Three positive solutions of a nonlinear Dirichlet problem with competing power nonlinearities

0.15 MB·English

by  Vladimir Lubyshev

Three positive solutions of a nonlinear Dirichlet problem with
competing power nonlinearities
Vladimir Lubyshev
January 19, 2015
Abstract
5
1 This paper studies a nonlinear Dirichlet problem for the p-Laplacian operator with nonlinearity
0 consistingofpowercomponents. Theproblemunderconsiderationcanbethoughofasaperturbation
2 of theAmbrosetti-Brezis-Cerami problem with concave-convex nonlinearity. The combined effect of
n power components in the perturbed nonlinearity allows to establish a higher order multiplicity of
a positive solutions. We study properties of the perturbed energy functional and prove the existence
J of threepositive solutions tothe problem at hand.

6
1
1 Introduction

]
P
In this paper, we study the following nonlinear Dirichlet problem:
A
h. −∆pu=λ |u|α−2u−|u|γ−2u +|u|β−2u in Ω,
(1)
at ( u=0(cid:0) (cid:1) on ∂Ω.
m
where Ω ⊂RN is a bounded smooth domain, λ is a real parameter, and ∆ u=∇·(|∇u|p−2∇u) is the
[ p
p-Laplacian operator. Problem (1) is studied under the following hypothesis on the exponents:

1
v (H0) 1<α<p<β <γ <p∗,
0
7
where p∗ is the critical Sobolev exponent defined by
8
3
0 pN/(N −p) if p<N,
p∗ :=
1. (∞ if p>N.
0
5 Our goalis to show that problem (1) has at least three positive solutions for any λ∈(0,λ∗) where λ∗ is
1 some positive real number.
:
v Multiplicity of positive solutions to nonlinear Dirichlet problem
i
X
−∆ u=f (·,u) in Ω,
p λ
r (2)
a ( u=0 on ∂Ω
has been extensively studied by many authors. In [1], problem (2) was studied with p = 2, bounded
Ω, and concave-convex nonlinearity f (x,u) = λ|u|α−2u+|u|β−2u, 1 < α < p < β 6 p∗, where the
λ
existence of two positive solutions for λ∈(0,λ∗), λ∗ >0, has been proved. This result was generalized
forp>2and/orothernonlinearoperators,andformoregeneralnonlinearitiesessentiallypreservingthe
concave-convexstructure (cf. [2,3,4,5]). Itis knownthat, atleastforthe casewhereΩ is the unit ball,
two positive solutions for the concave-convexproblem considered in [1] is the maximum one can expect
(cf. [6]).
Some recent papers that study higher order multiplicity of positive solutions to nonlinear problems
are [7, 8, 9]. In [7], a one-dimensional problem (2) with p=2 and f (x,u)=λg(u) has been considered
λ
for Ω=(−1,1). Under the assumption that g is concave on (0,γ) and convex on (γ,∞), γ >0, as well
as other assumptions such as g having a unique positive zero and lim g(u)/u=∞, the existence of
u→∞
1
threepositivesolutionshasbeenprovedforλ∈(λ ,λ∗),λ∗ >λ >0. In[8],problem(2)wasconsidered
∗ ∗
for p > N and f (x,u) = λa(x)u−γ +λg(x,u), γ > 0. Under some assumptions on the coefficients and
λ
the exponent γ, the existence of three positive solutions has been proved for λ belonging to an open
subinterval of (0,∞). Finally, in [9], a one-dimensional problem (2) (the authors considered a more
general form of the differential operator) with f (x,u)=λa(x)g(u) and Ω=(0,1) has been considered.
λ
Undervariousassumptions,includingg(u)6up−1 inapositiveneighborhoodof0,the existenceofthree
positive solutions has been shown for λ belonging to an open subinterval of (0,∞).
Problem (1), which we study in this paper, is different from that in [1] in the presence of the domi-
natingnegativeterm−λ|u|γ−2uwhichisresponsibleforproducinganextrapositivesolution. Thisleads
to the following main result of this paper.
