Some elliptic system and reduction method

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R E S E A R C H

Open Access

Some elliptic system and reduction method

Tacksun Jung

1*

_{and Q-Heung Choi}

2

*_{Correspondence:} [email protected] 1_{Department of Mathematics,} Kunsan National University, Kunsan, 573-701, Korea

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

Abstract

We get a theorem which shows the existence of at least three solutions for some elliptic system with Dirichlet boundary condition. We obtain this result by using the finite dimensional reduction method for the dimension of the system which reduces the infinite dimensional problem to the finite dimensional one. We also use critical point theory on the reduced finite dimensional subspace.

MSC: 35J50; 35J55

Keywords: elliptic system; reduction method; (PS) condition; critical point theory; variational method

1 Introduction

In this paper we are concerned with multiple solutions for a class of systems of elliptic equations with Dirichlet boundary condition

–u=Fu(x,u, . . . ,un) in,

–u=Fu(x,u, . . . ,un) in,

.. .

–un=Fun(x,u, . . . ,un) in,

ui(x) = , i= , . . . ,n, on∂,

(.)

whereis a bounded subset ofRn_{with smooth boundary}_∂,_n_≥_,_u

i(x)∈W,(),

F:Rn_×_Rn_→_R_{is a}_C_{function such that}_F₍_x_,_θ_{) = ,}_θ_{= (, . . . , ) and}_F

ui(x,u, . . . ,un) =

∂F(x,u,...,un)

∂ui ,i= , . . . ,n. LetU= (u, . . . ,un) and · Rndenote the Euclidean norm inR

n_{. Let}

us deﬁne

dUF(x,U) =FU(x,U) =gradUF(x,U) =

Fu(x,u, . . . ,un), . . . ,Fun(x,u, . . . ,un)

and

d_UF(U)·U=dFU(x,U)

·U ∀U∈E.

Letλ<λ≤ · · · ≤λk≤ · · ·be eigenvalues of the eigenvalue problem –u=λuin,u=  on∂, andφkbe an eigenfunction belonging to the eigenvalueλk,k≥.

We assume thatFsatisﬁes the following conditions:

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(F) F∈C₍_Rn_×_Rn_,_R₎_,_F₍_x_,_θ_{) = }_,_F

U(x,θ) =θ,x∈,θ= (, . . . , ).

(F) There exist constantsαandβ(α,βare not eigenvalues of the elliptic eigenvalue problem) such thatα<βand

αI≤d_UF(x,U)≤βI ∀(x,U)∈Rn×Rn,

and there existsk∈N∗such thatαI<λkI<dUF(x,U) <λk+I<βIfor everyU,

whereIis then×nidentity matrix.

(F) There exist eigenvaluesλh+, . . . ,λh+msuch that

λh<α<λh+<· · ·<λh+m<β<λh+m+,

whereh≥,m≥.

(F) There existγ andCsuch thatλh+m<γ <βand

F(x,U)≥ γU



Rn–C, ∀(x,U)∈Rn×Rn.

Some papers of Lee [–] concerning the semilinear elliptic system and some papers of the other several authors [, ] have treated the system of this like nonlinear elliptic equations. Some papers of Chang [] and Choi and Jung [] considered the existence and multiplicity of weak solutions for nonlinear boundary value problems with asymptotically linear term. The authors obtained some results for those problems by approaching the variational method, critical point theory and the topological method.

LetW_,(,R) be the Sobolev space with the norm

u

W_,(,R)=

|∇u|dx foru∈W,()

and the scalar product

(u,v)_W,

 (,R)= (∇u,∇v)L(,R).

LetEbe a cartesian product of the Sobolev spacesW_,(,R),i.e.,

E=W_,(,R)× · · · ×W_,(,R).

We endow the Hilbert spaceEwith the norm

U_E= n

i=

ui_W,  ()

,

whereui_W,  (,R)

=|∇ui(x)|dx. From now on we shall denote byW,() instead of

W_,(,R).

