Lipschitz continuous, strictly decreasing and asymptotic behavior and non- resonance with respect to the Fuˇcik spectrum

Electronic Journal of Differential Equations, Vol. 2018 (2018), No. 74, pp. 1–21.

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

FIRST CURVE OF FU ˇCIK SPECTRUM FOR THE

p-FRACTIONAL LAPLACIAN OPERATOR WITH NONLOCAL NORMAL BOUNDARY CONDITIONS

DIVYA GOEL, SARIKA GOYAL, KONIJETI SREENADH

Communicated by Vicentiu Radulescu

Abstract. In this article, we study the Fuˇcik spectrum of thep-fractional Laplace operator with nonlocal normal derivative conditions which is defined as the set of all (a, b)∈R² such that

Λn,p(1−α)(−∆)^α_pu+|u|^p−2u=χΩ

(a(u⁺)^p−1−b(u⁻)^p−1) in Ω, N_α,pu= 0 inRⁿ\Ω,

has a non-trivial solutionu, where Ω is a bounded domain inRⁿwith Lipschitz boundary,p≥2,n > pα,, α∈(0,1) and Ω:={x∈Ω :d(x, ∂Ω)≤}. We show existence of the first non-trivial curveCof the Fuˇcik spectrum which is used to obtain the variational characterization of a second eigenvalue of the problem defined above. We also discuss some properties of this curveC, e.g.

Lipschitz continuous, strictly decreasing and asymptotic behavior and non- resonance with respect to the Fuˇcik spectrum.

1. Introduction

The Fuˇcik spectrum ofp-fractional Laplacian with nonlocal normal derivative is defined as the set Σp of all (a, b)∈R² such that

Λn,p(1−α)(−∆)^α_pu+|u|^p−2u= χΩ

(a(u⁺)^p−1−b(u⁻)^p−1) in Ω, Nα,pu= 0 inRⁿ\Ω,

(1.1) has a non-trivial solution u, where Ω is a bounded domain in Rⁿ with Lipschitz boundary,p≥2,α, ∈(0,1) and Ω:={x∈Ω :d(x, ∂Ω)≤}. The (−∆)^α_p is the p-fractional Laplacian operator defined as

(−∆)^α_pu(x) := 2 p.v.

Z

Rⁿ

|u(x)−u(y)|^p−2(u(x)−u(y))

|x−y|^n+pα dy for allx∈Rⁿ, andNα,p is the associated nonlocal derivative defined in [8] as

Nα,pu(x) := 2 Z

Ω

|u(x)−u(y)|^p−2(u(x)−u(y))

|x−y|^n+pα dy for allx∈Rⁿ\Ω.

2010Mathematics Subject Classification. 35A15, 35J92, 35J60.

Key words and phrases. Nonlocal operator; Fuˇcik spectrum; Steklov problem; Non-resonance.

c

2018 Texas State University.

Submitted November 22, 2017. Published March 17, 2018.

1

Bourgain, Brezis and Mironescu [3] proved that for any smooth bounded domain Ω⊂Rⁿ,u∈W^1,p(Ω), there exist a constant Λn,p such that

lim

α→1⁻Λn,p(1−α) Z

Ω×Ω

|u(x)−u(y)|^p

|x−y|^n+pα dxdy= Z

Ω

|∇u|^pdx.

The constant Λn,p can be explicitly computed and is given by Λ_n,p= pΓ(^n+p₂ )

2πⁿ⁻¹² Γ(^p+1₂ ).

For a = b = λ, the Fuˇcik spectrum in (1.1) becomes the usual spectrum that satisfies

Λ_n,p(1−α)(−∆)^α_pu+|u|^p−2u=λ

χ_Ω|u|^p−2u in Ω, Nα,pu= 0 in Rⁿ\Ω.

(1.2) In [7], authors proved that there exists a sequence of eigenvalues λk,(Ω) of (1.2) such that λk,(Ω) → ∞as k → ∞. Moreover, 0 < λ1,(Ω)< λ2,(Ω) ≤ · · · ≤ λk,(Ω)≤. . ., and the first eigenvalueλ1,(Ω) of (1.2) is simple, isolated and can be characterized as follows

λ1,(Ω)

= inf

u∈W^α,p

nΛ_n,p(1−α) Z

Q

|u(x)−u(y)|^p

|x−y|^n+pα dx dy+ Z

Ω

|u|^pdx: Z

Ω

|u|^pdx=o . The Fuˇcik spectrum was introduced by Fuˇcik (1976) who studied the problem in one dimension with periodic boundary conditions. In higher dimensions, the non-trivial first curve in the Fuˇcik spectrum of Laplacian with Dirichlet boundary for bounded domain has been studied in [10]. Later in [6] Cuesta, de Figueiredo and Gossez studied this problem forp-Laplacian operator with Dirichlet boundary condition.

The Fuˇcik spectrum in the case of Laplacian,p-Laplacian operator with Dirichlet, Neumann and Robin boundary condition has been studied by many authors, for instance [2, 5, 18, 20, 7, 22]. Goyal and Sreenadh [14] extended the results of [6] to nonlocal linear operators which include fractional Laplacian. The existence of Fuˇcik eigenvalues for p-fractional Laplacian operator with Dirichlet boundary conditions has been studied by many authors, for instance refer [23, 24]. Also, in [15], Goyal discussed the Fuˇcik spectrum of ofp-fractional Hardy Sobolev-Operator with weight function. A non-resonance problem with respect to Fuˇcik spectrum is also discussed in many papers [6, 21, 16]. We also refer to the related papers [9, 12, 13, 17].

