Volume 2007, Article ID 43018,19pages doi:10.1155/2007/43018
Research Article
Multiple Positive Solutions of Nonhomogeneous Elliptic Equations in Unbounded Domains
Tsing-San Hsu
Received 7 July 2006; Revised 25 December 2006; Accepted 25 December 2006 Recommended by Martin J. Bohner
We will show that under suitable conditions on f andh, there exists a positive number λ∗ such that the nonhomogeneous elliptic equation−Δu+u=λ(f(x,u) +h(x)) inΩ, u∈H01(Ω), N≥2, has at least two positive solutions if λ∈(0,λ∗), a unique positive solution ifλ=λ∗, and no positive solution ifλ > λ∗, whereΩis the entire space or an exterior domain or an unbounded cylinder domain or the complement in a strip domain of a bounded domain. We also obtain some properties of the set of solutions.
Copyright © 2007 Tsing-San Hsu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Let 2∗=2N/(N−2) forN≥3, 2∗= ∞forN=2. In this paper, we study the existence, nonexistence, and multiplicity of solutions of the equation
−Δu+u=λf(x,u) +h(x)inΩ, uinH01(Ω),u >0 inΩ,N≥2, (1.1)λ whereλ >0,N=m+n≥2,n≥1, 0∈ω⊆Rmis a smooth bounded domain,S=ω×Rn, Dis a smooth bounded domain inRNsuch thatD⊂⊂S,Ω=S\Dis the exterior of this domain in the strip.
Associated to(1.1)λ, we consider the functionalI, foru∈H01(Ω), I(u)=1
2
Ω
|∇u|2+u2dx−λ
ΩFx,u+dx−λ
Ωh(x)u dx, (1.1) whereF(x,t)=t
0 f(x,s)ds.
It is assumed thath(x)∈L2(Ω)∩Lq0(Ω) for someq0> N/2 ifN≥4,q0=2 ifN=2, 3, h(x)≥0,h(x)≡0, and f(x,t) satisfies the following conditions:
(f1) f(x,·)∈C1([0, +∞),R+), f(x,t)≡0 forx∈S,t≤0, and limt→0(f(x,t)/t)=0 uniformly forx∈S;
(f2) there exists a positive constantCsuch that for allx∈Sandt∈R, 0< ∂
∂tf(x,t)≤C1 +|t|p−2
, (1.2)
where 2< p <2∗;
(f3) there exists a numberθ∈[1/ p, 1) such that θt∂
∂tf(x,t)≥ f(x,t)>0 ∀x∈S,t >0; (1.3) (f4) there exists f :R→Rsuch that lim|x|→∞f(x,t)= f(t) uniformly for bounded t >0, f(x,t)≥ f(t), for allx∈S,t≥0, and limt→∞(f(x,t)/t)= ∞uniformly for x∈S;
(f5) f(x,·)∈C2(0, +∞) and (∂2/∂t2)f(x,t)≥0 for allx∈S,t≥0.
Givenε >0, by (f1) and (f2), there exists aCε>0 such that
0≤ f(x,u)≤εu+Cε|u|p−1, (1.4) 0≤F(x,u)≤εu2+Cε|u|p. (1.5) IfΩ=RNorΩ=RN\D(m=0 in our case), then the homogeneous case of problem (1.1)λ (i.e., the caseh(x)≡0) has been studied by many authors; see Cao [1] and the references therein. For the nonhomogeneous case (h(x)≡0), Zhu-Zhou [2] have studied the multiplicity of positive solutions of equations similar to(1.1)λ. Recently, Chen [3]
showed that there exists a λ∗>0 such that (1.1)λ has exactly two positive solutions if λ∈(0,λ∗), and(1.1)λ has no positive solution whenλ∈(λ∗,∞). However, her method cannot determine whetherλ∗ is bounded or infinite (at least for general nonlinearity f(x,u)). In this paper, one of our results answers the question (seeTheorem 1.1). Now, we state our main results.