System (.) can be rewritten by

–U=grad_UF(x,U) in,

U=θ on∂,

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where –U= (–u, . . . , –un) andθ= (, . . . , ). In this paper we are looking for weak solutions of system (.) inE, that is,U= (u, . . . ,un)∈Esuch that

[–U·V]dx–

FU(x,U)·V=  for allV∈E.

Our main result is the following.

Theorem . Assume that F satisﬁes conditions(F)-(F).Then system(.)has at least

three nontrivial weak solutions.

The proof of Theorem . is organized as follows: We approach the variational method and use the finite dimensional reduction method for the dimension of the system, which reduces the infinite dimensional problem to the finite dimensional one, and we get criti-cal points of the functional on the infinite dimensional spaceEfrom that of the reduced functional on the finite dimensional subspace ofE. We also use critical point theory on the reduced finite dimensional subspace. In Section , we approach the variational method and the reduction method. We show that the reduced functional satisfies the (PS) condi-tion. In Section , we prove Theorem ..

2 Reduction approach

We assume thatF∈C₍_Rn_×_Rn_,_R_),_F₍_x_,_θ_{) = ,}_F

U(x,θ) =θ,θ= (, . . . , ) and there exist constantsαandβ(α,βare not eigenvalues of the elliptic eigenvalue problem) such that α<βand

αI≤d

UF(x,U)≤βI ∀(x,U)∈Rn×Rn,

and there existsk∈N∗ such thatαI<λkI<dUF(x,U) <λk+I<βI for everyU, where

U= (u, . . . ,un) and there exist eigenvaluesλh+, . . . ,λh+msuch that

λh<α<λh+<· · ·<λh+m<β<λh+m+,

whereh≥,m≥.

Lemma . Let Fui(x,U)∈L(),U= (u, . . . ,ui, . . . ,un),i= , . . . ,n.Then all the solutions

of

–U=grad_UF(x,U)

belong to E.

Proof LetFui(x,U)∈L

_{(). We note that}_{_λ

n:|λn|<|c|}is ﬁnite. ThenFui(x,u, . . . ,un)∈

L_(),_i_{= , . . . ,}_n_{, can be expressed by}

Fui(x,u, . . . ,un) =

∞

k=

hkφk,

∞

k=

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Then

(–)–Fui(x,u, . . . ,un) =



λk

hkφk.

Hence we have the inequality

(–)–Fui(x,u, . . . ,un)

 W_,()=

λk

 λ

k

h_k≤h_k<∞,

which means that

(–)–Fui(x,u, . . . ,un)_W,

 ()≤

Fui(x,u, . . . ,un)_L₍₎<∞,

souiW_,()<∞. Thus

UE= _n

i=

ui_W,  ()

 

<∞.

By the following Lemma ., weak solutions of system (.) coincide with critical points of the associated functionalI

I∈C,(E,R),

I(U) =

 |∇U|

_–_F₍_x_,_U₎

dx, (.)

whereU= (u, . . . ,un) and

|∇U|

_dx₌n i=

|∇ui|

_dx_,_n_≥_.

Lemma . Assume that F satisﬁes conditions(F)-(F).Then the functional I(U)is con-tinuous,Fréchet diﬀerentiable with Fréchet derivative

DI(U)·V=

(–U)·V–FU(x,U)·V

dx.

Moreover,DI∈C.That is,I∈C_.

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

I(U+V) –I(U)= 

(–U–V)·(U+V)dx–

F(x,U+V)dx

–  

(–U)·U dx+

F(x,U)dx

= 

(–U·V–V·U–V·V)dx

–

F(x,U+V) –F(x,U)dx.

We have

F(x,U+V) –F(x,U)dx≤

FU(x,U)·V+O

VE

dx=OVE

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Thus we have

I(U+V) –I(U)=OVE

, (.)

I(U+V) –I(U) –DI(U)·V=OV_E.

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

I(U+V) –I(U) –DI(U)·V

= 

(–U–V)·(U+V)dx–

F(x,U+V)dx

–  

(–U)·U dx+

F(x,U)dx–

–U–FU(x,U)

·V dx

= 

[–U·V–V·U–V·V]dx

–

F(x,U+V) –F(x,U)dx–

–U–FU(x,U)

·Vdx.

