Z. Angew. Math. Phys.c© 2014 Springer BaselDOI 10.1007/s00033-014-0486-6

Zeitschrift fur angewandteMathematik und Physik ZAMP

The Nehari manifold for nonlocal elliptic operators involving concave–convexnonlinearities

Wenjing Chen and Shengbing Deng

Abstract. In this paper, we study the multiplicity of solutions to equations driven by a nonlocal integro-differential operatorLK with homogeneous Dirichlet boundary conditions. In particular, using fibering maps and Nehari manifold, we obtainmultiple solutions to the following fractional elliptic problem

{(−�)su(x) = λuq + up, u > 0 in Ω;

u = 0, in RN\Ω,

where Ω is a smooth bounded set in Rn, n > 2s with s ∈ (0, 1), λ is a positive parameter, the exponents p and q satisfy

0 < q < 1 < p � 2∗s − 1 with 2∗

s = 2nn−2s

.

Mathematics Subject Classification. 35J20 · 35J60 · 47G20.

Keywords. Nonlocal integro-differential operator · Concave–convex nonlinearities · Nehari manifold.

1. Introduction

In this paper, we study the multiplicity of solutions to the boundary problem by the nonlocal operator{−LKu(x) = λuq + up, u > 0 in Ω;

u = 0, in RN\Ω,

(1.1)

where Ω is a smooth bounded set in Rn, n > 2s with s ∈ (0, 1), λ is a positive parameter, the exponents

p and q satisfy 0 < q < 1 < p � 2∗s − 1 with 2∗

s = 2nn−2s (n > 2s) the fractional Sobolev exponent, and LK

is an integro-differential operators of nonlocal fractional type defined as follows

LKu(x) =12

∫Rn

(u(x+ y) + u(x− y) − 2u(x))K(y)dy, x ∈ Rn.

Here, K : Rn\{0} → (0,+∞) is a function satisfying the following properties:

γK(x) ∈ L1(Rn) with γ(x) = min{|x|2, 1}; (1.2)

there exists θ > 0 such that K(x) � θ|x|−(n+2s) for any x ∈ Rn\{0}; (1.3)

K(x) = K(−x) for any x ∈ Rn\{0}. (1.4)

The authors have been partly supported by Fundamental Research Funds for the Central XDJK2015C042,XDJK2015C043, and Doctoral Fund of Southwest University SWU114040, SWU114041. The second author has been partlysupported by Postdoctoral Fondecyt Grant 3140403.

W. Chen and S. Deng ZAMP

A typical example for K is given by K(x) = |x|−(n+2s). In this case problem, (1.1) becomes{(−�)su(x) = λuq + up, u > 0 in Ω;

u = 0, in Rn\Ω,

(1.5)

where −(−�)s is the fractional Laplace operator, which may be defined as

− (−Δ)su(x) =12

∫Rn

u(x+ y) + u(x− y) − 2u(x)|y|n+2s

dy, x ∈ Rn. (1.6)

Recently, a great attention has been focused on the study of fractional and nonlocal operators ofelliptic type, see, for instance, [11,13–15,24] for the subcritical case and [8,12,16–18] for the criticalcase. In particular, Yu [24] used the Nehari manifold to establish the existence of solutions and multiplesolutions for elliptic problems involving the square root of the Laplacian. Brandle, E. Colorado, and A. dePablo [3] studied the fractional Laplace equation involving concave–convex nonlinearity for the subcriticalcase. The authors in [3] used the idea of the s−harmonic extension introduced by Caffarelli and Silvestre[7], which allows to express the fractional operator (−Δ)s as a Dirichlet to Neumann map.

Aim of this paper was to investigate multiple solutions to problems (1.1) and (1.5) for the subcriticalcase and the critical case. In the other words, we will use the Nehari manifold method to obtain themultiplicity of solutions to problems (1.1) and (1.5) for the following two cases:

• the subcritical case: 0 < q < 1, 1 < p < 2∗s ;

• the critical case: 0 < q < 1, p = 2∗s.

In problems (1.1) and (1.5), the parameter s ∈ (0, 1) is fixed and the set Ω ⊂ Rn, n > 2s, is open

bounded with Lipschitz boundary. The standard Dirichlet condition u = 0 in ∂Ω is replaced by thecondition that the function u vanishes outside Ω, consistently with the nonlocal character of the operatorLK . Problems (1.1) and (1.5) have variational structure, and solutions can be constructed as criticalpoints of an associated energy functional on some appropriate space. In order to study problems (1.1)and (1.5), it is important to encode the boundary condition u = 0 in R

n\Ω in the weak formulation,by considering also that in the norm ‖u‖Hs(Rn), the interaction between Ω and R

n\Ω gives positivecontribution.

