R E S E A R C H
Open Access
Existence of four solutions for the elliptic
problem with nonlinearity crossing one
eigenvalue
Tacksun Jung
1*and Q-Heung Choi
2*Correspondence: [email protected] 1Department of Mathematics, Kunsan National University, Kunsan, 573-701, Korea
Full list of author information is available at the end of the article
Abstract
We investigate the multiplicity of the weak solutions for the nonlinear elliptic boundary value problem. We get a theorem which shows the existence of at least four weak solutions for the asymptotically linear elliptic problem with Dirichlet boundary condition. We obtain this result by using the Leray-Schauder degree theory, the variational reduction method and critical point theory.
MSC: 35J15; 35J25
Keywords: elliptic boundary value problem; jumping nonlinearity; inverse compact operator; variational reduction method; Leray-Schauder degree theory
1 Introduction
Letbe a bounded open subset ofRnwith smooth boundary∂. LetLbe the self-adjoint
strongly elliptic partial differential operator
L=
≤i,j≤n
∂ ∂xi
aij(x) ∂ ∂xj
,
whereaij(x)∈L∞(). Letf :R→Rbe aCfunction. Letλkbe the eigenvalues andφk,k=
, , . . . , be the associated normalized eigenfunctions of the eigenvalue problemLu+λu= in,u= on∂. We note that <λ<λ≤ · · · ≤λk→ ∞andφ> .
In this paper, we consider the multiplicity of solutions of the following elliptic problem with Dirichlet boundary condition and jumping nonlinearity:
Lu+bu+–au–=sφ in, (.)
u= on∂.
The physical model for this kind of the jumping nonlinearity problem can be furnished by traveling waves in suspension bridges. The nonlinear equations with jumping nonlin-earity have been extensively studied by Lazer and McKenna [], McKenna and Walter [], Tarantello [], Micheletti and Pistoia [, ], Jung and Choi [–] and many other authors. McKenna and Walter [] proved that if –∞<f(–∞) <λandλn<f(+∞), then there
ex-istsssuch that for alls≥s, (.) has at least two solutions ifnis odd, and three solutions
ifnis even.
Our main result is as follows.
Theorem . Assume that–∞<a<λ<b<λ,s> .Then(.)has at least four
solu-tions.
For the proof of Theorem . we use the Leray-Schauder degree theory, the variational reduction method and the critical point theory. The outline of this paper is as follows: In Section , we show the existence of the third weak solution for (.) by the variational reduction method and critical point theory. In Section , we show the existence of the fourth weak solution for (.) by the Leray-Schauder degree theory.
2 Existence of the third weak solution
We note that (.) has a positive solutionv=bs–φλ and a negative solutionv= sφ a–λ. To
show the existence of the third solution of (.), we shall use the variational reduction method and the critical point theory. We assume that –∞<a<λ<b<λands> . Let
g(ξ) =bξ+–aξ–andH
() be the Sobolev space with the norm
u=
(–Lu)·u dx.
We consider the following functional associated with (.):
Ia,b(u,s) =
–
Lu·u–G(u) +sφu
dx, (.)
where
G(u) =b|u
+|
+
a|u–|
.
For simplicity, we shall writeI=Ia,bwhenaandbare fixed. By the condition –∞<a<λ<
b<λands> ,Iis well defined. By the following Lemma .,I(u,s)∈C(H(),R) and
Fréchetdifferentiable inH(), so the solutions of (.) coincide with the critical points of
I(u,s).
Lemma . Assume that–∞<a<λ<b<λand s> .Then I(u,s)is continuous and
Fréchet differentiable in H ()and
DuI(u,s)(h) =
–Lu·h–bu+–au–h+sφh dx
for h∈H
().Moreover,if we set
F(u,s) =
b|u+|
+
a|u–|
–sφu
dx,
then DuF(u,s)is continuous with respect to weak convergence,DuF(u,s)is compact,and
DuF(u,s)h=
bu+–au–h–sφh dx for all h∈H().
