R E S E A R C H
Open Access
Sign-changing solution and ground state
solution for a class of
(
p
,
q
)-Laplacian
equations with nonlocal terms on
R
N
Rui Li and Zhanping Liang
**Correspondence: [email protected] School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, P.R. China
Abstract
In the paper, we investigate the least energy sign-changing solution and the ground state solution of a class of (p,q)-Laplacian equations with nonlocal terms onRN. Applying the constraint variational method, the quantitative deformation lemma, and topological degree theory, we see that the equation has one least energy
sign-changing solutionu. Moreover, we regardc,das parameters and give a
convergence property of such a solutionuc,das (c,d)→0. Finally, using the Lagrange
multiplier method, we obtain a ground state solution of the equation and show that the energy ofuis strictly larger than two times the ground state energy.
Keywords: (p,q)-Laplacian equation; sign-changing solution; ground state solution; nonlocal term
1 Introduction
In this paper, we discuss the existence of a least energy sign-changing solution and a ground state solution of the following equation:
–
a+c
RN|∇u|
p
pu–
b+d
RN|∇u|
q
qu+h(x)|u|p–u+g(x)|u|q–u
=f(u), x∈RN, (.)
where ≤q<p<q∗,N< p,m=div(|∇u|m–∇u) is them-Laplacian operator,m∗=∞
forN≤m, andm∗=Nm/(N–m) forN>m.a,b are positive constants,c,d≥. We assume thath,gare continuous, coercive and positive functions.
Whenc=d= , equation (.) is the following (p,q)-Laplacian equation:
–apu–bqu+h(x)|u|p–u+g(x)|u|q–u=f(u), x∈RN. (.)
A special situation for (.) is the case wherep=q> ,i.e., a singlep-Laplacian equation. Whenp=q= , (.) becomes the nonlinear Laplacian type equation
–u+au=f(x,u), x∈RN. (.)
Equation (.) appears, for example, as the stationary version of a general reaction-diffusion equation
ut=divD(u)∇u+f(x,u),
whereudescribes a concentration,D(u) =|∇u|p–+|∇u|q–is the diffusion coefficient,
andf(x,u) is the reaction term connected with source and loss mechanisms. This equa-tion has extensive applicaequa-tions in physics and related sciences such as biophysics, plasma physics, and chemical reaction design. Typically, in chemical and biological applications, the reaction termf(x,u) is a polynomial ofuwith variable coefficients (see [–]).
The differential operator p +q is known as the (p,q)-Laplacian operator, ifp=q.
The singlep-Laplacian operator has been studied for at least four decades (see [, –]), whereas a deeper research involving the (p,q)-Laplacian operator has only arisen in the last decade (see [–, –]).
In [], the authors investigated the existence of a positive solution of equation (.) wherea> is a constant. In [], the authors proved the existence of sign-changing so-lutions of equation (.) where a∈L∞loc(RN) and ess infa> . We also refer the
inter-ested reader to more related results as regards equation (.) in [, ] and the references therein. In [], the authors proved the existence of least energy positive, negative, and sign-changing solutions for thep-Laplacian equation with potentials vanishing at infinity. In [], the author obtained multiplicity solutions of thep-Laplacian equation with a critical nonlinearity. Since the (p,q)-Laplacian operator is not homogeneous, some technical dif-ficulties appear when using the common methods of the elliptic equations. The existence of a nontrivial solution to equation (.) was obtained in [, , ]. In [], the authors dealt with the situation ≤q≤p<Nwithh∈LN+/p(RN) andg∈LN+/q(RN), whereas in
[] the authors considered the case <q<p<N, but thereh,gare positive constants. In [, ], the nonlinearityf(x,s) was suitably controlled by the variablesass→ and also as
|s| → ∞, uniformly with respect to the variablex. In [], the authors discussed the case that <q<p<q∗,p<Nwithh,gcontinuous, positive, and coercive functions onRNand f(x,s) a Carathéodory function satisfying some conditions.
To the best of our knowledge, there is little work researching the sign-changing solu-tion and the ground state of the (p,q)-Laplacian equations (.). Recently, Shuai in [] discussed the following Kirchhoff type problem:
⎧ ⎨ ⎩
–(a+b|∇u|)u=f(u), x∈,
u= , x∈∂.
Motivated by [], we investigate the sign-changing solution and the ground state solution of (p,q)-Laplacian equation with nonlocal terms.
In general, the working space to study (p,q)-Laplacian problems in a bounded domain
isW,p(), by taking advantage of the compact embeddingW,p()→Ls() for all s∈[,p∗). When the domain is the wholeRN, Sobolev’s embedding loses compactness. In
In this paper, we intend to choose an appropriate approach by taking into account the Banach space,
W=
u∈D,pRN∩D,qRN:
RNh|u|
p,
RNg|u|
q<∞
.