We will introduce the functional space that we will use in the following, which was introduced in [14]by Servadei and Valdinoci. We introduce this space as follows:

X ={u | u : R

n → R is measurable, u|Ω ∈ L2(Ω),

and (u(x) − u(y))√K(x− y) is in L2(Q,dxdy)

}where Q = R

2n\(CΩ × CΩ) with CΩ = Rn\Ω. The space X is endowed with the norm defined

‖u‖X = ‖u‖L2(Ω) +

⎛⎝∫

Q

|u(x) − u(y)|2K(x− y)dxdy

⎞⎠

1/2

. (1.7)

Then, we define

X0 = {u ∈ X : u = 0 a.e. in Rn\Ω} . (1.8)

with the norm

‖u‖X0 =

⎛⎝∫

Q

|u(x) − u(y)|2K(x− y) dxdy

⎞⎠

1/2

. (1.9)

Concave–convex nonlinearities

With this norm, X0 is a Hilbert space with scalar product defined as

〈u, v〉X0 =∫Q

(u(x) − u(y)

)(v(x) − v(y)

)K(x− y) dxdy. (1.10)

For this, see [14, Lemma 7]. For further details on X and X0 and also for their properties, we refer to[10], and the references therein.

Remark 1.1. (i) According to conditions of K, by [13, Lemma 11], we know that C20 (Ω) ⊆ X0, and so,

X and X0 are nonempty. Moreover, X ⊂ Hs(Ω) and X0 ⊂ Hs(Rn), where Hs(Ω) the usual fractionalSobolev space endowed with the norm (the so-called Gagliardo norm)

‖g‖Hs(Ω) = ‖g‖L2(Ω) +

⎛⎝ ∫

Ω×Ω

|g(x) − g(y)|2|x− y|n+2s

dxdy

⎞⎠

1/2

.

(ii) The embedding X0 ↪→ L2∗s (Ω) is continuous where 2∗

s = 2nn−2s .

Definition 1.1. We say that u is a weak solution of problem (1.1), if u satisfies∫Q

(u(x) − u(y)) (φ(x) − φ(y))K(x− y)dxdy =∫Ω

(λuq

+(x) + up+(x)

)φ(x)dx (1.11)

for all φ ∈ X0, where u+ = max{u, 0}.

In the sequel, we will omit the term weak when referring to solutions that satisfy the conditions ofDefinition 1.1. In fact, every weak solution of (1.1) is in L∞(Ω) by results of [19, Proposition 9] or [20,Theorem 3.1].

The fact that u is a weak solution is equivalent to being a critical point of the following functional

Jλ(u) = IK,p(u) − Iλ(u), (1.12)

with

IK,p(u) =12

∫Q

|u(x) − u(y)|2K(x− y)dxdy − 1p+ 1

∫Ω

up+1+ (x)dx, (1.13)

and

Iλ(u) =λ

q + 1

∫Ω

uq+1+ (x)dx. (1.14)

We can see that Jλ ∈ C1(X0,R) and

〈J ′λ(u), φ〉X0 =

∫Q

(u(x) − u(y)) (φ(x) − φ(y))K(x− y)dxdy − λ

∫Ω

uq+(x)φ(x)dx−

∫Ω

up+(x)φ(x)dx

for any φ ∈ X0.Our first result is about the subcritical case.

Theorem 1.1. Let s ∈ (0, 1), n > 2s, and Ω be an open bounded set of Rn with Lipschitz boundary. Let K

be a function satisfying conditions (1.2)–(1.4), if 0 < q < 1, 1 < p < 2∗s, then there exists λ1 > 0, such

that for λ ∈ (0, λ1), problem (1.1) has at least two solutions.

For the critical case, since the embedding X0 ↪→ L2∗s (Rn) (or, in the case of the fractional Laplacian,

Hs(Rn) ↪→ L2∗s (Rn) ) is not compact, then the energy functional does not satisfy the Palais–Smale

condition globally, but it is true for the energy functional in a suitable range related to the best fractional

W. Chen and S. Deng ZAMP

critical Sobolev constant in the embedding X0 ↪→ L2∗s (Rn). For this, let us define the best fractional

critical Sobolev constant SK as

SK = infv∈X0\{0}

∫R2n |v(x) − v(y)|2K(x− y)dxdy(∫

Ω|v(x)|2∗

s

)2/2∗s

for v ∈ X0\{0}. (1.15)

Theorem 1.2. Let s ∈ (0, 1), n > 2s, and Ω be an open bounded set of Rn with Lipschitz boundary. Let

K be a function satisfying conditions (1.2)–(1.4), if 0 < q < 1, p = 2∗s, assume that

there exists u0 ∈ X0\{0} with u0 � 0 a.e. in Rn, such that

supt�0

IK, 2∗s(tu0) <

s

nS

n2s

K , (1.16)

where IK, 2∗s

is given by (1.13). Then, there exists λ2 > 0, such that for λ ∈ (0, λ2), problem (1.1) has atleast two solutions.

