Electronic Journal of Differential Equations, Vol. 2017 (2017), No. 304, pp. 1–24.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

NON-HOMOGENEOUS PROBLEM FOR FRACTIONAL LAPLACIAN INVOLVING CRITICAL SOBOLEV EXPONENT

KUN CHENG, LI WANG

Communicated by Binlin Zhang

Abstract. In this article, we study the existence of positive solutions for the nonhomogeneous fractional equation involving critical Sobolev exponent

(−∆)^{s}u+λu=u^{p}+µf(x), u >0 in Ω,
u= 0, inR^{N}\Ω,

where Ω⊂R^{N} is a smooth bounded domain,N≥1, 0<2s <min{N,2},λ
andµ >0 are two parameters,p= ^{N+2s}_{N−2s} andf∈C^{0,α}( ¯Ω), whereα∈(0,1).

f≥0 andf6≡0 in Ω. For someλandN, by the barrier method and mountain
pass lemma, we prove that there exists 0<µ¯:= ¯µ(s, µ, N)<+∞such that
there are exactly two positive solutions ifµ∈(0,µ) and no positive solutions¯
forµ >µ. Moreover, if¯ µ= ¯µ, there is a unique solution (¯µ;u¯µ), which means
that (¯µ;uµ¯) is a turning point for the above problem. Furthermore, in case
λ >0 andN≥6sif Ω is a ball inR^{N}andfsatisfies some additional conditions,
then a uniqueness existence result is obtained forµ >0 small enough.

1. Introduction and main results

In this article, we focus our attention on the non-homogeneous fractional prob- lems. To be more precise, we consider the existence of multiple positive solutions for the following nonlinear elliptic equations involving the fractional Laplacian

(−∆)^{s}u+λu=u^{p}+µf(x), u >0 in Ω,

u= 0 inR^{N}\Ω, (1.1)

where s∈(0,1) is fixed, Ω ⊂R^{N} is a smooth bounded domain, λand µ >0 are
two parameters, p = 2^{∗}_{s} −1 where 2^{∗}_{s} = _{N−2s}^{2N} is the fractional critical Sobolev
exponent. Moreover,f(x) is a non-homogeneous perturbation satisfying following
assumption:

(A1) f ∈C^{0,α}( ¯Ω), whereα∈(0,1). f ≥0 andf 6≡0 in Ω.

The fractional Laplacian (−∆)^{s}is a classical linear integro-differential operator
of order 2swhich gives the standard Laplacian whens= 1.

2010Mathematics Subject Classification. 35A15,35J60, 46E35.

Key words and phrases. Non-homogeneous; fractional Laplacian; critical Sobolev exponent;

variational method.

c

2017 Texas State University.

Submitted September 23, 2017. Published December 11, 2017.

1

A range of powers of particular interest iss∈(0,1) and we can write the operator as

(−∆)^{s}u(x) =C_{N,s}P.V.

Z

R^{N}

u(x)−u(y)

|x−y|^{N}^{+2s}dy, x∈R^{N}, u∈ S(R^{N}), (1.2)
where P.V. is the principal value, C_{N,s} is a normalization constant and S(R^{N}) is
the Schwartz space of rapidly decaying C^{∞} functions in R^{N}. For an elementary
introduction to the fractional Laplacian and fractional Sobolev spaces we refer the
readers to [15, 20].

The motivation to study problem (1.1) comes from the nonlinear fractional Schr¨odinger equation

(−∆)^{s}u+V(x)u=f(x, u), x∈R^{N}. (1.3)
Solutions of (1.3) are standing wave solutions of the fractional Schr¨odinger equation
of the form

i∂ψ

∂t = (−∆)^{s}ψ+V(x)ψ−f(x,|ψ|), x∈R^{N}. (1.4)
that is solutions of the formψ(x, t) =e^{−iEt}u(x), whereE is a constant,u(x) is a
solution of (1.3). The fractional Schr¨odinger equation is a fundamental equation
in fractional quantum mechanics. It was discovered by Laskin ([18, 19]) as a re-
sult of extending the Feynman path integral, from the Brownian-like to L´evy-like
quantum mechanical paths, where the Feynman path integral leads to the classical
Schr¨odinger equation, and the path integral L´evy trajectories leads to the frac-
tional Schrdinger equation. Different to the classical Laplacian operator, the usual
analysis tools for elliptic PDEs can not be directly applied to (1.3) since (−∆)^{s}
is a nonlocal operator. In the remarkable work of Caffarelli-Silvestre [7], the au-
thors expressed the nonlocal operator (−∆)^{s} as a Dirichlet-Neumann map for a
certain elliptic boundary value problem with local differential operators defined on
the upper half space.

Since then, problems with the fractional Laplacian have been extensively studied,
especially on the existence and nonexistence of positive solutions, multiple solutions,
ground states and regularity, see for example, [1, 4, 6, 7, 10, 12, 22, 26, 27, 28, 29,
30, 32, 36] and the references therein. In particular, by using definition (1.2), the
Br´ezis-Nirenberg type problem was discussed in [1, 28]. On the other hand, by
adapting thes-harmonic extension introduced by Caffarelli and Silvestre [7], Cabr´e
and Tan [5] and Tan [32] investigated the Br´ezis-Nirenberg type problem for the
special cases=^{1}_{2}. For the general case 0< s <1, Colorado et al. in [1] studied the
concave-convex elliptic problem involving the fractional Laplacian. For the related
results about the nonhomogeneous fractional Laplacian equations, for example, we
refer to [25, 24, 35, 34] and the references therein.

In the local case thats= 1, (1.1) reduce to the equation

−∆u+λu=u^{2}^{∗}^{−1}+µf(x), u >0 in Ω,

u= 0 on∂Ω. (1.5)

By using variational methods, the existence of multiple positive solutions and nonexistence results for classical non-homogeneous elliptic equation like (1.5) have been studied, see [8, 9, 14, 21] and the references therein. Naito and Sato [21] con- sidered the problem (1.5) on the bounded domains. By using variational methods and Pohozaev identity, the authors investigate the multiplicity of positive solutions

to the problem and find the phenomenon depending on the space dimension N.

Precisely, they showed that the situation is drastically different between the cases N = 3,4,5 andN ≥6 ifµ >0.

It is nature to ask whether we can find multiple positive solutions of (1.5) if we
replace the Laplacian operator −∆ by the fractional Laplacian operator (−∆)^{s}?
As far as we know such a problem was not considered before. Firstly, Since (1.1)
has no trivial solutions, it presents specific mathematical difficulties. Secondly,
as we mention above, the fractional Laplacian operator (−∆)^{s} is nonlocal, and
this brings some essential difference with the elliptic equations with the classical
Laplacian operator, such as regularity, maximum principle, Pohozaev identity and
so on.