By (.),

I(U+V) –I(U) –DI(U)·V=OV_E.

ThusI∈C_.

LetE=W_,(,R)×· · ·×W_,(,R) and let{e,e, . . . ,en}be an orthonormal basis inRn. Then

L,Rn= m∈N

M(λm),

whereNis a natural number andM(λm) =span{φme, . . . ,φmen}is the eigenspace of – with eigenvalueλm,dimM(λm) =n,m= , , . . . . Let

L=

–∞<λm<α

M(λm), L=

α<λm<β

M(λm), L=

β<λm<∞

M(λm).

Then

L,Rn=L⊕L⊕L. (.)

For eachX∈L_(,_Rn_{), we have the composition}

X=X+X+X,

whereX∈L,X∈L,X∈L. LetPbe the orthogonal projection fromL(,Rn) onto

L,Pbe that fromL(,Rn) ontoLandPbe that fromL(,Rn) ontoL. Let

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ThenE=V⊕W⊕W, and forU∈E,U has the decompositionU=Y+Z+Z∈E, where

Y= (–)–X∈V, Z= (–)–X∈W, Z= (–)–X∈W. (.)

LetW=W⊕W. ThenWis the orthogonal complement ofVinE. LetP:E→Vbe the orthogonal projection ofEontoVandI–P:E→Wdenote that ofEontoW. Then every elementU∈Eis expressed byU=Y+Z,Y=PU,Z= (I–P)U. Then (.) is equivalent to the two systems in the two unknownsYandZ:

–Y=Pgrad_UF(x,Y+Z) in, (.)

–Z= (I–P)grad_UF(x,Y+Z) in, (.)

Y= (, . . . , ), Z= (, . . . , ) on∂.

LetY∈Vbe ﬁxed and consider the functionh:W×W→Rdeﬁned by

h(Z,Z) =I(Y+Z+Z).

The functionhhas continuous partial Fréchet derivativesDhandDhwith respect to its ﬁrst and second variables given by

Dih(Z,Z)·Xi=DI(Y+Z+Z)·Xi (.)

forXi∈Wi,i= , . By Lemma .,Iis a functional of classC.

By the following Lemma ., we can get critical points of the functionalI(U) on the inﬁ-nite dimensional spaceEfrom that of the functional on the ﬁnite dimensional subspaceV.

Lemma .(Reduction lemma) Assume that F satisﬁes conditions(F)-(F).Then

(i)there exists a unique solution Z∈W of the equation

–Z= (I–P)grad_UF(x,Y+Z) in,

Z= (, . . . , ) on∂.

If we put Z=(Y),thenis continuous on V and satisﬁes a uniform Lipschitz condition in V with respect to the L_norm₍_{also norm}_·_)._Moreover_,

DIY+(Y)·X=  for all X∈W.

(ii)There exists m< such that if Zand Xare in Wand Z∈W,then

Dh(Z,Z) –Dh(X,Z)

(Z–X)≤mZ–X.

(iii)There exists m> such that if Zand Xare in Wand Z∈W,then

Dh(Z,Z) –Dh(Z,X)

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(iv)If˜I:V→R is deﬁned byI˜(Y) =I(Y+(Y)),then˜I has a continuous Fréchet deriva-tive DI with respect to Y˜ ,and

D˜I(Y)·B=DIY+(Y)·B for all Y,B∈V. (.)

(v)Y∈V is a critical point ofI if and only if Y˜ +(Y)is a critical point of I.

Proof (i) Letδ=α+_β. Equation (.) is equivalent to

Z= (––δ)–(I–P)grad_UF(x,Y+Z) –δ(Y+Z). (.)

