R E S E A R C H
Open Access
Existence of solutions of elliptic boundary
value problems with mixed type
nonlinearities
Anmin Mao
*, Yan Zhu and Shixia Luan
*Correspondence: [email protected] School of Mathematical Sciences, Qufu Normal University, Shandong, 273165, P.R. China
Abstract
We study the existence of a nontrivial solution of the following elliptic boundary value problem with mixed type nonlinearities:
–u=f(x,u) in
,
u= 0 on
∂
,wheref(x,u) = –Ku+Wu. We consider the problem in a different case:
lim|u|→∞f(x,u)/u=∞, lim|u|→0f(x,u)/uis some constant. Assuming thatKsatisfies
the “pinching” condition, andWsatisfies a more general superquadratic growth condition than the well-known Ambrosetti-Rabinowitz condition usually used in literature, we obtain a nontrivial solution via the Mountain Pass Lemma.
MSC: 35J65; 35J20; 47J10
Keywords: pinching condition; Mountain Pass Lemma; Cerami condition
1 Introduction
In this paper, we shall be concerned with the elliptic boundary value problem in a different case
⎧ ⎨ ⎩
–u=f(x,u) in,
u= on∂, (P)
where⊂RN (N> ) is a bounded open domain with a smooth boundary∂andf ∈ C(×R,R).
The existence of nontrivial weak solutions for (P) have been studied in many papers, see [–]. Su and Zhao in [] considered problem (P) for resonance case at infinity,
lim|u|→∞f(xu,u)=λk, whereλkis an eigenvalue of the linear boundary value problem
⎧ ⎨ ⎩
–u=λu in,
u= on∂, (P)
the existence of multiple nontrivial solutions for (P) are obtained by minimax methods and Morse theory. Ambrosetti and Rabinowitz in [] established the existence of a nontrivial solution for problem (P) by assuming the following conditions:
(f) f(x, ) = ,limu→f(xu,u)= , uniformly in a.e.x∈. (f) There exist two positive constantsaandbsuch that
f(x,u)≤a+b|u|p for some≤p<N+ N– ,∀u∈R
,x∈.
And the following well-known Ambrosetti-Rabinowitz condition ((AR) for short):
∃θ> ,R> s.t. <θF(x,u)≤uf(x,u), for all|u| ≥R,x∈,
whereF(x,u) =uf(x,s)ds.
Since then, the (AR) condition has been used extensively in many literature sources, see [–]. It is well known that the (AR) condition is quite natural and convenient not only to ensure that the Euler-Lagrange functional associated to problem (P) has a mountain pass geometry but also to guarantee that the Palais-Smale sequence of the Euler-Lagrange functional is bounded. LetEbe a Hilbert space andG∈C(E,R). Recall that the sequence
{un}n∈N⊂Eis said to be a Palais-Smale sequence ofGprovided that{G(un)}is bounded andG(un)→ asn→+∞, the functionGsatisfies the Palais-Smale condition ((PS) for short) if and only if any Palais-Smale sequence forGcontains a convergent subsequence. The functionGsatisfies the Cerami condition ((C) for short) if any sequence{un}n∈N in E satisfyingG(un) is bounded andG(un)( +un)→ asn→+∞has a convergent subsequence.
Without (AR), it becomes more complicated. Indeed, there are many functions which are superlinear, but it is not necessary to satisfy (AR) even if <θ ≤. Willem and Zou stated the following examples:
f(x,u) =μ|u|μ–u+ (μ– )|u|μ–usinu+|u|μ–sinu, u∈R\ {},
whereμ> . Then it is easy to check that (AR) does not hold even for anyθ>μ– > . On the other hand, in order to verify (AR), it usually is an annoying task to compute a primitive function off and sometimes it is almost impossible. For example,
f(x,u) =|u|u +e(+|sinu|)α+|cosu|α , u∈R,
whereα> .
Some authors have tried to drop or weaken the above superlinear condition (AR) in recent years, see [–, , ]. Miyagaki and Souto [] adapted some monotonicity argu-ments studying the existence of nontrivial weak solutions of (P).
The aim of the manuscript is to consider the problem in a different case:
lim|u|→∞f(x,u)/u=∞, lim|u|→f(x,u)/u is some constant. We study this problem
main difficulty when dealing with this problem is the lack of compactness of the Sobolev embedding theorem.
In this paper, here,F(t,u) :=uf(t,s)dsreplaced by –K+W, satisfy
(F) F(x,u) = –K(x,u) +W(x,u),K,W:×R→RareC-maps. (K) There are two positive constantsbandbsuch that
b|u|≤K(x,u)≤b|u|, for all(x,u)∈×R.