Theorem 1.1. Under hypothesis (H0), there exists an λ∗ >0 such that for any λ∈(0,λ∗), problem (1)
has three distinct positive solutions u ,u ,u ∈C1,µ(Ω¯), 0<µ<1, such that
1 2 3
max(J (u ),J (u ))<0<J (u )
λ 1 λ 3 λ 2
where
1 λ 1 λ
J (v)= |∇v|p− |u|α− |u|β + |u|γ
λ
p α β γ
ZΩ ZΩ ZΩ ZΩ
is an energy functional for problem (1).
Many powerful methods exist in nonlinear analysis that help study the multiplicity of solutions to
nonlineardifferentialequations. ThesemethodsincludeMorsetheory,mountainpasslemma,fixedpoint
theorems, and the Pohozaev fibering method to name a few (cf. [10, 11, 12, 13] and references therein).
The fibering methodis especially usefulwhenthe nonlinearitycontainspolynomialcomponents andwill
serveastheunderlyingmethodinthispaper. Foritsapplicationstomultiplicityofsolutionstononlinear
equations and systems, cf. [14, 15, 2, 16] and references therein.
This paper is organized as follows. Section 2 contains some known results to be used in the paper.
Section 3 studies properties of the energy functional, J , necessary to prove Theorem 1.1. Sections 4–6
λ
study three optimization problems,eachleading to a positive solutionof problem(1). Finally, Section 7
is dedicated to proving our main result, Theorem 1.1.
2 Preliminaries
In this section, we list two known results that help studying critical points of a functional defined on a
Banach space. Throughout this section, differentiability always means Fr´echet differentiability.
Let(X,k·k)beaBanachspace,andletB andS berespectivelyitsclosedballandsphereofradius
ρ ρ
ρ. We are concerned with critical points of a differentiable functional J: X → R. In other words, we
want to study the equation
DJ(u)=0, u∈X. (3)
2.1 The fibering method
In this subsection, we present the Fibering Method of S. Pohozaev [17, 14, 13]. We will state its special
form, the spherical fibering, which is most suitable for our problem.
Suppose that the norm k·k of the Banach space X is differentiable away from the origin and that
1
J = k·kp+R
p
where p>1 and R: X →R is a differentiable functional.
Instead of studying critical points of the functional J directly, constrained critical points of the
fibering functional
I(v)=J(t(v)v)
where t=t(v) is a solution of
∂
J(tv)=0
∂t
2
are studied on S . Under certain assumptions, a constrained critical point v of I corresponds to a
1 ∗
critical point u =t(v )v of the original functional J.
∗ ∗ ∗
These constrainedcritical points typically arise as extrema of I on S . But because S is not weakly
1 1
closed, the limit of a maximizing orminimizing sequence,if it exists, may no longer be in S . There is a
1
regularity theorem that helps circumvent this difficulty. It reduces the problem to studying constrained
critical points of an extension of I to the unit ball. For reflexive X, this new problem is often more
tractable because of the weak compactness of B .
1
A rigorous formulation of the (spherical) fibering method combined with the regularity theorem is
presented below.
Theorem 2.1. Let U be an open subset of X with U ∩S 6=∅. Suppose that the equation
1
tp−1+DR(tv)v =0 (4)
has a solution t: B ∩U → [0,∞) that is differentiable on (B ∩U)\{0}. Consider the functional
1 1
I: (B ∩U)\{0}→R defined by
1
1
I(v)= t(v)p+R(t(v)v)
p
and let
M :={v∈B ∩U |t(v)6=0}.
1
Then for any critical point v ∈M of I on B ∩U,
∗ 1
(a) v ∈S ;
∗ 1
(b) u =t(v )v is a critical point of J, provided that DH(v )v 6=0 where H :=k·k.
∗ ∗ ∗ ∗ ∗
2.2 Mountain pass theorem
Another result in the critical point theory that we will use is the Mountain Pass Theorem by A. Am-
brosetti and P. Rabinowitz [18]. For that theorem to be applied, a certain compactness condition needs
to be satisfied, which is introduced is the definitions below.