The inspiring point of our work is [14, 15], where the existence of a nontrivial curve is studied only for p= 2 but the nature of the curve is left open for p6= 2.

In the present work, we extend the results obtained in [14] to the nonlinear case of p-fractional operator for any p ≥ 2 and also show that this curve is the first curve. We also showed the variational characterization of the second eigenvalue of the operator associated with (1.1). There is a substantial difference while handling the nonlinear nature of the operator. This difference is reflected while constructing the paths below a mountain-pass level (see the proof of Theorem 1.1). To the best of our knowledge, no work has been done on the Fuˇcik spectrum for nonlocal operators with nonlocal normal derivative. We would like to remark that the main

result obtained in this paper is new even for the following p-fractional Laplacian equation with Dirichlet boundary condition:

(−∆)^α_pu+|u|^p−2u=a(u⁺)^p−1−b(u⁻)^p−1 in Ω, u= 0 onRⁿ\Ω.

With this introduction, we state our main result.

Theorem 1.1. Lets≥0 then the point(s+c(s), c(s))is the first nontrivial point of Σp in the intersection betweenΣp and the line(s,0) +t(1,1)of (1.1).

This article is organized as follows: In section 2 we give some preliminaries. In section 3 we construct a first nontrivial curve in Σp, described as (s+c(s), c(s)).

In section 4 we prove that the linesλ_1,(Ω)×R andR×λ_1,(Ω) are isolated in Σ_p, the curve that we obtained in section 3 is the first nontrivial curve and give the variational characterization of second eigenvalue of (1.1). In section 5 we prove some properties of the first curve and non resonance problem.

2. Preliminaries

In this section we assemble some requisite material. By [8] we know the nonlocal analogue of divergence theorem which states that for any bounded functionsuand v∈C², it holds that

Z

Ω

(−∆)^α_pu(x)dx=− Z

Ω^c

Nα,pu(x)dx.

More generally, we have following integration by parts formula Hα,p(u, v) =

Z

Ω

v(x)(−∆)^α_pu(x)dx+ Z

Ω^c

v(x)Nα,pu(x)dx, whereHα,p(u, v) is defined as

Hα,p(u, v) :=

Z

Q

|u(x)−u(y)|^p−2(u(x)−u(y))(v(x)−v(y))

|x−y|^n+pα dy, Q:=R²ⁿ\(Ω^c)². Now, given a measurable functionu:Rⁿ→R, we set

kukα,p := (kuk^p_Lp(Ω)+ [u]^p_α,p)^1/p, where [u]α,p:= (Hα,p(u, u))^1/p. (2.1) Thenk · kα,p defines a norm on the space

W^α,p :={u:Rⁿ→Rmeasurable :kukα,p<∞}.

Clearly W^α,p ⊂ W^α,p(Ω), where W^α,p(Ω) denotes the usual fractional Sobolev space endowed with the norm

kukW^α,p =kukL^p+Z

Ω×Ω

(u(x)−u(y))^p

|x−y|^n+pα dx dy^1/p . To study the fractional Sobolev space in detail see [19].

Definition 2.1. A function u ∈ W^α,p is a weak solution of (1.1), if for every v∈ W^α,p,usatisfies

Λ_n,p(1−α)H_α,p(u, v) + Z

Ω

|u|^p−2uv−a Z

Ω

(u⁺)^p−1v+b

Z

Ω

(u⁻)^p−1v= 0.

Now, we define the functionalJ associated to problem (1.1) asJ :W^α,p →R such that

J(u) = Λn,p(1−α) Z

Q

|u(x)−u(y)|^p

|x−y|^n+pα dx dy+ Z

Ω

|u|^pdx

−a Z

Ω

(u⁺)^pdx+b

Z

Ω

(u⁻)^pdx.

ThenJ is Fr´echet differentiable inW^α,p and for allv∈ W^α,p. hJ⁰(u), vi= Λn,p(1−α)Hα,p(u, v) +

Z

Ω

|u|^p−2uv−a Z

Ω

(u⁺)^p−1v+b

Z

Ω

(u⁻)^p−1v.

For the sake of completeness, we describe the Steklov problem (−∆)pu+|u|^p−2u= 0 in Ω,

|∇u|^p−2∂u

∂ν =λ|u|^p−2u on∂Ω,

(2.2) where Ω is a bounded domain and p >1. By [7], (1.1) is related to (2.2) in the sense that if Ω be a bounded smooth domain in Rⁿ with Lipschitz boundary and p∈(1,∞). For a fixedu∈W^1,p(Ω)\W₀^1,p(Ω), we have

lim

→0⁺

1 Z

Ω

|u|^pdx= Z

∂Ω

|u|^pdS and lim

α→1⁻Λn,p(1−α)[Eu]^p_α,p=k∇uk^p_Lp(Ω), where E is a bounded linear extension operator fromW^1,p(Ω) to W₀^1,p(BR) such that Eu =uin Ω and Ω is relatively compact in BR, the ball of radius R in Rⁿ. This leads to the following Lemma in [20].

Lemma 2.2. Let Ω be a smooth domain in Rⁿ with Lipschitz boundary and p∈ (1,∞). For a fixedu∈W^1,p(Ω)\W₀^1,p(Ω), it holds

lim

α→1⁻

Λ_n,p(1−α)[Eu]^p_α,p+kEuk^p_Lp(Ω)

1

1−αkEuk^p_Lp(Ω1−α)

=

k∇uk^p_Lp(Ω)+kuk^p_Lp(Ω) kuk^p_Lp(∂Ω)

.