We recall that the spaceD,m(RN) is a reflexive Banach space which is characterized by
(see [])
D,mRN=
u∈Lm∗RN: ∂u
∂xi ∈L
mRN
and its norm is equivalent to the norm∇uLm(RN). We denote the norm ofLm(RN) as | · |mhereafter. Moreover,W,m(RN)⊂D,m(RN)→Lm
∗ (RN).
We takeh,gas continuous, coercive, and positive functions onRN and define normed
spaces (Wp,a,h, · ) and (Wq,b,g, · ), respectively, by
Wp,a,h=
u∈D,pRN:
RNh|u|
p<∞
,
Wq,b,g=
u∈D,qRN:
RNg|u|
q<∞
,
with norms
u=
RN
a|∇u|p+h|u|p
/p
,
u=
RN
b|∇u|q+g|u|q
/q
.
ThenWp,a,h andWq,b,g are reflexive Banach spaces. The embeddingWp,a,h→Ls(RN) is
continuous for all s∈[p,p∗] and compact for all s∈[p,p∗). Similarly, the embedding
Wq,b,g→Ls(RN) is continuous ifs∈[q,q∗] and compact ifs∈[q,q∗) (see []).
Now we can define our working spaceW:
W=Wp,a,h∩Wq,b,g
endowed with the norm
u=u+u.
Then it is easy to see thatWis a reflexive Banach space and the embeddingW→Ls(RN)
is continuous ifs∈[q,p∗] and compact ifs∈[q,p∗).
For brevity, we omit the integral domainRNwhen no confusion arises hereafter.
We assume thatf∈C(R,R) satisfies the following hypotheses:
(f) lims→f(s)/|s|q–= ;
(f) for some constantr∈(p,p∗),lim|s|→∞f(s)/|s|r–= ;
(f) lim|s|→∞F(s)/|s|p=∞, whereF(s) =
t
f(t) dtfor alls∈R;
Define the energy functionalI:W→Rof (.) by
I(u) =
pu p
+
qu q
+
c
p
|∇u|p
+ d q
|∇u|q
–
F(u), u∈W. (.)
Then the functionalIis well defined onW and belongs toC(W,R). Moreover, for any
u,ϕ∈W, we have
I(u),ϕ= a|∇u|p–∇u· ∇ϕ+h|u|p–uϕ+ b|∇u|q–∇u· ∇ϕ+g|u|q–uϕ
+c
|∇u|p
|∇u|p–∇u· ∇ϕ
+d
|∇u|q
|∇u|q–∇u· ∇ϕ–
f(u)ϕ. (.)
A critical point ofIcorresponds to a solution of (.). Furthermore, ifu∈Wis a solution of (.) withu±= , thenuis a sign-changing solution of (.), where
u+(x) =maxu(x), , u–(x) =minu(x), .
Obviously, the energy functionalI:W→Rof (.) is given by
I(u) =
pu p
+
qu q
–
F(u), u∈W.
Foru∈W,
I(u) =I
u++I
u–, I(u),u±=Iu±,u±. (.)
Whenc,d> , the nonlocal terms (|∇u|p)pu, (|∇u|q)quare involved in equation
(.), for the functionalIgiven by (.) it is apparent that
I(u) =Iu++Iu–+c
p ∇u
+p
∇u–p+d
q ∇u
+q
∇u–q, (.)
I(u),u±=Iu±,u±+c ∇u+p ∇u–p+d ∇u+q ∇u–q. (.)
Clearly, the functionalIdoes no longer satisfy (.), since it contains two nonlocal terms. Hence, there may be some differences in investigating the sign-changing solution of equa-tion (.) betweenc,d> andc=d= .
In order to obtain a sign-changing solution of equation (.), we try to seek a minimizer of the functionalIover the following constraint:
M=u∈W:u±= ,I(u),u+=I(u),u–= (.)
and
Then we show that the minimizer is indeed a sign-changing solution of (.). As we have mentioned before, the functionalI no longer satisfies the properties (.), so it is more difficult to prove thatM=∅. Actually, we will obtainM=∅by using the Brouwer fixed point theorem, which is different from the approach in [].
In order to get the ground state solution of equation (.), let
N =u∈W\ {}:I(u),u= , (.)
and consider the ground state energy
˜
m=infI(u) :u∈N. (.)
Now, we state our main results.
Theorem . If the assumptions(f)-(f)hold, then equation(.)has one least energy
sign-changing solution.
Theorem . Suppose the assumptions (f)-(f)hold. For any sequence {(cn,dn)} with
cn,dn≥,as(cn,dn)→,there exists a subsequence,still denoted by{(cn,dn)},such that ucn,dn→uin W,and uis a least energy sign-changing solution of equation(.).
Theorem . Suppose the assumptions(f)-(f)hold.
(i) There exists a ground state solutionvof equation(.).
(ii) m> m˜.In particular,the ground state solution must maintain the sign unchanged.