Before presenting our main results, we first give some notation. Letλ1be the first
eigenvalue of the non-local operator (−∆)^{s} with homogeneous Dirichlet condition
on Ω (see [28]). We denote byH^{s}(R^{N}) the usual fractional Sobolev space endowed
with the so-calledGagliardo norm

kgk_{H}s(R^{N})=kgk_{L}2(R^{N})+Z

R^{N}×R^{N}

|g(x)−g(y)|^{2}

|x−y|^{N}^{+2s} dx dy1/2

, (1.6)

andX_{0}^{s}(Ω) is the function space defined as
X_{0}^{s}(Ω) =

u∈H^{s}(R^{N}) :u= 0 a.e. inR^{N}\Ωo

. (1.7)

We refer to [28, 29] for a general definition ofX_{0}^{s}(Ω) and its properties and to [15]

for an account of the properties ofH^{s}(R^{N}). InX_{0}^{s}(Ω) we can consider the norm
kvk_{X}^{s}

0(Ω)=Z

R^{N}×R^{N}

|v(x)−v(y)|^{2}

|x−y|^{N}^{+2s} dx dy1/2

.
The pair (X_{0}^{s}(Ω),k · k_{X}^{s}

0(Ω)) yields a Hilbert space (see for instance [15]) with scalar product

hu, viX_{0}^{s}(Ω)=Z

R^{N}×R^{N}

(u(x)−u(y))(v(x)−v(y))

|x−y|^{N}^{+2s} dx dy^{1/2}

. (1.8)
We also consider another norm inX_{0}^{s}(Ω),

kukλ=Z

R^{N}×R^{N}

|u(x)−u(y)|^{2}

|x−y|^{N}^{+2s} dx dy+λ
Z

Ω

|u|^{2}dx^{1/2}

. (1.9)

Ifλ >−λ1,k · kλis equivalent withk · k_{X}^{s}

0(Ω), see [15] for more details.

Observe that by [15], ifu, v∈X_{0}^{s}(Ω), then
Z

Ω

v(−∆)^{s}udx=
Z

R^{N}

(−∆)^{s/2}u(−∆)^{s/2}vdx=hu, viX_{0}^{s}(Ω). (1.10)
This leads us to define the solutions to our problem (1.1) in a variational framework.

In this paper, we also suppose thatf ∈(X_{0}^{s}(Ω))^{0}, where (X_{0}^{s}(Ω))^{0} denote the dual
space ofX_{0}^{s}(Ω).

Definition 1.1. We say thatu∈X_{0}^{s}(Ω) is a positive solution of (1.1) ifu >0 a.e.

in Ω such that Z

R^{N}×R^{N}

(u(x)−u(y))(ϕ(x)−ϕ(y))

|x−y|^{N}^{+2s} dx dy+λ
Z

Ω

uϕdx

= Z

Ω

u^{p}ϕdx+µ
Z

Ω

f ϕdx

(1.11)

for everyϕ∈X_{0}^{s}(Ω).

Definition 1.2. For any fixedµ >0, we say thatu_{µ}is a positiveminimal solution
of (1.1) ifu_{µ} satisfies 0< u_{µ}≤u_{µ} in Ω for any positive solutionu_{µ} of (1.1).

In our context, the fractional Sobolev constant is given by S(N, s) := inf

v∈H^{s}(R^{N})\{0}

Q_{N,s}(v)>0, (1.12)
where

QN,s(v) :=

R

R^{N}×R^{N}

|v(x)−v(y)|^{2}

|x−y|^{N+2s} dx dy
(R

R^{N}|v(x)|^{2}^{∗}^{s}dx)^{2/2}^{∗}^{s} , v∈H^{s}(R^{N})

is the associated Rayleigh quotient. The constant S(N, s) is well defined and in- dependent of the domain (see for instance [1]). By [13], S(N, s) is attained by a family of functions

uε(x) = ε^{(N−2s)/2}

(|x|^{2}+ε^{2})^{(N−2s)/2}, ε >0, (1.13)
that is

(−∆)^{s/2}u_{ε}

2

L^{2}(R^{N})=
Z

R^{N}×R^{N}

|u_{ε}(x)−u_{ε}(y)|^{2}

|x−y|^{n+2s} dx dy=S(N, s)kuεk^{2}_{L}2(R^{N}).
(1.14)
The main goal of this paper is to exhibit the existence and nonexistence results
for (1.1) with more general nonlinear termf under some weaker assumptions. Our
main results are as follows:

Theorem 1.3. Let (A1) hold andλ >−λ1. Then, there exists µ¯ ∈(0,+∞)such that

(i) if0< µ <µ, the problem¯ (1.1)has a positive minimal solutionu_{µ}∈X_{0}^{s}(Ω).

Furthermore,u_{µ} is increasing inµforµ∈(0,µ), and¯ u_{µ}→0 inX_{0}^{s}(Ω)as
µ→0;

(ii) ifµ= ¯µ, the problem (1.1)has a unique positive solution in X_{0}^{s}(Ω);

(iii) ifµ >µ, the problem¯ (1.1)has no positive solution in X_{0}^{s}(Ω).

Remark 1.4. There is no positive solution of (1.1) if λ≤ −λ_{1}. In fact, assume
to the contrary that there exists a positive solution uof (1.1) withλ≤ −λ1. Let
ϕ1 be the eigenfunction corresponding to the first eigenvalueλ1with ϕ1>0 in Ω.

Then, we have 0 =

Z

R^{N}

(−∆)^{s/2}u(−∆)^{s/2}ϕ1dx−λ1

Z

Ω

uϕ1dx

≥ Z

R^{N}

(−∆)^{s/2}u(−∆)^{s/2}ϕ1dx+λ
Z

Ω

uϕ1dx

= Z

Ω

(u^{p}ϕ1+µf ϕ1)dx >0.

This is a contradiction.

Theorem 1.3 indicates that equation (1.1) has a minimal solution u_{µ} ∈ X_{0}^{s}(Ω)
for 0< µ≤µ, unique positive solution for¯ µ= ¯µ, and has no solution forµ >µ.¯
A natural questions is whether there are more solutions for some 0 < µ ≤µ, or¯
analogous to Theorem 1.3, the uniqueness result hold for some specialµ. Our main
results in this direction can be stated as follows.

Theorem 1.5. Assume (A1) holds. Then

(i) if 0 < µ < µ,¯ (1.1) has a second positive solution u¯µ ∈ X_{0}^{s}(Ω) satisfies

¯

uµ > u_{µ} in Ωfor λ∈(−λ1,0] andN >2s; or λ >0 and 2s < N < 6s.