Remark 1.2. When K(x) = |x|−(n+2s), LK is fractional Laplacian −(−Δ)s, condition (1.16) can beguaranteed by results of [16, Section 4.2], or [2, Lemma 2.11].

When s = 1, problem (1.5) reduces to a standard semilinear laplace partial differential equation. Moreprecisely, {−�u(x) = λuq + up, u > 0 in Ω;

u = 0, on ∂Ω,(1.17)

where Ω is a bounded domain in Rn (n > 2), 0 < q < 1 < p < n+2

n−2 , and λ > 0 is a parameter.Ambrosetti–Brezis–Cerami [1] studied problem (1.17). They established that there exists λ0 > 0 suchthat problem (1.17) attains at least two solutions for λ ∈ (0, λ0), has a solution for λ = λ0, and nosolution for λ > λ0. There are some results for elliptic equation involving concave–convex nonlinearities,see for instance [5,6,22,23,25] and the references therein.

This paper is organized as follows. In Sect. 2, we give some notations and preliminaries about Neharimanifold and fibering maps. In Sect. 3, we consider subcritical case for nonlocal elliptic operators LK

and give the proof of Theorem 1.1. In Sect. 4, we deal with the critical case and prove Theorem 1.2.

2. Some preliminary results

We will consider critical points of the function Jλ on the Hilbert space X0. We denote the Nehari manifoldby

Nλ = {u ∈ X0\{0} : 〈J ′λ(u), u〉X0 = 0} ,

where 〈, 〉X0 is the scalar product on X0 given by (1.8). It is clear that all critical points of Jλ must lieon Nλ, and Nλ is much smaller set than X0. So, it is easy to study functional Jλ on Nλ.

It is easy to see that u ∈ Nλ if and only if

‖u‖2X0

− λ

∫Ω

uq+1+ (x)dx−

∫Ω

up+1+ (x)dx = 0. (2.1)

The Nehari manifold Nλ is closely linked to the behavior of the function of the form ϕu : t �→ Jλ(tu)for t > 0 defined by

ϕu(t) := Jλ(tu) =t2

2‖u‖2

X0− λ

tq+1

q + 1

∫Ω

uq+1+ (x)dx− tp+1

p+ 1

∫Ω

up+1+ (x)dx.

Concave–convex nonlinearities

Such maps are known as fibering maps and were introduced by Drabek and Pohozaev in [9] and are alsodiscussed by Brown and Zhang [6], Brown and Wu [5].

Lemma 2.1. Let u ∈ X0\{0}, then tu ∈ Nλ if and only if ϕ′u(t) = 0.

Proof. The result is a consequence of the fact that

ϕ′u(t) = 〈J ′

λ(tu), u〉X0 =1t〈J ′

λ(u), tu〉X0 .

�

Thus, the elements in Nλ correspond to stationary points of the maps ϕu. We note that

ϕ′u(t) = t‖u‖2

X0− λtq

∫Ω

uq+1+ (x)dx− tp

∫Ω

up+1+ (x)dx, (2.2)

and

ϕ′′u(t) = ‖u‖2

X0− qλtq−1

∫Ω

uq+1+ (x)dx− ptp−1

∫Ω

up+1+ (x)dx. (2.3)

By Lemma 2.1, u ∈ Nλ if and only if ϕ′u(1) = 0. Hence, for u ∈ Nλ, by (2.2) we get

ϕ′′u(1) = ‖u‖2

X0− λq

∫Ω

up+1+ (x)dx− p

∫Ω

up+1+ (x)dx

= (1 − p)∫Ω

up+1+ (x)dx+ λ(1 − q)

∫Ω

uq+1+ (x)dx

= (1 − q)‖u‖2X0

− (p− q)∫Ω

up+1+ (x)dx

= (1 − p)‖u‖2X0

+ λ(p− q)∫Ω

uq+1+ (x)dx.