This implies that I∈C(H
Proof Letu∈H
(). First, we will prove thatI(u,s) is continuous with respect to the
variableu. We consider
I(u+v,s) –I(u,s)=
u·(–Lv) +
v·(–Lv) –G(u+v) +G(u)
dx.
Letu=hmφm,v=
˜
hmφm. Then we have
u·(–Lv)dx=
λmhmhm˜ ≤ uv,
v·(–Lv)dx=λmh˜m≤ v.
On the other hand,
(u+v)+–u+≤u+|v|+|v|, (u+v)––u–≤u–|v|+|v|,
and hence
(u+v)+–u+dx
≤
u+|v|+|v| dx≤uv+v,
(u+v)––u–dx
≤
u–|v|+|v| dx≤uv+v.
With the above results, we see thatI(u,s) is continuous atu. To proveI(u,s) isFréchet
differentiable atu∈H(), it is enough to compute the following: I(u+v,s) –I(u,s) –DuI(u,s)v=
v(–Lv) –G(u+v) +G(u) –g(u)v
≤
v
+Cγvu+v+Mv
≤Cvv+u+v+ .
LetVbe the one-dimensional space spanned by the eigenfunctionφwhose eigenvalue
isλ. LetW be the orthogonal complement ofV inH(). LetP:H()→V be the
orthogonal projection of H
() onto V andI–P:H()→W denote that of H()
ontoW. Then every elementu∈H
() is expressed byu=v+z,v∈Pu,z= (I–P)u.
Then (.) is equivalent to the two systems in the two unknownsvandz:
Lv+Pb(v+w)+–a(v+w)–=sφ in, (.)
Lz+ (I–P)b(v+w)+–a(v+w)–= in, (.)
The subspaceWis spanned by eigenfunctions corresponding to the eigenvaluesλk,k≥.
Letv∈Vbe fixed and consider the functionh:W→Rdefined by
h(w) =I(v+w,s).
The functionhhas continuousFréchetderivativesDhgiven by
Dh(w)(y) =DuI(v+w,s)(y) (.)
fory∈W. By Lemma .,Iis a function of classC.
By the following Lemma ., we can get the critical points of the functionalI(u,s) on the infinite dimensional spaceH
() from that of the reduced functionalI˜(v,s) on the finite
dimensional subspaceV.
Lemma .(Reduction Method) Assume that–∞<a<λ<b<λand s> .Then
(i) there exists a unique solutionz∈W of the equation
Lz+ (I–P)b(v+w)+–a(v+w)–= in,
z= on.
If we putz=θ(v,s),thenθ is continuous onVand satisfies a uniform Lipschitz condition invwith respect toLnorm(also norm · ).Moreover,
DuI
v+θ(v,s),s(w) = for allw∈W.
(ii) There existsm> such that ifwandyare inW,then
Dh(w) –Dh(y)(w–y)≥mw–y.
(iii) IfI˜:V→Ris defined by˜I(v,s) =I(v+θ(v,s),s),thenI˜has a continuous Fréchet derivativeDv˜Iwith respect tov,and
DvI˜(v,s)(h) =DuIv+θ(v,s),s(h) for allv,h∈V. (.)
(iv) Ifv∈Vis a critical point ofI˜if and only ifv+θ(v,s)is a critical point ofI.
(v) LetS⊂Vand ⊂H
()be open bounded regions such that
v+θ(v,s);v∈S= ∩v+θ(v,s);v∈V.
IfD˜I(v,s)= forv∈∂S,then
deg(D˜I,S, ) =deg(DI, , ),
whereddenote the Leray-Schauder degree.
(vi) Ifu=v+θ(v)is a critical point of mountain pass type ofI,thenvis a critical
Proof (i) Letδ=a+b. Ifg(ξ) =g(ξ) –δξ, then equation (.) is equivalent to
z= (–L–δ)–(I–P)g(v+z)
. (.)