System (.) can be rewritten as

zi= (––δ)–(I–P)

Fui(x,y+z, . . . ,yn+zn) –δ(y+z, . . . ,yn+zn)

,

i= , , . . . ,n, (.)

y=· · ·=yn= , z=· · ·=zn=  on∂,

whereui=yi+zi,i= , , . . . ,n,U= (u, . . . ,un),Y= (y, . . . ,yn),Z= (z, . . . ,zn). The oper-ator (––δ)–(I–P) is a self-adjoint, compact and linear map from (I–P)L(,R) into itself, and by condition (F) itsLnorm ismin{λh+m+–δ,δ–λh}–. LetU,U∈E. Since

Fui(x,U) –δU

–Fui(x,U) –δU

≤max|α–δ|,|β–δ|U–URn=α+β

 U–URn,

it follows that the right-hand side of (.) defines, for fixedY∈V, a Lipschitz mapping of (I–P)L(,R) into itself with Lipschitz constantr= (min{λh+m+–δ,δ–λh})–×α+_β<  becauseλh+m+–δ> α_+β andδ–λh>α+_β. Therefore, by the contraction mapping prin-ciple, for each givenY∈V,i= , , . . . ,n, there exists a uniquezi∈(I–P)L(,R) which satisfies (.). Thus, for fixedY∈V, there exists a uniqueZ∈(I–P)L_(,_Rn_{) which} sat-isfies (.). If(Y) denotes the uniqueZ∈(I–P)L_(,_Rn_{) which solves (.), then}_is continuous and satisfies a uniform Lipschitz condition inYwith respect to theLnorm (also norm · E). In fact, ifZ=(Y) andZ=(Y), then

Z–ZL₍_,_Rn₎ =(––δ)–(I–P)

grad_UF(x,Y+Z) –δ(Y+Z)

–grad_UF(x,Y+Z) –δ(Y+Z)_L₍_,_Rn₎

≤r(Y+Z) – (Y+Z)_L₍_,_Rn₎

≤rY–YL₍_,_Rn₎+Z_–Z__L₍_,_Rn₎

≤rY–YE+rZ–ZE.

Hence

Z–ZL₍_,_Rn₎≤CY_–Y__L₍_,_Rn₎, C= r

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LetU=Y+Z,Y∈VandZ=(Y). IfX∈(I–P)L_(,_Rn₎_∩_E_,

DIY+(Y)·X=

–Y+(Y)·X–Pgrad_UF(x,Y+Z) –δ(Y+Z)·X

– (I–P)grad_UF(x,Y+Z) –δ(Y+Z)·Xdx.

It follows from (.) that

–Z(x)·X(x) –grad_UFx,Y(x) +Z(x)·X(x)dx= .

Since

–Y(x)·X(x)dx= ,

we have

DIY+(Y)·X= . (.)

(ii) IfZandXare inWandZ∈W, then

Dh(Z,Z) –Dh(X,Z)

(Z–X) =

_∇(Z–X)–

grad_UF(x,Y+Z+Z)

–grad_UF(x,Y+X+Z)

·(Z–X)

dx.

Since|∇(Z–X)|=Z–XE≤λhZ–X_L₍_,_Rn₎and

grad_UF(x,Y+Z+Z) –gradUF(x,Y+X+Z)

·(Z–X)

≥αZ–XL₍_,_Rn₎≥

α λh

Z–XE,

Dh(Z,Z) –Dh(X,Z)

(Z–X)≤

 – α λh

Z–XE,

where  –_λα

h < .

(iii) Similarly, using the fact that |∇(Z –X)|dx= Z –XE ≥λh+m+Z –

X_L₍_,_Rn₎and

grad_UF(x,Y+Z+Z) –gradUF(x,Y+Z+X)

·(Z–X)

≤βZ–XL₍_,_Rn₎≤

β λh+m+

Z–XE,

we see that ifZandXare inWandZ∈W, then

Dh(Z,Z) –Dh(Z,X)

(Z–X)≥

 – β λh+m+

Z–XE,

where ( –_λ β

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(iv) Since the functionalIhas a continuous Fréchet derivativeDI,˜Ihas a continuous Fréchet derivativeD˜Iwith respect toY.

