For someλandN, by the barrier method and mountain pass lemma, we prove that there exists 0<µ

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

(−∆)^su+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^Nandfsatisfies 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

(−∆)^su+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 (−∆)^sis 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

(−∆)^su(x) =C_N,sP.V.

Z

R^N

u(x)−u(y)

|x−y|^N+2sdy, 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

(−∆)^su+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 = (−∆)^sψ+V(x)ψ−f(x,|ψ|), x∈R^N. (1.4) that is solutions of the formψ(x, t) =e^−iEtu(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=¹₂. 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_Hs(R^N)=kgk_L2(R^N)+Z

R^N×R^N

|g(x)−g(y)|²

|x−y|^N+2s dx dy1/2

, (1.6)

andX₀^s(Ω) is the function space defined as X₀^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 ofX₀^s(Ω) and its properties and to [15]

for an account of the properties ofH^s(R^N). InX₀^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 (X₀^s(Ω),k · k_X^s

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

hu, viX₀^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 inX₀^s(Ω),

kukλ=Z

R^N×R^N

|u(x)−u(y)|²

|x−y|^N+2s dx dy+λ Z

Ω

|u|²dx^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∈X₀^s(Ω), then Z

Ω

v(−∆)^sudx= Z

R^N

(−∆)^s/2u(−∆)^s/2vdx=hu, viX₀^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 ∈(X₀^s(Ω))⁰, where (X₀^s(Ω))⁰ denote the dual space ofX₀^s(Ω).

Definition 1.1. We say thatu∈X₀^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^pϕdx+µ Z

Ω

f ϕdx

(1.11)

for everyϕ∈X₀^s(Ω).

Definition 1.2. For any fixedµ >0, we say thatu_µis 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∗sdx)^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+ε²)^(N−2s)/2, ε >0, (1.13) that is

(−∆)^s/2u_ε

2

L²(R^N)= Z

R^N×R^N

|u_ε(x)−u_ε(y)|²

|x−y|^n+2s dx dy=S(N, s)kuεk²_L2(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∈X₀^s(Ω).

Furthermore,u_µ is increasing inµforµ∈(0,µ), and¯ u_µ→0 inX₀^s(Ω)as µ→0;

(ii) ifµ= ¯µ, the problem (1.1)has a unique positive solution in X₀^s(Ω);

(iii) ifµ >µ, the problem¯ (1.1)has no positive solution in X₀^s(Ω).

Remark 1.4. There is no positive solution of (1.1) if λ≤ −λ₁. 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/2u(−∆)^s/2ϕ1dx−λ1

Z

Ω

uϕ1dx

≥ Z

R^N

(−∆)^s/2u(−∆)^s/2ϕ1dx+λ Z

Ω

uϕ1dx

= Z

Ω

(u^pϕ1+µf ϕ1)dx >0.

This is a contradiction.

Theorem 1.3 indicates that equation (1.1) has a minimal solution u_µ ∈ X₀^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 ∈ X₀^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 ∈ X₀^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

(−∆)^sv+λv= (v+u_µ)^p−u^p_µ in Ω, v∈X₀^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 (−∆)^su ≥ 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∈X₀^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−1ϕdx+µ Z

Ω

f ϕdx≥0.

Thenuis a super-solution of

(−∆)^su=−λu+u^2∗s−1+µf in Ω,

u∈X₀^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

(−∆)^su+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

u²dx− 1 p+ 1

Z

Ω

(u+)^p+1dx

−µ Z

Ω

f udx.

The functionalIλ,µis well-defined for everyu∈X₀^s(Ω) and belongs toC¹(X₀^s(Ω),R).

Moreover, for anyu, ϕ∈X₀^s(Ω), we have hI_λ,µ⁰ (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₊)^pϕdx−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 ∈X₀^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 inX₀^s(Ω) forµ >0 small enough.

Proof. Define Φ : [0,+∞)×X₀^s(Ω)→(X₀^s(Ω))⁰ by

Φ(µ, u) = (−∆)^su+λu−(u+)^p−µf. (2.3) Then Φ is a continuous operator, and forw∈X₀^s(Ω), we have

Φ_u(µ, u)w= (−∆)^sw+λw−p(u₊)^p−1w. (2.4) In particular, Φu(0,0)w = (−∆)^sw+λw. It is clear that Φu(0,0) : X₀^s(Ω) → (X₀^s(Ω))⁰ is invertible forλ >−λ1. Then, by the implicit function theorem, there exists a functionuµ∈X₀^s(Ω) forµ∈(0, µ0] with someµ0>0 such that Φ(µ, uµ) = 0 andkuµkX₀^s(Ω)→0 asµ→0. Furthermore, there is no other solution of Φ(µ, u) = 0 in a neighborhood of the origin inX₀^s(Ω) forµ >0 sufficiently small. Then, u_µ solves the problem

(−∆)^su+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 inX₀^s(Ω) for µ >0 sufficiently small.