Moreover,(¯µ;u_{µ}_{¯})is a bifurcation point for problem (1.1);

(ii) there exists µ^{∗} =µ^{∗}(λ) ∈ (0,µ)¯ such that (1.1) has a second positive so-
lution u¯_{µ} ∈ X_{0}^{s}(Ω) satisfies u¯_{µ} > u_{µ} for every µ^{∗} ≤ µ < µ¯ if λ > 0,
N ≥6s.

Theorem 1.6. Assume (A1)holds, λ >0, and N ≥6s. Ω ={x∈R^{N} :|x|< R}

with some R > 0, and let f = f(|x|) be radially symmetric about the origin and
f(r)is decreasing in r∈[0, R]. Then, there exists µ_{∗}∈(0, µ^{∗})such that (1.1) has
a unique positive solutionu_{µ} forµ∈(0, µ_{∗}].

By Theorems 1.5 and 1.6, it is obviously that the existence of the second solution depend on λ and the space dimension N. To prove Theorems 1.5 and 1.6, we consider the auxiliary equation

(−∆)^{s}v+λv= (v+u_{µ})^{p}−u^{p}_{µ} in Ω, v∈X_{0}^{s}(Ω) (1.15)
by classical Mountain-Pass Lemma and variational methods.

The rest of this article is organized as follows. In Section 2, we first present variational framework to deal with problem (1.1), Then we show the existence of positive minimal solutions to (1.1) and prove Theorem 1.3. In Section 3, by studying the auxiliary equation (1.15), we give the proof of Theorem 1.5. At last, in Section 4, we prove Theorem 1.6.

2. Existence and properties of minimal solutions

In this section, we show the existence of positive minimal solutions to (1.1) and present some properties of the solutions which will be used in the sequel. Now we give a Maximum Principle which will be used frequently in our text.

Proposition 2.1(Maximum principle). If(A1)holds, u≥0is a solution of (1.1), then either u≡0 inΩ oruis strictly positive inΩ.

Proof. Letk(x, u) =−λu+u^{2}^{∗}^{s}^{−1}+µf, then there existsC >0 which is independent
withusuch that|k(x, u)| ≤C(1 +|u^{2}^{∗}^{s}^{−1}|). Then by [1, Proposition 2.2], we have
u∈ L^{∞}(Ω). Moreover, similar as the proof of [30, Proposition 2.1.9], we deduce
thatu∈C^{0,γ}(Ω) for any 0< γ <2sif 2s≤1, oru∈C^{1,γ}(Ω) for any 0< γ <2s−1
if 2s >1.

We will discuss this problem into following two cases.

Case 1: −λ1 < λ ≤ 0. In this case, we will get (−∆)^{s}u ≥ 0. Then by [30,
Proposition 2.1.7], we haveu >0.

Case 2: λ >0. Since u≥0 is a solution of (1.1), for anyϕ≥0 andϕ∈X_{0}^{s}(Ω),
we have

Z

R^{N}×R^{N}

(u(x)−u(y))(ϕ(x)−ϕ(y))

|x−y|^{N+2s} dx dy+λ
Z

Ω

uϕ

= Z

Ω

u^{2}^{∗}^{s}^{−1}ϕdx+µ
Z

Ω

f ϕdx≥0.

Thenuis a super-solution of

(−∆)^{s}u=−λu+u^{2}^{∗}^{s}^{−1}+µf in Ω,

u∈X_{0}^{s}(Ω).

Thus by [23, Theorem 1.2], we conclude thatu >0 in Ω.

Taking into account that we are looking for positive solutions for problem (1.1), we will consider the Dirichlet problem

(−∆)^{s}u+λu= (u+)^{p}+µf(x) in Ω,

u= 0 inR^{N}\Ω, (2.1)

where u+ := max{u,0}. The crucial observation here is that, by Proposition 2.1, ifuis a solution of (2.1) thenuis strictly positive in Ω and, therefore, it is also a solution of (1.1).

The energy functional related to the problem (2.1) is given by Iλ,µ(u) = 1

2 Z

R^{N}×R^{N}

|u(x)−u(y)|^{2}

|x−y|^{N+2s} dx dy+λ
2
Z

Ω

u^{2}dx− 1
p+ 1

Z

Ω

(u+)^{p+1}dx

−µ Z

Ω

f udx.

The functionalIλ,µis well-defined for everyu∈X_{0}^{s}(Ω) and belongs toC^{1}(X_{0}^{s}(Ω),R).

Moreover, for anyu, ϕ∈X_{0}^{s}(Ω), we have
hI_{λ,µ}^{0} (u), ϕi=

Z

R^{N}×R^{N}

u(x)−u(y)

ϕ(x)−ϕ(y)

|x−y|^{N}^{+2s} dx dy
+λ

Z

Ω

uϕdx− Z

Ω

(u_{+})^{p}ϕdx−µ
Z

Ω

f ϕdx.

(2.2)

Clearly, critical points ofI_{λ,µ} are the weak solutions for the problem (1.1).

Lemma 2.2. Assume that (A1) holds. There exists µ0 > 0 such that, for µ ∈
(0, µ0], the (1.1) has a positive solutionuµ ∈X_{0}^{s}(Ω) satisfying kuµk_{X}^{s}

0(Ω) →0 as
µ→0. Furthermore, (1.1) has a unique positive solutionuµ in a neighborhood of
the origin inX_{0}^{s}(Ω) forµ >0 small enough.

Proof. Define Φ : [0,+∞)×X_{0}^{s}(Ω)→(X_{0}^{s}(Ω))^{0} by

Φ(µ, u) = (−∆)^{s}u+λu−(u+)^{p}−µf. (2.3)
Then Φ is a continuous operator, and forw∈X_{0}^{s}(Ω), we have

Φ_{u}(µ, u)w= (−∆)^{s}w+λw−p(u_{+})^{p−1}w. (2.4)
In particular, Φu(0,0)w = (−∆)^{s}w+λw. It is clear that Φu(0,0) : X_{0}^{s}(Ω) →
(X_{0}^{s}(Ω))^{0} is invertible forλ >−λ1. Then, by the implicit function theorem, there
exists a functionuµ∈X_{0}^{s}(Ω) forµ∈(0, µ0] with someµ0>0 such that Φ(µ, uµ) =
0 andkuµkX_{0}^{s}(Ω)→0 asµ→0. Furthermore, there is no other solution of Φ(µ, u) =
0 in a neighborhood of the origin inX_{0}^{s}(Ω) forµ >0 sufficiently small. Then, u_{µ}
solves the problem

(−∆)^{s}u+λu= (u_{+})^{p}+µf in Ω

for eachµ∈(0, µ0] and the local uniqueness of the solution holds. By Proposition
2.1, we obtain uµ > 0 in Ω. Thus, (1.1) has a unique positive solution uµ in a
neighborhood of the origin inX_{0}^{s}(Ω) for µ >0 sufficiently small.