Remark . The three results above are also valid for (p,q)-Laplacian problems in a bounded domain. Consider the following two problems:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
–(a+c|∇u|p)pu– (b+d
|∇u|q)qu
+h(x)|u|p–u+g(x)|u|q–u=f(u), x∈,
u= , x∈∂
(.)
and
⎧ ⎨ ⎩
–apu–bqu+h(x)|u|p–u+g(x)|u|q–u=f(u), x∈,
u= , x∈∂,
where is a bounded domain inRN, h,g are continuous and non-negative functions,
including the caseh≡g≡. Because the embeddingW,m()→Ls() is continuous if s∈[,m∗] and compact ifs∈[,m∗), we find solutions in the spaceW,p()∩W,q(), and then can also obtain the same conclusions as Theorems .-. for (.).
Both the conclusions of (.) and of (.) are true whenp=q,i.e., these results are true for a singlep-Laplacian equation with nonlocal term.
constrained problem (.) is a sign-changing solution. Then we prove the convergence property of solutions of (.). Finally, we prove the existence of the ground state solution and give the energy comparison.
Throughout this paper,C andCk denote various positive constants, which may vary from line to line.
2 Preliminaries
We use constraint minimization onMto seek a critical point ofI. We begin this section by doing some preparation work.
Lemma . Assume that(f)-(f)hold.If u∈W with u= ,then
(i) lims→ f(su)u
|s|q– = ;
(ii) lim|s|→∞ |sf|(sup–)us=∞;
(iii) lim|s|→∞ F|s(|sup) =∞;
(iv) moreover,ifu±= ,thenlim|(s,t)|→∞
F(su+)+F(tu–)
|s|p+|t|p =∞.
Proof (i) By the conditions (f) and (f), for any givenε> , there existsCε> such that
f(s)≤ε|s|q–+Cε|s|r–, s∈R, (.)
F(s)≤ε
q|s| q+Cε
r |s|
r, s∈R. (.)
By the condition (f), we have, for eachη∈R,
lim
s→
f(sη)
|s|q–= . (.)
Thus, by (.), (.) and the Lebesgue dominated convergence theorem, the conclusion (i) holds.
(ii) By the conditions (f) and (f), we have
lim
|s|→∞ f(s)
|s|p–s=∞. (.)
It follows from (.) that, for any givenM> , there existsR> such that
f(s)s
|s|p ≥M, |s|>R. (.)
By the condition (f), we have
lim
s→
f(s)s–M|s|p |s|q = .
Then there existsCM> such that
f(s)s–M|s|p
It follows from (.) and (.) that
f(s)s≥M|s|p–CM|s|q, s∈R. (.)
It follows from (.) that
lim inf
|s|→∞
f(su)u
|s|p–s≥M
|u|p– lim
|s|→∞ CM
|s|p–q
|u|q=M
|u|p.
Then, by the arbitrariness ofM, the conclusion (ii) is true.
(iii) By the condition (f), for any givenM> , there existsR> such that
F(s)
|s|p ≥M, |s|>R. (.)
By the condition (f), we have
lim
s→
F(s) –M|s|p |s|q = .
Then there existsCM> such that
F(s) –M|s|p
|s|q ≥–CM, |s| ∈(,R]. (.)
It follows from (.) and (.) that
F(s)≥M|s|p–CM|s|q, s∈R. (.)
Then it follows from (.) that
lim inf
|s|→∞
F(su)
|s|p ≥M
|u|p– lim
|s|→∞ CM
|s|p–q
|u|q=M
|u|p.
Thus, by the arbitrariness ofM, the conclusion (iii) is also true.
(iv) For convenience, we denote functionsψ(s) =
F(su+) andψ(s) =
F(su–) for all
s∈R. Then, by (iii), we have
ψ(s)→ ∞, ψ(s)→ ∞, |s| → ∞. (.)
Because of (.) and the continuity ofψ,ψ, there existsC> such that
ψ(s)≥–C, ψ(s)≥–C, s∈R. (.)
By (iii), for any givenM> , there existsR> such that
ψ(s) +C |s|p ≥M,
ψ(s) +C
When|(s,t)|=√s+t≥√R, by the inequality √
s+t≤√max|s|,|t|,
max{|s|,|t|} ≥R. We may suppose that|s| ≥ |t|, so that|s| ≥R. Combining with (.) and (.), we have
ψ(s) +C+ψ(t) +C |s|p+|t|p ≥
ψ(s) +C
|s|p ≥M.
Then we have
lim
|(s,t)|→∞
ψ(s) +C+ψ(t) +C
|s|p+|t|p =∞. (.)
Therefore, it follows from (.) that (iv) holds.
Remark . By the condition (f), for eachη∈R\ {}, we see thatf(sη)η/|s|p–is
in-creasing on (–∞, ) and (,∞), respectively. Therefore, for eachu∈Wwithu= , we see that fs(sup–)uis increasing on (,∞).
Now we start to check that the setMis nonempty.