Taking= 1−α, by Lemma 2.2 the eigenvalueλ_1,1−α(Ω_1−α)→λ1 asα→1⁻, whereλ1is the first eigenvalue of the operator associated with (2.2). Similarly, we obtain that as α→1⁻ the Fuˇcik Spectrum of the operator associated with (1.1) tends to Fuˇcik Spectrum of the Steklov problem.

We shall throughout use the function spaceW^α,p with the normk · kand we use the standard L^p(Ω) space whose norms are denoted bykukL^p(Ω). Also, we denote λn,(Ω) byλn,. Hereφ1,is the eigenfunction corresponding toλ1,.

3. The Fuˇcik spectrumΣ_p

In this section, we study existence of the first nontrivial curve in the Fuˇcik spectrum Σp of (1.1). We find that the points in Σpare associated with the critical value of some restricted functional. For this, for fixeds∈Rands≥0, we consider the functionalJs:W^α,p→Rdefined by

Js(u) = Λn,p(1−α) Z

Q

|u(x)−u(y)|^p

|x−y|^n+pα dx dy+ Z

Ω

|u|^pdx−s

Z

Ω

(u⁺)^pdx.

ThenJ_s∈C¹(W^α,p,R) and for anyφ∈ W^α,p hJ_s⁰(u), φi=pΛ_n,p(1−α)H_α,p(u, φ) +p

Z

Ω

|u|^p−2uφ dx−ps

Z

Ω

(u⁺)^p−1φ dx.

Also ˜Js:=Js|S isC¹(W^α,p,R), whereS is defined as S :=

u∈ W^α,p:I(u) := 1 Z

Ω

|u|^p= 1 .

We first note that u∈ S is a critical point of ˜Js if and only if there exists t ∈R such that

Λn,p(1−α)Hα,p(u, v)−s Z

Ω

(u⁺)^p−1v dx= t

Z

Ω

|u|^p−2uv dx, (3.1) for allv∈ W^α,p. Henceu∈ S is a nontrivial weak solution of the problem

Λn,p(1−α)(−∆)^α_p +|u|^p−2u=χΩ

(s+t)(u⁺)^p−1−t(u⁻)^p−1 in Ω, Nα,pu= 0 inRⁿ\Ω,

which exactly means (s+t, t)∈Σ_p. Substitutingv=uin (3.1), we obtaint= ˜J_s(u).

Thus we obtain the following Lemma which links the critical point of ˜Js and the spectrum Σp.

Lemma 3.1. Fors≥0, (s+t, t)∈R² belongs to the spectrum Σp if and only if there exists a critical pointu∈ S ofJ˜s such that t= ˜Js(u), a critical value.

Proposition 3.2. The first eigenfunction φ1, is a global minimum for J˜s with J˜_s(φ_1,) =λ_1,−s. The corresponding point in Σ_p is(λ_1,, λ_1,−s) which lies on the vertical line through(λ_1,, λ_1,).

Proof. We have

J˜s(u) =Λn,p(1−α) Z

Q

|u(x)−u(y)|^p

|x−y|^n+pα dx dy+ Z

Ω

|u|^pdx−s

Z

Ω

(u⁺)^pdx

≥λ1,

Z

Ω

|u|^pdx−s

Z

Ω

(u⁺)^pdx≥λ1,−s.

Thus ˜Js is bounded below byλ1,−s. Moreover, J˜s(φ1,) =λ1,−s

Z

Ω

(φ⁺_1,)^pdx=λ1,−s.

Thusφ1,is a global minimum of ˜Jswith ˜Js(φ1,) =λ1,−s.

Proposition 3.3. The negative eigenfunction−φ1, is a strict local minimum for J˜_s with J˜_s(−φ_1,) =λ_1,. The corresponding point inΣ_p is(λ_1,+s, λ_1,), which lies on the horizontal line through (λ_1,, λ_1,).

Proof. Suppose by contradiction that there exists a sequenceu_k ∈ S,u_k 6=−φ1,

with ˜Js(uk) ≤ λ1,, uk → −φ1, in W^α,p. We claim that uk changes sign for sufficiently large k. Sinceuk → −φ1,, uk must be <0 for sufficiently largek. If uk≤0 for a.ex∈Ω, then

J˜s(uk) = Λn,p(1−α) Z

Q

|uk(x)−uk(y)|^p

|x−y|^n+pα dx dy+ Z

Ω

|uk|^pdx > λ1,, sinceu_k 6≡ ±φ1,and we obtain contradiction as ˜J_s(u_k)≤λ_1,. Therefore the claim is proved.

Now, definewk :=

1/pu⁺_k ku⁺_kkLp(Ω)

and rk := Λn,p(1−α)

Z

Q

|wk(x)−w_k(y)|^p

|x−y|^n+pα dx dy+ Z

Ω

|wk|^pdx.

We claim that rk → ∞ as k→ ∞. Assume by contradiction thatrk is bounded.

Then there exists a subsequence (still denoted by {wk}) of {wk} and w ∈ W^α,p such that wk * w weakly in W^α,p and wk → w strongly in L^p(Ω). It implies wk→wstrongly inL^p(Ω). Therefore ¹R

Ωw^pdx= 1,w≥0 a.e. in Ω and so for some η >0,δ=|{x∈Ω:w(x)≥η}|>0. Since,u_k → −φ_1,in W^α,p and hence in L^p(Ω). Therefore, for each η >0, |{x∈Ω : u_k(x)≥η}| → 0 ask → ∞ and

|{x∈Ω:wk(x)≥η}| →0 as k→ ∞, which is a contradiction toη > 0. Hence, rk→ ∞. Clearly, one can have

|uk(x)−uk(y)|^p

= (|uk(x)−uk(y)|²)^p/2= [((u⁺_k(x)−u⁺_k(y))−(u⁻_k(x)−u⁻_k(y)))²]^p/2

= [(u⁺_k(x)−u⁺_k(y))²+ (u⁻_k(x)−u⁻_k(y))²−2(u⁺_k(x)−u⁺_k(y))(u⁻_k(x)−u⁻_k(y))]^p/2