Lemma 2.3. Assume that there exists a positive functionuˆ∈X₀^s(Ω) satisfying (−∆)^suˆ+λˆu≥uˆ^p+ ˆµf inΩ (2.5) for someµ >ˆ 0. Then, for anyµ∈(0,µ], there exists a positive solutionˆ u∈X₀^s(Ω) of (1.1) satisfying 0 < u(x) ≤ u(x)ˆ for x ∈ Ω. Furthermore, for any positive solution ˜u∈X₀^s(Ω) of (1.1), the solutionusatisfiesu(x)≤u(x)˜ forx∈Ω.

Proof. Letµ∈(0,µ], and putˆ u₀≡0. Inductively, we can define{u_n}, by a solution of the problem

(−∆)^sun+λun= (u_n−1)^p+µf, un∈X₀^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 thatu₁∈X₀^s(Ω) and 0< u₁<uˆ in Ω by Proposition 2.1. Then, it follows thatu^p₁ ∈L^(p+1)0(Ω) ⊂(X₀^s(Ω))⁰, where (p+ 1)⁰ = 2N /(N+ 2s) is the conjugate exponent ofp+ 1 = 2N /(N−2s). By induction, we obtainun ∈X₀^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/2un(−∆)^s/2ψdx+λ Z

Ω

unψdx= Z

Ω

u^p_n−1ψdx+µ Z

Ω

f ψdx (2.7) for anyψ∈X₀^s(Ω). Puttingψ=un, we obtain

Z

R^N

|(−∆)^s/2un|²dx+λ Z

Ω

u²_ndx= Z

Ω

u^p_n−1undx+µ Z

Ω

f undx

≤ Z

Ω

ˆ

u^p+1dx+µ Z

Ω

fudxˆ

Thus,{un}is bounded inX₀^s(Ω). Hence, there exist a subsequence, denoted again {un}, andu∈X₀^s(Ω) satisfying, asn→ ∞, un * u in X₀^s(Ω) weakly, un →uin L²(Ω) strongly, andun→ua.e. in Ω. By the monotone convergence theorem, we

have Z

Ω

u^p_n−1ψdx→ Z

Ω

u^pψdx asn→ ∞.

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

R^N

(−∆)^s/2u(−∆)^s/2ψdx+λ Z

Ω

uψdx= Z

Ω

u^pψdx+µ Z

Ω

f ψdx (2.8) for anyψ∈X₀^s(Ω). This implies thatu∈X₀^s(Ω) is a solution to (1.1). From (2.6), we have 0< u≤uˆ in Ω.

Let ˜u∈X₀^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∈X₀^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_µ06=∅ for someµ₀>0. Then,S_µ 6=∅ for allµ∈(0, µ₀).

(ii) If S_µ6=∅, 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, µ₀) and u₀∈S_µ0. Applying Lemma 2.2 and Lemma 2.3 with ˆ

u=u₀ and ˆµ=µ₀, we obtain a positive solutionu∈X₀^s(Ω) of (1.1). This implies thatS_µ6=∅ for allµ∈(0, µ₀).

(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_µˆ in Ω. Put z =u_µˆ−u_µ ≥ 0. Then, z satisfies (−∆)^sz+λz ≥(ˆµ−µ)f ≥0, 6≡0 in Ω. By Proposition 2.1, we obtainz >0 in Ω, that is,u_µˆ> 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 ∈X₀^s(Ω). Putting u=ψ =u_µ in (2.8), we have

ku_µk²_λ=ku_µk^p+1_Lp+1+µ Z

Ω

f u_µdx≤ kuµk^p+1_Lp+1+µ Z

Ω

f u_µdx.

By the Sobolev inequality, we obtain ku_µk²_λ≤Ckuµk^p+1_Xs

0(Ω)+µkfk(X₀^s(Ω))⁰kuµkX^s₀(Ω), with some constant C > 0. Since k · kλ is equivalent with k · k_X^s

0(Ω), Lemma 2.2 implies that ku_µkX₀^s(Ω) → 0 as µ → 0. By the local uniqueness of the solution uµ in a neighborhood of the origin inX₀^s(Ω), we obtainuµ ≡u_µ forµsufficiently

small.