Thus, it is natural to split Nλ into three parts corresponding to local minima, local maxima, and pointsof inflection, i.e.,

N+λ = {u ∈ Nλ : ϕ′′

u(1) > 0};

N−λ = {u ∈ Nλ : ϕ′′

u(1) < 0};

N0λ = {u ∈ Nλ : ϕ′′

u(1) = 0}.We will prove the existence of solutions of problem (1.1) by investigating the existence of minimizers offunctional Jλ on Nλ. Although Nλ is a small subset of X0, we will see that local minimizers on Neharimanifold Nλ are usually critical points of Jλ. We have the following result.

Lemma 2.2. Suppose that u0 is a local minimizer of Jλ on Nλ and u0 �∈ N0λ. Then, u0 is a critical point

of Jλ.

Proof. The proof is the same as that in Brown–Zhang [6, Theorem 2.3]. �

In order to understand the Nehari manifold and fibering maps, let us consider the function ψu : R+ →

R defined by

ψu(t) = t1−q‖u‖2X0

− tp−q

∫Ω

up+1+ dx. (2.4)

W. Chen and S. Deng ZAMP

Fig. 1. The graphs of ψu(t) and ϕu(t)

It is clear that for t > 0, tu ∈ Nλ if and only if

ψu(t) = λ

∫Ω

uq+1+ dx. (2.5)

Moreover,

ψ′u(t) = (1 − q)t−q‖u‖2

X0− (p− q)tp−q−1

∫Ω

up+1+ dx (2.6)

and so, we can see that if tu ∈ Nλ, then

tqψ′u(t) = ϕ′′

u(t). (2.7)

Hence, tu ∈ N+λ (or N−

λ ) if and only if ψ′u(t) > 0 (or < 0).

Suppose u ∈ X0 and u+ �≡ 0. From (2.6), ψu satisfies the following properties:

(a) ψu has a unique critical point at t = tmax(u) =(

(1−q)‖u‖2X0

(p−q)∫Ω up+1

+ dx

) 1p−1

> 0;

(b) ψu is strictly increasing on (0, tmax(u)) and strictly decreasing on (tmax(u),+∞);(c) lim

t→+∞ψu(t) = −∞.

Moreover, by the fact∫Ωuq+1

+ dx > 0, we know that (2.5) has no solution if λ satisfies the followingcondition

λ

∫Ω

uq+1+ dx > ψu(tmax(u)) =

[(1 − q

p− q

) 1−qp−1

−(

1 − q

p− q

) p−qp−1

]‖u‖

2(p−q)p−1

X0

(∫Ωup+1

+ dx)1−qp−1

. (2.8)

Moreover, from (2.2) and (2.5) and if λ satisfies (2.8), then ϕ′u(t) > 0. It seems ϕ′

u(t) < 0 when λ is large.Hence, tu /∈ Nλ for any t > 0. On the other hand, if λ satisfies

0 < λ

∫Ω

uq+1+ dx < ψu(tmax(u)),

then there are t1 and t2 with t1 < tmax(u) < t2, such that

ψu(t1) = ψu(t2) = λ

∫Ω

uq+1+ dx, and ψ′

u(t1) > 0, ψ′u(t2) < 0,

and (2.2) and (2.5) implies that ϕ′u(t1) = ϕ′

u(t2) = 0. By (2.7), we have that ϕ′′u(t1) > 0, ϕ′′

u(t2) < 0.These facts imply that the fibering map ϕu has a local minimum at t1 and a local maximum at t2 suchthat t1u ∈ N+

λ and t2u ∈ N−λ . It can be seen that the graphs of ψu(t) and ϕu(t) can be seen in Fig. 1.

Concave–convex nonlinearities

3. The subcritical case: 0 < q < 1 < p < 2∗s − 1

In this section, we consider multiple solutions of (1.1) for the subcritical case.

Lemma 3.1. There exists λ1 > 0, such that for any λ ∈ (0, λ1), we have N0λ = ∅.

Proof. Suppose not, that is, N0λ �= ∅ for any λ > 0. Then, for u ∈ N0

λ, we have 〈J ′λ(u), u〉X0 = 0, and

ϕ′′u(1) = 0. Namely,

‖u‖2X0

= λ

∫Ω

uq+1+ ds+

∫Ω

up+1+ dx, and ‖u‖2

X0= λq

∫Ω

uq+1+ ds+ p

∫Ω

up+1+ dx.

Then, we have

(1 − q)‖u‖2X0

= (p− q)∫Ω

up+1+ dx, and (p− 1)‖u‖2

X0= λ(p− q)

∫Ω

uq+1+ dx. (3.1)

By Holder inequality and Remark 1.1 (ii), we get that there exist two positive constants C1, C2 such that

‖u‖2X0

� C1‖u‖p+1X0

and ‖u‖2X0

� λC2‖u‖q+1X0

.