= [(u⁺_k(x)−u⁺_k(y))²+ (u⁻_k(x)−u⁻_k(y))²+ 2u⁺_k(x)u⁻_k(y) + 2u⁻_k(x)u⁺_k(y)]^p/2

≥ |u⁺_k(x)−u⁺_k(y)|^p+|u⁻_k(x)−u⁻_k(y)|^p. Using the above inequality, we have

J˜_s(u_k) = Λ_n,p(1−α) Z

Q

|uk(x)−u_k(y)|^p

|x−y|^n+pα dx dy+ Z

Ω

|uk|^p−s Z

Ω

(u⁺_k)^pdx

≥

Λ_n,p(1−α) Z

Q

|u⁺_k(x)−u⁺_k(y)|^p

|x−y|^n+pα dx dy+ Z

Ω

|u⁺_k|^p

+h

Λn,p(1−α) Z

Q

|u⁻_k(x)−u⁻_k(y)|^p

|x−y|^n+pα dx dy+ Z

Ω

|u⁻_k|^p

−s

Z

Ω

(u⁺_k)^pdxi

≥ (r_k−s)

Z

Ω

(u⁺_k)^pdx+λ_1,

Z

Ω

(u⁻_k)^pdx.

(3.2)

On the other hand, sinceu_k ∈ S, we obtain J˜s(uk)≤λ1,=λ1,

Z

Ω

(u⁺_k)^pdx+λ1,

Z

Ω

(u⁻_k)^pdx. (3.3) From (3.2) and (3.3), we have

(rk−s−λ1,)

Z

Ω

(u⁺_k)^pdx≤0,

and this impliesrk−s≤λ1,, which contradicts thatrk →+∞. Therefore,−φ1,

is the strict local minimum.

Proposition 3.4 ([1]). Let Y be a Banach space, g, f ∈C¹(Y,R), M ={u∈Y : g(u) = 1} andu0,u1∈M. Let >0 such that ku1−u0k> and

inf{f(u) :u∈M andku−u₀kY =}>max{f(u₀), f(u₁)}.

Assume thatf satisfies the (PS) condition onM and that

Γ ={γ∈C([−1,1], M) :γ(−1) =u0 andγ(1) =u1}

is non empty. Thenc= inf_γ∈Γmax_{u∈γ[−1,1]}f(u)is a critical value of f|M. We now find the third critical point via mountain pass Theorem as stated above.

A norm of derivative of the restriction ˜Js ofJsat u∈ S is defined as kJ˜_s⁰(u)k_∗= min{kJ˜_s⁰(u)−tI⁰(u)k:t∈R}.

Lemma 3.5. Js satisfies the (PS) condition on S.

Proof. LetJs(uk) andtk ∈Rbe a sequences such that for someK >0,

|Js(uk)| ≤K, (3.4)

Λn,p(1−α)Hα,p(uk, v) + Z

Ω

|uk|^p−2ukv−s Z

Ω

(u⁺_k)^pv dx

−tk

Z

Ω

|uk|^p−2ukv dx

≤η_kkvk

(3.5)

for all v ∈ W^α,p, η_k → 0. From (3.4), using fractional Sobolev embedding, we obtain{u_k} is bounded in W^α,p which implies there is a subsequence denoted by u_k and u₀ ∈ W^α,p such that u_k * u₀ weakly in W^α,p, and u_k → u₀ strongly in L^p(Ω) for all 1≤p < p^∗_α. Substitutingv=u_k in (3.5), we obtain

|tk| ≤Λn,p(1−α) Z

Q

|uk(x)−uk(y)|^p

|x−y|^n+pα dx dy+ Z

Ω

|uk|^p+s

Z

Ω

(u⁺_k)^pdx+ηkkukk

≤C.

Hence,t_k is a bounded sequence so has a convergent subsequence sayt_k that con- verges tot. Next, we claim thatuk →u0strongly in W^α,p. Sinceuk* u0 weakly inW^α,p, we obtain

Z

Q

|u0(x)−u0(y)|^p−2(u0(x)−u0(y))(uk(x)−uk(y))

|x−y|^n+pα dx dy

→ Z

Q

|u0(x)−u0(y)|^p

|x−y|^n+pα dx dy as k→ ∞.

(3.6)

AlsohJ˜_s⁰(uk),(uk−u0)i=o(ηk). This implies

Λn,p(1−α) Z

Q

1

|x−y|^n+pα

|uk(x)−uk(y)|^p−2

×(uk(x)−uk(y))((uk−u0)(x)−(uk−u0)(y)) dx dy

≤o(η_k) +kukk^p−1_Lp(Ω)kuk−u₀kL^p(Ω)+sku⁺_kk^p−1_Lp(Ω)kuk−u₀kL^p(Ω)

+|tk|kukk^p−1_Lp(Ω)kuk−u0k_Lp(Ω)→0 ask→ ∞. Thus,

Z

Q

|uk(x)−uk(y)|^p

|x−y|^n+pα dx dy

− Z

Q

|uk(x)−uk(y)|^p−2(uk(x)−uk(y))(u0(x)−u0(y))

|x−y|^n+pα dx dy→0,

(3.7)

ask→ ∞. As we know that|a−b|^p≤2^p(|a|^p−2a− |b|^p−2b)(a−b) for alla, b∈R. Therefore, from (3.6) and (3.7) we obtain

Z

Q

|(uk−u0)(x)−(uk−u0)(y)|^p

|x−y|^n+pα dx dy→0 ask→ ∞

Hence,uk converges strongly to u0 inW^α,p.