Theorem 1.1. LetΩ=S\DorΩ=RN\DorΩ=SorΩ=RN. Supposeh(x)≥0,h(x)≡ 0,h(x)∈L2(Ω)∩Lq0(Ω) for someq0> N/2 ifN≥4,q0=2 ifN=2, 3, and f(x,t) satisfies (f1)–(f5). Then there existsλ∗>0, 0< λ∗<∞, such that
(i) equation(1.1)λhas at least two positive solutionsuλ,Uλanduλ< Uλifλ∈(0,λ∗);
(ii) equation(1.1)λ∗has a unique positive solutionuλ∗; (iii) equation(1.1)λhas no positive solutions ifλ > λ∗,
whereuλis the minimal solution of(1.1)λandUλis the second solution of(1.1)λconstructed inSection 4.
Theorem 1.2. Under the assumptions ofTheorem 1.1, then
(i)uλis strictly increasing with respect toλ,uλis uniformly bounded inL∞(Ω)∩H01(Ω) for allλ∈(0,λ∗] and
uλ−→0 inL∞(Ω)∩H01(Ω) asλ−→0+; (1.6)
(ii)Uλis unbounded inL∞(Ω)∩H01(Ω) forλ∈(0,λ∗), that is,
λlim→0+
Uλ=lim
λ→0+
Uλ∞= ∞, (1.7)
whereUλ =(Ω(|∇U|2+U2)dx)1/2andUλ∞=supx∈Ω|U(x)|.
First of all, we list some properties of f(x,t). The proof can be found in Zhu-Zhou [2, Lemma 2.1].
Lemma 1.3. Assume (f1), (f3), and (f5) hold, then
(i)t f(x,t)≥νF(x,t) for allx∈S,t >0 andν=1 +θ−1∈(2,p+ 1];
(ii)t−1/θf(x,t) is monotone nondecreasing andt−1f(x,t) is strictly monotone increasing for allx∈S,t >0;
(iii) f(x,t1+t2)≥f(x,t1) +f(x,t2) and f(x,t1+t2)≡ f(x,t1) +f(x,t2) for allx∈S, t1,t2>0.
2. Asymptotic behavior of solutions
Throughout this paper, letx=(y,z) be the generic point ofRN withy∈Rm,z∈Rn, N=m+n≥2,n≥1. We denote byCandCi(i=1, 2,. . .) universal constants, maybe the constants here should be allowed to depend onnandp, unless some statement is given, and denote (∂/∂t)f(x,t) and (∂2/∂t2)f(x,t) by f(x,t) andf(x,t), respectively, in what follows.
We define
u =
Ω
|∇u|2+u2dx 1/2
, up=
Ω|u|pdx 1/ p
, 2≤p <∞, u∞=sup
x∈Ω
u(x) .
(2.1)
Now, we introduce the equation at infinity associated with(1.1)λon an unbounded cylin- der domainS,
−Δu+u=λ f(u) inS,
u∈H01(S), N≥2. (2.1)λ
P. L. Lions has studied the following minimization problem closely related to(2.1)λ: S∞=infI∞(u) :u∈H01(S),u≡0,I∞(u)=0>0, (2.2) whereI∞(u)=(1/2)S(|∇u|2+u2)dx−λSF(u+)dx,F(t)=t
0 f(s)ds. For this problem, also a minimum exists and is realized by a ground state solutionw >0 inSsuch that
S∞=I∞(w)=sup
t≥0
I∞(tw). (2.3)
In order to get the asymptotic behavior of solutions of(1.1)λand(2.1)λ, we need the following Lemmas2.3 and2.5. First, we quote two regularity lemmas (see Hsu [4] for the proof). Now, letXbe aC1,1domain inRN(typically the domains considered in the introduction).
Lemma 2.1. Letf :X×R→Rbe a Carath ´eodory function such that for almost everyx∈X, there holds
f(x,u) ≤C|u|+|u|p−1
uniformly inx∈X, (2.4) where 2< p <2∗. Ifu∈H01(X) is a weak solution of equation−Δu= f(x,u) +h(x) inX, whereh∈LN/2(X)∩L2(X), thenu∈Lq(X) forq∈[2,∞).
Lemma 2.2. Letg∈L2(X)∩Lq(X) for someq∈[2,∞) and letu∈H01(X) be a weak solu- tion of the equation−Δu+u=ginX. Thenu∈W2,q(X) satisfies
uW2,q(X)≤CuLq(X)+gLq(X)
, (2.5)
whereC=C(N,q,∂X).