Definition 2.1. Given c∈R, a sequence (u )⊂X is called a Palais-Smale sequence at level c ((PS) -
n c
sequence, in short) for the differentiable functional J: X →R if
J(u )→c and DJ(u )→0 in X∗.
n n
Definition 2.2. A differentiable functional J: X → R is said to satisfy the Palais-Smale condition if
given c∈R, every (PS) -sequence for J contains a (strongly) convergentsubsequence.
c
The theorem is question is presented below.
Theorem 2.2. Let J ∈C1(X,R) satisfy the Palais-Smale condition and let
max(J(0),J(w))<infJ
Sρ
for some kwk>ρ>0. Then J has a critical point u such that
∗
J(u )= inf max J(γ(t))
∗
γ∈Γt∈[0,1]
where
Γ:={γ ∈C([0,1],X)|γ(0)=0 and γ(1)=w}.
3
3 Some properties of the energy functional
It will be convenient for us to represent J as
λ
1 λ 1 λ
J (v)= |∇v|p− A(v)− B(v)+ C(v)
λ
p α β γ
ZΩ
where
A(u):= |u|α, B(u):= |u|β, C(u):= |u|γ.
ZΩ ZΩ ZΩ
Lemma 3.1. The energy functional J is coercive for any λ>0.
λ
Proof. It suffices to show that the functional
λ 1 λ
R (v):=− A(v)− B(v)+ C(v) (5)
λ
α β γ
is bounded from below on W1,p(Ω). It is readily seen that
0
λ
R (v)> kvkγ −c kvkβ −λc kvkα
λ γ Lγ 1 Lγ 2 Lγ
for some constants c and c independent of v. We complete the proof by noticing that the function
1 2
λ
s7→ sγ −c sβ −λc sα
1 2
γ
is bounded from below on [0,∞).
Let
1 ∂
Jˆ (t,v):= tp+R (tv) and Φ (t,v):=t−α Jˆ (t,v).
λ λ λ λ
p ∂t
Clearly, Jˆ (t,v)=J (tv) for v ∈S . Consider the equation
λ λ 1
Φ (t,v)=tp−α−tβ−αB(v)+λtγ−αC(v)−λA(v)=0, t>0. (6)
λ
Because of our assumption of the exponents, (H0), this equation has at most three solutions. Set
U :={v ∈W1,p(Ω)| equation (6) has three solutions in t}.
λ 0
Given v ∈ W1,p(Ω), the ith solution of (6) will be denoted t (v) = t (v), where the symbol λ will be
0 i i,λ
dropped for simplicity.
Lemma 3.2. There exists an Λ>0 such that for any λ∈(0,Λ), the following hold.
(a) U ∩S 6=∅.
λ 1
(b) U is open in W1,p(Ω).
λ 0
(c) For any v ∈U ∩B ,
λ 1
Jˆ (t (v),v)<0<Jˆ (t (v),v).
λ 1 λ 2
There exists an w ∈U ∩S such that
λ 1
J (t (w)w)<0.
λ 3
4
Proof. (a): By the Rellich-Kondrachov theorem, there is an v∗ ∈S such that
1
B(v∗)=maxB(v).
v∈S1
Choose Λ>0 so that
max tp−α−tβ−αB(v∗) >ΛmaxA(v) (7)
t>0 v∈S1
and (cid:2) (cid:3)
min tp−α−tβ−αB(v∗)+Λtγ−αC(v∗) <0.
t>0
Then it is clear that v∗ ∈U for a(cid:2)ll λ∈(0,Λ). (cid:3)
λ
(b): Consider the map
∂
T (t,v):= Φ (t,v), Φ (t,v) .
λ λ λ
∂t
(cid:18) (cid:19)
We will write T (t,v)> 0 (resp., T (t,v) <0) to mean that the two components of T (t,v) are strictly
λ λ λ
positive (resp., strictly negative).
It is readily seen that v ∈U if and only if
λ
T (s ,v)>0 and T (s ,v)<0
λ 1 λ 2
for some 0<s <s <∞. Now fix any v ∈U and choose 0<s0 <s0 <∞ with
1 2 0 λ 1 2
T (s0,v )>0 and T (s0,v )<0.
