Weak solutions for the singular potential wave system

(1)

R E S E A R C H

Open Access

Weak solutions for the singular potential

wave system

Tacksun Jung

1

_{and Q-Heung Choi}

2*

*_{Correspondence:}

[email protected] 2_{Department of Mathematics} Education, Inha University, Incheon, 402-751, Korea

Full list of author information is available at the end of the article

Abstract

We investigate the existence of weak solutions for a class of the system of wave equations with singular potential nonlinearity. We obtain a theorem which shows the existence of nontrivial weak solution for a class of the wave system with singular potential nonlinearity and the Dirichlet boundary condition. We obtain this result by using the variational method and critical point theory for indeﬁnite functional. MSC: 35L51; 35L70

Keywords: class of wave system; singular potential nonlinearity; Dirichlet boundary condition; variational method; critical point theorem for indeﬁnite functional; (PS)c

condition

1 Introduction

LetDbe an open subset inRn_{with compact complement}_C₌_Rn_\_D_,_n_≥_{. In this paper} we investigate the multiplicity of the solutions for a class of the system of nonlinear wave equations with the Dirichlet boundary condition and periodic condition:

(u)tt– (u)xx=

∂ ∂u

Gx,t,u(x,t), . . . ,un(x,t)

in

–π ,

π



×R,

(u)tt– (u)xx=

∂ ∂u

Gx,t,u(x,t), . . . ,un(x,t)

in

–π ,

π



×R,

.. .

(un)tt– (un)xx= ∂

∂un

Gx,t,u(x,t), . . . ,un(x,t)

in

–π ,

π



×R,

ui

±π ,t

= ,

ui(x,t) =ui(–x,t) =ui(x, –t) =ui(x,t+π), i= , . . . ,n,

(.)

whereG∈C_([–π ,

π ]×R

_×_D_,_R_{). Let}_U_{= (}_u

, . . . ,un). We assume thatGsatisﬁes the following conditions:

(G) There existsR> such that

supG(x,t,U)+grad_UG(x,t,U)_Rn(x,t,U)∈ –

π

,

π



×R×Rn\BR

< +∞.

(2)

(G) There is a neighborhoodZofCinRn_{such that}

G(x,t,U)≥ A

d₍_U_,_C₎ for(x,t,U)∈

–π ,

π



×R×Z,

whered(U,C)is the distance function fromUtoCandA> is a constant. The system (.) can be rewritten as

⎧ ⎪ ⎨ ⎪ ⎩

Utt–Uxx=gradUG(x,t,U(x,t)),

U(±π_,t) = (, . . . , ),

U(x,t) =U(x,t+π) =U(–x,t) =U(x, –t),

(.)

whereUtt–Uxx= ((u)tt– (u)xx, . . . , (un)tt– (un)xx).

Remark We have a simple example satisfying the above conditions (G)-(G):

G(x,t,U) = cost U

R

, U= (u,u),u=x,u=x+t.

Our main result is the following.

Theorem . Assume that the nonlinear term G satisﬁes conditions(G)-(G).Then sys-tem(.)has at least one nontrivial weak solution.

For the proof of Theorem ., we approach the variational method and use the critical point theory for indefinite functional. In Section , we introduce a Banach space and the associated functionalIof (.), and recall the critical point theory for indefinite functional. In Section , we prove thatIsatisfies the geometric assumptions of the critical point the-orem for indefinite functional and prove Thethe-orem ..

2 Variational approach The eigenvalue problem

vtt–vxx=λv in

–π ,

π



×R,

v

±π ,t

= , (.)

v(x,t) =v(–x,t) =v(x, –t) =v(x,t+π) has inﬁnitely many eigenvalues

λmn= (n+ )– m (m,n= , , , . . .)

and corresponding normalized eigenfunctionsφmn(x,t),m,n> , given by

φn= √



π cos(n+ )x forn≥, φmn=



(3)

Letbe the square [–π_,π_]×[–π_,π_], andEbe the Hilbert space deﬁned by

E=

v∈L()vis even inxandt,

∂ v= 

.

The set of functions{φmn}is an orthonormal basis inE. Let us denote an elementv, inE, as

v=hmnφmn,

and we deﬁne a subspaceEofEas

E=

v∈E |λmn|hmn<∞

.

This is a complete normed space with a norm

v=|λmn|hmn

 

.

Since{λmn|m,n= , , , . . .}is unbounded from above and from below and has no ﬁnite accumulation point, it is convenient for the following to rearrange the eigenvaluesλmnby increasing magnitude: from now on we denote by (ρ–

i)i≥the sequence of negative

eigen-values of (.), by (ρ+

i)i≥the sequence of positive ones, so that

· · · ≤ρ_i–≤ · · · ≤ρ_–≤ρ_–≤ρ_+≤ρ_+≤ · · · ≤ρ_i+≤ · · ·.