Lemma 3.6. Let η₀>0 be such that

J˜s(u)>J˜s(−φ1,) (3.8) for allu∈B(−φ1,, η₀)∩ S withu6≡ −φ1,, where the ball is taken in W^α,p. Then for any 0< η < η₀,

inf{J˜_s(u) :u∈ S and ku−(−φ_1,)k=η}>J˜_s(−φ_1,). (3.9) Proof. If possible, let infimum in (3.9) is equal to ˜Js(−φ1,) =λ1,for someη with 0< η < η0. It implies there exists a sequenceuk ∈ S withkuk−(−φ1,)k=ηsuch that

J˜_s(u_k)≤λ_1,+ 1

2k². (3.10)

Consider the set V ={u∈ S :η−δ≤ ku−(−φ1,)k ≤η+δ}, whereδ is chosen such that η−δ >0 and η+δ < η0. From (3.9) and given hypotheses, it follows that inf{J˜s(u) : u ∈ V} = λ1,. Now for each k, we apply Ekeland’s variational principle to the functional ˜J_sonV to get the existence ofv_k∈V such that

J˜_s(v_k)≤J˜_s(u_k), kv_k−u_kk ≤ 1

k, (3.11)

J˜_s(v_k)≤J˜_s(u) +1

kku−v_kk, for allu∈V. (3.12) We claim thatv_kis a Palais-Smale sequence for ˜J_sonS. That is, there existsM >0 such that |J˜_s(v_k)| < M and kJ˜_s⁰(v_k)k∗ → 0 ask→ ∞. Once this is proved then by Lemma 3.5, there exists a subsequence denoted byv_k of v_k such that v_k →v strongly in W^α,p. Clearly, v ∈ S and satisfies kv−(−φ_1,)k ≤ η +δ < η₀ and J˜s(v) =λ1,which contradicts (3.8).

Now, the boundedness of ˜J_s(v_k) follows from (3.10) and (3.11). So, we only need to prove that kJ˜_s⁰(v_k)k∗ → 0. Let k > ¹_δ and takew ∈ W^α,p tangent to S at v_k. That is, ¹R

Ω|vk|^p−2v_kw dx= 0. Then by takingu_t:= _kv^1/p(vk+tw)

k+twk_Lp(Ω) fort∈R, we obtain

limt→0kut−(−φ1,)k=kvk−(−φ1,)k ≤ kvk−ukk+kuk−(−φ1,)k

≤ 1

k+η < δ+η, and

limt→0kut−(−φ1,)k=kvk−(−φ1,)k ≥ kuk−(−φ1,)k − kvk−ukk

≥η−1

k > η−δ.

Hence, fort small enoughut∈V and replacingubyut in (3.12), we obtain J˜_s(v_k)≤J˜_s(u_t) +1

kku_t−v_kk.

Letr(t) :=^1/pkvk+twkL^p(Ω), then J_s(v_k)−J_s(v_k+tw)

t

≤ J_s(u_t) +¹_kku_t−v_kk −J_s(v_k+tw) t

= 1

k t r(t)kv_k(1−r(t) +tw)k+1 t

1 r(t)^p −1

J(v_k+tw).

Now since

d

dtr(t)^p|t=0=p Z

Ω

|vk|^p−2vkw= 0,

we obtain ^r(t)_t^p−1 → 0 as t → 0, and then ^1−r(t)_t → 0 as t → 0. Therefore, we obtain

|hJ_s⁰(v_k), wi| ≤ 1

kkwk. (3.13)

Sincewis arbitrary inW^α,p, we choosea_ksuch that ¹R

Ω|v_k|^p−2v_k(w−a_kv_k)dx= 0. Replacingwbyw−akvk in (3.13), we obtain

hJ_s⁰(vk), wi −akhJ_s⁰(vk), vki ≤ 1

kkw−akvkk.

Sincekakvkk ≤Ckwk, we obtain

hJ_s⁰(vk), wi−tkR

Ω|vk|^p−2vkw dx

≤^C_kkwk, where tk =hJ_s⁰(vk), vki. Hence, kJ˜_s⁰(vk)k∗→0 ask→ ∞, as we required.

Proposition 3.7. Let W^α,p be a Banach Space. Let η >0 such that kφ1,−(−φ1,)k> η and

inf{J˜s(u) :u∈ S andku−(−φ1,)k=η}>max{J˜s(−φ1,),J˜s(φ1,)}.

ThenΓ ={γ∈C([−1,1],S) :γ(−1) =−φ_1, andγ(1) =φ_1,} is non empty and c(s) = inf

γ∈Γ max

u∈γ[−1,1]Js(u) (3.14)

is a critical value ofJ˜_s. Moreoverc(s)> λ_1,.

Proof. We prove that Γ is non-empty. To end this, we take φ ∈ W^α,p such that φ6∈Rφ1,and consider the pathtφ1,+ (1− |t|)φthen

w= ^1/p(tφ1,+ (1− |t|)φ) ktφ1,+ (1− |t|)φk_Lp(Ω)

.

Moreover the (PS) condition and the geometric assumption are satisfied by the Lemmas 3.5 and 3.6. Then by Proposition 3.4,c(s) is a critical value of ˜Js. Using the definition ofc(s) we have c(s)>max{J˜s(−φ1,),J˜s(φ1,)}=λ1,.

Thus we have proved the following result.