λ 1 0 λ 2 0
Since T (t,·) is continuousonW1,p(Ω) for anyt>0,there is a neighborhoodN ofv inW1,p(Ω) such
λ 0 v0 0 0
that
T (s0,v)>0 and T (s0,v)<0 for all v ∈N .
λ 1 λ 2 v0
This implies that N ⊂U . Since v was arbitrary, we conclude that U is open.
v0 λ 0 λ
(c): Reduce Λ, if necessary, so that
tp−α tβ−α Λ
max − B(v∗) > maxA(v) (8)
t>0 (cid:20) p β (cid:21) α v∈S1
and
tp tβ tγ
min − B(v∗)+Λ C(v∗) <0. (9)
t>0 p β γ
(cid:20) (cid:21)
Fix any v ∈U ∩B . Then the inequality
λ 1
Jˆ (t (v),v)<0
λ 1
follows from the fact that α<min(p,β,γ). It is clear that
t7→Jˆ (t,v) is increasing on [t (v),t (v)]. (10)
λ 1 2
By (7), for any v ∈B \{0}, the equation
1
Φ˜ (t,v):=tp−α−tβ−αB(v)−λA(v)=0
λ
has exactly two solutions, 0<t˜(v)<t˜(v)<∞.
1 2
Let us show that
t (v)<t˜(v)<t˜(v)<t (v). (11)
1 1 2 2
Since Φ (t,v)>Φ˜ (t,v) for all t>0,
λ λ
Φ (·,v)>0 on [t˜ (v),t˜(v)].
λ 1 2
Therefore, either [t˜(v),t˜(v)] ⊂ (t (v),t (v)) or [t˜ (v),t˜ (v)] ⊂ (t (v),∞). We want to verify that the
1 2 1 2 1 2 3
second option is impossible. Indeed, since ∂ Φ (t,v) > ∂ Φ˜ (t,v) for t > 0 and ∂ Φ˜ (t,v) > 0 on
∂t λ ∂t λ ∂t λ
(0,t˜(v)],
1
Φ (·,v) is increasing on (0,t˜(v)].
λ 1
5
This implies that t˜(v)<t (v), finishing the proof of (11).
1 2
Entertaining (10), (11), we deduce that
Jˆ (t (v),v)>Jˆ (t˜ (v),v)>H(t˜(v),v)
λ 2 λ 2 2
where
tp tα tβ
H(t,v):= −λ A(v)− B(v).
p α β
But by (8),
H(t˜(v),v)=maxH(t,v)>0,
2
t>0
yielding that Jˆ (t (v),v)>0.
λ 2
Finally, take w = v∗. The minimum point for the left hand side of (9) is bounded from below by a
positive constant as Λ ↓ 0, whereas the first zero of t 7→ J (tw) is o(1) as λ ↓ 0. This means that we
λ
can decrease Λ, if necessary, so that the third critical value of t 7→ J (tw) is negative or, equivalently,
λ
J (t (w)w)<0.
λ 3
4 First critical point
Let Λ > 0 be as in part (b) of Lemma 3.2. In view of inequality (7), equation (6) has the minimal
solution t=t (v) for any v ∈B \{0}. Moreover,
1 1
∂
Φ (t,v)>0. (12)
λ
∂t
(cid:12)t=t1(v)
(cid:12)
(cid:12)
Lemma 4.1. The functional t is continuo(cid:12)usly differentiable and weakly continuous on B \{0}.
1 1
Proof. The statement of the lemma follows from (12), the Implicit Function Theorem, and the weak
continuity of A,B, and C.
Define the functional I : B \{0}→R by
1 1
I (v):=Jˆ (t (v),v)<0.
1 λ 1
Theorem 4.1. For any λ ∈ (0,Λ), there is a nonnegative v ∈ S such that u = t (v )v is a critical
∗ 1 ∗ 1 ∗ ∗
point of J with J (u )<0.
λ λ ∗
Proof. Consider the problem
c (λ):= inf I (v). (13)
1 1
v∈B1\{0}
Let(v )be its minimizing sequence,whichwemayassumetobe nonnegative. AccordingtoLemma3.1,
n
t :=t (v ) is a bounded sequence. Therefore, without loss of generality we can assume that
1,n 1 n
v ⇀v weakly in W1,p(Ω),
n ∗
t →t
1,n ∗
for some nonnegative v ∈B and 06t <∞.