Lemma 2.3. Assume that there exists a positive functionuˆ∈X_{0}^{s}(Ω) satisfying
(−∆)^{s}uˆ+λˆu≥uˆ^{p}+ ˆµf inΩ (2.5)
for someµ >ˆ 0. Then, for anyµ∈(0,µ], there exists a positive solutionˆ u∈X_{0}^{s}(Ω)
of (1.1) satisfying 0 < u(x) ≤ u(x)ˆ for x ∈ Ω. Furthermore, for any positive
solution ˜u∈X_{0}^{s}(Ω) of (1.1), the solutionusatisfiesu(x)≤u(x)˜ forx∈Ω.

Proof. Letµ∈(0,µ], and putˆ u_{0}≡0. Inductively, we can define{u_{n}}, by a solution
of the problem

(−∆)^{s}un+λun= (u_{n−1})^{p}+µf, un∈X_{0}^{s}(Ω)
forn= 1,2, . . .. Furthermore,{u_{n}}satisfies

0< u1(x)< u2(x)<· · ·<u(x),ˆ forx∈Ω. (2.6)
In fact, it is clear thatu_{1}∈X_{0}^{s}(Ω) and 0< u_{1}<uˆ in Ω by Proposition 2.1. Then,
it follows thatu^{p}_{1} ∈L^{(p+1)}^{0}(Ω) ⊂(X_{0}^{s}(Ω))^{0}, where (p+ 1)^{0} = 2N /(N+ 2s) is the
conjugate exponent ofp+ 1 = 2N /(N−2s). By induction, we obtainun ∈X_{0}^{s}(Ω)
andun−1< un<uˆin Ω for eachn= 1,2, . . .. Thus, (2.6) holds.

By the definition ofun, it follows that Z

R^{N}

(−∆)^{s/2}un(−∆)^{s/2}ψdx+λ
Z

Ω

unψdx= Z

Ω

u^{p}_{n−1}ψdx+µ
Z

Ω

f ψdx (2.7)
for anyψ∈X_{0}^{s}(Ω). Puttingψ=un, we obtain

Z

R^{N}

|(−∆)^{s/2}un|^{2}dx+λ
Z

Ω

u^{2}_{n}dx=
Z

Ω

u^{p}_{n−1}undx+µ
Z

Ω

f undx

≤ Z

Ω

ˆ

u^{p+1}dx+µ
Z

Ω

fudxˆ

Thus,{un}is bounded inX_{0}^{s}(Ω). Hence, there exist a subsequence, denoted again
{un}, andu∈X_{0}^{s}(Ω) satisfying, asn→ ∞, un * u in X_{0}^{s}(Ω) weakly, un →uin
L^{2}(Ω) strongly, andun→ua.e. in Ω. By the monotone convergence theorem, we

have Z

Ω

u^{p}_{n−1}ψdx→
Z

Ω

u^{p}ψdx asn→ ∞.

Then, lettingn→ ∞in (2.7), we obtain Z

R^{N}

(−∆)^{s/2}u(−∆)^{s/2}ψdx+λ
Z

Ω

uψdx= Z

Ω

u^{p}ψdx+µ
Z

Ω

f ψdx (2.8)
for anyψ∈X_{0}^{s}(Ω). This implies thatu∈X_{0}^{s}(Ω) is a solution to (1.1). From (2.6),
we have 0< u≤uˆ in Ω.

Let ˜u∈X_{0}^{s}(Ω) be a positive solution of (1.1). Then, ˜u > u0 ≡0, and ˜u > un

for eachn= 1,2, . . ., by induction. Thus, we obtain ˜u≥uin Ω.

For eachµ >0, define the solution set S_{µ} by
Sµ=

u∈X_{0}^{s}(Ω) :uis a positive solution of (1.1) .
Lemma 2.2 implies thatSµ6=∅for sufficient smallµ >0.

Lemma 2.4. Let (A1) hold.

(i) Assume thatS_{µ}_{0}6=∅ for someµ_{0}>0. Then,S_{µ} 6=∅ for allµ∈(0, µ_{0}).

(ii) If S_{µ}6=∅, then there exists a minimal solution u_{µ} ∈S_{µ}.

(iii) Assume that uµ ∈Sµ anduµˆ ∈Sµˆ are minimal solutions with0< µ <µ.ˆ Then, uµ< uµˆ inΩ.

(iv) Let uµ be the solution of (1.1)obtained in Lemma 2.2, and letu_{µ}∈Sµ be
the minimal solution. Then, u_{µ} ≡u_{µ} forµ >0 sufficiently small.

Proof. (i) Letµ∈(0, µ_{0}) and u_{0}∈S_{µ}_{0}. Applying Lemma 2.2 and Lemma 2.3 with
ˆ

u=u_{0} and ˆµ=µ_{0}, we obtain a positive solutionu∈X_{0}^{s}(Ω) of (1.1). This implies
thatS_{µ}6=∅ for allµ∈(0, µ_{0}).

(ii) Assume that u∈ S_{µ}. Applying Lemma 2.3 with ˆu= u and ˆµ =µ, there
exists u_{µ} ∈S_{µ} such thatu_{µ}≤uin Ω. By the latter part of Lemma 2.3,u_{µ} is the
minimal solution ofSµ.

(iii) Applying Lemma 2.3 with ˆu = uµˆ, we deduce that u_{µ} ≤ u_{µ}_{ˆ} in Ω. Put
z =u_{µ}_{ˆ}−u_{µ} ≥ 0. Then, z satisfies (−∆)^{s}z+λz ≥(ˆµ−µ)f ≥0, 6≡0 in Ω. By
Proposition 2.1, we obtainz >0 in Ω, that is,u_{µ}_{ˆ}> u_{µ} in Ω.

(iv) Sinceu_{µ}∈Sµis the minimal solution, we haveu_{µ}≤uµin Ω. Note that the
solution of (1.1) satisfies (2.8) for anyψ ∈X_{0}^{s}(Ω). Putting u=ψ =u_{µ} in (2.8),
we have

ku_{µ}k^{2}_{λ}=ku_{µ}k^{p+1}_{L}p+1+µ
Z

Ω

f u_{µ}dx≤ kuµk^{p+1}_{L}p+1+µ
Z

Ω

f u_{µ}dx.