It yields that C1

1−p

1 � ‖u‖X0 � (λC2)1

1−q . This is impossible if λ is sufficiently small. Thus, we obtainthat there exists λ1 > 0 such that for any λ ∈ (0, λ1), we have N0

λ = ∅. �

Lemma 3.2. Jλ is coercive and bounded from below on Nλ for λ ∈ (0, λ1).

Proof. If u ∈ Nλ, (1.12) and (2.1) imply that

Jλ(u) =(

12

− 1p+ 1

)‖u‖2

X0− λ

(1

q + 1− 1p+ 1

)∫Ω

uq+1+ dx.

By Holder inequality and Remark 1.1 (ii), we get∫Ω

uq+1+ dx � Cn,q,s,θ,|Ω|‖u‖q+1

X0,

where Cn,q,s,θ,|Ω| is a positive constant depending on n, s, q, θ, |Ω|. Since 0 < q < 1 < p, then we obtainthat the functional Jλ is coercive and bounded from below on Nλ. �

By Lemmas 3.1 and 3.2, for any λ ∈ (0, λ1), we know that Nλ = N+λ ∪ N−

λ and Jλ is coercive andbounded from below on N+

λ and N−λ . Therefore, we may define

α+λ = inf

u∈N+λ

Jλ(u), α−λ = inf

u∈N−λ

Jλ(u).

We have the following result.

Proposition 3.1. If 0 < λ < λ1, then the functional Jλ has a minimizer u1 in N+λ and satisfies

(1) Jλ(u1) = infu∈N+

λ

Jλ(u) < 0;

(2) u1 is a solution of problem (1.1).

Proof. Since Jλ is bounded from below on N+λ , there exists a minimizing sequence {uk} ⊂ N+

λ such that

limk→∞

Jλ(uk) = infu∈N+

λ

Jλ(u).

W. Chen and S. Deng ZAMP

Thus, by Lemma 3.2, the sequence {uk} is bounded in X0. From [14, Lemma 7], (X0, ‖ · ‖X0) is a Hilbertspace, and then, there exists u1 ∈ X0 such that, up to a subsequence,∫

Q

(uk(x) − uk(y)) (φ(x) − φ(y))K(x− y)dxdy

→∫Q

(u1(x) − u1(y)) (φ(x) − φ(y))K(x− y)dxdy for ∀φ ∈ X0

as k → ∞. Moreover, by [14, Lemma 8], up to a subsequence,

uk → u1 in Lr(Rn), uk → u1 a.e. in Rn, (3.2)

as k → ∞, and by [4, Theorem IV-9], there exists � ∈ Lr(Rn) such that

|uk(x)| � �(x) a.e. in Rn

for any 1 � r < 2∗s = 2n

n−2s (n > 2s). Therefore, by dominated convergence theorem, we have that∫Ω

(uk)q+1+ dx →

∫Ω

(u1)q+1+ dx, and

∫Ω

(uk)p+1+ dx →

∫Ω

(u1)p+1+ dx

as k → ∞.Moreover, there exists t1 such that t1u1 ∈ N+

λ and Jλ(t1u1) < 0. Hence, we have infu∈N+

λ

Jλ(u) < 0.

Next, we show that uk → u1 strongly in X0. If not, then ‖u1‖X0 < lim infk→∞

‖uk‖X0 . Thus, for {uk} ∈ N+λ ,

we get

limk→∞

ϕ′uk

(t1) = limk→∞

⎛⎝t1‖uk‖2

X0− λtq1

∫Ω

(uk)q+1+ dx− tp1

∫Ω

(uk)p+1+ dx

⎞⎠

> t1‖u1‖2X0

− λtq1

∫Ω

(u1)q+1+ dx− tp1

∫Ω

(u1x)p+1+ dx = ϕ′

u1(t1) = 0.

That is, ϕ′uk

(t1) > 0 for k large enough. Since uk = 1 · uk ∈ N+λ , we can see that ϕ′

uk(t) < 0 for t ∈ (0, 1)

and ϕ′uk

(1) = 0 for all k. Then, we must have t1 > 1. On the other hand, ϕu1(t) is decreasing on (0, t1),and so

Jλ(t1u1) � Jλ(u1) < limk→∞

Jλ(uk) = infu∈N+

λ

Jλ(u),

which is a contradiction. Hence, uk → u1 strongly in X0. This implies

Jλ(uk) → Jλ(u1) = infu∈N+

λ

Jλ(u) as k → ∞.

Namely, u1 is a minimizer if Jλ on N+λ . Using Lemma 2.2, u1 is a solution to (1.1). �

Next, we establish the existence of a local minimum for Jλ on N−λ .