We will denote by the sequence (ρi) all the sequences (ρ+_i) and (ρ–

i). Let (e–i,e+i,i≥) be an orthonormal system of the eigenfunctions associated with the eigenvalues{ρ_i–,ρ_i+,i≥}. We will denote by the sequence (ei) the sequences (e+_i), (e–_i). LetE+be the span of closure of eigenfunctions associated with positive eigenvalues andE– be the span of closure of eigenfunctions associated with negative eigenvalues. LetH be thenCartesian product space ofE,i.e.,

H=E×E× · · · ×E.

LetH+_and_H–_{be the subspaces on which the functional}

U →Q(U) =

–|Ut|+|Ux|

dx dt, U= (u, . . . ,un)

is positive deﬁnite and negative deﬁnite, respectively. Then

H=H+⊕H–.

LetP+_{be the projection from}_H_onto_H+_and_P–_{be the projection from}_H_onto_H–_{. The}

norm inHis given by

U=P+U+P–U, U= (u, . . . ,un),

whereP+_U₌n

i=P+ui,P–U=

n

(4)

Let (Hn)nbe a sequence of closed ﬁnite dimensional subspace ofHwith the following assumptions:Hn=Hn–⊕Hn+, whereHn+⊂H+,Hn–⊂H–for alln(Hn+andHn–are subspaces ofH),dimHn< +∞,Hn⊂Hn+,

n∈NHnis dense inH.

In this paper we are trying to ﬁnd the weak solutionsU∈C₍_,_D₎_∩_H_{of system (.),}

that is,U= (u, . . . ,un)∈C(,D)∩Hsuch that

–(u)t·(φ)t– (u)t·(φ)t–· · ·– (un)t·(φn)t + (u)x·(φ)x+· · ·+ (un)x·(φn)x

dx dt

–

∂ ∂u

Gx,t,U(x,t)·φ–

∂ ∂u

Gx,t,U(x,t)·φ–· · ·

–

∂ ∂un

Gx,t,U(x,t)·φn= 

for allφ= (φ, . . . ,φn)∈C(,D)∩H,i.e.,

[–Ut·φt+Ux·φx]dx dt–

grad_UGx,t,U(x,t)·φ=  for allφ∈C(,D)∩H.

Let us introduce an open set of the Hilbert spaceHas follows:

X=U∈H|U(x,t)∈D⊂Rn, (x,t)∈. Let us consider the functional onX

I(U) =  

–|Ut|+|Ux|

dx dt–

G(x,t,U)dx dt

=Q(U) –

G(x,t,U)dx dt

=  P

+_U

– P

–_U

–

G(x,t,U)dx dt, (.)

whereQ(U) = _[–|Ut|+|Ux|]dx dt andU=

n

i=ui. The Euler equation for (.) is (.). By the following Lemma .,I∈C(X,R), and so the weak solutions of system (.) coincide with the critical points of the associated functionalI(U).

Lemma . Assume that G satisﬁes conditions(G)-(G).Then I(U)is continuous and Fréchet diﬀerentiable in X with Fréchet derivative

DI(U)V=

–Ut·Vt+Ux·Vx–gradUG

x,t,U(x,t)·V(x,t)dx dt ∀V∈X.

Moreover,DI∈C.That is,I∈C.

Proof First we prove thatI(U) is continuous. ForU,V∈X,

I(U+V) –I(U)

= 

(U+V)tt– (U+V)xx

(5)

–

G(x,t,U+V)dx dt

–  

(Utt–Uxx)·U dx dt+

G(x,t,U)dx dt

= 

(Utt–Uxx)·V+ (Vtt–Vxx)·U+ (Vtt–Vxx)·V

dx dt

–

G(x,t,U+V) –G(x,t,U)dx dt.

We have

G(x,t,U+V) –G(x,t,U)dx dt

≤

grad_UGx,t,U(x,t)·V+OVRndx dt=OV_Rn. (.)

Thus we have

I(U+V) –I(U)=OVRn.

Next we shall prove thatI(U) is Fréchet diﬀerentiable inX. ForU,V∈X,

I(U+V) –I(U) –∇I(U)V

= 

(U+V)tt– (U+V)xx·(U+V)dx dt–

G(x,t,U+V)dx dt

–  

(Utt–Uxx)·U dx dt+

G(x,t,U)dx dt

–

Utt–Uxx–gradUG

x,t,U(x,t)·V dx dt

= 

(Vtt–Vxx)·U+ (Vtt–Vxx)·V

dx

–

G(x,t,U+V) –G(x,t,U)dx dt+

grad_UGx,t,U(x,t)·V dx dt.