By the Sobolev inequality, we obtain
ku_{µ}k^{2}_{λ}≤Ckuµk^{p+1}_{X}s

0(Ω)+µkfk(X_{0}^{s}(Ω))^{0}kuµkX^{s}_{0}(Ω),
with some constant C > 0. Since k · kλ is equivalent with k · k_{X}^{s}

0(Ω), Lemma 2.2
implies that ku_{µ}kX_{0}^{s}(Ω) → 0 as µ → 0. By the local uniqueness of the solution
uµ in a neighborhood of the origin inX_{0}^{s}(Ω), we obtainuµ ≡u_{µ} forµsufficiently

small.

Next, let us consider the eigenvalue problem

(−∆)^{s}φ+λφ=κa(x)φ, φ∈X_{0}^{s}(Ω), (2.9)
whereλ∈R,a(x)∈L^{N/2s}(Ω), anda(x)>0 in Ω. We assume thatλ >−λ1, where
λ1is the first eigenvalue of (−∆)^{s}with zero Dirichlet boundary condition on Ω. In
order to find the first eigenvalue of (2.9), we consider the following minimization
problem

κ1= inf

ψ∈X_{0}^{s}(Ω)\{0}

R

R^{N}|(−∆)^{s/2}ψ|^{2}dx+λR

Ωψ^{2}dx
R

Ωa(x)ψ^{2}dx . (2.10)

Lemma 2.5. Let(A1) hold. The infimumκ1 in (2.10)is positive and achieved by
some φ_{1}∈X_{0}^{s}(Ω) with φ_{1}>0 in Ω. In particular, (κ_{1}, φ_{1}) is the first eigenvalue
and the first eigenfunction to the problem (2.9).

Proof. Let{ψn} ⊂X_{0}^{s}(Ω) be a minimizing sequence of (2.10) satisfying
Z

Ω

a(x)ψ^{2}_{n}dx= 1,
Z

R^{N}

|(−∆)^{s/2}ψn|^{2}dx+λ
Z

Ω

ψ^{2}_{n}dx→κ1 as n→ ∞.

Since{ψ_{n}}is bounded inX_{0}^{s}(Ω), there exists a subsequence, still denoted by{ψ_{n}},
and a function φ1 ∈ X_{0}^{s}(Ω) such that, as n → ∞, ψn → φ1 weakly in X_{0}^{s}(Ω),
ψn→φ1 strongly inL^{2}(Ω),ψn→φ1 a.e. in Ω. Then, it follows that

κ1= lim inf

n→∞

Z

R^{N}

|(−∆)^{s/2}ψn|^{2}dx+λ
Z

Ω

ψ_{n}^{2}dx≥
Z

R^{N}

|(−∆)^{s/2}φ1|^{2}dx+λ
Z

Ω

φ^{2}_{1}dx.

Sincea(x)∈L^{N/2s}(Ω),{ψ_{n}^{2}}is bounded inL^{N/(N}^{−2s)}(Ω), we obtain
Z

Ω

a(x)ψ_{n}^{2}dx→
Z

Ω

a(x)φ^{2}_{1}dx= 1 as n→ ∞.

Hence,φ16≡0 achieves the infimumκ1>0. Clearly,|φ1|also achievesκ1, since
kφ1k^{2}_{X}s

0(Ω)= Z

R^{N}×R^{N}

|(φ^{+}_{1}(x)−φ^{+}_{1}(y))−(φ^{−}_{1}(x)−φ^{−}_{1}(y))|^{2}

|x−y|^{N}^{+2s} dx dy

≥ Z

R^{N}×R^{N}

|(φ^{+}_{1}(x)−φ^{+}_{1}(y)) + (φ^{−}_{1}(x)−φ^{−}_{1}(y))|^{2}

|x−y|^{N}^{+2s} dx dy

=k|φ1|k^{2}_{X}s
0(Ω).

Then, we assume thatφ_{1}≥0 a.e. in Ω. Note thatφ_{1}satisfies
(−∆)^{s}φ1+λφ1=κ1a(x)φ1 in Ω.

Thus,φ1>0 in Ω by Proposition 2.1.

Defineg_{0} by the unique solution of the problem

(−∆)^{s}g0+λg0=f in Omega, g0∈X_{0}^{s}(Ω). (2.11)
By Proposition 2.1, we find thatg0>0 in Ω. Let us consider the eigenvalue problem
(−∆)^{s}φ+λφ=κ(g0)^{p−1}φ in Ω, φ∈X_{0}^{s}(Ω). (2.12)
since g0 ∈ X_{0}^{s}(Ω) ⊂L^{2N/N−2s}(Ω), we have (g0)^{p−1} ∈ L^{N/2s}(Ω). By Lemma 2.5,
there exist the first eigenvalue κ1>0 and the corresponding eigenfunctionφ1>0
in Ω.

Proof of Theorem 1.3 (i) and (iii). Put ¯µ= sup{µ >0 : Sµ 6=∅}. By Lemma 2.2
implies that ¯µ >0. Now we show that ¯µ <∞. Letµ >0 such thatSµ 6=∅, and
letu∈ Sµ. Put v =u−µg0, where g0 is the solution of (2.11). then,v satisfies
(−∆)^{s}v+λv=u^{p} >0 in Ω. By Proposition 2.1, we have v >0 in Ω, and hence,
u > µg0. Then, it follows that

(−∆)^{s}u+λu > µ^{p−1}(g0)^{p−1}u in Ω. (2.13)
Let φ_{1} > 0 be the eigenfunction corresponding to the first eigenvalue κ_{1} to the
problem (2.12); that is,

(−∆)^{s}φ_{1}+λφ_{1}=κ_{1}(g_{0})^{p−1}φ_{1} in Ω. (2.14)
Multiply (2.13) byφ1and (2.14) byu, respectively, and integrating them on Ω, we
have

µ^{p−1}
Z

Ω

(g0)^{p−1}uφ1dx <

Z

R^{N}

(−∆)^{s/2}u(−∆)^{s/2}φ1dx+λ
Z

Ω

uφ1dx

=κ1

Z

Ω

(g0)^{p−1}uφ1dx.

Thenµ < κ^{1/p−1}_{1} ifSµ6=∅, and hence ¯µ≤κ^{1/p−1}_{1} <+∞.

By the definition of ¯µ, (1.1) has no positive solution forµ >µ, so (iii) of Theorem¯ 1.3 holds. From Lemma 2.4, we obtain (i) of Theorem 1.3.

Forµ∈(0,µ), let¯ u_{µ} be the minimal solution of (1.1) obtained in Theorem 1.3.