Proposition 3.2. If 0 < λ < λ1, then the functional Jλ has a minimizer u2 in N−λ and satisfies

(1) Jλ(u2) = infu∈N−

λ

Jλ(u) > 0;

(2) u2 is a solution of problem (1.1).

Concave–convex nonlinearities

Proof. Since Jλ is bounded from below on N−λ , there exists a minimizing sequence {uk} ⊂ N−

λ such that

limk→∞

Jλ(uk) = infu∈N−

λ

Jλ(u).

By the same argument given in the proof of Proposition 3.1, there exists u2 ∈ X0 such that, up to asubsequence, ∫

Q

(uk(x) − uk(y)) (φ(x) − φ(y))K(x− y)dxdy

→∫Q

(u2(x) − u2(y)) (φ(x) − φ(y))K(x− y)dxdy for ∀φ ∈ X0

as k → ∞, and ∫Ω

(uk)q+1+ dx →

∫Ω

(u2)q+1+ dx, and

∫Ω

(uk)p+1+ dx →

∫Ω

(u2)p+1+ dx

as k → ∞. Moreover, from the analysis of fibering maps ϕu(t), we know that there exist t1, t2 witht1 < tmax(u) < t2 such that t1u ∈ N+

λ , t2u ∈ N−λ , and Jλ(t1u) � Jλ(tu) � Jλ(t2u).

Next, we show that uk → u2 strongly in X0. If not, then ‖u2‖X0 < lim infk→∞

‖uk‖X0 . Thus, for {uk} ∈N−

λ , we have Jλ(uk) � Jλ(tuk) for all t � tmax(u), and

Jλ(t2u2) =t222

‖u2‖2X0

− λtq+12

q + 1

∫Ω

(u2)q+1+ dx− tp+1

2

p+ 1

∫Ω

(u2)p+1+ dx

< limk→∞

⎛⎝ t22

2‖uk‖2

X0− λ

tq+12

q + 1

∫Ω

(uk)q+1+ dx− tp+1

2

p+ 1

∫Ω

(uk)p+1+ dx

⎞⎠

= limk→∞

Jλ(t2uk) � Jλ(uk) = infu∈N−

λ

Jλ(u),

which is a contradiction. Hence, uk → u2 strongly in X0. This implies

Jλ(uk) → Jλ(u2) = infu∈N−

λ

Jλ(u) as k → ∞.

Namely, u2 is a minimizer if Jλ on N−λ . Using Lemma 2.2, u2 is a solution to (1.1). �

Proof of Theorem 1.1: By Propositions 3.1 and 3.2 and Lemma 2.2, we get that problem (1.1) has twosolutions u1 ∈ N+

λ and u2 ∈ N−λ in X0. Since N+

λ ∩ N−λ = ∅, then those two solutions are distinct. This

finishes the proof. �

4. The critical case: 0 < q < 1, p = 2∗s − 1

In this section, we consider the multiplicity of solutions to problem (1.1) for the critical case, i.e., p =2∗

s − 1 = n+2sn−2s .

Lemma 4.1. There exists λ∗ > 0 such that N0λ = ∅ for each λ ∈ (0, λ∗).

Proof. The proof is analogous to the proof of Lemma 3.1. �

Lemma 4.2. Jλ is coercive and bounded from below on Nλ for λ ∈ (0, λ∗).

Proof. The proof is analogous to the proof of Lemma 3.2. �

W. Chen and S. Deng ZAMP

By Lemmas 4.1 and 4.2, for any λ ∈ (0, λ∗), we know that Nλ = N+λ ∪ N−

λ and Jλ is coercive andbounded from below on N+

λ and N−λ . Therefore, we may define

α+λ = inf

u∈N+λ

Jλ(u), α−λ = inf

u∈N−λ

Jλ(u).

We have the following result.

Proposition 4.1. Let {uk} ⊂ X0 be a (PS)c sequence for Jλ with

c <s

nS

n2s

K −Mλ2∗

s2∗

s−q

then there exists a subsequence of {uk}, which converges strongly in X0, where SK is defined in (1.15)and M is a positive constant given by

M =(2n− (n− 2s)(q + 1))(1 − q)

4(q + 1)

((1 − q)(n− 2s)

4s

) q+12∗

s−(q+1)

|Ω|. (4.1)

Proof. From Lemma 4.2, we see that {uk} is bounded in X0. Then, up to a sequence, still denoted byuk, there exists u∞ ∈ X0 such that uk → u∞ weakly in X0, that is∫