Theorem 3.8. For each s ≥ 0, the point (s+c(s), c(s)), where c(s) > λ_1, is defined by the minimax formula (3.14), then the point(s+c(s), c(s))belongs toΣ_p. It is a trivial fact that Σp is symmetric with respect to diagonal. The whole curve, that we obtain using Theorem 3.8 is denoted by

C:={(s+c(s), c(s)),(c(s), s+c(s)) :s≥0}.

4. First nontrivial curve

We start this section by establishing that the lines R× {λ1,} and {λ1,} ×R are isolated in Σp. Then we state some topological properties of the functional ˜Js

and some Lemmas. Finally, we prove that the curveC constructed in the previous section is the first non trivial curve in the spectrum Σp. As a consequence of this, we also obtain a variational characterization of the second eigenvalueλ2,.

Proposition 4.1. The linesR× {λ1,}and{λ1,} ×Rare isolated in Σp. In other words, there exists no sequence(ak, bk)∈Σp withak > λ1,andbk > λ1,such that (ak, bk)→(a, b)with a=λ1, orb=λ1,.

Proof. Suppose by contradiction that there exists a sequence (ak, bk)∈Σpwithak, bk > λ1, and (ak, bk)→(a, b) withaorb=λ1,. Letuk ∈ W^α,p be a solution of

Λ_n,p(1−α)(−∆)^α_pu_k+|uk|^p−2u_k =χ_Ω

(a_k(u⁺_k)^p−1−b_k(u⁻_k)^p−1) in Ω, Nα,puk= 0 inRⁿ\Ω,

(4.1) with ¹R

Ω|uk|^pdx= 1. Multiplying byuk in (4.1) and integrate, we have Λn,p(1−α)

Z

Q

|uk(x)−u_k(y)|^p

|x−y|^n+pα dx dy+ Z

Ω

|uk|^pdx

= ak

Z

Ω

(u⁺_k)^pdx−bk

Z

Ω

(u⁻_k)^pdx≤ak.

Thus{uk} is a bounded sequence inW^α,p. Therefore up to a subsequenceuk* u weakly in W^α,p and uk →ustrongly inL^p(Ω). Then taking limitk→ ∞in the weak formulation of (4.1), we obtain

Λn,p(1−α)(−∆)^α_pu+|u|^p−2u= χΩ

(λ1,(u⁺)^p−1−b(u⁻)^p−1) in Ω, Nα,pu= 0 inRⁿ\Ω.

(4.2) Takingu⁺ as test function in (4.2) we obtain

Λn,p(1−α)Hα,p(u, u⁺) + Z

Ω

(u⁺)^pdx= λ_1,

Z

Ω

(u⁺)^pdx. (4.3) Observe that

((u(x)−u(y))(u⁺(x)−u⁺(y)) = 2u⁻(x)u⁺(y) + (u⁺(x)−u⁺(y))², (4.4) and

|u(x)−u(y)|^p−2= (|u(x)−u(y)|²)^p−22

= (|u⁺(x)−u⁺(y)|²+|u⁻(x)−u⁻(y)|²+ 2u⁺(x)u⁻(y) + 2u⁺(y)u⁻(x))^p−22

≥ |u⁺(x)−u⁺(y)|^p−2.

(4.5)

Using (4.4) and (4.5) in (4.3) and the definition ofλ1,, we obtain λ1,

Z

Ω

(u⁺)^pdx≤Λ_n,p(1−α) Z

Q

|u⁺(x)−u⁺(y)|^p

|x−y|^n+pα dx dy+ Z

Ω

(u⁺)^pdx

≤ λ_1,

Z

Ω

(u⁺)^pdx.

Thus

Λn,p(1−α) Z

Q

|u⁺(x)−u⁺(y)|^p

|x−y|^n+pα dx dy+ Z

Ω

(u⁺)^pdx= λ1,

Z

Ω

(u⁺)^pdx, so either u⁺ ≡0 or u=φ_1,. Ifu⁺ ≡0 thenu≤0 and (4.2) implies thatuis an eigenfunction withu≤0 so thatu=−φ1,. So, in any caseuk converges to either φ1,or−φ1,in L^p(Ω). Thus

either|{x∈Ω:u_k(x)<0}| →0 or|{x∈Ω:u_k(x)>0}| →0 (4.6) ask→ ∞. On the other hand, takingu⁺_k as test function in (4.1), we obtain

Λn,p(1−α)Hα,p(uk, u⁺_k) + Z

Ω

|uk|^p−2uku⁺_k = ak

Z

Ω

(u⁺_k)^p. (4.7) Using H¨olders inequality, fractional Sobolev embeddings and (4.7), we obtain

Λn,p(1−α) Z

Q

|u⁺_k(x)−u⁺_k(y)|^p

|x−y|^n+pα dx dy+ Z

Ω

(u⁺_k)^pdx

≤Λn,p(1−α) Z

Q

|uk(x)−uk(y)|^p−2(uk(x)−uk(y))(u⁺_k(x)−u⁺_k(y))

|x−y|^n+pα dx dy

+ Z

Ω

|uk|^p−2uku⁺_kdx

= Λn,p(1−α)Hα,p(uk, u⁺_k) + Z

Ω

|uk|^p−2uku⁺_kdx

=ak

Z

Ω

(u⁺_k)^pdx

≤a_k

C|{x∈Ω:u_k(x)>0}|^1−pqku⁺_kk^p

with a constantC >0,p < q≤p^∗= _n−pα^np . Then we have

|{x∈Ω :uk(x)>0}|^1−pq ≥a⁻¹_k C⁻¹min{Λn,p(1−α),1}.

Similarly, one can show that

|{x∈Ω :uk(x)<0}|^1−pq ≥b⁻¹_k C⁻¹min{Λn,p(1−α),1}.