(v) Suppose that there existsY∈Vsuch thatDI˜(Y) = . FromDI˜(Y)·B=DI(Y+(Y))·

Bfor allY,B∈V,DI(Y+(Y))(B) =DI˜(Y)(B) =  for allB∈V. SinceDI(Y+(Y))·B=  for allB∈W andEis the direct sum ofV andW, it follows thatDI(Y+(Y)) = . ThusY+(Y) is a solution of (.). Conversely, ifUis a solution of (.) andY=PU, thenDI˜(Y) = .

Remark We note that ifY∈V, then(Y) = .

3 Proof of Theorem 1.1

Lemma .((PS) condition) Assume that F satisﬁes conditions(F)-(F).Then –I˜(v)is bounded from below andI˜(v)satisﬁes the Palais-Smale condition.

Proof We have

˜

I(Y) =IY+(Y) +(Y)

= 

–Y+(Y) +(Y)·Y+(Y) +(Y)dx

–

Fx,Y+(Y) +(Y)dx

= 

–Y+(Y)·Y+(Y)dx

–

Fx,Y+(Y)dx+ 

–(Y)·(Y)dx

–

Fx,Y+(Y) +(Y)–Fx,Y+(Y)dx.

We claim that

 

–(Y)·(Y)dx–

Fx,Y+(Y) +(Y)–Fx,Y+(Y)dx

≤.

In fact, we note that

 

–(Y)

·(Y)dx

= – 

–(Y)·(Y)dx+

–(Y)·(Y)dx

= – 

–(Y)·(Y)dx+

FU

x,Y+(Y) +(Y)·(Y)dx

by (.). We also note that

Fx,Y+(Y) +(Y)–Fx,Y+(Y)

= 



FU

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Thus we have  

–(Y)·(Y)dx–

Fx,Y+(Y) +(Y)–Fx,Y+(Y)dx

= – 

–(Y)·(Y)dx+

FU

x,Y+(Y) +(Y)·(Y)dx

–   FU

x,Y+(Y) +t(Y)

·(Y)dx dt.

We note that

d dt

FU

x,Y+(Y) +t(Y)·t(Y)

=d_UFx,Y+(Y) +t(Y)·t(Y)·(Y)

+FU

x,Y+(Y) +t(Y)·(Y),

which leads to   d dt FU

x,Y+(Y) +t(Y)·t(Y)dt

= 



d_UFx,Y+(Y) +t(Y)·t(Y)·(Y)dt

+ 



FU

x,Y+(Y) +t(Y)

·(Y)dt.

That is,

FU

x,Y+(Y) +(Y)·(Y)

= 



d_UFx,Y+(Y) +t(Y)

·t(Y)

·(Y)dt

+ 



FU

x,Y+(Y) +t(Y)·(Y)dt.

Thus we have

FU

x,Y+(Y) +(Y)

·(Y) – 



FU

x,Y+(Y) +t(Y)

·(Y)dt

= 



d_UFx,Y+(Y) +t(Y)·t(Y)·(Y)dt.

Thus we have

 

–(Y)

·(Y)dx–

Fx,Y+(Y) +(Y)

–Fx,Y+(Y) dx = – 

–(Y)·(Y)dx+

FU

x,Y+(Y) +(Y)·(Y)dx

–   FU

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= – 

–(Y)·(Y)dx

+ 



d_UFx,Y+(Y) +t(Y)·t(Y)·(Y)dx dt≤

by condition (F). Thus by condition (F) we have

˜

I(Y)≤  

–Y+(Y)·Y+(Y)dx–

Fx,Y+(Y)dx

≤ 

(λh+m–γ)Y+(Y) 

E+C→–∞ asY+(Y)E→ ∞. (.)

Thus –I(˜Y) is bounded from below and satisﬁes the (PS) condition.

Lemma . Assume that F satisﬁes conditions(F)-(F).Then Y=θ,θ= (, . . . , )is nei-ther minimum nor degenerate.

Proof By (.) we have

˜

I(Y)≤  

–Y+(Y)·Y+(Y)dx–

Fx,Y+(Y)dx. (.)

