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Multiplicity solutions of a class fractional Schrödinger equations

  • Li-Jiang Jia , Bin Ge EMAIL logo , Ying-Xin Cui and Liang-Liang Sun
Published/Copyright: August 8, 2017
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Open Mathematics
From the journal Open Mathematics Volume 15 Issue 1

Abstract

In this paper, we study the existence of nontrivial solutions to a class fractional Schrödinger equations

(−Δ)su+V(x)u=λf(x,u)inRN,

where (−Δ)su(x)=2limε→0∫RN∖Bε(X)u(x)−u(y)|x−y|N+2sdy,x∈RN is a fractional operator and s ∈ (0, 1). By using variational methods, we prove this problem has at least two nontrivial solutions in a suitable weighted fractional Sobolev space.

Keywords: Fractional Laplacian; Variational methods; Nontrivial solution
MSC 2010: 35P15; 35P30; 35R11

1 Introduction

There are a lot of interesting problems in the standard framework of the Laplacian (and, more generally, of uniformly elliptic operators), widely studied in the literature. A natural question is whether or not the existence results got in this classical context can be extended to the non-local framework of the fractional Laplacian type operators.

First, we focus on the so-called fractional Schrödinger equation

i∂ψ∂t=(−Δ)sψ+V(x)ψ−|ψ|p−1ψ,(1)

where (x, t) ∈ ℝ × (0, +∞), and V : ℝN → ℝ an external potential function. The fractional Laplacian operator (−Δ)su with 0 < s < 1 of a function ϕ ∈ ℓ is defined by

ℑ((−Δ)sϕ)(ξ)=|ξ|2sℑ(ϕ)ξ,

where ℓ denotes the Schwartz space of rapidly decreasing C∞ functions in ℝN ℑ is the Fourier transform, i.e.,

ℑ((−Δ)sϕ)(ξ)=|ξ|2sℑ(ϕ)ξ=1(2π)N2∫RNe2πiξ.xϕ(x)dx.

This equation was introduced by Laskin ([21, 22]), and comes from an expansion of the Feynman path integral from Brownian-like to Lévy-like quantum mechanical paths. When s = 1, the Lévy dynamics becomes the Brownian dynamics, and (1) reduces to the classical Schrödinger equation

i∂ψ∂t=−Δψ+V(x)ψ−|ψ|p−1ψ.

Standing wave solutions to this equation are solutions of the form

ψ(x,t)=e−iωtu(x),

where u solves the elliptic equation

−Δψ+V(x)ψ−|ψ|p−1ψ=0.

In this paper we study the following fractional Schrödinger equation

(−Δ)su+V(x)u=λf(x,u)inRN(2)

where λ is a parameter.

The fractional Schrödinger equations are an important model in the study of the fractional quantum mechanics. Recently, this has been widely investigated by many authors in the last decades, see [3-15, 18-20, 22-26] and references therein. In most of the papers mentioned above the existence of positive solutions has been considered under different assumptions on V and f. We refer the reader to [16, 17] and to the references included for a selfcontained overview of the basic properties of fractional Sobolev spaces.

In [3], the author used the Ekeland variational principle and the mountain pass theorem to obtain a nontrivial solution for (2) with the Ambrosetti-Rabinowitz condition:

there is a constant μ > 2 such that

0≤μ∫0sf(x,t)dt≤sf(x,s),forallx∈RN,s∈R∖{0}.(AR)

In [4-6], the authors used variant fountain theorems and the ℤ2 version of mountain pass theorem to establish the existence of infinitely many nontrivial high-energy or small-energy solutions for (2). In [7], the authors used the concentration compactness principle to show that (2) (V(x) = 1) has at least two nontrivial radial solutions without the (AR) condition.

In [25], Bisci and Rǎdulescu studied the following equation

(−Δ)su+V(x)u=λ(f(x,u)+κg(x,u))inRN

when the potential V ∈ C(ℝN) satisfies

(V)infRNV(x)>0

and

(V0)lim|y|→+∞μ{x∈B(y,r):V(x)≤M}<+∞,∀M>0,

where μ denotes the Lebesgue measure in ℝN, B(y, r) denotes the open ball in ℝN with center y and radius r > 0, and established two existence theorems for two nontrivial solutions when the nonlinearity f and g satisfy

(h)max{|f(x,t)|,|g(x,t)|}≤W(x)|t|q,

where W ∈ L1(ℝN)∩L∞(ℝN) and q ∈ (0, 1).

Motivated by the above papers, we shall assume that f(x, t) satisfies the following conditions:

(f1) f ∈ C(ℝN+1, ℝ), and

limt→0f(x,t)t=0.

(f2)

lim|t|→∞|f(x,t)|t=0.

(f3)

supt∈RF(t)>0.

where F(x,u)=∫0uf(x,s)ds.

And on the potential function V we assume

(V1) V ∈ C(ℝN, ℝ) is a positive weight and there exists a constant V0 > 0 such that V(x) ≥ V0 for all x ∈ ℝN.

(V2)

1V(x)∈L1(RN).

The main purpose of this paper is to generalize the main results of [24, 25]. Now we state our main results:

Theorem 1.1

Assume that f and V satisfy (f1) (f2), (f3), (V1), and (V2). Then problem (1.2) admits at least two nontrivial nonnegative solutions.

This paper is organized as follows. In Section 2, we will give some notation and introduce our main idea. In Section 3, we prove Theorem 1.1.