For each u ∈ W with u± = , for convenience, we denote the positive numbers
A,u= (
|∇u+|p),A ,u= (
|∇u–|p),A ,u=
|∇u+|p|∇u–|p;B
,u= (
|∇u+|q),B ,u=
(|∇u–|q),B ,u=
|∇u+|q|∇u–|q.
Lemma . Assume that(f)-(f)hold.If u∈W with u±= ,then there is a unique pair
(su,tu)of positive numbers such that suu++tuu–∈M.
Proof For any givenu∈W with u±= , we define a function u:R+×R+ →Rby
u(s,t) =I(su++tu–), whereR+= [,∞), that is,
u(s,t) =
ps pu+p
+
qs qu+q
+
c
pA,us
p+c pA,us
ptp
+ d qB,us
q+d qB,us
qtq–
Fsu+
+
pt pu–p
+
qt qu–q
+
c
pA,ut
p
+ d qB,ut
q–
Ftu–. (.)
Fors,t> , since
∇u(s,t) =
∂u
∂s (s,t),
∂u
∂t (s,t)
=Isu++tu–,u+,Isu++tu–,u–
=
s
Isu++tu–,su+,
t
we havesu++tu–∈Mif and only if (s,t) is a critical point of
u. Next we will prove the
existence of a critical point ofu.
For any givent∈R+, we have, fors> ,
∂
∂su(s,t)
=sp–u+p+sq–u+q+cA,usp–+cA,usp–tp
+dB,usq–+dB,usq–tq–
fsu+u+
=sq–
sp–qu+p+u+q+cA,usp–q+cA,usp–qtp
+dB,usq+dB,utq–
f(su+)u+
sq–
(.)
=sp–
spu
+p +
sp–qu
+q
+cA,u+
tp spcA,u
+ dB,u
sp–q + tqdB,u
sp–q –
f(su+)u+
sp–
. (.)
Sinceu+= , it follows from (.) and Lemma .(i) that ∂
∂su(s,t) > fors> small. It
follows from (.) and Lemma .(ii) that∂∂su(s,t) < fors> large. Thus there exists
s> such that∂∂su(s,t) = .
Suppose that there exist s,s with <s<s such that ∂∂su(s,t) = ∂
∂su(s,t) = .
Then (.) implies that
spiu
+p
+
sip–q
u+q+cA,u+ tp spicA,u+
dB,u sip–q+
tqdB,u sip–q
=
f(siu+)u+
sip– , i= , .
Hence
sp –
sp
u+p+
sp–q –
sp–q
u+q+
sp –
sp
tpcA,u
+
sp–q –
sp–q
dB,u+
sp–q –
sp–q
tqdB,u
= f(su
+)u+
sp– –
f(su+)u+
sp–
. (.)
But according to Remark ., the right side of (.) is negative and (.) is absurd. There-fore there exists a uniques=s(t) > such that ∂∂su(s,t) = .
Now we can define a mapϕ:R+→(,∞) byϕ(t) =s(t), wheres(t) satisfies the
prop-erties just mentioned previously, withtin the place oft. By definition, we have
∂u
∂s
ϕ(t),t
that is, fort≥,
ϕp–(t)u+p+ϕq–(t)u+q+cA,uϕp–(t) +cA,uϕp–(t)tp
+dB,uϕq–(t) +dB,uϕq–(t)tq
=
fϕ(t)u+
u+. (.)
We will prove some properties of the functionϕ.
(a)ϕhas a positive lower bound.
In fact, suppose there exists {tn} ⊂R+ such that ϕ(tn)→ . Then, by (.) and
Lemma .(i), we have
u+q≤ lim
n→∞
f(ϕ(tn)u+)u+
ϕq–(tn)
= .
This is absurd. Thus there existsC> such thatϕ(s)≥Cfor alls∈R+.
(a)ϕis continuous.
In fact, lettn→tinR+. We firstly prove that{ϕ(tn)}is bounded. Suppose, by
contra-diction, that there is a subsequence{tnk}of{tn}such thatϕ(tnk)→ ∞. It follows from
(.) that
ϕp(tnk)u
+p +
ϕp–q(tnk)
u+q+cA,u+ tpnk
ϕp(tnk)cA,u+
dB,u
ϕp–q(tnk)
+ t
q nkdB,u
ϕp–q(tnk)
=
f(ϕ(tnk)u +)
ϕp–(tnk) u
+. (.)
Lettingk→ ∞in (.), according to Lemma .(ii), we have a contradictioncA,u=∞.
Thus, {ϕ(tn)} is bounded. For any subsequence {ϕ(tn)} of {ϕ(tn)}, since {ϕ(tn)} is
bounded, there exists a subsequence{ϕ(tn)}of{ϕ(tn)}such thatϕ(tn)→sand it follows
from (a) thats> . Passing to the limit asn→ ∞in (.) witht=tn, we get
sp–u+p+sq–u+q+cA,usp–+cA,usp–t
p
+dB,usq–+dB,usq–t
q
=
fsu+
u+. (.)