The operator (–L–δ)–(I–P) is self adjoint, compact and linear map from (I–P)L()
into itself and itsLnorm is|λ–δ|–. Since|g(ξ) –g(ξ)| ≤max{|a–δ|,|b–δ|}|ξ–ξ|=
δ|ξ–ξ|, it follows that the right-hand side of (.) defines, for fixedv∈V, a Lipschitz
mapping of (I–P)L() into itself with Lipschitz constantr= δ
|λ–δ| < . Therefore, by the contraction mapping principle, for givenv∈V, there exists a uniquez= (I–P)L()
which satisfies (.). Ifθ(v,s) denote the uniquez∈(I–P)L() which solves (.), then θ is continuous and satisfies a uniform Lipschitz condition invwith respect toLnorm
(also norm · ). In fact, ifz=θ(v,s) andz=θ(v,s), then
z–zL() =(–L–δ)–(I–P)
g(v+z) –g(v+z)L() ≤r(v+z) – (v+z)L()
≤rv–vL()+z–zL()
≤rv–v+rz–z.
Hence,
z–z ≤Cv–v, C=
r
–r. (.)
Letu=v+z,v∈Vandz=θ(v,s). Ifw∈(I–P)L()∩H (),
DuIv+θ(v,s),s(w) =
–L(v+z)·w–Pg(v+z)w
– (I–P)g(v+w)w+sφw dx.
From (.), we see that
(–Lz)·w– (I–P)g(v+z)w dx= .
Since
Lv·w= and
Pg(v+z)w–sφw dx= ,
we have
DuIv+θ(v,s),s(w) = . (.)
(ii) Ifwandyare inW, then
Dh(w) –Dh(y)(w–y) =
Since[–L(w–y)(w–y)]dx=w–yand (g(ξ
) –g(ξ))(ξ–ξ) <b(ξ–ξ), we see that
ifwandyare inW, thenw–yL()≤λ
w–y
and
g(v+w) –g(v+y)(w–y) dx≤ b
λw
–y
and
Dh(w) –Dh(y)(w–y)≥
+b b
λ
w–y,
where ( +bλb ) > .
(iii) Since the functionalIhas a continuousFréchetderivativeDuI,˜Ihas a continuous
FréchetderivativeDv˜Iwith respect tov.
(iv) Suppose that there existsv∈Vsuch thatDvI˜(v,s) = . FromDv˜I(v,s)(h) =DuI(v+
θ(v,s),s)(h) for allv,h∈V,DuI(v+θ(v,s),s)(h) =Dv˜I(v,s)(h) = for allh∈V. Since
DuI(v+θ(v,s),s)(w) = for allw∈WandEis the direct sum ofV andW, it follows that
DuI(v+θ(v,s),s) = . Thus,v+θ(v,s) is a solution of (.). Conversely, ifuis a solution
of (.) andv=Pu, thenDvI˜(v) = .
(v) The proof of part (v) follows by arguing as in Lemma . of [].
(vi) Supposevis not of mountain pass type of˜I. LetSbe an open neighborhood ofvin
V such that˜I–(–∞,˜I(v))∩Sis empty or path connected. IfI˜–(–∞,I˜(v))∩Sis empty,
by part (i) we see that{v+w:v∈V,w∈W} ∩I–(–∞,I(u)) is also empty. Thusuis not
of mountain pass type forI. IfI˜–(–∞,I˜(v
))∩Sis path connected, LettingT={v+w:
v∈V,|w–θ(v)|< }and using again (i) it is seen thatT∩I–(–∞,I(u
)) is also path
connected.
We define the functional onH()
Ia∗,b(u) =Ia,b(u, ) =
(–Lu)·u–
b
u
+–a
u
–
dx.
The critical points ofIa∗,b(u) coincide with the solutions of the equation
Lu+bu+–au–= inH(). (.)
Under the assumption –∞<a<λ<b<λ, (.) has only the trivial solution and hence
Ia∗,b(u) has only one critical pointu= . In fact, from (.), we obtain
(L+λ)u·φ(x) + (b–λ)u+φ(x) –
a–λ(x)
u–·φ(x) dx= (.)
under the assumption –∞<a<λ<b<λthe left-hand side of (.) is nonnegative, so
the only possibility to hold (.) is thatu= . Givenv∈V, letθ∗(v) =θ(v, ) be the unique solution of the equation
Lz+ (I–P)b(v+z)+–a(v+z)– = inW.