Since

Fx,Y+(Y)

= 



dF dt

x,tY+(Y)

dt

= 



FU

x,tY+(Y)·Y+(Y)dt,

we have that

Fx,Y+(Y)

– 

d_UF(x,θ)·Y+(Y)

·Y+(Y)

dx

= 



FU

x,tY+(Y)

–d_UF(x,θ)·tY+(Y)

·Y+(Y)

dx dt

≤

<supt<

d_UFx,tY+(Y)

–d_UF(x,θ)_L(_V_,_V₎Y+(Y)  E.

Thus we have

–

Fx,Y+(Y)

≤– 

d_UF(x,θ)·Y+(Y)·Y+(Y)+oY+(Y)_E.

Since  ∈C(V,W), it follows that if Y →, then (Y)E=O(YE) because (θ) =θ. Thus

Y+(Y)_E=OYE

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SinceFU(x,θ) =θ, there exists a bounded self-adjoint operatorA∈L(E,E) which com-mutes withPoandPsuch that

λh+I≤A≤dUF(x,θ).

Thus we have

˜

I(Y)≤  

–Y+(Y)

·Y+(Y) dx – 

d_UF(x,θ)·Y+(Y)·Y+(Y)dx+oY_E

≤ 



–Y+(Y)

·Y+(Y) dx – 

AY+(Y)·Y+(Y)+oY_E

=  

–(Y)·(Y)

dx– 

A(Y)

·(Y) +  

(–Y·Y)dx

– 

A(Y)·Y+oY_E

asYE→. Sinceλh+I≤A, it follows that

 

–(Y)·(Y)dx– 

A(Y)·(Y)

≤



–(Y)·(Y)dx– 

λh+(Y)·(Y)≤.

Therefore we have

˜

I(Y)≤  

(–Y·Y)dx– 

A(Y)·Y+oY_E

≤ 



(–Y)·Y–λh+Y

dx+oY_E

asYE→. Similarly we can choose a bounded self-adjoint operatorB∈L(E,E) which commutes withPoandPsuch that

d_UF(x,θ)≤B≤λh+m+I.

This leads to

˜

I(Y)≥  

(–Y·Y)dx– 

B(Y)·Y+oY_E

≥ 



(–Y)·Y–λh+m+Y

dx+oY_E

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Lemma . Assume that F satisﬁes conditions(F)-(F).Then

˜

I(Y)→–∞ asYE→ ∞.

Proof The proof can be found in the proof of Lemma ..

Proof of Theorem. By Lemma .,I˜(Y) is continuous and Fréchet diﬀerentiable inV. By Lemma ., –˜I(Y) satisﬁes the (PS) condition. By Lemma .,Y=θ is neither minimum nor degenerate. By Lemma .,I˜(Y)→–∞asYE→ ∞. We note thatmaxY∈V˜I(Y) >  is another critical value ofI˜. Thus there exists the third critical point of˜I(Y). Thus (.) has at least three nontrivial solutions.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript and read 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: 21 February 2015 Accepted: 27 February 2015 References

1. Ha, KS, Lee, YH: Existence of multiple positive solutions of singular boundary value problems. Nonlinear Anal. TMA28, 1429-1438 (1997)

2. Lee, YH: Existence of multiple positive radial solutions for a semilinear elliptic system on an unbounded domain. Nonlinear Anal. TMA47, 3649-3660 (2001)

3. Lee, YH: A multiplicity result of positive radial solutions for a multiparameter elliptic system on an exterior domain. Nonlinear Anal. TMA45, 597-611 (2001)

4. Lee, YH: Multiplicity of positive radial solutions for multiparameter semilinear elliptic systems on an annulus. J. Diﬀer. Equ.174, 420-441 (2001)

5. Dunninger, DR, Wang, H: Multiplicity of positive radial solutions for an elliptic system on an annulus domain. Nonlinear Anal. TMA42(5), 803-811 (2000)

6. Lan, K, Webb, RL: Positive solutions of semilinear equation with singularities. J. Diﬀer. Equ.148, 407-421 (1998) 7. Chang, KC: Inﬁnite Dimensional Morse Theory and Multiple Solution Problems. Birkhäuser, Basel (1993)