Since (a_k, b_k) does not belong to the trivial lines λ_1,×R andR×λ_1, of Σ_p, by (4.1) we conclude that uk changes sign. Hence, from the above inequalities, we obtain a contradiction with (4.6). Therefore, the trivial linesλ1,×RandR×λ1,

are isolated in Σp.

Lemma 4.2 ([6]). Let S={u∈ W^α,p : ¹R

Ω|u|^pdx= 1} then (1) S is locally arcwise connected.

(2) Any open connected subset Oof S is arcwise connected.

(3) IfO⁰ is any connected component of an open setO ⊂ S, then∂O⁰∩ O=∅.

Lemma 4.3. Let O={u∈ S : ˜J_s(u)< r}, then any connected component of O contains a critical point of J˜_s.

Proof. Let O1 be any connected component of O, let d = inf{J˜s(u) : u ∈ O1}, whereO1 denotes the closure ofO1inW^α,p. We show that there existsu₀∈ W^α,p such that ˜J_s(u₀) = d. For this let u_k ∈ O1 be a minimizing sequence such that

J˜s(uk) ≤ d+ _2k¹2. For each k, by applying Ekeland’s Variational principle, we obtain a sequencev_k ∈ O1 such that

J˜s(vk)≤J˜s(uk), kvk−ukk ≤ 1

k, J˜s(vk)≤J˜s(v) +1

kkv−vkk ∀v∈ O1. Forklarge enough, we have

J˜s(vk)≤J˜s(uk)≤d+ 1 2k² < r,

thenv_k ∈ O. By Lemma 4.2, we obtainv_k 6∈∂O₁ so v_k ∈ O₁. On the other hand, fortsmall enough andwsuch that ¹R

Ω|vk|^p−2vkw dx= 0, we have ut:= ^1/p(vk+tw)

kv_k+twk_Lp(Ω)

∈ O1.

Then ˜J_s(v_k)≤J˜_s(u_t) + ¹_kku_t−v_kk. Following the same calculation as in Lemma 3.6, we have thatvk is a Palais-Smale sequence for ˜Js onS i.e ˜Js(vk) is bounded and kJ˜s(vk)k∗ →0. Again by Lemma 3.5, up to a subsequence vk →u0 strongly in W^α,p and hence ˜Js(u0) = d < r and moreover u0 ∈ O. By part 3 of Lemma 4.2, u0 6∈∂O1 so u0 ∈ O1. Hence u0 is a critical point of ˜Js, which completes the

proof.

Before proving the main Theorem 1.1, we state some Lemmas and the details of the proof can be found in [4] and [11].

Lemma 4.4 ([4, Lemma B.1]). Let 1≤p≤ ∞ andU, V ∈Rsuch thatU.V ≤0.

Define the following function

g(t) =|U−tV|^p+|U−V|^p−2(U−V)V|t|^p, t∈R. Then we have

g(t)≤g(1) =|U−V|^p−2(U−V)U, t∈R.

Lemma 4.5 ([11, Lemma 4.1]). Let α∈(0,1) and p >1. For any non-negative functions u, v ∈ W^α,p, consider the function σt := [(1−t)v^p(x) +tu^p(x)]^1/p for allt∈[0,1]. Then

[σt]α,p≤(1−t)[v]α,p+t[u]α,p, for allt∈[0,1], where[u]α,p is defined in (2.1).

Proof of Theorem 1.1. Assume by contradiction that there existsµsuch thatλ_1,<

µ < c(s) and (s+µ, µ)∈Σ_p. Using the fact that {λ1,} ×R andR× {λ1,} are isolated in Σ_pand Σ_pis closed we can choose such a point withµminimum. In other words, ˜Jshas a critical valueµ withλ1,< µ < c(s), but there is no critical value in (λ_1,, µ). If we construct a path connecting fromφ_1,to−φ1,such that ˜J_s≤µ, then we obtain a contradiction with the definition ofc(s), which wiil complete the proof.

Letu∈ S be a critical point of ˜J_sat levelµ. Thenusatisfies Λ_n,p(1−α)

Z

Q

|u(x)−u(y)|^p−2(u(x)−u(y))(v(x)−v(y))

|x−y|^n+pα dx dy+

Z

Ω

|u|^p−2uv dx

=(s+µ)

Z

Ω

(u⁺)^p−1v dx−µ

Z

Ω

(u⁻)^p−1v dx (4.8)

for allv∈ W^α,p. Substitutingv=u⁺in (4.8), we have Λn,p(1−α)

Z

Q

|u(x)−u(y)|^p−2(u(x)−u(y))(u⁺(x)−u⁺(y))

|x−y|^n+pα dx dy+

Z

Ω

(u⁺)^pdx

=(s+µ)

Z

Ω

(u⁺)^pdx. (4.9)

Since,|u⁺(x)−u⁺(y)|^p≤ |u(x)−u(y)|^p−2(u(x)−u(y))(u⁺(x)−u⁺(y), we obtain Λn,p(1−α)

Z

Q

|u⁺(x)−u⁺(y)|^p

|x−y|^n+pα dx dy+ Z

Ω

(u⁺)^pdx−s Z

Ω

(u⁺)^pdx≤µ.