Let us defineI˜a∗,b(v) =Ia∗,b(v+θ∗(v)). We note thatI˜a∗,b(v) has only the trivial critical point
Lemma . Let–∞<a<λ<b<λ.Then we have
˜
Ia∗,b(v) > for all v∈V with v= .
Proof To prove the conclusion, it suffices to show thatI˜a∗,b(v) does not satisfies the follow-ing cases:
(i) I˜a∗,b(v)≤and˜Ia∗,b(v) = for somev∈Vwithv= .
(ii) I˜a∗,b(v)≥and˜Ia∗,b(v) = for somev∈Vwithv= .
(iii) I˜a∗,b(v) < for allv∈Vwithv= .
(iv) There existvandvinV such thatI˜a∗,b(v) < and˜Ia∗,b(v) > .
Suppose that (i) holds. It follows thatI˜a∗,b(v) has an absolute maximum atv, and hence
D˜Ia∗,b(v) = . Therefore, by Lemma .,u=v+θ∗(v) is a nontrivial solution of (.),
which is a contradiction. A similar argument show that it is impossible that (ii) holds. Suppose that (iii) holds. Then there existst∈(, ) such that for alltwith <t≤t,
t˜Ia∗,b(v) + ( –t)I˜,∗b(v)≤ for allv= .
We note that there existv= andtwith <t≤tsuch thatt˜Ia∗,b(v) + ( –t)I˜,∗b(v) =
for somev= andt. Lettbe the greatest number such that
t˜Ia∗,b(v) + ( –t)˜I,∗b(v) = .
Then <t≤t. SincetI˜a∗,b(v) + ( –t)I˜,∗b(v)≤ for allv= , and hencevis a point of
maximum oft˜Ia∗,b(v) + ( –t)I˜,∗b(v), we have
DtI˜a∗,b(v) + ( –t)˜I∗,b(v) = .
Letv∈Vbe given and <t< . Letθt∗(v) be the unique solution of the equation
Lz+ (I–P)bu+–tau–= inW.
By Lemma .,v+θt∗
(v)is a nontrivial solution of the equation t
Lu+bu+–au–+ ( –t)
Lu+bu+–au–= inH(),
that is,
Lu+bu+–tau–= inH(),
which contradicts the fact that the above equation has only the trivial solution because –∞<ta<λ<b<λ. Suppose that (iv) holds. Since the equation (.) has only the trivial
solution,Ia∗,b(u) has only one critical pointu= . By the assumption (iv), the trivial solution
u= is a point of inflexion, and hence by Lemma .,v= is the critical point of˜Ia∗,b(v), which is a point of inflexion. By the assumption (iv), there existv,vinVandt∈(, )
We note that there existstwith <t≤tsuch thattI˜a∗,b(v)+(–t)˜I,∗b(v) < andt˜Ia∗,b(v)+
( –t)˜I,∗b(v) > . Lettbe the greatest number such that
t˜Ia∗,b(v) + ( –t)I˜,∗b(v) < and t˜I∗a,b(v) + ( –t)˜I,∗b(v) > .
Then <t≤t. SincetI˜a∗,b(v) + ( –t)˜I,∗b(v) < andtI˜a∗,b(v) + ( –t)˜I,∗b(v) > . Thus,
there exists a point of inflexionv= oft˜Ia∗,b(v) + ( –t)˜I,∗b(v) and we have
Dt˜Ia∗,b(v) + ( –t)I˜,∗b(v) = .
Letv∈Vbe given and <t< . Letθt∗(v) be the unique solution of the equation
Lz+ (I–P)bu+–tau–= inW.
By Lemma .,v+θt∗(v) is a nontrivial solution of the equation
t
Lu+bu+–au–+ ( –t)
Lu+bu+–au–= inH(),
that is,
Lu+bu+–tau–= inH(),
which contradicts the fact that the above equation has only the trivial solution because
–∞<ta<λ<b<λ.
