorem (See [], Theorem .) Let X be a Banach space,∈C₍_X_,_R_)._Assume satisﬁes the(PS)condition,is even and bounded from below,and() = .If for any k∈N,

there exists a k-dimensional subspace Xk_{of X and}_ρ

k> such thatsupXk_∩_S

ρ_k< ,where

Sρ={u∈X| u=ρ},then at least one of the following conclusions holds.

(i) There exists a sequence of critical points{uk}satisfying(uk) < for allkand

uk →ask→ ∞.

(ii) There existsr> such that for any <a<rthere exists a critical pointusuch that

u=aand(u) = .

In order to apply this theorem to prove our main result, we set our working spaceE=

H₍_Ω₎_∩_H

(Ω) as the spaceXin Theorem . 3 Proof of Theorem 1

Now we prove our main result, Theorem .

Proof of Theorem Chooseg∈C(Ω×R,R) so thatgis odd inu∈R,

g(x,u) =

⎧ ⎪ ⎨ ⎪ ⎩

g(x,u), x∈Ω,|u| ≤δ/, odd, x∈Ω,δ/ <|u| ≤δ, , x∈Ω,|u|>δ.

()

Sincegis bounded, together with (), we see that there exists a constantcsuch that g(x,u)≤c for allx∈Ωandu∈R. ()

Here and in the sequel, we denoteci>  (i= , , . . .) for diﬀerent positive constants.

In order to obtain solutions of (), we study the system

_u₊_c_u₌_g₍_x_,_u₎ _in_Ω_,

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with the functional deﬁned by

(u) =  

Ω

|u|–c|∇u|dx–

Ω

G(x,u)dx= u

_–

Ω

G(x,u)dx, ()

for allu∈E. Here,G(x,u) :=_ug(x,s)dsis the primitive of the nonlinearityg. It is easy to ﬁnd that∈C₍_E_,_R_),_{() =  and}_{is even. In view of the boundedness of}_G₍_x_,_u_),_is

bounded from below. At the same time

(u)v= (u,v)E–

Ω

g(x,u)v dx ()

for allu,v∈E. It is well known that:E→Eis compact.

Now the proof of conclusion of Theorem  is divided into three steps.

Step .satisﬁes the (PS) condition.

We assume the sequence{un} ⊂Esatisﬁes the requirement that{(un)}is bounded and

(un)→ asn→ ∞. Now we claim that the sequence{un}is bounded inEand possesses

a strong convergent subsequence.

First, we claim that{un}is bounded inE. By (), we setun=u–n+un+u+n∈E–⊕E⊕E+.

Considering(un)→ asn→ ∞, we have(un)<  asnis large enough. Then

(un)u+n≤ (un)u+n<u+n. ()

By (), (), (), (), and (), we have

u+_n> (un)u+n

=un,u+n

E–

Ω

g(x,un)u+ndx ≥u+_n,u+_n_E–

Ω

g(x,un)u+ndx ≥u+_n–cu+n

≥u+_n–cu+n. ()

From (), we learn that{u+

n}is bounded inE. Similarly, we see that{u–n}is bounded inE,

too.

Now we consider{u

n}. By the assumption that{(un)}is bounded, there exists a

con-stantK>  such that|(un)| ≤K. Thus, by (), we have

K≥ 

un

_–

Ω

G(x,un)dx ≥ 

u

–

n



+u_n+u+_n–

Ω

G(x,un)dx ≥ 

u

 n  – Ω

G(x,un) –G

x,u_ndx–

Ω

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Here,

G(x,un) –G

x,u_n= g(x,ξ)u+n+u–n,

≤ g(x,ξ)u+n+ g(x,ξ)u–n, ξis betweenunandun, ()

and

Gx,u_n=

un



g(x,s)ds= g(x,ξ)un, ξis between  andun. ()

By (), (), ()-(), together with the boundedness of{u+

n}and{u–n}, we have

K≥ 

u



n



–

Ω

g(x,ξ)u+n+ g(x,ξ)u–ndx–

Ω

g(x,ξ)undx ≥ 

u



n



–cu+n+u –

n

–cun

≥ 

u



n



–c–cun. ()

From (), we can see that{u

n}is bounded inE. Thus,{un}is bounded inE.

Second, we claim that{un}possesses a strong convergent subsequence inE.

Now, deﬁne an operatorT:E→Eas

(Tu,v)E=

Ω

g(x,u)v dx.

It is obvious thatT is a compact operator. Then () changes into

(u)v= (u,v)E– (Tu,v)E= (u–Tu,v)E.

So,(u) =u–Tu. Setu=un, we have

(un) =un–Tun. ()

Considering the boundedness of{un}and the compactness of the operatorT,{Tun}has

a strong convergent subsequence. Combining (un)→ asn→ ∞, we see that{un}

possesses a strong convergent subsequence inE.

Step .Construction of spaceXk_∩_S

ρk and proof ofsupXk_∩_S

ρk< .

In view of (), we learn that for anyM>  there exists a constantβ>  such that

G(x,u)

|u| >M, |u|<β, uniformly forx∈Ω. ()

For anyk∈N, we choosek-dimensional subspace ofEasXk₍_e.g._{the subspace spanned}

by eigenfunctions {en}). By the equivalence of any two norms on ﬁnite-dimensional

space Xk_{, we can choose a suﬃcient small constant} _ρ

k>  such that, for any u∈Xk, u∞<β, asu=ρkuniformly forx∈Ω. That is for anyu∈Xk,

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Now we can construct the spaceXk_∩_S

ρ_k whereSρ_k ={u∈X| u=ρk}. For anyu∈

Xk_∩_S

ρk, by the equivalence of any two norms on ﬁnite-dimensional spaceX

k_{, together}

with () and (), we have

(u) =  u

_–

Ω

G(x,u)

|u| |u| _dx

<  u

_–_M_u 

≤ 

u

_–_Mc

u. ()

For anyMlarge enough, we can choose a suﬃciently smallρksuch that(u) < . Thus,

sup

Xk_∩_S

ρ_k

< . ()

Now we appeal to Theorem  to obtain inﬁnitely many solutionsukfor () such that uk → ask→ ∞.

Step .Solutions of () are solutions of ().

By virtue of the boundedness ofg(x,u), we have the solutions of ()uk∈W,q(Ω) for

anyq∈[, +∞). Then we have

uk∈W,q(Ω)∩H(Ω)→C,

α

(Ω)⊂C(Ω) () (the embedding theorem is given in [], Theorem .).

Sinceuk →, thenuk∞→. Whenkis large enough, there exists a constantδ> 

such that|uk(x)|<δ/. Theng(x,uk(x)) =g(x,uk(x)). Therefore,ukare the solutions of ()

withksuﬃciently large anduk∞→ ask→ ∞.

Competing interests

The authors declare that they have no ﬁnancial or non-ﬁnancial competing interests.

Authors’ contributions

HG wrote the first draft and TA corrected and improved the final version. All authors read and approved the final draft.

Acknowledgements

The authors would like to thank the referee for his/her careful reading of the manuscript and the useful suggestions and comments. Supported by the Fundamental Research Funds for the Central Universities of China (No. 2016B08614).

Received: 17 July 2016 Accepted: 7 December 2016

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