Thus by (.), we have

I(U+V) –I(U) –DI(U)V=OVRn. (.)

Similarly, it is easily checked thatI∈C_.

Let

X+=X∩H+, X–=X∩H–.

Lemma . Assume that G satisﬁes conditions(G)-(G).Let{Uk} ⊂X– and UkU

(6)

Proof To prove the conclusion, it suﬃces to prove that

Gx,t,Uk(x,t)dx dt→+∞.

SinceG(x,t,U(x,t)) is bounded from below, it suﬃces to prove that there is a subset˜ of

such that

˜

Gx,t,Uk(x,t)

dx dt→+∞.

U∈∂Xmeans that there exists (x∗,t∗)∈such thatU(x∗,t∗)∈∂D. Let us set

δ

x∗,t∗=

(x,t)∈x–x∗_Rn+t–t∗ 

Rn<δ

.

By (G) and (G), there exists a constantBsuch that

G(x,t,U)≥ A

d₍_U_,_C₎–B.

Thus we have

δ(x∗,t∗)

Gx,t,U(x,t)dx dt≥

δ(x∗,t∗)

A

U(x,t) –U(x∗,t∗)_Rn

–B

dx dt

for allδ> . By Schwarz’s inequality, we have

U(x,t) –Ux∗,t∗_Rn≤

x–x∗

 

Rn+t–t∗  

Rn

UxRn+UtRn





≤δ

UxRn+UtRn





.

Thus we have

δ(x∗,t∗)

Gx,t,U(x,t)dx dt≥

δ(x∗,t∗)

A

δ((Ux_Rn+Ut_Rn))

–B

dx dt→ ∞.

Hence

δ(x∗,t∗)

Gx,t,U(x,t)dx dt=∞.

Since the embeddingX→C(,Rn_{) is compact, we have}

maxU(x,t) –Uk(x,t)_Rn|(x,t)∈

→ ask→ ∞.

Thus by Fatou’s lemma, we have

lim inf

Gδ(x∗,t∗)

Gx,t,Uk(x,t)≥

Gδ(x∗,t∗)

lim infGx,t,Uk(x,t)

=

Gδ(x∗,t∗)

(7)

Thus

lim inf

Gδ(x∗,t∗)

Gx,t,Uk(x,t)

= +∞.

Thus

I(Uk) =

 

–(Uk)t+(Uk)x–Gx,t,Uk(x)dx dt

=  P

+_U

k



– P

–_U

k



–

Gx,t,Uk(x)

dx dt

= – P

–_U

k



–

Gx,t,Uk(x)dx dt→–∞,

so we prove the lemma.

We recall the critical point theorem for the indeﬁnite functional (cf.[]). Let

Br=

u∈X| u ≤r,

Sr=

u∈X| u=r.

Theorem .(Critical point theorem for the indeﬁnite functional) Let X be a real Hilbert space with X=X⊕Xand X=X⊥.Suppose that I∈C(X,R)satisﬁes(PS),and

(I) I(u) =_(Lu,u) +bu,whereLu=LPu+LPuandLi:Xi→Xiis bounded and

self-adjoint,i= , , (I) bis compact,and

(I) there exists a subspaceX˜ ⊂Xand setsS⊂X,Q⊂ ˜Xand constantsα>ωsuch that

(i) S⊂XandI|S≥α, (ii) Qis bounded andI|∂Q≤ω, (iii) Sand∂Qlink.

Then I possesses a critical value c≥α. 3 Proof of Theorem 1.1

We shall show that the functionalI(U) satisﬁes the geometric assumptions of the critical point theorem for indeﬁnite functional.

Lemma .(Palais-Smale condition) Assume that G satisﬁes conditions(G)and(G).

Then I(u)satisﬁes the(PS)condition in X.

Proof We shall prove the lemma by contradiction. We suppose that there exists a sequence {Uk} ⊂XsatisfyingI(Uk)→γ and

DI(Uk) = (Uk)tt– (Uk)xx–grad_UGx,t,Uk(x,t)→θ inX, (.) or equivalently

Uk– (Dtt–Dxx)–

(8)

whereθ= (, . . . , ) and (Dtt–Dxx)–is a compact operator. We claim that the sequence {Uk}, up to a subsequence, converges. It suﬃces to prove that the sequence{Uk}is bounded inX. By contradiction, we suppose thatUkRn→ ∞. Then, for largek, we have

UkRn≥R_. (.)

It follows from (.) that

G(x,t,Uk)dx dt≤ ||supG(x,t,Uk)|(x,t,Uk)∈×Rn\BR

. (.)