∗ 1 ∗
Since c (λ) < 0, we must have 0 < t < ∞. In particular, equation (6) implies that v 6≡ 0. By
1 ∗ ∗
Lemma 4.1 and the weak closedness of B , the infimum in (13) is attained at v .
1 ∗
Entertaining Theorem 2.1 and Lemma 4.1, we conclude that v ∈ S and that u = t (v )v is a
∗ 1 ∗ 1 ∗ ∗
critical point of J . Since v ∈S , J (u )=I (v )<0.
λ ∗ 1 λ ∗ 1 ∗
6
5 Second critical point
Consider the continuously differentiable functional J¯ : W1,p(Ω)→R defined by
λ 0
1
J¯ (v):= |∇v|p+R (v+)
λ λ
p
ZΩ
where v+ :=max(v,0).
Lemma 5.1. The following statements are true.
(a) J¯ satisfies the Palais-Smale condition for any λ>0.
λ
(b) There exists an ρ>0 such that for any λ∈(0,Λ),
max infJ¯ , inf J¯ <0<infJ¯ .
λ λ λ
Bρ W01,p\Bρ ! Sρ
Proof. (a): Fix any c∈R and a (PS) -sequence (u ) for J¯ :
c n λ
J¯ (u )→c, (14)
λ n
DJ¯ (u )→0 in W−1,p′(Ω). (15)
λ n
MimickingtheproofofLemma3.1,oneverifiesthatthefunctional J¯ iscoercive. Therefore(14)implies
λ
that (u ) is bounded.
n
Since ∆ : W1,p(Ω)→W−1,p′(Ω) is a homeomorphism, we can write
p 0
(−∆ )−1DJ¯ (u )=u +(−∆ )−1DR (u+).
p λ n n p λ n
By our assumptions (H0) on the exponents, DR : W1,p(Ω) → W−1,p′(Ω) is a compact operator, and
λ 0
hencesoistheoperatorK :=(−∆ )−1DR : W1,p(Ω)→W1,p(Ω). Takingintoaccount(15),weobtain
λ p λ 0 0
o(1)=u +K (u+).
n λ n
Since (u ) is bounded and K is compact, we conclude that (u ) contains a (strongly) convergent
n λ n
subsequence in W1,p(Ω).
0
(b): Denote by ρ > 0 the value of t where the left hand side of (8) attains its maximum. It is clear
that
infJ¯ <0<infJ¯ .
λ λ
Bρ Sρ
Since the value of t where the left hand side of (9) attains its minimum is >ρ, we also deduce that
inf J¯ <0.
λ
W01,p(Ω)\Bρ
Theorem 5.1. For any λ∈(0,Λ), the functional J has a critical point u >0 with J (u )>0.
λ ∗ λ ∗
Proof. According to statement (b) of Lemma 5.1, there is an ρ>0 and an w∈W1,p(Ω)\B such that
0 ρ
max(J¯ (0),J¯ (w))=0<infJ¯ .
λ λ λ
Sρ
Since, by statement (a) of the same Lemma, J¯ satisfies the Palais-Smale condition, we conclude from
λ
the Mountain Pass Theorem 2.2 that J¯ has a critical point u such that
λ ∗
J¯ (u )= inf max J¯ (γ(t))>0
λ ∗ λ
γ∈Γt∈[0,1]
where
Γ:={γ ∈C([0,1],W1,p(Ω))|γ(0)=0andγ(1)=w}.
0
7
To finish the proof, it suffices to show that u > 0 because then u will be a critical point of the
∗ ∗
original functional J . Since
λ
0=DJ¯ (u )u− = |∇u |p−2∇u ·∇u− = |∇u−|p
λ ∗ ∗ ∗ ∗ ∗ ∗
ZΩ ZΩ
and u ∈W1,p(Ω), we must have u− =0 or, equivalently, that u >0.
∗ 0 ∗ ∗
6 Third critical point
Define the functional I : U →R by
3 λ
I (v):=Jˆ (t (v),v)= min Jˆ (t,v).