(K) There exists∈(, ]such that
K(x,u)≤Ku(x,u)u≤K(x,u), for all(x,u)∈×R.
(W) W(x,u)≥andWu(x,u) =o(|u|)as|u| →uniformly inx.
(W) W(x,u)/u→ ∞as|u| → ∞uniformly inx.
(W) SetW˜(x,u) := Wu(x,u)u–W(x,u),W˜ (x,u) > ifu= , W˜(x,u)→ ∞as|u| → ∞ uniformly in x, and there exist r > and σ > N/ such that |Wu(x,u)|σ ≤ cW˜(x,u)|u|σ if|u| ≥r.
We will prove the following results.
Theorem . If assumptions (F), (K), (K) and (W)-(W) are satisfied, then problem (P)
has a nontrivial weak solution.
Remark
(i) Our assumptions (W), (W) are weaker than (AR), and there is no monotone condition;
(ii) The condition (K) can be written in the form≤KKu((xx,,uu))u≤,∈(, ]which is weaker than the condition≤KKu((xx,,uu))u≤in [, ].
Example Consider the functions
K(x,u) = +exp–|x| u, W(x,u) =
– +|x|
uln +u .
A straightforward computation shows thatKandW(x,u) satisfy the assumptions of The-orem ., but neitherF(x,u) norW(x,u) satisfy the (AR) condition.
Example Consider the more general functions
Ku(x,u) =V(x)u, Wu(x,u) =g(x,u),
whereg(x,u) is of superlinear growth as|u| → ∞. A straightforward computation shows thatKandWsatisfy the assumptions of Theorem ..
2 Preliminary results
LetH:=H
() be the Sobolev space equipped with the inner product and the norm
(u,v) =
(∇u· ∇v+uv)dx, u= (u,u), u,v∈H.
And we denote the usualLp()-norm
up=
u(x)pdx
p
.
Our approach will be the variational techniques. Define the Euler-Lagrange functional associated to problem (P) given by
(u) =
|∇u|dx–
–K(x,u) +W(x,u)dx, for allu∈H.
From the assumptions onf, it is standard to check that ∈Cwhose Gateaux derivative
is
(u)v=
∇u· ∇v dx–
–Ku(x,u)v+Wu(x,u)v
dx, for allu,v∈H.
Letη:H→[, +∞) be given by
η(u) :=
|∇u|+ K(x,u)dx
.
Hence
(u) = η
(u) –
W(x,u)dx.
By (K),
(u)u≤
|∇u|dx+
K(x,u)dx–
Wu(x,u)u dx.
By (K) and setb,:=min{, b},b,:=max{, b},
b,u≤η(u)≤b,u.
It is worth pointing out that if the functionK(x,u) is of the form V(x)uwithV(x)∈
C(,R) andinf
V(x)≥V> thenηin a Hilbert spaceX={u∈H();
V(x)u<∞}
is equivalent to the norm · ; however, if the functionK(x,u) is not of the form V(x)u
,
ηis not a norm because of the lack of norm’s linear property.
Lemma . (see []) Let H be a real Banach space, ∈C(H,R), satisfying () = .
Moreover,
(ii) there existse∈H\Bρ()such that (e)≤.
Then there exists a sequence{un} ∈H such that (un)(+un)→and (un)→c≥α as n→ ∞.
Lemma .(see []) Assume that||<∞,≤p, r<∞, f ∈C(¯ ×R)and|f(x,u)| ≤ c( +|u|pr). Then, for every u∈Lp(), f(x,u)∈Lr()and the operator A:Lp()→Lr(),
u→f(x,u)is continuous.
3 Proofs of theorems
First of all, we recall a property of the functionK(x,u), which is necessary to the proof of the geometric structure of theCfunctional .
Fact Assume that (K) holds, then
K(x,u)≤K
x, u |u|
|u|, for all x∈and|u| ≥.
Proof DefineG:s→K(x,s–u)s,s∈(, +∞)
G(s) = –Ku
x,s–u u ss
+
Kx,s–u s–
= –Ku
x,s–u s–us–+Kx,s–u s–
=s––Ku
x,s–u s–u+Kx,s–u .
By (K),G(s)≥, which impliesG(s) is non-decreasing. So, we have
K(x,u) =G()≤G(s) =K
x, u |u|
|u|, if|u|=s≥.
Next we discuss the geometric structure of theCfunctional onH.
Lemma . Under the assumptions of Theorem ., there are constantsρ,α> such that
|∂Bρ()≥α.