Let us setWk=U_Uk_k. ThenWk= , and hence the subsequence{Wk}, up to a subse-quence, converges weakly toWwithW= . By (.), we have

←DI(Uk)Uk UkH

=

(Wk)tt– (Wk)xx·Wkdx dt–

G(x,t,Uk) Uk

= P+Wk



–P–Wk



–

G(x,t,Uk) Uk

. (.)

Lettingk→ ∞in (.), by (.), we have

 = lim

k→∞P +_W

k



– lim

k→∞P –_W

k



=

(Wtt–Wxx)·W dx dt

=P+W–P–W. (.)

Thus we have

lim

k→∞

P+Wk



=P+W, lim

k→∞

P–Wk



=P–W.

Thus

lim

k→∞Wk=W

and by (.),Wis the weak solution of the equation

Wtt–Wxx=  inX.

We claim thatN(A)∩X={(, . . . , )}, whereAU=Utt–UxxandN(A) is the kernel ofA. In fact, letW∈N(A)∩X,W= (w, . . . ,wn). Then

wi=

m,n

(wi)mncosmtcos(n+ )x, i= , . . . ,n.

We note that

(9)

ThusN(A)∩X={(, . . . , )}. ThusW= (, . . . , ), which is absurd to the fact thatW= . Thus{Uk}is bounded. Thus the subsequence, up to a subsequence,Ukconverges weakly toUinX. By Lemma .,U∈Xand thatgrad_UG(·,Uk)is bounded. Since (Dtt–Dxx)– is compact and (.) holds,{Uk}converges strongly toU. Thus we prove the lemma.

Let

Q=B¯r∩X–

⊕re|e∈B∩X+,  <r<R

.

Lemma . Assume that G satisﬁes conditions(G)and(G).Then there exist sets Sρ⊂ X+_{with radius}_ρ_{> ,}_Q_⊂_{X and constants}_α_{> }_{such that}

(i) Sρ⊂X+andI|Sρ≥α,

(ii) Qis bounded andI|∂Q≤, (iii) Sρand∂Qlink.

Proof (i) Let us chooseU∈X+_⊂_X_{. Then}_U₍_x_,_t₎_∈_D_{. By (G),}_G₍_x_,_t_,_U_{) is bounded above}

and there exists a constantC> 

I(U) =  P

+_U_–

P

–_U_–

G(x,t,U)dx dt≥ 

P

+_U_–_C

forC> . Then there exist constants ρ>  andα >  such that if U∈Sρ∩X+, then I(U)≥α.

(ii) Let us choosee∈B∩X+. LetU∈ ¯Br∩X–⊕ {re| <r}. ThenU=V +W,V ∈ ¯

Br∩X–,W=re. We note that:

IfV∈ ¯Br∩X–, then

–VtRn+VxRn

dx dt= –P–U≤.

By (G),G(x,t,V+re) is bounded from below. Thus by Lemma ., there exists a constant

A>  such that ifU=V+re, then we have

I(U) =  r

_–

P

–_V_–

G(x,t,V+re)dx dt

≤  r

_–

P

–_V_–

A

d₍_V₊_re_,_C₎dx dt.

We can choose a constantR>rsuch that ifU=V+re∈Q= (B¯r∩X–)⊕ {re|e∈B∩ X+_{,  <}_r_<_R_}_{, then}_I₍_U_{) < . Thus we prove the lemma.}

Proof of Theorem. By Lemma .,I(U) is continuous and Fréchet diﬀerentiable inX

and, moreover,DI∈C. By Lemma ., if{Uk} ⊂X–andUkUweakly inXwithU∈∂X, thenI(Uk)→–∞. By Lemma .,I(u) satisﬁes the (PS) condition. By Lemma ., there exist setsSρ⊂X+ with radiusρ> ,Q⊂Xand constantα>  such thatI|Sρ≥α,Qis

bounded andI|∂Q≤, andSρand∂Qlink. By the critical point theorem,I(U) possesses

a critical valuec≥α. Thus (.) has at least one nontrivial weak solution. Thus we prove

Theorem .

Competing interests

(10)

Authors’ contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read, checked and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Kunsan National University, Kunsan, 573-701, Korea.}2_{Department of Mathematics}

Education, Inha University, Incheon, 402-751, Korea.

Acknowledgements

This work (Tacksun Jung) 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-2010-0023985).

Received: 13 February 2013 Accepted: 9 December 2013 Published:07 Jan 2014 References

1. Benci, V, Rabinowitz, PH: Critical point theorems for indeﬁnite functionals. Invent. Math.52, 241-273 (1979)

10.1186/1687-2770-2014-7