Again substitutingv=u⁻ in (4.8), we have Λ_n,p(1−α)

Z

Q

|u(x)−u(y)|^p−2(u(x)−u(y))(u⁻(x)−u⁻(y))

|x−y|^n+pα dx dy−

Z

Ω

(u⁻)^pdx

=−µ Z

Ω

(u⁻)^pdx. (4.10)

Therefore, Λn,p(1−α)

Z

Q

|u(x)−u(y)|^p−2((u⁻(x)−u⁻(y))²+ 2u⁺(x)u⁻(y))

|x−y|^n+pα dx dy

+ Z

Ω

(u⁻)^pdx

= µ Z

Ω

(u⁻)^pdx

Since |u⁻(x)−u⁻(y)|^p ≤ |u(x)−u(y)|^p−2[(u⁻(x)−u⁻(y))²+ 2u⁺(x)u⁻(y)], ti follows that

Λn,p(1−α) Z

Q

|u⁻(x)−u⁻(y)|^p

|x−y|^n+pα dx dy+ Z

Ω

|u⁻|^pdx≤µ.

Therefore, from all above relations, one can easily verify that J˜s(u) =µ, J˜s

^p1u⁺ ku⁺k_Lp(Ω)

≤µ,J˜s

^1pu⁻ ku⁻k_Lp(Ω)

≤µ−s,J˜s

−^1pu⁻ ku⁻k_Lp(Ω)

≤µ.

Since,uchanges sign (see Proposition 3.3), the following paths are well-defined on S:

u1(t) = u⁺−(1−t)u^{− −1p} ku⁺−(1−t)u⁻k_Lp(Ω)

,

u2(t) = [(1−t)(u⁺)^p+t(u⁻)^p]^{1/p −1p} k(1−t)(u⁺)^p+t(u⁻)^pkL^p(Ω)

,

u3(t) = (1−t)u⁺−u^{− −1p} k(1−t)u⁺−u⁻kL^p(Ω)

.

Then, using the above calculations and Lemma 4.4 for U = u⁺(x)−u⁺(y) and V =u⁻(x)−u⁻(y), one can easily obtain that for all t∈[0,1],

J˜_s(u₁(t))≤Λn,p(1−α)R

Q

|U−V|^p−2(U−V)U

|x−y|^n+pα +R

Ω(u⁺)^p−^sR

Ω(u⁺)^{p −1}ku⁺−(1−t)u⁻k^p_Lp(Ω)

+|1−t|^p

−Λ_n,p(1−α)R

Q

|U−V|^p−2(U−V)V

|x−y|^n+pα +R

Ω(u⁻)^{p −1}ku⁺−(1−t)u⁻k^p_Lp(Ω

)

=µ, by using (4.9) and (4.10). Now using Lemma 4.5 we have

J˜s(u2(t))≤(1−t)

Λ_n,p(1−α)R

Q

|u⁺(x)−u⁺(y)|^p

|x−y|^n+pα +R

Ω(u⁺)^p−^sR

Ω(u⁺)^{p −1}k(1−t)(u⁺)^p+t(u⁻)^pk^p_Lp(Ω

)

+t

Λn,p(1−α)R

Q

|u⁻(x)−u⁻(y)|^p

|x−y|^n+pα +R

Ω(u⁻)^p−^sR

Ω(u⁻)^{p −1}k(1−t)(u⁺)^p+t(u⁻)^pk^p_Lp(Ω)

≤µ− stR

Ω(u⁻)^p

⁻¹k(1−t)(u⁺)^p+t(u⁻)^pk^p_Lp(Ω)

≤µ.

Again, by Lemma 4.4, forU =u⁻(y)−u⁻(x) andV =u⁺(y)−u⁺(x), we obtain J˜s(u3(t))

≤Λn,p(1−α)R

Q

|U−V|^p−2(U−V)U

|x−y|^n+pα +R

Ω(u⁻)^{p −1}k(1−t)u⁺−u⁻k^p_Lp(Ω)

+

|1−t|^p

−Λn,p(1−α)R

Q

|U−V|^p−2(U−V)V

|x−y|^n+pα +R

Ω(u⁺)^p−^sR

Ω(u⁺)^{p −1}k(1−t)u⁺−u⁻k^p_Lp(Ω)

=µ, by using (4.9) and (4.10).

Let O = {v ∈ S : ˜J_s(v) < µ−s}. Then clearly φ_1, ∈ O, while −φ1, ∈ O if µ−s > λ1,. Moreoverφ1,and−φ1,are the only possible critical points of ˜Js in Obecause of the choice ofµ. We note that

J˜s

^1/pu⁻ ku⁻k_Lp(Ω)

≤µ−s,

^1/pu⁻/ku⁻k_Lp(Ω)does not change sign and vanishes on a set of positive measure, it is not a critical point of ˜J_s. Therefore, there exists a C¹ path η : [−δ, δ] → S with η(0) = ^1/pu⁻/ku⁻k_Lp(Ω) and _dt^dJ˜_s(η(t))|_t=0 6= 0. Using this path we can move from^1/pu⁻/ku⁻k_Lp(Ω)to a pointvwith ˜Js(v)< µ−s. Taking a connected component of Ocontainingvand applying Lemma 4.3 we have that eitherφ1, or

−φ1,is in this component. Let us assume that it isφ1,. So we continue by a path u4(t) from ^1/pu⁻/ku⁻k_Lp(Ω)to φ1,which is at level less than µ. Then the path

−u4(t) connects −^1/pu⁻/ku⁻k_Lp(Ω)to −φ1,. We observe that

|J˜s(u)−J˜s(−u)| ≤s.

Then it follows that

J˜_s(−u₄(t))≤J˜_s(u₄(t)) +s≤µ−s+s=µ for allt.

Connectingu1(t),u2(t) andu4(t), we obtain a path fromutoφ1,and joiningu3(t) and−u4(t) we obtain a path fromuto−φ1,. These yields a pathγ(t) onS joining fromφ1,to −φ1,such that ˜Js(γ(t))≤µfor allt, which concludes the proof.

As a consequence of Theorem 1.1, we give a variational characterization of the second value of (1.2).