By Lemmas2.1and2.2, we obtain the first asymptotic behavior of solution of(1.1)λ. Lemma 2.3 (asymptotic lemma 1). Let (f1), (f2) hold and letu be a weak solution of (1.1)λ, thenu(y,z)→0 as|z| → ∞uniformly fory∈ω. Moreover, there exist positive con- stantsC1andC2such that
u∞≤C1uq0+λC2
up(p−−11)q0+hq0
. (2.6)
Proof. Suppose that u is a solution of (1.1)λ, then −Δu+u=λ(f(x,u) +h(x)) in Ω.
Since h∈L2(Ω)∩Lq0(Ω) for some q0> N/2 ifN≥4,q0=2 ifN=2, 3, this implies h∈L2(Ω)∩LN/2(Ω) forN≥2. By (1.4) andLemma 2.1, we conclude that
u∈Lq(Ω) forq∈[2,∞). (2.7)
Henceλ(f(x,u) +h(x))∈L2(Ω)∩Lq0(Ω) and byLemma 2.2, we have
u∈W2,2(Ω)∩W2,q0(Ω), q0> N/2 ifN≥4, q0=2 ifN=2, 3. (2.8) Now, by the Sobolev embedding theorem, we obtain thatu∈Cb(Ω). It is well known that the Sobolev embedding constants are independent of domains (see Adams [5]). Thus there exists a constantCsuch that forR >0,
uL∞(Ω\BR)≤CuW2,q0(Ω\BR) forN≥2, (2.9)
whereBR= {x=(y,z)∈Ω| |z| ≤R}. From this, we conclude thatu(y,z)→0 as|z| → ∞ uniformly fory∈ω. ByLemma 2.2and (1.4), we also have that
u∞≤CuW2,q0(Ω)
≤Cuq0+λ f(x,u) +λh(x)q0
≤C1uq0+λC2
u(pp−−11)q0+hq0
,
(2.10)
whereC1,C2are constants independent ofλ.
Remark 2.4. Letwbe a positive solution of(2.1)λ. Ifh(x)≡0 and f(x,t)≡ f(t) for all x∈S,t∈R, byLemma 2.3, then we have thatw(y,z)→0 as|z| → ∞uniformly for y∈ω.
We useLemma 2.3, and modify the proof in Hsu [6], we obtain a precise asymptotic behavior of solutions of(2.1)λat infinity and the second asymptotic behavior of solutions of(1.1)λ.
Lemma 2.5 (asymptotic lemma 2). Letwbe a positive solution of(2.1)λ, letube a positive solution of(1.1)λand letϕbe the first positive eigenfunction of the Dirichlet problem−Δϕ= μ1ϕinω, then for anyε >0 with 0< ε <1 +μ1, there exist constantsC,Cε>0 such that
w(y,z)≤Cεϕ(y) exp−
1 +μ1−ε|z| , w(y,z)≥Cϕ(y) exp−
1 +μ1|z|
|z|−(n−1)/2 u(y,z)≥Cϕ(y) exp−
1 +μ1|z|
|z|−(n−1)/2.
as|z| −→ ∞,y∈ω, (2.11)
Proof. (i) First, we claim that for anyε >0 with 0< ε <1 +μ1, there existsCε>0 such that w(y,z)≤Cεϕ(y) exp−
1 +μ1−ε|z|
as|z| −→ ∞, y∈ω. (2.12) Without loss of generality, we may assumeε <1. Now givenε >0, by (f1), (f4), and Remark 2.4, we may chooseR0large enough such that
λ fw(y,z)≤λ fx,w(y,z)≤εw(y,z) for|z| ≥R0. (2.13) Letq=(qy,qz),qy∈∂ω,|qz| =R0, andBa small ball inΩsuch thatq∈∂B. Sinceϕ(y)>
0 for x=(y,z)∈B,ϕ(qy)=0,w(x)>0 for x∈B,w(q)=0, by the strong maximum principle (∂ϕ/∂y)(qy)<0, (∂w/∂x)(q)<0. Thus
limx→q
|z|=R0
w(x) ϕ(y)=
(∂w/∂x)(q) (∂ϕ/∂y)qy
>0. (2.14)
Note thatw(x)ϕ−1(y)>0 for x=(y,z), y∈ω,|z| =R0. Thusw(x)ϕ−1(y)>0 forx= (y,z),y∈ω,|z|=R0. Sinceϕ(y) exp(−
1 +μ1−ε|z|) andw(x) belong toC1(ω×∂BR0(0)), if set
Cε= sup
y∈ω,|z|=R0
w(x)ϕ−1(y) exp1 +μ1−εR0
, (2.15)
thenCε>0 and
Cεϕ(y) exp−
1 +μ1−εR0
≥w(x) fory∈ω,|z| =R0. (2.16) LetΦ1(x)=Cεϕ(y) exp(−
1 +μ1−ε|z|) forx∈Ω. Then for|z| ≥R0, we have Δw−Φ1
(x)− w−Φ1
(x)= −λ fw(x)+
ε+
1 +μ1−ε(n−1)
|z|
Φ1(x)
≥ −εw(x) +εΦ1(x)
=εΦ1−w(x).