We consider the following linearized eigenvalue problem

(−∆)^{s}φ+λφ=κp(u_{µ})^{p−1}φ in Ω, φ∈X_{0}^{s}(Ω). (2.15)
Since u_{µ} ∈X_{0}^{s}(Ω)⊂L^{2N/N−2s}(Ω), we have (u_{µ})^{p−1} ∈L^{N/2s}(Ω). By Lemma 2.5,
there exists the first eigenvalueκ1(µ)>0 of the problem (2.15), and it holds

Z

R^{N}

|(−∆)^{s/2}ψ|^{2}dx+λ
Z

Ω

ψ^{2}dx≥κ_{1}(µ)
Z

Ω

p(u_{µ})^{p−1}ψ^{2}dx (2.16)
for anyψ∈X_{0}^{s}(Ω).

To show the existence and uniqueness of solution for (1.1) withµ= ¯µ, we need the following lemmas.

Lemma 2.6. If µ∈(0,µ), then¯ κ_{1}(µ)>1.

Proof. For 0< µ < µ <ˆ µ, let¯ u_{µ} andu_{µ}_{ˆ} be the minimal solution ofSµ and Sµˆ,
respectively. Put z=u_{µ}_{ˆ}−u_{µ}. We find that z >0 from Lemma 2.4(iii), and that
z satisfies

(−∆)^{s}z+λz > p(u_{µ})^{p−1}z in Ω. (2.17)
Letφ1>0 be the eigenfunction corresponding to the first eigenvalue κ1(µ) to the
problem (2.15), that is,

(−∆)^{s}φ1+λφ1=κ1(µ)p(u_{µ})^{p−1}φ1 in Ω. (2.18)
Multiplying (2.17) and (2.18) by φ_{1} and z, respectively, and integrating them on
Ω, we obtain

κ_{1}(µ)
Z

Ω

p(u_{µ})^{p−1}φ_{1}zdx=
Z

R^{N}

(−∆)^{s/2}φ_{1}(−∆)^{s/2}zdx+λ
Z

Ω

φ_{1}zdx

>

Z

Ω

p(u_{µ})^{p−1}φ1zdx.

This implies thatκ1(µ)>1.

Lemma 2.7. Forµ ∈(0,µ), let¯ u_{µ} be the minimal solution of (1.1) obtained in
Theorem 1.3. Then, there exists a constant M > 0 independent of µ such that
ku_{µ}kX_{0}^{s}(Ω)≤M for allµ∈(0,µ).¯

Proof. Put vµ = u_{µ}−µg0, where g0 is the solution of problem (2.11). Then,
vµ∈X_{0}^{s}(Ω) and satisfies (−∆)^{s}vµ+λvµ= (vµ+µg0)^{p} in Ω; that is,

Z

R^{N}

(−∆)^{s/2}v_{µ}(−∆)^{s/2}ψdx+λ
Z

Ω

v_{µ}ψdx=
Z

Ω

(v_{µ}+µg_{0})^{p}ψdx
for anyψ∈X_{0}^{s}(Ω). Puttingψ=vµ, we have

kvµk^{2}_{λ}=
Z

Ω

(vµ+µg0)^{p}vµdx.

Since k · kλ is equivalent with k · kX^{s}_{0}(Ω), it suffices to show that there exists a
constantM^{0} >0 independent ofµsuch thatkv_{µ}k_{λ}≤M^{0} forµ∈(0,µ).¯

For anyε >0, there exists a constantC=C(ε)>0 such that
(t+s)^{p}≤(1 +ε)(t+s)^{p−1}t+Cs^{p} fort, s≥0.

Then, we have

kv_{µ}k^{2}_{λ}≤(1 +ε)
Z

Ω

(u_{µ})^{p−1}v^{2}_{µ}dx+Cµ^{p}
Z

Ω

(g_{0})^{p}v_{µ}dx.

From (2.16) and Lemma 2.6 it follows that Z

Ω

(u_{µ})^{p−1}v^{2}_{µ}dx < 1
pkvµk^{2}_{λ}.
By using H¨older and Sobolev inequalities, we obtain

Z

Ω

(g0)^{p}vµdx≤ kg0k^{p}_{L}p+1kvµk_{L}p+1≤Ckg0k^{p}_{L}p+1kvµk_{X}^{s}

0(Ω)≤C^{0}kg0k^{p}_{L}p+1kvµkλ

with some constantC,C^{0} >0. Then, it follows that
kvµk^{2}_{λ}≤(1 +ε)

p kvµk^{2}_{λ}+ ¯µ^{p}C^{0}kg0k^{p}_{L}p+1kvµkλ.
This implies thatkvµkλ is bounded forµ∈(0,µ), and hence¯ kvµk_{X}^{s}

0(Ω)is bounded

forµ∈(0,µ).¯

Lemma 2.8. For µ= ¯µ, the problem (1.1) has a positive minimal solutionu_{µ}_{¯} ∈
X_{0}^{s}(Ω), and there hold u_{µ}< u_{µ}_{¯} inΩ forµ <µ¯ andu_{µ}→u_{µ}_{¯} a.e. inΩas µ↑µ.¯
Proof. Let{µn}be sequence such thatµ_{n} < µ_{n+1}andµ_{n}→µ¯asn→ ∞. Sinceu_{µ}
is increasing inµ∈(0,µ) by Lemma 2.4 (iii), we have¯ u_{µ}_{n} < u_{µ}_{n+1} in Ω. Lemma
2.7 implies that{u_{µ}

n}is bounded inX_{0}^{s}(Ω). Then, there exists a positive function

¯

u∈X_{0}^{s}(Ω) such that, asn→ ∞,u_{µ}_{n} *u¯ weakly inX_{0}^{s}(Ω),u_{µ}_{n} →u¯ strongly in
L^{2}(Ω). andu_{µ}_{n}→u¯ a.e. in Ω. We note here thatu_{µ}_{n} satisfies

Z

R^{N}

(−∆)^{s/2}u_{µ}_{n}(−∆)^{s/2}ψdx+λ
Z

Ω

u_{µ}_{n}ψdx=
Z

Ω

u^{p}_{µ}_{n}ψdx+µn

Z

Ω

f ψdx (2.19)
for anyψ∈X_{0}^{s}(Ω), and that ¯usatisfies

Z

Ω

¯

u^{p}ψdx≤ kuk¯ ^{p}_{L}_{p+1}kψk_{L}p+1<∞.