Thus (.) and (.) imply
∂u
∂s (s,t) = .
Consequently, by the uniqueness,s=ϕ(t). Thereforeϕis continuous.
(a)ϕ(t)≤tfortlarge.
In fact, if there exists a sequence{tn}withtn→ ∞such thatϕ(tn) >tnfor alln∈N,
thenϕ(tn)→ ∞and it follows from (.) that∞ ≤cA,u+cA,u. This is a contradiction.
Thusϕ(t)≤tfortlarge.
Similarly, for eachs∈R+, we consider the functionu(s,·) and consequently, we can
define a mapϕ:R+→(,∞) which satisfies
∂u
∂t
s,ϕ(s)
that is, fors≥,
ϕp–(s)u–p+ϕq–(s)u–q+cA,uϕp–(s) +cA,uϕp–(s)sp
+dB,uϕq–(s) +dB,uϕq–(s)sq
=
fϕ(s)u–
u–, (.)
and it also satisfies (a), (a), and (a) above.
Now we prove the existence of a critical point ofuby the Brouwer fixed point theorem.
By (a), there existsC> such thatϕ(t)≤tfor allt>Candϕ(s)≤sfor alls>C. Let
C=max
max
t∈[,C]ϕ(t),s∈max[,C]ϕ(s)
.
Letξ=max{C,C}. We defineT: [,ξ]→R+asT(s) =ϕ(ϕ(s)). Now we showT(s)∈
[,ξ] for alls∈[,ξ]. In fact, let ≤s≤ξ=max{C,C}. Ift=ϕ(s) >C, then
T(s) =ϕ(t)≤t=ϕ(s)≤ ⎧ ⎨ ⎩
s, s>C,
maxs∈[,C]ϕ(s), s≤C,
so
T(s)≤max{C,C}.
Ift=ϕ(s)≤C, then
T(s) =ϕ(t)≤ max
t∈[,C]ϕ(t)≤C.
Note thatT is continuous. Then, by the Brouwer fixed point theorem, there existssu∈
[,ξ] such thatϕ(ϕ(su)) =su. Lettu=ϕ(su). Then we have
su=ϕ(tu), tu=ϕ(su). (.)
Sinceϕi> , (.) impliessu,tu> . By the definition we have
∂u
∂s (su,tu) =
∂u
∂t (su,tu) = .
Thus, (su,tu) is a critical point ofu.
Now we prove the uniqueness of (su,tu). In fact, consideringw∈Mwe have
∇w(, ) =
∂w
∂s (, ),
∂w
∂t (, )
=Iw++w–,w+,Iw++w–,w–= (, ),
which implies that (, ) is a critical point ofw. Now we prove that (, ) is the unique
a critical point ofwwith <t≤s. Then it follows from (.) and (.) that
spw+p+sqw+q+cA,wsp+cA,wspt
p
+dB,wsq+dB,wsqt
q
=
fsw+
sw+, (.)
tpw–p+tqw–q+cA,wtp+cA,wspt
p
+dB,wtq+dB,wsqt
q
=
ftw–
tw–. (.)
From (.) andt≤s, we have
spw+p+sqw+q+c(A,w+A,w)sp+d(B,w+B,w)sq≥
fsw+
sw+. (.)
On the other hand, sincew∈M, we have
w+p+w+q+cA,w+cA,w+dB,w+dB,w=
fw+w+. (.)
Hence, from (.) and (.), we get
–
sp
w+p+
–
sp–q
w+q+
–
sp–q
d(B,w+B,w)
≤ fw+w+–f(sw
+)w+
sp–
.
From the above inequality and Remark . we conclude thats≤ and then <t≤s≤.
Now we prove thatt≥. In fact, from (.) and <t≤s, we have
tpw–p+tqw–q+c(A,w+A,w)tp+d(B,w+B,w)tq≤
ftw–
tw–. (.)
On the other hand, sincew∈M, we get
w–p+w–q+cA,w+cA,w+dB,w+dB,w=
fw–w–. (.)
Now from (.) and (.), we obtain
–
tp
w–p+
–
tp–q
w–q+
–
tp–q
d(B,w+B,w)
≥ fw–w––f(tw
–)w–
tp–
.
By Remark ., we conclude thatt≥. Consequently,t=s= , this shows that (, ) is
the unique critical point ofwwith positive coordinates.
Now we assume thatu∈W withu±= and (s,t), (s,t) are both critical points with
positive coordinates for the mapu. Then
Therefore,
w=
s
s
su++
t
t
tu–=
s
s
w++
t
t
w– ∈M.
Sincew∈Mand (ss,tt) is a critical point of the mapwwith positive coordinates, by
the uniqueness we have
s
s
=t
t
= ,
which implies that (s,t) = (s,t).