3 λ 3 λ
t>t2(v)
Lemma 6.1. The functional t is continuously differentiable and weakly continuous on U .
3 λ
Proof. The statement of the lemma follows from the inequality
∂
Φ (t,v)>0, v ∈U
λ λ
∂t
(cid:12)t=t3(v)
(cid:12)
(cid:12)
the Implicit Function Theorem, and t(cid:12)he weak continuity of A,B, and C.
Theorem 6.1. For any λ ∈ (0,Λ), there is a nonnegative v ∈ S ∩U such that u = t (v )v is a
∗ 1 λ ∗ 3 ∗ ∗
critical point of J with J (u )<0.
λ λ ∗
Proof. By statement (c) of Lemma 3.2,
c (λ):= inf I (v)<0. (16)
3 3
v∈Uλ∩B1
Let (v ) be a minimizing sequence for problem (16) which we may assume to be nonnegative. Let
n
t :=t (v ). Then we can assume that
3,n 3 n
v ⇀v weakly in W1,p(Ω), (17)
n ∗ 0
t →t (18)
3,n 3
for some nonnegative v ∈B and 06t 6∞.
∗ 1 3
Since, by Lemma 3.1, J is coercive and I (v ) > J (t v ), we must have t < ∞. Entertaining
λ 3 n λ 3,n n 3
(16),we alsoobtainthat 0<t . Inother words,0<t <∞. Taking into accountequation(6), we infer
3 3
that v 6≡0.
∗
Let us show that v ∈U . Let t :=t (v ). Without loss of generality, we may assume that
∗ λ 2,n 2 n
t →t
2,n 2
for some 06t 6t . Since v 6≡0, we deduce from (6) that t 6=0 and hence
2 3 ∗ 2
0<t 6t .
2 3
It is also clear that both t and t are solutions to equation (6) for v =v :
2 3 ∗
Φ (t ,v )=0, i=2,3.
λ i ∗
By statement (c) of Lemma 3.2,
tp
0< 2,n +R (t v ), n>1,
λ 2,n n
p
8
yielding that
tp
06 2 +R (t v ). (19)
λ 2 ∗
p
Since
tp
3 +R (t v )=c (λ)<0,
λ 3 ∗ 3
p
we deduce that
0<t <t .
2 3
Since α is the smallest exponent, we deduce from (19) that equation (6) for v = v∗ has a third
solution, t ∈(0,t ), satisfying
1 2
tp
1 +R (t v )<0.
λ 1 ∗
p
Therefore, equation (6) for v = v has three solutions, yielding that v ∈ U . So, we deduce that the
∗ ∗ λ
infimum in (16) is attained at v .
∗
Applying Theorem 2.1 and Lemma 6.1, we conclude that v ∈S and that u =t (v )v is a critical
∗ 1 ∗ 3 ∗ ∗
point of J . Since v ∈S , J (u )=I (v )<0.
λ ∗ 1 λ ∗ 3 ∗
7 Proof of the main result
In this section, we prove the main result of this paper, Theorem 1.1. We will also need the following
regularity result (cf. [19]).
Theorem 7.1. Let Ω ⊂ RN be a bounded domain with C2 boundary and let u ∈ W1,p(Ω) be a weak
0
solution of the problem
−∆ u=f(·,u) in Ω,
p
( u=0 on ∂Ω.
where f: Ω×R is a Carath´eodory function that satisfies the inequality
|f(x,u)|6c(1+|u|m−1) for a.e. x∈Ω and all u∈R,
where m=p∗ if p<N and m>1 if p>N. Then u∈C1,µ(Ω¯) for some µ∈(0,1).
We now move on to proving the main result.
Proof of Theorem 1.1. Denote by λ∗ the number Λ in Lemma 3.2 and fix any λ∈(0,λ∗).
Letu ,u ,u befunctionsu inTheorem4.1,Theorem5.1,andTheorem6.1,respectively. Sincethey
1 2 3 ∗
are critical points of the energy functional, they are weak solutions of problem (1). Since J (u )< 0 <
λ i
J (u ) for i=1,3, u 6=u and u 6= u . Let us show that u 6=u . If this was not the case, we would
λ 2 2 1 2 3 1 3
have v =v and t (v )=t (v ). But then that would imply t (v )=t (v )=t (v ), contradicting the
1 3 1 1 3 3 1 3 1 1 3 3
definition of U .