Lettingn→ ∞in (2.19), by the monotone convergence theorem, we obtain Z

R^{N}

(−∆)^{s/2}u(−∆)¯ ^{s/2}ψdx+λ
Z

Ω

¯ uψdx=

Z

Ω

¯

u^{p}ψdx+ ¯µ
Z

Ω

f ψdx

Thus, ¯u ∈ X_{0}^{s}(Ω) is a positive solution of (1.1); i.e., ¯u ∈ S_{µ}_{¯}. From Lemma 2.4
(ii), there exists a minimal solution u_{µ}_{¯} ∈ Sµ¯ Then, u_{µ}_{¯} ≤ u. We will verify that¯
u_{µ}_{¯} ≡ u. In fact, from Lemma 2.4 (iii), we have¯ u_{µ}_{n} < u_{µ}_{¯} in Ω for n= 1,2, . . ..
It follows that ¯u≤u_{µ}_{¯}, and hence u_{µ}_{¯} ≡u. Since¯ u_{µ} is increasing inµ∈(0,µ), we¯
haveu_{µ}< u_{¯}_{µ} in Ω forµ <µ¯ andu_{µ}→u_{µ}_{¯} a.e. in Ω asµ↑µ.¯
Denote byκ_{1}(¯µ), the first eigenvalue of the linearized problem (2.15) withµ= ¯µ.

By Lemma 2.5, the first eigenvalueκ1(¯µ) is given by κ1(¯µ) = inf

ψ∈X_{0}^{s}(Ω)\{0}

R

R^{N}|(−∆)^{s/2}ψ|^{2}dx+λR

Ωψ^{2}dx
R

Ωp(u_{µ}_{¯})^{p−1}ψ^{2}dx . (2.20)
Sinceu_{µ}< u_{µ}_{¯} by Lemma 2.4, we haveκ_{1}(µ)≥κ_{1}(¯µ) forµ∈(0,µ).¯

Lemma 2.9. Assume that the problem (2.9)has the first eigenvalueκ1>1. Then,
for any f ∈(X_{0}^{s}(Ω))^{0}, the problem

(−∆)^{s}u+λu=a(x)u+f, inΩ. (2.21)
has a unique solution in X_{0}^{s}(Ω).

Proof. DefineI_{f}(u), foru∈X_{0}^{s}(Ω), by
I_{f}(u) = 1

2 Z

R^{N}

|(−∆)^{s/2}u|^{2}dx+λ
2
Z

Ω

u^{2}dx−1
2

Z

Ω

a(x)u^{2}dx−
Z

Ω

f udx.

From (2.10), it follows that Z

R^{N}

|(−∆)^{s/2}ψ|^{2}dx+λ
Z

Ω

ψ^{2}dx≥κ1

Z

Ω

a(x)ψ^{2}dx (2.22)
for anyψ∈X_{0}^{s}(Ω). Then, we have

If(u)≥ 1 2 − 1

2κ1

kuk^{2}_{λ}− kfk(X^{s}_{0}(Ω))^{0}kukX_{0}^{s}(Ω),

Since k · kλ is equivalent with k · kX_{0}^{s}(Ω), we obtain If(u) → ∞ as kukλ → ∞.

Thus, I_{f} is coercive and bounded from below inX_{0}^{s}(Ω). SinceI_{f} is weakly lower
semicontinuous onX_{0}^{s}(Ω), there existsu∈X_{0}^{s}(Ω) which attains the infimum, and
hence (2.21) has a solution in X_{0}^{s}(Ω). To show the uniqueness of the solution of
(2.21), it suffices to show that (2.21) has only trivial solution whenf ≡0. Assume
to the contrary that there exists a non-trivial solution u ∈ X_{0}^{s}(Ω). Then, from
(2.21) we have

Z

Ω

a(x)u^{2}dx=
Z

R^{N}

|(−∆)^{s/2}u|^{2}dx+λ
Z

Ω

u^{2}dx≥κ_{1}
Z

Ω

a(x)u^{2}dx.

This contradictsκ_{1}>1. Thus, (2.21) has a unique solution inX_{0}^{s}(Ω).

Lemma 2.10. We haveκ1(µ)→κ1(¯µ)asµ↑µ¯ andκ1(¯µ) = 1.

Proof. First, we will show that κ1(µ)→ κ1(¯µ) as µ↑ µ. By Lemma 2.5, we find¯ that

κ1(¯µ) = inf

ψ∈X_{0}^{s}(Ω)\{0}

R

R^{N}|(−∆)^{s/2}ψ|^{2}dx+λR

Ωψ^{2}dx
R

Ωp(u_{µ}_{¯})^{p−1}ψ^{2}dx

= R

R^{N}|(−∆)^{s/2}φ1|^{2}dx+λR

Ωφ^{2}_{1}dx
R

Ωp(u_{µ}_{¯})^{p−1}φ^{2}_{1}dx ,

whereφ1is the eigenfunction corresponding to the first eigenvalueκ1(¯µ). Let{µn} be a sequence such that µn < µn+1 and µn → µ¯ as n → ∞. By the monotone convergence theorem, we have

Z

Ω

p(u_{µ}

n)^{p−1}φ^{2}_{1}dx→
Z

Ω

p(u_{µ}_{¯})^{p−1}φ^{2}_{1}dx

asn→ ∞. Then, for anyε >0, there existsδ >0 such that, if 0<µ¯−µ < δthen R

R^{N}|(−∆)^{s/2}φ1|^{2}dx+λR

Ωφ^{2}_{1}dx
R

Ωp(u_{µ})^{p−1}φ^{2}_{1}dx −
R

R^{N}|(−∆)^{s/2}φ1|^{2}dx+λR

Ωφ^{2}_{1}dx
R

Ωp(u_{µ}_{¯})^{p−1}φ^{2}_{1}dx >0.

Put

˜ κ(µ) =

R

R^{N}|(−∆)^{s/2}φ1|^{2}dx+λR

Ωφ^{2}_{1}dx
R

Ωp(u_{µ})^{p−1}φ^{2}_{1}dx .

It follows from the above inequality that 0 < κ(µ)˜ −κ1(¯µ) < ε. Since κ1(¯µ) ≤ κ1(µ)≤˜κ(µ), we have

0≤κ_{1}(µ)−κ_{1}(¯µ)≤κ(µ)˜ −κ_{1}(¯µ)< ε if 0<µ¯−µ < δ.

This implies thatκ1(µ)→κ1(¯µ) asµ↑µ. Since¯ κ1(µ)>1 forµ∈(0,µ) by Lemma¯
2.5, we haveκ1(¯µ)≥1. Finally, we will show κ1(¯µ) = 1. Assume to the contrary
that κ1(¯µ)>1. Define Φ : (0,∞)×X_{0}^{s}(Ω)→(X_{0}^{s}(Ω))^{0} by (2.3). Foru∈X_{0}^{s}(Ω),
we have (2.4), and, in particular,

Φu(¯µ, u_{µ}_{¯})w= (−∆)^{s}w+λw−p(u_{µ}_{¯})^{p−1}w.