Lemma . For a fixed u∈W with u±= ,the vector (su,tu),which was obtained in Lemma.,is the unique maximum point of the functionu(s,t).
Proof From the proof of Lemma ., (su,tu) is the unique critical point ofuin (,∞)×
(,∞). By (.), we have
u(s,t) =
sp+tp p
sp sp+tpu
+p
+
q sq sp+tpu
+q
+
p tp sp+tpu
–p
+
q tq sp+tpu
–q
+sp+tpd
qB,u sq sp+tp +
d qB,u
sqtq sp+tp +
d
qB,u tq sp+tp
+sp+tp c
pA,u sp sp+tp+
c pA,u
sptp sp+tp+
c
pA,u tp sp+tp
–sp+tp F(su
+) +F(tu–)
sp+tp
:=sp+tp(s,t) +(s,t) +(s,t) –
F(su+) +F(tu–)
sp+tp
.
It is clear that (s,t),(s,t)→ as |(s,t)| → ∞ and(s,t) is bounded. Then, by
Lemma .(iv), we deduce thatu(s,t)→–∞as|(s,t)| → ∞. So it is sufficient to check
that a maximum point cannot be obtained on the boundary ofR+×R+. Without loss of
generality, we may assume that (,¯t) is a maximum point ofu. Similar to (.), we can
get ∂
∂su(s,¯t) > forssmall. Thenu(s,t¯) is an increasing function with respect tosifsis
small enough, the pair (,t¯) is not a maximum point ofuinR+×R+.
Lemma . Let(f)-(f)hold.Suppose that u∈W with u±= such thatI(u),u+ ≤, I(u),u– ≤.Then the unique pair(s
u,tu)of positive numbers obtained in Lemma. satisfies <su,tu≤.
Proof We may suppose thatsu≥tu> . Sincesuu++tuu–∈M,
spuu+p+suqu+q+cA,usup+cA,usup+dB,usuq+dB,usuq
≥spuu+p+squu+q+cA,usup+cA,usputup+dB,usuq+dB,usqutuq
=
The assumptionI(u),u+ ≤ gives
u+p+u+q+cA,u+cA,u+dB,u+dB,u≤
fu+u+. (.)
Combining (.) and (.), we get
spu
– u+p+
sup–q
– u+q+
sup–q
–
(dB,u+dB,u)
≥ f(suu+)u+
sup–
–fu+u+
.
Ifsu> , then the left side of this inequality is negative. But by Remark ., the right side is positive, thus we must havesu≤. Then the proof is completed. From Lemma . andM⊂N, we know thatN is nonempty andm,m˜ is well defined. Now we prove the following lemma.
Lemma . Assume that(f)-(f)hold.If v∈W with v= ,then there is a unique sv∈
(,∞)such that svv∈N.Moreover,ifI(v),v ≤,then sv∈(, ].
Proof For fixedv∈Wwithv= ands∈(,∞),sv∈N if and only ifI(sv),sv= , where
I(sv),sv=sq
sp–qvp+vq+csp–q
|∇v|p
+dsq
|∇v|q
–
f(sv)v sq–
(.)
=sp
spv p
+
sp–qv q
+c
|∇v|p
+ d
sp–q
|∇v|q
–
f(sv)v sp–
. (.)
Sincev= , it follows from (.) and Lemma . (i) thatI(sv),sv> fors> small. On the other hand, it follows from (.) and Lemma .(ii) thatI(sv),sv< fors> large. Thus there existssv> such thatI(svv),svv= .
Now we prove the uniqueness ofsv. Suppose that there exists,swith <s<ssuch
thatI(sv),sv=I(sv),sv= . Then (.) implies that
spiv p
+
sip–qv q
+c
|∇v|p
+ d
sip–q
|∇v|q
=
f(siv)v
sip– , i= , .
Hence,
sp –
sp
vp+
sp–q –
sp–q
vq+
d sp–q –
d sp–q
|∇v|q
= f(sv)v
sp– – f(sv)v
sp–
.
Now we claim thatsv∈(, ]. It follows from (.) that
spv
vp+
svp–q
vq+c
|∇v|p
+ d
svp–q
|∇v|q
=
f(svv)v svp–
. (.)
The assumptionI(v),v ≤ gives
vp+vq+c
|∇v|p
+d
|∇v|q
≤ f(v)v. (.)
Combining (.) and (.), we have
–
spv
vp+
–
svp–q
vq+
–
svp–q
d
|∇v|q
≤ f(v)v–f(svv)v
svp–
.
According to Remark ., it is absurd ifsv> . Thussv∈(, ]. Lemma . Assume that(f)-(f)hold.Then m≥ ˜m> ,and m,m can both be obtained˜ .
Proof (i) For any givenε> , by (.), we have
f(u)u≤ε
|u|q+Cε
|u|r, u∈W. (.)
Hence, for some ε> small, by the continuous embedding of Wp,a,h →Lr(RN) and Wq,b,g→Lq(RN), we get
f(u)u≤
u
q
+Cur, u∈W. (.)