Q

(uk(x) − uk(y)) (φ(x) − φ(y))K(x− y)dxdy

→∫Q

(u∞(x) − u∞(y)) (φ(x) − φ(y))K(x− y)dxdy for ∀φ ∈ X0

as k → ∞. Moreover, by [16, Lemma 9], we have that

uk → u∞ weakly in L2∗s (Rn); uk → u∞ in Lr(Rn); uk → u∞ a.e. in R

n,

as k → ∞, and by [4, Theorem IV-9], there exists � ∈ Lr(Rn) such that

|uk(x)| � �(x) a.e. in Rn

for any 1 � r < 2∗s = 2n

n−2s (n > 2s). Therefore, by dominated convergence theorem, we have that∫Ω

(uk)q+1+ dx →

∫Ω

(u∞)q+1+ dx.

By Brezis-Lieb Lemma [21, Lemma 1.32], we get∫Q

|uk(x) − uk(y)|2K(x− y)dxdy →∫Q

|uk(x) − u∞(x) − uk(y) + u∞(y)|2K(x− y)dxdy

+∫Q

|u∞(x) − u∞(y)|2K(x− y)dxdy + o(1)

and ∫Ω

(uk(x))2∗s

+ dx =∫Ω

((uk − u∞)(x))2∗s

+ dx+∫Ω

(u∞(x))2∗s

+ dx+ o(1)

Concave–convex nonlinearities

as k → ∞. Then,

〈J ′λ(uk), uk〉X0 =

∫Q

|uk(x) − uk(y)|2K(x− y)dxdy − λ

∫Ω

(uk(x))q+1+ dx−

∫Ω

(uk(x))2∗s

+ dx

=∫Q

|uk(x) − u∞(x) − uk(y) + u∞(y)|2K(x− y)dxdy +∫Q

|u∞(x) − u∞(y)|2K(x− y)dxdy

−λ∫Ω

(uk(x))q+1+ dx−

⎛⎝∫

Ω

((uk − u∞)(x))2∗s

+ dx+∫Ω

(u∞)2∗s

+ dx+ o(1)

⎞⎠ + o(1)

=∫Q

|(uk−u∞)(x)−(uk−u∞)(y)|2K(x− y)dxdy−∫Ω

((uk − u∞)(x))2∗s

+ dx+〈J ′λ(u∞), u∞〉X0 + o(1).

By 〈J ′λ(u∞), u∞〉X0 = 0 and 〈J ′

λ(uk), uk〉X0 → 0 as k → ∞, we know that

‖uk − u∞‖2X0

=∫Q

|(uk − u∞)(x) − (uk − u∞)(y)|2K(x− y)dxdy → b, (4.2)

and

∫Ω

((uk − u∞)(x))2∗s

+ dx → b, as k → ∞. (4.3)

If b = 0, the proof is complete. Assuming b > 0, by the definition of SK in (1.15), we get

‖uk − u∞‖2X0

� SK

⎛⎝∫

Ω

((uk − u∞)(x))2∗s

+ dx

⎞⎠

22∗

s

.

Thus, we have b � SKb22∗

s . That is, b � Sn2s

K . On the other hand, we have

c = limk→∞

Jλ(uk) = limk→∞

⎛⎝1

2‖uk‖2

X0− λ

1q + 1

∫Ω

(uk(x))q+1+ dx− 1

p+ 1

∫Ω

(uk(x))2∗s

+ dx

⎞⎠

� Jλ(u∞) +s

nS

n2s

K . (4.4)

By the assumption that c < snS

n2s

K , we have Jλ(u∞) < 0. In particular, u∞ �= 0 and

0 <12‖u∞‖2

X0<

12∗

s

∫Ω

(u∞(x))2∗s

+ dx+ λ1

q + 1

∫Ω

(u∞(x))q+1+ dx.

W. Chen and S. Deng ZAMP

Then,

c = limk→∞

Jλ(uk) = limk→∞

(Jλ(uk) − 1

2〈J ′

λ(uk), uk〉X0

)

= limk→∞

⎛⎝ s

n

∫Ω

((uk − u∞)(x))2∗s

+ dx+s

n

∫Ω

(u∞(x))2∗s

+ dx+ λ

(12

− 1q + 1

)∫Ω

(uk(x))q+1+ dx

⎞⎠

=s

nb+

s

n

∫Ω

(u∞(x))2∗s

+ dx+ λ

(12

− 1q + 1

)∫Ω

(u∞(x))q+1+ dx

� s

nS

n2s

K +s

n

∫Ω

(u∞(x))2∗s

+ dx+ λ

(12

− 1q + 1

)∫Ω

(u∞(x))q+1+ dx.