λ
Applying Theorem 7.1, we infer that u ,u ,u ∈ C1,µ(Ω¯) for some µ ∈ (0,1), which completes our
1 2 3
proof.
References
[1] A. Ambrosetti, H. Brezis, and G. Cerami, “Combined effects of concave and convex nonlinearities
in some elliptic problems,” J. Funct. Anal., vol. 122, pp. 519–543,1994.
[2] Y.Il’yasov,“Onnonlocalexistenceresultsforellipticequationswithconvex-concavenonlinearities,”
Nonlinear Anal., vol. 61, pp. 211–236,2005.
[3] S. Li, S. Wu, and H.-S. Zhou, “Solutions to semilinear elliptic problems with combined nonlineari-
ties,” J. Diff. Equat., vol. 185, pp. 200–224,2002.
9
[4] S. T. Kyritsi and N. S. Papageorgiou, “Pairs of positive solutions for p-laplacian equations with
combined nonlinearities,” Commun. Pure and Appl. Anal., vol. 8, pp. 1031–1051,2009.
[5] A. Ambrosetti, A. Garcia, and I. Peral,“Multiplicity results for some nonlinear elliptic equations,”
J. Funct. Anal., vol. 137, pp. 219–242,1996.
[6] M.Tang,“Exactmultiplicity forsemilinearelliptic dirichletproblemsinvolvingconcaveandconvex
nonlinearities,” Proc. Roy. Soc. Edinburgh Sect. A, vol. 133, pp. 705–717,2003.
[7] K.-C.Hung,“Exactmultiplicityofpositivesolutionsofasemipositoneproblemwithconcave-convex
nonlinearity,” J. Differential Equations, vol. 255, pp. 3811–3831,2013.
[8] L. Zhao, Y. He, and P. Zhao, “The existence of three positive solutions of a singular p-laplacian
problem,” Nonlinear Anal., vol. 74, pp. 5745–5753,2011.
[9] D. Bai and Y. Chen, “Three positive solutions for a generalized laplacian boundary value problem
with a parameter,” Appl. Math. Comput., vol. 219, pp. 4782–4788,2013.
[10] K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems. Progress in
Nonlinear Differential Equations and Their Applications, Birkhauser, 1993.
[11] K.C.Chang,Methods inNonlinear Analysis. SpringerMonographsinMathematics,Springer,2005.
[12] M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and
Hamiltonian Systems. Springer, 2008.
[13] S. I. Pohozaev, Nonlinear variational problems via the fibering method, vol. 5 of Handbook of Dif-
ferential Equations. Stationary Partial Differential Equations, pp. 49–209. Amsterdam: Elsevier,
2008.
[14] S.I. Pohozaev,“The fibrationmethod for solvingnonlinear boundary value problems,” Trudy Mat.
Inst. Steklov., vol. 192, pp. 146–163,1990.
[15] Y. Bozhkov and E. Mitidieri, “Existence of multiple solutions for quasilinear systems via fibering
method,” vol. 190, pp. 239–267,2003.
[16] V. F. Lubyshev, “Multiple solutions of an even-order nonlinear problem with convex-concave non-
linearity,” Nonlinear Anal., vol. 74, pp. 1345–1354,2011.
[17] S. I.Pohozaev,“An approachto nonlinear equations,” Dokl. Akad. Nauk SSSR,vol.247,pp. 1327–
1331,1979.
[18] A. Ambrosetti and P. H. Rabinowitz, “Dual variational methods in critical point theory and appli-
cations,” J. Functional Analysis, vol. 14, pp. 349–381,1973.
[19] D. A. Kandilakis and A. N. Lyberopoulos, “Multiplicity of positive solutions for some quasilin-
ear dirichlet problems on bounded domains in rn,” Commentationes Mathematicae Universitatis
Carolinae, vol. 44, pp. 645–658,2003.
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