From now on, we shall denote thatIa,b=I.
Lemma . Let–∞<a<λ<b<λ.Then we have
˜
Ia,b(v,s)→ ∞ asv → ∞.
Proof We shall prove the lemma by contradiction. We suppose that there exists a sequence
{vn}∞
inVand a numberM> such thatvn → ∞asn→ ∞and˜I(vn,s)≤M. For given
vn∈V, letwn=θ(vn,s) be the unique solution of the equation
Lw+ (I–P)b(vn+w)+–a(vn+w)––sφ–h(x)
= inW.
By Lemma ., we have that for some constantk,
θ(vn) –θ()≤kvn.
From this the sequence{vn+wn
vn }is bounded inH. Letun=vn+wn,v
∗
n=vvnn,w
∗ n=wvnn
andu∗n=v∗n+w∗nforn≥. Forwn=θ(vn,s) andw∗n= wn
vn, we have
w∗n=L–
(I–P)
–b(vn+wn)
+
vn +a
(vn+wn)–
vn –s
φ
vn– h(x)
vn
Since{vn+wn
vn }is bounded and φ
vn→, –b
(vn+wn)+
vn +a
(vn+wn)–
vn –s φ vn –
h(x)
vn is bounded
inH. SinceL– is a compact operator, by passing to a subsequence, we get that{w∗
n}∞
converges tow∗. SinceVis -dimensional subspace, we may assume that{v∗n}∞ converges tov∗∈V withv∗= . Therefore,{u∗n}∞ converges tou∗inH
(). Since˜I(vn,s)≤Mfor
alln, we have, for alln,
–
Lun·un–
b
u
+
n
–a u
–
n
+sφun+h(x)un
dx≤M.
Dividing the above inequality byvn, we obtain
– Lu ∗ n·u∗n–
b
u
∗ n
+
–a u
∗ n
–
+sφ
u∗n vn+h(x)
u∗n vn
dx
≤ M
vn. (.)
From the definition ofwn=θ(vn,s), we have that for anyy∈W,n≥,
Lun·y+bu+ny–au–ny–sφy–h(x)y dx= . (.)
Let us sety=wnin (.) and divide byvn. Then we have
Lu∗n·w∗n+bu∗n+wn∗–au∗n–nw∗n–sφ
w∗n vn–h(x)
w∗n vn
dx= (.)
for alln≥. Lety∈W. Dividing (.) byvnand lettingn→ ∞, we obtain
Lu∗·y+bu∗+y–au∗–y–sφy–h(x)y dx= . (.)
(.) can be rewritten in the form
D˜Ia∗,bv∗+w∗(y) = for ally∈W.
Byw∗=θ(v∗), lettingn→ ∞in (.), we have
lim
n→∞
Lw∗n·w∗ndx= –
bu∗w∗–au∗w∗ dx
=
Lu∗·w∗ dx=
Lw∗·w∗ dx.
Hence,
lim
n→∞
Lu∗n·u∗ndx=
Lu∗·u∗ dx.
Lettingn→ ∞in (.), we obtain
˜ Ia∗,bv∗=
–
Lu
∗·u∗–b
u
∗+
–a u
∗–
Sincev∗= , this contradicts the fact that˜I∗(v) > for allv= . Thus, ˜I(v,s)→ ∞as
v → ∞.
Lemma . Assume that–∞<a<λ<b<λand s> .Then there exists a small open
neighborhood B of vin V such that vis a strict local point maximum of˜I(v,s)and there
exists a small open neighborhood D of vin V such that vis a strict point of minimum of
˜
I(v,s),where v=bs–φλ is a positive solution and v= sφ
a–λ is a negative solution of(.). Proof Since the positive solutionv=bs–φλis inV,θ(v,s) = andI+θ, whereIis an
iden-tity map onV, is continuous onV, it follows that there exists a small open neighborhoodB
ofvinVsuch that ifv+v∈B, thenv+v+θ(v+v,s) > . Here,θ(v+v,s) =θ(v,s) = .