Next, let us consider the eigenvalue problem

(−∆)^sφ+λφ=κa(x)φ, φ∈X₀^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 (−∆)^swith 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∈X₀^s(Ω)\{0}

R

R^N|(−∆)^s/2ψ|²dx+λR

Ωψ²dx R

Ωa(x)ψ²dx . (2.10)

Lemma 2.5. Let(A1) hold. The infimumκ1 in (2.10)is positive and achieved by some φ₁∈X₀^s(Ω) with φ₁>0 in Ω. In particular, (κ₁, φ₁) is the first eigenvalue and the first eigenfunction to the problem (2.9).

Proof. Let{ψn} ⊂X₀^s(Ω) be a minimizing sequence of (2.10) satisfying Z

Ω

a(x)ψ²_ndx= 1, Z

R^N

|(−∆)^s/2ψn|²dx+λ Z

Ω

ψ²_ndx→κ1 as n→ ∞.

Since{ψ_n}is bounded inX₀^s(Ω), there exists a subsequence, still denoted by{ψ_n}, and a function φ1 ∈ X₀^s(Ω) such that, as n → ∞, ψn → φ1 weakly in X₀^s(Ω), ψn→φ1 strongly inL²(Ω),ψn→φ1 a.e. in Ω. Then, it follows that

κ1= lim inf

n→∞

Z

R^N

|(−∆)^s/2ψn|²dx+λ Z

Ω

ψ_n²dx≥ Z

R^N

|(−∆)^s/2φ1|²dx+λ Z

Ω

φ²₁dx.

Sincea(x)∈L^N/2s(Ω),{ψ_n²}is bounded inL^N/(N−2s)(Ω), we obtain Z

Ω

a(x)ψ_n²dx→ Z

Ω

a(x)φ²₁dx= 1 as n→ ∞.

Hence,φ16≡0 achieves the infimumκ1>0. Clearly,|φ1|also achievesκ1, since kφ1k²_Xs

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|k²_Xs 0(Ω).

Then, we assume thatφ₁≥0 a.e. in Ω. Note thatφ₁satisfies (−∆)^sφ1+λφ1=κ1a(x)φ1 in Ω.

Thus,φ1>0 in Ω by Proposition 2.1.

Defineg₀ by the unique solution of the problem

(−∆)^sg0+λg0=f in Omega, g0∈X₀^s(Ω). (2.11) By Proposition 2.1, we find thatg0>0 in Ω. Let us consider the eigenvalue problem (−∆)^sφ+λφ=κ(g0)^p−1φ in Ω, φ∈X₀^s(Ω). (2.12) since g0 ∈ X₀^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 (−∆)^sv+λv=u^p >0 in Ω. By Proposition 2.1, we have v >0 in Ω, and hence, u > µg0. Then, it follows that

(−∆)^su+λu > µ^p−1(g0)^p−1u in Ω. (2.13) Let φ₁ > 0 be the eigenfunction corresponding to the first eigenvalue κ₁ to the problem (2.12); that is,

(−∆)^sφ₁+λφ₁=κ₁(g₀)^p−1φ₁ 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−1uφ1dx <

Z

R^N

(−∆)^s/2u(−∆)^s/2φ1dx+λ Z

Ω

uφ1dx

=κ1

Z

Ω

(g0)^p−1uφ1dx.

Thenµ < κ^1/p−1₁ ifSµ6=∅, and hence ¯µ≤κ^1/p−1₁ <+∞.

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

(−∆)^sφ+λφ=κp(u_µ)^p−1φ in Ω, φ∈X₀^s(Ω). (2.15) Since u_µ ∈X₀^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/2ψ|²dx+λ Z

Ω

ψ²dx≥κ₁(µ) Z

Ω

p(u_µ)^p−1ψ²dx (2.16) for anyψ∈X₀^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¯ κ₁(µ)>1.

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

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

(−∆)^sφ1+λφ1=κ1(µ)p(u_µ)^p−1φ1 in Ω. (2.18) Multiplying (2.17) and (2.18) by φ₁ and z, respectively, and integrating them on Ω, we obtain

κ₁(µ) Z

Ω

p(u_µ)^p−1φ₁zdx= Z

R^N

(−∆)^s/2φ₁(−∆)^s/2zdx+λ Z

Ω

φ₁zdx

>

Z

Ω

p(u_µ)^p−1φ1zdx.