By Lemma 2.9, for every f ∈(X_{0}^{s}(Ω))^{0}, there exists a unique solution w∈X_{0}^{s}(Ω)
of Φu(¯µ, u_{µ}_{¯})w=f; that is, Φu:X_{0}^{s}(Ω)→(X_{0}^{s}(Ω))^{0} is invertible at (¯µ, u_{µ}_{¯}). Then,
by the implicit function theorem, there exist ε > 0 such that Φ(µ, u) = 0 has a
solutionu_{µ}∈X_{0}^{s}(Ω) for µ∈(¯µ−ε,µ¯+ε). From Lemma 2.2, we obtain a positive
solution u_{µ} of (1.1) for µ ∈ (¯µ−ε,µ¯+ε). This contradicts the definition of ¯µ.

Thus, we obtainκ1(¯µ) = 1.

Proof of Theorem 1.3 (ii). Letu_{µ}_{¯}∈Sµ¯be the minimal solution obtained in Lemma
2.2, we will show the uniqueness ofu_{µ}_{¯}∈Sµ¯. Assume u∈Sµ¯, and putz=u−u_{µ}_{¯}.
Sinceu_{µ}_{¯} is the minimal solution,z satisfiesz≥0 and

(−∆)^{s}z+λz=u^{p}−(u_{µ}_{¯})^{p} in Ω. (2.23)
Letφ1∈X_{0}^{s}(Ω) be the first eigenfunction of the linearized problem of (2.15) with
µ= ¯µ. Sinceκ1(¯µ) = 1 from Lemma 2.10, we have

(−∆)^{s}φ1+λφ1=p(u_{µ}_{¯})^{p−1}φ1 in Ω. (2.24)
Multiplying (2.23) and (2.24) by φ1 and z, respectively, and integrating them on
Ω, we have

Z

Ω

(u^{p}−u^{p}_{µ}_{¯})φ_{1}dx=
Z

R^{N}

(−∆)^{s/2}φ_{1}(−∆)^{s/2}zdx+λ
Z

Ω

φ_{1}zdx

=p Z

Ω

(u_{µ}_{¯})^{p−1}(u−u_{µ}_{¯})φ_{1}dx.

Hence, it follows that

Z

Ω

F(u, u_{µ}_{¯})φ1dx= 0

where F(σ, τ) = σ^{p} −τ^{p} −pτ^{p−1}(σ−τ). We note here that, for σ ≥ τ ≥ 0,
F(σ, τ)≥ 0 andF(σ, τ) = 0 holds if and only ifσ = τ. Then, from φ1 >0, we
conclude thatF(u, u_{µ}) = 0 a.e. in Ω, and henceu=u_{µ} a.e. in Ω. Thus, Theorem

1.3(ii) is obtained.

3. Existence of the second solution: Proof of Theorem 1.5
Letu_{µ} be the minimal solution of (1.1) forµ∈(0,µ) obtained in Theorem 1.3.¯
To find a second solution of (1.1), we introduce the problem

(−∆)^{s}v+λv= (v+u_{µ})^{p}−u^{p}_{µ} in Ω, v∈X_{0}^{s}(Ω). (3.1)
Assume that (3.1) has a positive solution v, and put ¯uµ = v+u_{µ}. Then,

¯

u_{µ} ∈ X_{0}^{s}(Ω) solves (1.1) and satisfies ¯u_{µ} > u_{µ} in Ω. We will show the existence

of solutions of (3.1) by using Mountain-Pass Lemma. To this end, we define the corresponding variational functional of (3.1) by

I¯_{λ,µ}(v) = 1
2

Z

R^{N}

|(−∆)^{s/2}v|^{2}dx+λ
2
Z

Ω

v^{2}dx−
Z

Ω

G(v, u_{µ})dx, (3.2)
forv∈X_{0}^{s}(Ω), where

G(σ, τ) := 1

p+ 1(σ++τ)^{p+1}− 1

p+ 1τ^{p+1}−τ^{p}σ+. (3.3)
Obviously, ¯Iλ,µ:X_{0}^{s}(Ω)→RisC^{1}; Ifv∈X_{0}^{s}(Ω) is a critical point, thenvsatisfies

Z

R^{N}

(−∆)^{s/2}v(−∆)^{s/2}ψ+
Z

Ω

λvψ− Z

Ω

g(v, u_{µ})ψdx= 0 (3.4)
for anyψ∈X_{0}^{s}(Ω), where

g(σ, τ) := (σ_{+}+τ)^{p}−τ^{p}, (3.5)
By Proposition 2.1, we havev >0, and hencev is a positive solution to (3.1).

The following lemma gives the properties of the functionsG(σ, τ),g(σ, τ) defined in (3.3) and (3.5). For a proof we refer the reader to [21, Appendix B.1], so we omit it here.

Lemma 3.1. (i) There exists a constant C=C(p)>0 such that
g(σ, τ)≤C(σ^{p}+τ^{p−1}σ) forσ, τ ≥0.

(ii) Forσ, τ ≥0, 1

p+ 1σ^{p+1}≤G(σ, τ)≤ 1

2g(σ, τ)σ.

(iii) For any ε >0, there is a constant C=C(ε)>0such that G(σ, τ)−p

2τ^{p−1}σ^{2}≤ετ^{p−1}σ^{2}+Cσ^{p+1} forσ, τ ≥0.

(iv) Putc_{p}= min{1, p−1}. Then,

g(σ, τ)σ−(2 +cp)G(σ, τ)≥ −c_{p}p

2 τ^{p−1}σ^{2} forσ, τ ≥0.

(v) If N≥6s, that is1< p≤2, then (p+ 1)G(σ, τ)−g(σ, τ)σ≤p(p−1)

2 τ^{p−1}σ^{2} forσ, τ ≥0.

To use Mountain-Pass Lemma to find critical point of ¯Iλ,µ, we first define H(σ, τ) :=G(σ, τ)− 1

p+ 1(σ+)^{p+1} and h(σ, τ) :=g(σ, τ)−(σ+)^{p}, (3.6)
and the following two lemmas show the properties ofH(σ, τ) andh(σ, τ), the reader
can refer [21, Appendix B.2] for the proof.

Lemma 3.2. (i) There exists a constant C >0 such that, fors, t≥0,
H(σ, τ)≤C(σ^{p}τ+τ^{p}σ),

h(σ, τ)σ≤C(σ^{p}τ+τ^{p}σ).

(ii) For σ_{0}≥0 andτ_{0}>0, there is a constant C=C(σ_{0}, τ_{0})>0 such that
H(σ, τ)≥Cσ^{p} forσ≥σ_{0}, τ≥τ_{0}.