Therefore, ifv+v∈B, then forz=θv+v,s=θ(v) = , we have
˜
I(v+v,s) =I(v+v,s)
=
–
L(v+v)·(v+v) –
b|v+v|
+sφ(v+v) dx = –
Lv·v–
b|v|
dx+C,
where C= –
Lv·v–
b|v|
+sφv
dx=I(v,s) =˜I(v,s).
Eachv∈Vhas the formv=cφTherefore, we have, inB,
˜
I(v+v,s) –˜I(v+v,s)
=
–
Lv·v–
b|v|
dx
= (λ–b)
v< .
Sinceλ<b<λ,vis a strict local point maximum of˜I(v,s). Since the negative solution
v=as–φλis inV,θ(v,s) = andI+θ, whereIis an identity map onV, is continuous onV,
it follows that there exists a small open neighborhoodDofvinV such that ifv+v∈B,
thenv+v+θ(v+v,s) < . Here,θ(v+v,s) =θ(v,s) = . Therefore, ifv+v∈B, then for
z=θv+v,s=θ(v,s) = , we have
˜
I(v+v,s) =I(v+v,s) =
–
L(v+v)·(v+v) –
a|v+v|
+sφ(v+v) dx = –
Lv·v–
a|v|
dx+C,
where
C=
–
Lv·v–
a|v|
+sφv
dx
Eachv∈Vhas the formv=cφTherefore, we have, inB,
˜
I(v+v,s) –I˜(v,s)
=
–
Lv·v–
a|v|
dx
= (λ–a)
v>
since –∞<a<λ. Thus,vis a strict local point minimum ofI˜(v,s).
Lemma .(Existence of the Third Solution) Assume that–∞<a<λ<b<λand s> .
Then(.)has the third weak solution which is a strict point of minimum of Ia,b(u,s). Proof We know that (.) has a positive solutionv=bs–φλand a negative solutionv=
sφ a–λ.
By Lemma . (reduction method), we shall show the existence of the third weak solu-tion of (.). By Lemma .,˜I(v,s) is continuous andFréchetdifferentiable. By Lemma .,
˜
I(v,s)→ ∞asv → ∞, so˜I(v,s) is bounded from below, satisfies the (P.S.) condition. By Lemma ., there exists a small open neighborhoodBofvinV such thatvis a strict
local point of maximum of˜I(v,s) and there exists a small open neighborhoodDofvinV
such thatvis a strict local point of minimum ofI˜(v,s). By the shape of the graph of the
functionalI˜on the -dimensional subspaceV, there exists the third critical pointv=
which is a strict local point of minimum of˜I(v,s). By Lemma .,v+θ(v,s) is a solution
of (.) and a strict local point of minimum ofIa,b(u,s).
3 Existence of the fourth weak solution
In this section, we shall consider the Leray-Schauder degree of the elliptic operator for the multiplicity of the solutions of (.).
Lemma . Assume that–∞<a<λ<b<λ.Then there exists s> ,> such that the
Leray-Schauder degree
degu–L––bu++au–+sφ
,B∗s(v),
= – (.)
for s≥s.Here,B∗τ denotes a ball of radiusτ in H()and v=bs–φλ is a positive solution of(.).
The proof of this lemma has the similar process to that of Theorem in [].
Lemma . If–∞<a<λ<b<λ,then there exist positive constants s,such that
degu–L––bu++au–+sφ
,B∗s(v),
=
for s≥s,where v=as–φλis a negative solution of(.).
Lemma . If–∞<a<λ<b<λ,then there exist positive constants s,γ > such that
degu–L––bu++au–+sφ
,B∗γ(v),
=
for s≥s,where vis the third solution of(.).
Proof We know that the third solutionvof (.) is an isolated local minimum ofIa,b(u,s).
By [], there exist positive constantss,γ> such that
degD˜I,B∗γ(v)∩V,
=
fors≥s, whereB∗γ(v)∩Vis a ball centered at containing no other critical point. By (v)
of Lemma .,
degu–L––bu++au–+sφ
,B∗γ(v),
=degDI˜,B∗γ(v)∩V,
= .