Proof From (W) and (W), as|u|>r, we have
Wu(x,u)
σ
≤c
Wu(x,u)u–W(x,u)
|u|σ
≤cWu(x,u)|u|σ+,
wherecis a positive constant. Hence as|u|>r,
Wu(x,u)≤c|u|σ+/(σ–).
Fromσ>N/, we knowσ+ /(σ– ) < *– , so we can choose σ/(σ– )≤p< *. And
using (W) again, we observe that for any given> there isc> such that
and
W(x,u)≤|u|+c|u|p. (.)
It follows from (.) and the Sobolev embedding theorem that for allu∈H
W(x,u)dx≤u+cupp≤u+ccup, (.)
wherecis a positive constant. Then combining (K) and (.), we obtain
(u)≥
|∇u|dx+bu–
u+ccup
=
|∇u|dx+ (b–)u–ccup≥min
, (b–)
u–ccup
set b=min{, (b–)}, it is clear thatb> . We chooseu=ρ= ( b
cc)
p– andα=
ρ b
, then
(u)≥α.
Lemma . Under the assumptions of Theorem ., there exists e∈H\Bρ()such that
(e) < .
Proof Lete∈H\,M=maxx∈,|u|≤K(x,u) and A> (M+)e
e
. By (W), there exists
B> such that
W(x,u)≥A|u|–B, for allx∈,u∈H. (.)
As< , by Fact , we have
η(ξe) =
|∇ξe|+ K(x,ξe)
dx
≤
|∇e|ξdx+
{x∈;|ξe|≤}K(x,
ξe)dx+
{x∈;|ξe|≥}K(x,
ξe)dx
≤
|∇e|ξdx+ M||+ M
{x∈;|ξe|≥}
|ξe|dx
≤
|∇e|ξdx+ M||+ M
{x∈;|ξe|≥}|
ξe|dx
≤ξ( + M)e+ M||. (.)
Then, by inequalities (.) and (.), we get
(ξeo) = η
(ξe ) –
W(x,ξe)dx≤
+ M ξ
e
+M||–Aξe–B||
=
+ M e
–Ae
By the choice ofA, we have (+Me–Ae) < , so there existsξ∈Rsuch that if
e=ξe, then
(e) < .
Suppose that the assumptions of Theorem . hold, we have Lemma . and Lemma .. Now it follows from Lemma . that there is a sequence{un} ⊂Hsuch that
(u
n) +un → and (un)→c≥α asn→ ∞. (.)
Lemma . Under the assumptions of Theorem ., the functional satisfies the (C)
con-dition.
Proof Let{un} ⊂Hbe such that
(un) is bounded and
+un (un)→. (.)
By (K) we observe that for largen,
c≥ (un) –
(u
n)un
=
K(x,un)dx–
Ku(x,un)undx+
Wu(x,un)un–W(x,un)dx
≥
K(x,un) –
K(x,un)dx+
Wu(x,un)un–W(x,un)dx
≥
˜
W(x,un)dx. (.)
Arguing indirectly, assume as a contradiction thatun → ∞. Settingνn=un/un, then νn= and since the embedding H→Ls fors∈[, *), we haveνns≤γsνn=γs. Observe that, from (.), (K) and (K)
(u
n)un=
|∇un|dx+
Ku(x,un)undx–
Wu(x,un)undx
≥
|∇un|dx+
K(x,un)dx–
Wu(x,un)undx
≥
|∇un|dx+
b|un|
dx–
Wu(x,un)undx
≥
|∇un|
dx+
b|un|
dx–
Wu(x,un)undx
≥ un
b,–
Wu(x,un)νn un
dx
.
It follows that for any> andnlarge enough,
Wu(x,un)νn un
dx≥b,
Set forr≥
h(r) :=infW˜ (x,u) :x∈andu∈Rwith|u| ≥r.
By (W),h(r) > for allr> andh(r)→ ∞asr→ ∞. For ≤a<b, let
n(a,b) =
x∈:a≤un(x)<b
and
Cab=inf
˜ W(x,u)
u
x∈andu∈Rwitha≤un(x)<b
.
SinceW˜(x,u) > ifu= , one hasCb a> and
˜
Wx,un(x) ≥Cabun(x)
for allx∈n(a,b).
It follows from (.) that
c≥
˜
W(x,un)dx
=
n(,a)
˜
W(x,un)dx+
n(a,b)
˜
W(x,un)dx+
n(b,∞)
˜
W(x,un)dx
≥
n(,a)
˜
W(x,un)dx+Cab
n(a,b)
un(x)dx+h(b)n(b,∞). (.)