(2.17)
HenceΔ(w−Φ1)(x)−(1−ε)(w−Φ1)(x)≥0 for|z| ≥R0.
The strong maximum principle implies thatw(x)−Φ1(x)≤0 forx=(y,z), y∈ω,
|z| ≥R0, and therefore we get this claim.
(ii) Let Ψ(y,z)=
1 +1
|z|
ϕ(y) exp−
1 +μ1|z|
|z|−(n−1)/2 for (y,z)∈Ω. (2.18) It is very easy to show that
−ΔΨ+Ψ≤0 fory∈ω,|z|large. (2.19) Therefore, by means of the maximum principle, there exists a constantC >0 such that
w(y,z)≥Cϕ(y) exp−
1 +μ1|z|
|z|−(n−1)/2 u(y,z)≥Cϕ(y) exp−
1 +μ1|z|
|z|−(n−1)/2 as|z| −→ ∞, y∈ω. (2.20)
This completes the proof ofLemma 2.5.
3. Existence of the minimal solution
We now prove the existence of minimal positive solutions of(1.1)λ.
Lemma 3.1. If (f1) and (f2) hold, then for any givenρ >0, there existsλ0>0 such that for λ∈(0,λ0), one hasI(u)>0 for allu∈Sρ= {u∈H01(Ω)| u =ρ}. Moreover, for anyε≥ 0, there existsδ >0 (δ≤ρ) such thatI(u)≥ −εfor allu∈ {u∈H01(Ω)|ρ−δ≤ u =ρ}. Proof. By (1.5), the Sobolev embedding theorem, and the H¨older inequality, we have that, for allu∈Sρ,
I(u)=1
2u2−λ
ΩFx,u+dx−λ
Ωhu dx
≥1
2u2−λ
Ω
ε|u|2+Cε|u|p
dx−λh2u
≥1
2u2−λCu2+up
dx−λh2u
≥ρ 1
2ρ−λCρ+ρp−1−λh2
,
(3.1)
whereC >0 is a constant which is independent ofλ,ρ. Hence by (3.1), there existsλ0>0 such that forλ∈(0,λ0), we haveI(u)>0 for allu∈Sρ.
Moreover, we can chooseλ0>0 small enough such that
∂
∂ρ 1
2ρ−λCρ+ρp−1=1
2−λ1 + (p−1)ρp−2>0 forλ∈ 0,λ0
. (3.2)
Then for anyε≥0, there existsδ >0 (δ≤ρ) such thatI(u)≥ −εfor allu∈ {u∈H01(Ω)|
ρ−δ≤ u ≤ρ}.
Lemma 3.2. Assume (f1) and (f2) hold. Ifλ0is chosen as inLemma 3.1andλ∈(0,λ0), then there exists au0∈Bρsuch thatu0is a positive solution of(1.1)λ.
Proof. Sinceh≡0 andh≥0, we can choose a functionϕ∈H01(Ω) such thatΩhϕ >0.
Fort∈(0, +∞), then by (1.5), I(tϕ)=t2
2
Ω
|∇ϕ|2+ϕ2−λ
IRN+
Fx,tϕ+−λt
Ωhϕ
≤t2
2ϕ2+λCt2
Ω
|ϕ|2+tp−2|ϕ|p
−λt
Ωhϕ.