So we prove the lemma.
Lemma . Assume that–∞<a<λ<b<λand s is bounded.Then there exist C and
s∗> such that any solution to(.)satisfiesu ≤C for any s with s≤s∗.
Proof If not, then there exist (un,sn) withun → ∞,sn→s∗,snis bounded, which satisfy equation (.). Now letvn= un
un, andvnsatisfies
Lvn+
bv+n–av–n–snφ(x)
un = . (.)
Taking the inner product of both sides of (.) withφ, we have
=–Lvn–λvn,φ=
(b–λ)v+n+ (λ–a)v–n,φ
–
snφ(x)
un ,φ
. (.)
Thus, snφ(x)
un ,φ
≥
|vn|φ(x)≥
vnφ(x)
.
Thus, ifvnis a solution of (.), then
vn,φ(x)≤
snφ(x)
un ,φ(x)
. (.)
Sincevn’s are precompact inH
(), there existsvwithv= such thatvn→v. Taking
the limit of both sides of (.), we have v,φ(x)≤.
Thus, we have v,φ(x)= ,
We have the following no solvability condition for (.).
Lemma . Assume that–∞<a<λ<b<λ.Then there exists a constant s< so small
enough that if s≤s,the problem
–Lu=bu+–au––sφ in H()
has no solution.
Proof We can rewrite (.) as
(–L–λ)u= –λu++bu++ (λ–a)u––sφ. (.)
Taking the inner product of both sides of (.) withφ, we have
=–(λ–b)u++ (λ–a)u––sφ,φ
≥–s.
Thus, there is no solution fors≤sifs< . This completes the proof.
Lemma . Let–∞<a<λ<b<λ.Then there existsβ> ,depending on C and s∗such
that
degu–L––bu++au–+sφ
,B∗β(), =
for s≤s∗andβ>C.
Proof By Lemma ., there exists a constants< so small enough that ifs≤s, (.) has
no solution. By Lemma ., there exist a constantCands∗> such that ifuis a solution of (.) withs≤s∗, thenu ≤C. Let us chooseβso large thatβ>C. We note that
u–L––bu++au–+sφ
= on∂B∗β()
and
u–L––bu++au–+sφ+λ(s–s)φ(x)
= on∂B∗β()
for ≤λ≤. By the homotopy invariance property, we have that the Leray-Schauder degree
dLS
u–L––bu++au–+sφ
,B∗β(),
=dLSu–L––bu++au–+sφ+λ(s–s)φ(x)
,B∗β(),
=dLS
u–L––bu++au–+sφ
,B∗β(), = ,
Proof of Theorem. By Lemma . and Lemma ., there exists a large numberβ> (depending onCands∗) such that all solutions of (.) are contained in the ballBβ() and the Leray-Schauder degree
degu–L––bu++au–+sφ
,B∗β(),
=
forβ>Cand <s<s∗. Let us choose a positive numbers∗such thats∗=max{s,s,s}
ands∗<s∗. Then by Lemma ., Lemma . and Lemma ., for allswiths∗≤s≤s∗, there exist three disjoint ballsB∗s(v),B∗s(v) andB∗γ(v) such that
deg(u–L––bu++au–+sφ
,B∗s(v, ) = –,
degu–L––bu++au–+sφ
,B∗s(v),
= ,
degu–L––bu++au–+sφ
,B∗γ(v),
= .
By the excision property of Leray-Schauder degree,
deg(u–L––bu++au–+sφ
,B∗β()\
B∗s
v∪B∗s(v)∪B∗γ(v)
, = –.
Thus, (.) has at least four solutions.
Abbreviation
EFSEPNCOE: Existence of Four Solutions for the Elliptic Problem with Nonlinearity Crossing One Eigenvalue.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
TJ carried out (EFSEPNCOE) studies, participated in the sequence alignment and drafted the manuscript. QC participated in the sequence alignment. All 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 (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: 20 August 2012 Accepted: 4 April 2013 Published: 19 April 2013
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doi:10.1186/1029-242X-2013-187