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_µkX₀^s(Ω)≤M for allµ∈(0,µ).¯

Proof. Put vµ = u_µ−µg0, where g0 is the solution of problem (2.11). Then, vµ∈X₀^s(Ω) and satisfies (−∆)^svµ+λvµ= (vµ+µg0)^p in Ω; that is,

Z

R^N

(−∆)^s/2v_µ(−∆)^s/2ψdx+λ Z

Ω

v_µψdx= Z

Ω

(v_µ+µg₀)^pψdx for anyψ∈X₀^s(Ω). Puttingψ=vµ, we have

kvµk²_λ= Z

Ω

(vµ+µg0)^pvµdx.

Since k · kλ is equivalent with k · kX^s₀(Ω), it suffices to show that there exists a constantM⁰ >0 independent ofµsuch thatkv_µk_λ≤M⁰ forµ∈(0,µ).¯

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

Then, we have

kv_µk²_λ≤(1 +ε) Z

Ω

(u_µ)^p−1v²_µdx+Cµ^p Z

Ω

(g₀)^pv_µdx.

From (2.16) and Lemma 2.6 it follows that Z

Ω

(u_µ)^p−1v²_µdx < 1 pkvµk²_λ. By using H¨older and Sobolev inequalities, we obtain

Z

Ω

(g0)^pvµdx≤ kg0k^p_Lp+1kvµk_Lp+1≤Ckg0k^p_Lp+1kvµk_X^s

0(Ω)≤C⁰kg0k^p_Lp+1kvµkλ

with some constantC,C⁰ >0. Then, it follows that kvµk²_λ≤(1 +ε)

p kvµk²_λ+ ¯µ^pC⁰kg0k^p_Lp+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¯ ∈ X₀^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+1andµ_n→µ¯asn→ ∞. Sinceu_µ is increasing inµ∈(0,µ) by Lemma 2.4 (iii), we have¯ u_µn < u_µn+1 in Ω. Lemma 2.7 implies that{u_µ

n}is bounded inX₀^s(Ω). Then, there exists a positive function

¯

u∈X₀^s(Ω) such that, asn→ ∞,u_µn *u¯ weakly inX₀^s(Ω),u_µn →u¯ strongly in L²(Ω). andu_µn→u¯ a.e. in Ω. We note here thatu_µn satisfies

Z

R^N

(−∆)^s/2u_µn(−∆)^s/2ψdx+λ Z

Ω

u_µnψdx= Z

Ω

u^p_µnψdx+µn

Z

Ω

f ψdx (2.19) for anyψ∈X₀^s(Ω), and that ¯usatisfies

Z

Ω

¯

u^pψdx≤ kuk¯ ^p_Lp+1kψk_Lp+1<∞.

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

R^N

(−∆)^s/2u(−∆)¯ ^s/2ψdx+λ Z

Ω

¯ uψdx=

Z

Ω

¯

u^pψdx+ ¯µ Z

Ω

f ψdx

Thus, ¯u ∈ X₀^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_µn < 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 byκ₁(¯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∈X₀^s(Ω)\{0}

R

R^N|(−∆)^s/2ψ|²dx+λR

Ωψ²dx R

Ωp(u_µ¯)^p−1ψ²dx . (2.20) Sinceu_µ< u_µ¯ by Lemma 2.4, we haveκ₁(µ)≥κ₁(¯µ) forµ∈(0,µ).¯

Lemma 2.9. Assume that the problem (2.9)has the first eigenvalueκ1>1. Then, for any f ∈(X₀^s(Ω))⁰, the problem

(−∆)^su+λu=a(x)u+f, inΩ. (2.21) has a unique solution in X₀^s(Ω).

Proof. DefineI_f(u), foru∈X₀^s(Ω), by I_f(u) = 1

2 Z

R^N

|(−∆)^s/2u|²dx+λ 2 Z

Ω

u²dx−1 2

Z

Ω

a(x)u²dx− Z

Ω

f udx.

From (2.10), it follows that Z

R^N

|(−∆)^s/2ψ|²dx+λ Z

Ω

ψ²dx≥κ1

Z

Ω

a(x)ψ²dx (2.22) for anyψ∈X₀^s(Ω). Then, we have

If(u)≥ 1 2 − 1

2κ1

kuk²_λ− kfk(X^s₀(Ω))⁰kukX₀^s(Ω),

Since k · kλ is equivalent with k · kX₀^s(Ω), we obtain If(u) → ∞ as kukλ → ∞.