(3.3)
Then fortsmall enough,I(tϕ)<0. Soα=inf{I(u)|u∈Bρ}. Clearlyα >−∞. ByLemma 3.1, there exists ρ such that 0< ρ< ρ andα=inf{I(u)|u∈Bρ}. By Ekeland’s vari- ational principle [7], there exists a (PS)α-sequence{uk} ⊂Bρ, that is,I(uk)=α+o(1) andI(uk)=o(1) strongly inH−1(Ω) ask→ ∞. Then there exists a subsequence{uk} and u0∈H01(Ω) such that uku0 weakly inH01(Ω), uk→u0 strongly in Lqloc(Ω) for 2≤q <2∗anduk→u0a.e. inΩ. SinceI(uk)=o(1) strongly inH−1(Ω) ask→ ∞, and by (f1) and (f2), we haveI(u0)=0 inH−1(Ω), that is,u0is a weak nonnegative solution of(1.1)λ; and sinceh≡0, by the maximum principle for weak solutions, we haveu0>0
inΩ.
By the standard barrier method, we prove the following lemma.
Lemma 3.3. If (f1) and (f2) hold, then there existsλ∗∈(0, +∞] such that
(i) for anyλ∈(0,λ∗),(1.1)λhas a minimal positive solutionuλ anduλ is strictly in- creasing inλ;
(ii) ifλ > λ∗,(1.1)λhas no positive solution.
Proof. SettingQλ= {0< λ <+∞ |(1.1)λis solvable}, byLemma 3.2, we haveQλis non- empty. Denotingλ∗=supQλ>0, we claim that(1.1)λ has at least one solution for all λ∈(0,λ∗). In fact, for anyλ∈(0,λ∗), by the definition ofλ∗, we know that there exists λ>0 and 0< λ < λ< λ∗such that(1.1)λhas a solutionuλ>0, that is,
−Δuλ+uλ=λfx,uλ
+h(x)≥λfx,uλ
+h(x). (3.4)
Thenuλis a supersolution of(1.1)λ. Fromh(x)≥0 andh(x)≡0, it is easy to see that 0 is a subsolution of(1.1)λ. By the standard barrier method, there exists a solutionuλ>0 of (1.1)λsuch that 0≤uλ≤uλ. Since 0 is not a solution of(1.1)λandλ> λ, the maximum
principle implies that 0< uλ< uλ. Again using a result of Amann [8, Theorem 9.4], we
can choose a minimal positive solutionuλof(1.1)λ.
Letuλbe the minimal positive solution of(1.1)λforλ∈(0,λ∗), we study the following eigenvalue problem
−Δv+v=σλfx,uλ
v inΩ,
v∈H01(Ω), v >0 inΩ, (3.5)
then we have the following.
Lemma 3.4. Assume (f1)–(f5) hold, and let the first eigenvalueσλof (3.5) be defined by σλ=inf
Ω
|∇v|2+v2dx|v∈H01(Ω),
Ωfx,uλv2dx=1
. (3.6)
Then
(i)σλis achieved;
(ii)σλ> λand is strictly decreasing inλ,λ∈(0,λ∗);
(iii)λ∗<+∞and(1.1)λ∗has a minimal positive solutionuλ∗.
Proof. (i) Indeed, recall assumption (f3), by the definition ofσλ, we know that 0< σλ<
+∞. Let{vk} ⊂H01(Ω) be a minimizing sequence ofσλ, that is,
Ωfx,uλv2kdx=1,
Ω
∇vk 2+v2kdx−→σλ ask−→ ∞. (3.7)
This implies that{vk}is bounded inH01(Ω), then there exists a subsequence, still denoted by{vk}and somev0∈H01(Ω) such that
vk v0 weakly inH01(Ω), vk−→v0 almost everywhere inΩ,
vk−→v0 strongly inLsloc(Ω) for 2≤s <2∗.
(3.8)
Thus
Ω
∇v0 2+v20dx≤lim inf
Ω
∇vk 2+vk2dx=σλ. (3.9)
ByLemma 2.3and (f1), we have f(x,uλ)→0 as|x| → ∞, it is standard to show thatv0
achievesσλ. Clearly|v0|also achievesσλ. By (3.5) and the maximum principle, we may assumev0>0 inΩ.