For everyv∈N, we haveI(v),v= , that is,
vp+vq+c
|∇v|p
+d
|∇v|q
=
f(v)v. (.)
For everyu∈M, we haveI(u),u±= , that is,
u±p+u±q+c
|∇u|p ∇u±p+d
|∇u|q ∇u±q=
fu±u±. (.)
Hence, for someε> small, it follows from (.), (.), (.), and (.) that
wp+wq≤
w
q
+Cwr, w=v,u±, (.)
wp+wq≤ε
|w|q+Cε
|w|r, w=v,u±. (.)
So, by (.), there exists a constantα> , which is not dependent onc,d, such that
w≥α, w=v,u±, (.)
By the condition (f) andf ∈C(R,R), we have
f(s)s– (p– )f(s)s≥, s∈R. (.)
By (.), we have
f(s)s– pF(s)≥, s∈R. (.)
Then
I(u) =I(u) – p
I(u),u
= pu
p
+
q –
p
uq+
q–
p
d
|∇u|q
+
p f(u)u– pF(u)
> pu
p
+
q –
p
uq≥
pα
p. (.)
This implies thatm˜ ≥αp/(p).
(ii) Let{vn} ⊂N such thatI(vn)→ ˜m. Then it follows from (.) that{vn}is bounded
inWand there existsv∈Wsuch thatvnvinW.
Let {un} ⊂M⊂N such that I(un)→m. Then it follows from (.) that {un} is bounded inWand there existsu∈Wsuch thatu±nu±inW(see Lemma A.).
Sincevn∈N,un∈M, it follows from (.) that
αp≤ wnp ≤ε
|wn|q+Cε
|wn|r, wn=vn,u±n.
Using the boundedness of{wn}, there isC> such that
αp≤εC+Cε
|wn|r.
Choosingε=αp/(C
), we get
|wn|r≥
αp
C
,
whereC=Cε. By the compactness of the embeddingW→Lr(RN), we get
|w|r≥ α p
C
, w=v,u±. (.)
Thusw= . Equations (.) and (.) combined with the Lebesgue dominated convergence theorem give
lim
n→∞
f(wn)wn=
f(w)w, lim
n→∞
F(wn) =
(iii) By the weak lower semi-continuity of the norm, we have
vp+vq+c
|∇v|p
+d
|∇v|q
≤lim inf
n→∞
vnp+vnq+c
|∇vn|p
+d
|∇vn|q
.
Then from (.) we get
vp+vq+c
|∇v|p
+d
|∇v|q
≤
f(v)v. (.)
From (.) and Lemma ., there existssv∈(, ] such thatv¯=svv∈N.
It follows from (.) thatG(s) =f(s)s– pF(s) is a non-negative function, increasing on [,∞), and decreasing on (–∞, ]. Then we have
˜
m≤I(v¯) – p
I(¯v),v¯
= p¯v
p + q– p ¯vq+
q–
p
d
|∇¯v|q
+
p f(v¯)v¯– pF(v¯)
= ps
p vv
p + q– p
sqvvq+
q –
p
dsvq
|∇v|q
+
p f(svv)svv– pF(svv)
≤
pv p + q– p vq+
q–
p
d
|∇v|q
+
p f(v)v– pF(v)
≤lim inf
n→∞
I(vn) – p
I(vn),vn=m˜.
We then deduce thatsv= . Thus,v¯=vandI(v) =m˜.
(iv) By the weak lower semi-continuity of the norm, we have
u±p+u±q+c
|∇u|p ∇u±p+d
|∇u|q ∇u±q
≤lim inf
n→∞
u±np+u±nq+c
|∇un|p ∇u±np+d
|∇un|q ∇u±nq
.
Then from (.) we get
u±p+u±q+c
|∇u|p ∇u±p+d
|∇u|q ∇u±q≤
fu±u±. (.)
From (.) and Lemma ., there exists (su,tu)∈(, ]×(, ] such that
¯
Note thatG(s) =f(s)s– pF(s) is a non-negative function, increasing on [,∞) and de-creasing on (–∞, ]. Then we have
m≤I(u¯) – p
I(u¯),u¯
= p¯u
p + q– p ¯uq+
q–
p
d
|∇ ¯u|q
+
p f(u¯)u¯– pF(u¯)
= ps
p uu+
p + q– p
squu+q+
q–
p
dsuq ∇u+q
+
p f
suu+suu+– pFsuu++
q– p
dsqutqu ∇u+q ∇u–q
+ pt
p uu–
p + q – p
tquu–q+
q–
p
dtuq ∇u–q
+
p f
tuu–tuu–– pFtuu–
≤
pu p + q– p uq+
q–
p
d
|∇u|q
+
p f(u)u– pF(u)
≤lim inf
n→∞
I(un) –
p
I(un),un
=m.