Moreover, by Holder inequality, we have

∫Ω

(u∞(x))q+1+ dx � |Ω|

2∗s −(q+1)

2∗s

⎛⎝∫

Ω

(u∞(x))2∗s

+ dx

⎞⎠

q+12∗

s

Thus,

c � s

nS

n2s

K +s

n

⎛⎝∫

Ω

(u∞(x))2∗s

+ dx

⎞⎠ + λ

(12

− 1q + 1

)|Ω|

2∗s −(q+1)

2∗s

⎛⎝∫

Ω

(u∞(x))2∗s

+ dx

⎞⎠

q+12∗

s

:=s

nS

n2s

K + h(η)

where

h(η) =s

nη2∗

s + λ

(12

− 1q + 1

)|Ω|

2∗s −(q+1)

2∗s ηq+1 with η =

⎛⎝∫

Ω

(u∞(x))2∗s

+ dx

⎞⎠

12∗

s

.

We note that h(η) attains its minimum at η0 =(

λ(1−q)(n−2s)4s

) 12∗

s −(q+1) |Ω| 12∗

s and

h(η0) = − (2n− (n− 2s)(q + 1))(1 − q)4(q + 1)

((1 − q)(n− 2s)

4s

) q+12∗

s −(q+1)

|Ω|λ2∗

s2∗

s−(q+1) = −Mλ2∗

s2∗

s−(q+1) .

with M given by (4.1). Therefore,

c � s

nS

n2s

K −Mλ2∗

s2∗

s−(q+1) .

Then, we get a contradiction with our hypothesis. Hence, b = 0 and, we conclude that uk → u∞ stronglyin X0. This completes the proof. �

Proposition 4.2. There exists λ2 > 0 and u0 ∈ X0 such that

supt>0

Jλ(tu0) <s

nS

n2s

K −Mλ2∗

s2∗

s −(q+1) (4.5)

for λ ∈ (0, λ2). In particular,

α−λ <

s

nS

n2s

K −Mλ2∗

s2∗

s−(q+1)

where the constant M depends on q, n, s and |Ω|, which given is by (4.1)

Concave–convex nonlinearities

Proof. Let λ∗∗ > 0 be such that snS

n2s

K −Mλ2∗

s2∗

s−(q+1) > 0 for all λ ∈ (0, λ∗∗). By condition (1.16), we havethat there is u0 ∈ X0\{0} such that

Jλ(tu0) � supt�0

IK,2∗s(tu0) − λ

tq+1

q + 1

∫Ω

(u0)q+1+ dx <

s

nS

n2s

K − λtq+10

q + 1

∫Ω

(u0)q+1+ dx. (4.6)

Let λ∗∗∗ :=(

tq+10

∫Ω(u0)

q+1+ dx

M(q+1)

) 2∗s−(q+1)

q+1

with M is a positive constant defined in (4.1). Then, for λ ∈(0, λ∗∗∗), we find

− tq+10

q + 1λ

∫Ω

(u0)q+1+ dx < −Mλ

2∗s

2∗s−(q+1) .

Thus, we obtain that (4.5) holds. Finally, we set λ2 = min{λ∗, λ∗∗, λ∗∗∗}; by the analysis of fibering mapsϕu(t) = Jλ(tu), we get

α−λ <

s

nS

n2s

K −Mλ2∗

s2∗

s−(q+1)

for λ ∈ (0, λ2). This completes the proof. �

Now, we are ready to prove Theorem 1.2.

Proof of Theorem 1.2: By Propositions 4.1 and 4.2, there exists two sequences {u+k } and {u−

k } in X0

such that

Jλ(u+k ) → α+

λ , J ′λ(u+

k ) → 0 and Jλ(u−k ) → α−

λ , J ′λ(u−

k ) → 0

as k → ∞. We observe that from the analysis of fibering maps ϕu(t), we have α+λ < 0. Similar to the proof

of Propositions 3.1 and 3.2 and Theorem 1.1, problem (1.1) has two solutions u1 ∈ N+λ and u2 ∈ N−

λ inX0. Since N+

λ ∩ N−λ = ∅, then these two solutions are distinct. This finishes the proof. �

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Wenjing Chen and Shengbing DengSchool of Mathematics and StatisticsSouthwest UniversityChongqing 400715People’s Republic of Chinae-mail: wjchen1102@gmail.com

Shengbing DengDepartamento de MatematicaPontificia Universidad Catolica de ChileAvda. Vicuna Mackenna 4860Macul, Chilee-mail: shbdeng65@gmail.com

(Received: February 3, 2014; revised: October 8, 2014)