Setτ:= σ/(σ– ), sinceσ>N/, one seesτ∈(, *). Fix arbitrarilyτˆ∈(τ, *), using (.), n(b,∞)≤
c
h(b)→ asb→ ∞uniformly inn,
which implies by the Hölder inequality that
n(b,∞)
|νn|τdx≤
n(b,∞)
dx –ττ
n(b,∞)
|νn|τ τ τ dx
τ τ
≤γττn(b,∞)
–τ τ
→ (.)
asb→ ∞uniformly inn. Using (.) again, for any fixed <a<b,
n(a,b)
|νn|dx= un
n(a,b)
|un|dx≤ c
Cb
aun →
asn→ ∞. Let <<b,
, by (W), there existsa> such that
Wu(x,u)≤ γ|
Consequently,
n(,a)
Wu(x,un) |un| |
νn|dx≤
n(,a) γ|
νn|dx≤ γ
νn≤ (.)
for alln. By (W) and (.), we can take largeb≥rso that
n(b,∞)
Wu(x,un) |un| |
νn|dx≤
n(b,∞)
|Wu(x,un)|σ |un|σ
dx
σ
n(b,∞) |νn|σ
dx
σ
≤
cW˜(x,un)dx
σ
n(b,∞) |νn|τdx
τ
. (.)
Hence combining (.), (.) and (.), there isnsuch that
n(b,∞)
|Wu(x,un)||νn| |un|
dx≤(cc)
σ
n(b,∞) |νn|τdx
τ
< (.)
forn≥n. Note that there isγ =γ() > independent ofnsuch that
Wu(x,un)≤γ|un| forx∈n(a,b).
By (.)
n(a,b)
|Wu(x,un)||νn| |un|
dx≤γ
n(a,b)
|νn|dx< (.)
for alln≥n. Therefore, combining (.)-(.), we obtain forn≥n
|Wu(x,un)||νn| |un|
dx≤<b, –
which contradicts (.). Hence{un}is bounded inH. Going if necessary to a subsequence, we assume that
unu inHfor someu∈H,
which impliesun→ua.e. in, because the imbeddingH()→L() is compact. Hence
we haveun–u→ and|( (un) – (u))(un–u)| →. Using the Hölder inequality
fx,un(x) –f
x,u(x) un(x) –u(x) dx
≤
fx,un(x) –f
x,u(x) q
q
un(x) –u(x)p
p
(p+q = ) forun→uinLp(), and by (K), (K), (W), (W) we have
f(x,u)≤c +|u|p– =c +|u|
Then, by Lemma ., we havef(x,un(x))→f(x,u(x)) inLq(). Thus
fx,un(x) –f
x,u(x) un(x) –u(x) dx→
asn→+∞. Moreover, a straightforward computation shows that
(un) – (u) (un–u) =∇(un–u)
–
fx,un(x) –f
x,u(x) un(x) –u(x) dx
it is clear that
∇(un–u)
→. (.)
Finally,
un–u→ inH.
This completes the proof.
Now, we are ready to prove Theorem ..
We will obtain a critical point of λby the use of a standard version of the Mountain
Pass Lemma (see []). It provides the minimax characterization for the critical value which is important for what follows. Therefore, we state this lemma precisely.
Lemma .(see []) Let H be a real Banach space and λ:H→R be a C-smooth
functional. If satisfies the following conditions:
(i) () = ,
(ii) every sequence{un}n∈N inHsuch that{ (un)}n∈Nis bounded inRand (un)→
inH∗asn→+∞, contains a convergent subsequence ((PS) condition), (iii) there are constantsρ,α> such that |∂Bρ()≥α,
(iv) there is a constante∈H\Bρ()such that (e)≤,
where Bρ()is an open ball in H of radiusρcentered at , then possesses a critical value
c≥αgiven by
c=inf
g∈smax∈[,]
g(s) ,
where
=g∈C[, ],H :g() = ,g() =e}.
Now we are ready to give the proofs of Theorem ..
Proof of Theorem . Under conditions (F), (K), (K), (W)-(W), as shown in [], a
infg∈maxs∈[,] (g(s)). Hence,uis a nontrivial solution of problem (P) satisfying (u) =c,
(u) = . The proof is done.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The paper is the result of joint work of all authors who contributed equally to the final version of the paper. All authors read and approved the final manuscript.
Acknowledgements
The authors are grateful to anonymous referees for detailed reading of the manuscript and valuable comments, which helped us improve this work. This work was supported by The National Natural Science Foundation of China (No. 11101237) and SNSFC ZR2012AM006.
Received: 14 May 2012 Accepted: 20 August 2012 Published: 31 August 2012 References
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