We then deduce thatsu=tu= . Thus,u¯=uandI(u) =m.
3 Proof of the main results
The purpose of this section is to prove our main results. We start to prove that the min-imizerufor the minimization problem (.) is indeed a sign-changing solution of (.), that is,I(u) = .
Proof of Theorem. Using the quantitative deformation lemma and topological degree theory, we prove thatI(u) = .
It is clear thatI(u),u+=I(u),u–= . It follows from Lemma . that, for (s,t)∈ R+×R+and (s,t)= (, ),
Isu++tu–<Iu++u–=m. (.)
It follows from (.) that|u±|r≥αp/(C
) :=τr. Then|u±|r≥τ. We denote byγrthe
embedding constant ofW→Lr(RN).
IfI(u)= , then there existr,ρ> such that
I(v)≥ρ, v–u ≤r. (.)
Letδ∈(,min{τ/(γr),r/}) and letσ∈(,min{/,δ/(u),δ/(u)}). LetD= ( –
σ, +σ)×( –σ, +σ) andϕ(s,t) =su++tu–for all (s,t)∈D. It follows from (.) that
¯
m= max
(s,t)∈∂DI
Letε=min{(m–m¯)/,ρδ/}andS=B(u,δ). Then it follows from (.) that
I(v)≥ε/δ, v∈I–[m– ε,m+ ε]∩Sδ. (.)
Applying (.) and Lemma . in [], p., there exists a deformation η∈C([, ]×
W,W) such that
(b) η(,v) =vifv∈/I–([m– ε,m+ ε])∩Sδ;
(b) η(,Im+ε∩S)⊂Im–ε;
(b) η(,v) –v ≤δfor allv∈W.
By Lemmas . and ., for (s,t)∈D, we knowI(ϕ(s,t))≤m<m+ε, that is,ϕ(s,t)∈Im+ε. Since
ϕ(s,t) –up =su++tu––u+–u–p
=|s– |pu+p+|t– |pu–p
≤σpup
< (δ/)p,
and similarlyϕ(s,t) –uq< (δ/)q, we haveϕ(s,t) –u=ϕ(s,t) –u
+ϕ(s,t) –u<δ.
Thusϕ(s,t)∈S. By (b), we haveI(η(,ϕ(s,t))) <m–ε. Then it is clear that
max (s,t)∈D
Iη,ϕ(s,t)≤m–ε<m. (.)
We will prove that η(,ϕ(D))∩M=∅, which is a contradiction with (.). Therefore,
I(u) = , that is,uis a sign-changing solution for equation (.). In fact, onDwe also defineψ(s,t) =η(,ϕ(s,t)) and
(s,t) =
Iϕ(s,t),su+,Iϕ(s,t),tu–
=Isu++tu–,su+,Isu++tu–,tu–,
(s,t) =
Iψ(s,t),ψ+(s,t),Iψ(s,t),ψ–(s,t).
Let
P(s,t) =Isu++tu–,su+
=spu+p+squ+q+cA,usp+cA,usptp
+dB,usq+dB,usqtq–
fsu+su+,
Q(s,t) =Isu++tu–,tu–
=tpu–p+tqu–q+cA,utp+cA,usptp
+dB,utq+dB,usqtq–
By direct calculation, we have
∂P(s,t)
∂s
(,)
= (p– )u+p+ (q– )u+q+ (p– )cA,u+ (p– )cA,u
+ (q– )dB,u+ (q– )dB,u–
fu+u+,
∂P(s,t)
∂t
(,)
=pcA,u+qdB,u,
∂Q(s,t)
∂s
(,)
=pcA,u+qdB,u,
∂Q(s,t)
∂t
(,)
= (p– )u–p+ (q– )u–q+ (p– )cA,u+ (p– )cA,u
+ (q– )dB,u+ (q– )dB,u–
fu–u–.
By (.), we get
∂P(s,t)
∂s
(,)
< –(pcA,u+qdB,u)
and
∂Q(s,t)
∂t
(,)
< –(pcA,u+qdB,u).
Set the matrix
M=
∂P(,)
∂s
∂P(,)
∂t ∂Q(,)
∂s
∂Q(,)
∂t
.
Then we get
J(, ) =detM> .
SinceisC, (, ) is the unique isolated zero of, by Lemmas . and . in [], p.,
we have
deg(,D, ) =ind
, (, )
=sgnJ(, ) = .
It follows from (.),m¯ <m– ε, and (b) above thatϕ=ψon∂D. Hencedeg(,D, ) = deg(,D, ) = . Thus there exists a pair (s,t)∈Dsuch that(s,t) = . Since|u±|r≥
τ, (s,t)∈D, we have|ϕ+(s,t)|r=s|u+|r≥τ/ and|ϕ–(s,t)|r=t|u–|r≥τ/. By (b),
we have
ψ(s,t) –ϕ(s,t)r≤γrψ(s,t) –ϕ(s,t)≤γrδ.
This implies that