Thus, I_f is coercive and bounded from below inX₀^s(Ω). SinceI_f is weakly lower semicontinuous onX₀^s(Ω), there existsu∈X₀^s(Ω) which attains the infimum, and hence (2.21) has a solution in X₀^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 ∈ X₀^s(Ω). Then, from (2.21) we have

Z

Ω

a(x)u²dx= Z

R^N

|(−∆)^s/2u|²dx+λ Z

Ω

u²dx≥κ₁ Z

Ω

a(x)u²dx.

This contradictsκ₁>1. Thus, (2.21) has a unique solution inX₀^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∈X₀^s(Ω)\{0}

R

R^N|(−∆)^s/2ψ|²dx+λR

Ωψ²dx R

Ωp(u_µ¯)^p−1ψ²dx

= R

R^N|(−∆)^s/2φ1|²dx+λR

Ωφ²₁dx R

Ωp(u_µ¯)^p−1φ²₁dx ,

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−1φ²₁dx→ Z

Ω

p(u_µ¯)^p−1φ²₁dx

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

R^N|(−∆)^s/2φ1|²dx+λR

Ωφ²₁dx R

Ωp(u_µ)^p−1φ²₁dx − R

R^N|(−∆)^s/2φ1|²dx+λR

Ωφ²₁dx R

Ωp(u_µ¯)^p−1φ²₁dx >0.

Put

˜ κ(µ) =

R

R^N|(−∆)^s/2φ1|²dx+λR

Ωφ²₁dx R

Ωp(u_µ)^p−1φ²₁dx .

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

0≤κ₁(µ)−κ₁(¯µ)≤κ(µ)˜ −κ₁(¯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,∞)×X₀^s(Ω)→(X₀^s(Ω))⁰ by (2.3). Foru∈X₀^s(Ω), we have (2.4), and, in particular,

Φu(¯µ, u_µ¯)w= (−∆)^sw+λw−p(u_µ¯)^p−1w.

By Lemma 2.9, for every f ∈(X₀^s(Ω))⁰, there exists a unique solution w∈X₀^s(Ω) of Φu(¯µ, u_µ¯)w=f; that is, Φu:X₀^s(Ω)→(X₀^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∈X₀^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

(−∆)^sz+λz=u^p−(u_µ¯)^p in Ω. (2.23) Letφ1∈X₀^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

(−∆)^sφ1+λφ1=p(u_µ¯)^p−1φ1 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¯)φ₁dx= Z

R^N

(−∆)^s/2φ₁(−∆)^s/2zdx+λ Z

Ω

φ₁zdx

=p Z

Ω

(u_µ¯)^p−1(u−u_µ¯)φ₁dx.

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

(−∆)^sv+λv= (v+u_µ)^p−u^p_µ in Ω, v∈X₀^s(Ω). (3.1) Assume that (3.1) has a positive solution v, and put ¯uµ = v+u_µ. Then,

¯

u_µ ∈ X₀^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/2v|²dx+λ 2 Z

Ω

v²dx− Z

Ω

G(v, u_µ)dx, (3.2) forv∈X₀^s(Ω), where

G(σ, τ) := 1

p+ 1(σ++τ)^p+1− 1

p+ 1τ^p+1−τ^pσ+. (3.3) Obviously, ¯Iλ,µ:X₀^s(Ω)→RisC¹; Ifv∈X₀^s(Ω) is a critical point, thenvsatisfies

Z

R^N

(−∆)^s/2v(−∆)^s/2ψ+ Z

Ω

λvψ− Z

Ω

g(v, u_µ)ψdx= 0 (3.4) for anyψ∈X₀^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−1σ) 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−1σ²≤ετ^p−1σ²+Cσ^p+1 forσ, τ ≥0.

(iv) Putc_p= min{1, p−1}. Then,

g(σ, τ)σ−(2 +cp)G(σ, τ)≥ −c_pp

2 τ^p−1σ² 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−1σ² 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^pτ+τ^pσ),

h(σ, τ)σ≤C(σ^pτ+τ^pσ).

(ii) For σ₀≥0 andτ₀>0, there is a constant C=C(σ₀, τ₀)>0 such that H(σ, τ)≥Cσ^p forσ≥σ₀, τ≥τ₀.