(ii) We now proveσλ> λ. Settingλ> λ >0 andλ∈(0,λ∗), byLemma 3.3,(1.1)λhas a positive solutionuλ. Sinceuλis the minimal positive solution of(1.1)λ, thenuλ> uλas λ> λ. By virtue of(1.1)λand(1.1)λ, we see that
−Δuλ−uλ+uλ−uλ=λfx,uλ−λ fx,uλ+ (λ−λ)h. (3.10)
Applying the Taylor expansion and noting thatλ> λ,h(x)≥0, and f(x,t)≥0,f(x,t)>
0 for allt >0, we get
−Δuλ−uλ
+uλ−uλ
≥(λ−λ)fx,uλ
+λfx,uλ
uλ−uλ
> λ fx,uλ
uλ−uλ
. (3.11)
Letv0∈H01(Ω) andv0>0 solves (3.5). Multiplying (3.11) byv0and noting (3.5), then we get
σλ
Ωfx,uλ
uλ−uλ
v0dx > λ
Ωfx,uλ
uλ−uλ
v0dx, (3.12) henceσλ> λ. Now, letvλbe a minimizer ofσλ, then
Ωfx,uλv2λdx >
Ωfx,uλv2λdx=1, (3.13) and there existst, with 0< t <1 such that
Ωfx,uλ tvλ2
dx=1. (3.14)
Therefore
σλ≤t2vλ2<vλ2=σλ (3.15) showing thatσλis strictly decreasing inλforλ∈(0,λ∗).
(iii) We show next thatλ∗<+∞. Letλ0∈(0,λ∗) be fixed. For anyλ≥λ0, we have σλ> λand by (3.15), then
σλ0≥σλ> λ (3.16)
for allλ∈[λ0,λ∗). Thusλ∗<+∞. By (3.5) andσλ> λ, we have
Ω
∇uλ 2+ uλ 2dx >
Ωλ fx,uλu2λdx, (3.17) and also we have
Ω
∇uλ 2+ uλ 2 dx−
Ωλ fx,uλ
uλdx−
Ωλh(x)uλdx=0. (3.18) By (f3) and (3.17), we have that
Ω
∇uλ 2+ uλ 2 dx=
Ωλ fx,uλ
uλdx+
Ωλh(x)uλdx
≤θ
Ωλ fx,uλu2λdx+λh2uλ
≤θuλ2+λh2uλ.
(3.19)
This implies that
uλ≤ λ
1−θh2 (3.20)
for allλ∈(0,λ∗). ByLemma 3.3(i), the solutionuλis strictly increasing with respect to λ; we may suppose that
uλ uλ∗ weakly inH01(Ω) asλ−→λ∗, (3.21) and by (1.4), we obtain that
Ω
∇uλ· ∇ϕ+uλϕdx−→
Ω
∇uλ∗· ∇ϕ+uλ∗ϕdx,
λ
Ω
fx,uλ+hϕ dx−→λ∗
Ω
fx,uλ∗+hϕ dx
asλ−→λ∗ (3.22)
for allϕ∈H01(Ω). Henceuλ∗ is a minimal positive solution of(1.1)λ∗. This completes the
proof ofLemma 3.4.
4. Existence of second solution
Whenλ∈(0,λ∗), we know that(1.1)λhas a minimal positive solutionuλbyLemma 3.3, then we need only to prove that(1.1)λhas another positive solution in the form ofUλ= uλ+v, wherevis a solution of the following equation:
−Δv+v=λfx,uλ+v−fx,uλ inΩ,
v >0 inΩ, v∈H01(Ω). (4.1)
We define the energy functionalJ:H01(Ω)→Ras follows:
J(v)=1 2
Ω
|∇v|2+v2dx−λ
Ω
Fx,uλ+v+−Fx,uλ
−fx,uλ
v+dx. (4.2)
Using the monotonicity of f and the maximum principle, we know that the nontrivial critical points of energy functionalJare the positive solutions of (4.1).
First, we give an inequality about concerning f anduλ.
Lemma 4.1. If (f1) and (f2) hold, then for anyε >0, there existsCε>0 such that
fx,uλ+s−fx,uλ−fx,uλs≤εs+Cεsp−1, s≥0, uniformly∀x∈S, (4.3) where 1< p <2∗−1 anduλis the minimal solution of(1.1)λ.