Remark 1.2

If g(x, u) = 0, then (h) ⇒ (f2). However, there are many functions which satisfy (f2), but do not satisfy the condition (h). For example, the function

f(x,t)={ t2,M112−M02M1−M0(t−M0)+M02,t12,0,if 0≤t≤M0,M0>1if M0≤t≤M1,if t≥M1,if x<0,

does not satisfy (h), but it satisfies our conditions (f2).

2 Preliminary

In this section, for the reader’s convenience, we collect some basic results that will be used in the forthcoming sections. In the following, we denote the N-dimensional Lebesgue measure of a set A ⊂ ℝN by meas(A). We use " ⇀ " and " → " to denote weak and strong convergence in the related function space. For any Eucliden space (ℝN ; |·|) we will denote

Bρ={u∈RN:|u−u0|≤ρ}(u0∈RN,ρ>0).

2.1 Variational formulation of the problem

First we introduce a variational setting for problem (2). The Gagliardo seminorm is defined for all measurable function u : ℝN → ℝ by

u:[u]2,s=(∫R2N|u(x)−u(y)|2|x−y|N+2sdxdy)12,

we can define the fractional Sobolev space

Hs(RN)={u∈L2(RN):umeasurable,[u]2,s<∞},

with

||u||Hs=(||u||22+[u]2,s2)12,

where ||u||ϑ=(∫RN|υ|ϑds)1ϑ, which is the norm of Lϑ(ℝN). The space Hs(ℝN) is a Hilbert space with the inner product

〈u,υ〉=∫R2N(u(x)−u(y))(υ(x)−υ(y))|x−y|N+2xdxdy+∫RNuυdx.

The space Hs (ℝN) can be described by means of the Fourier transform. Indeed, it is defined by

Hs(RN)={u∈L2(RN):∫RN(1+|ξ|2)s|ℑu(ξ)|2dξ<+∞}.

In this case, the inner product and the norm are defined as

〈u,υ〉=∫RN(1+|ξ|2)sℑu(ξ)ℑυ(ξ)dξ,||u||Hs=(∫RN(1+|ξ|2)s|ℑu(ξ)|2dξ)12.

In order to give the relationship of the above two norms, we introduce the definition of Schwartz function ℓ, that is, the rapidly decreasing C∞ function on ℝN . If u ∈ ℓ, the fractional Laplacian (−Δ)s acts on u as

(−Δ)su(x)=C(N,s)limε→0+∫RN∖B(0,ε)u(x)−u(y)|x−y|N+2sdy,

where C(N, s) is the following constant

C(N,s)=(∫RN1−cos⁡(ζ1)|ζ|n+2s)−1.

In [17], it is proved that

(−Δ)su=ℑ−1(|ξ|2sℑu),[u]2,s=2C(N,s)∫RN|ξ|2s|ℑu|2dξ,

and that

[u]2,s=2C(N,s)||(−Δ)s2u||22.

As a consequence, the norms on Hs(ℝN) defined above

u→(∫R2N(u(x)−u(y))2|x−y|N+2sdxdy+∫RNu2dx)12,u→(∫RN(1+|ξ|2)s|ℑu(ξ)|2dξ)12,u→(∫RN|ξ|2s|ℑu(ξ)|2dξ+∫RNu2dx)12,u→(||(−Δ)s2u||22+∫RNu2dx)12.

are all equivalent.

Moreover, it is easy to see that (Hs(ℝN), ||u||2,s) is a uniformly convex Banach space and the embedding Hs(ℝN) ↪ Lϑ(ℝN) is continuous for any ϑ∈[2,2s∗] by Theorem 6.7 of [17], that is, there exists a positive constant C* such that

||u||Lϑ(RN)≤C∗||u||2,sforallu∈Hs(RN).

We will work in the following linear subspace

E={u∈Hs(RN),∫RNV(x)|u|2dx<+∞},

which denotes the completion of C0∞(RN) with respect to the norm

||u||E=[u]2,s2+||u||2,V212,||u||2,V2=∫RNV(x)|u|2dx.

We also know that (E, ||·||E) is a uniformly convex Banach space, see ([27] Lemma 10). The dual space of (E, ||·||E) is denoted by (E*, ||·||E*). 〈·, ·〉 denotes the pairing between E and its dual space E*. We define the nonlinear operator Φ : E → E* as

〈Φ(u),υ〉=∫R2N(u(x)−u(y))(υ(x)−υ(y))|x−y|N+2sdxdy+∫RNV(x)u(x)υ(x)dx.

It can be seen that a weak solution of problem (2) is a function u ∈ E such that

〈Φ(u),υ〉=∫RNf(x,u)υdx.(3)

for all υ ∈ E. Clearly, for all u ∈ E, 〈Φ(u),υ〉=||u||E2.

Now we introduce the minimal hypotheses on the reaction term of (2):

(H)f : ℝN × ℝ → ℝ is a Carathéodory mapping, F(x,t)=∫0tf(x,s)ds for all (x,t) ∈ ℝN × ℝ and q∈(2,2s∗) such that

|f(x,u)|≤a(1+|u|q−1),

a.e. in ℝN and for all t ∈ ℝ(a > 0).

We set for all u ∈ E

I(u)=||u||E22−∫RNF(x,u)dx.(4)

defined on E, and for any u ∈ E, it holds that:

〈I′(u),υ〉=〈Φ(u),υ〉−∫RNf(x,u)υdx.(5)

By hypotheses (H), we have I ∈ C1(E). We denote by K(I) the set of all critical points of I. If u ∈ K(I), then (4) hold for all υ ∈ E, i.e., u is a weak solution to (2).

2.2 Some preliminary lemmas

We first recall some embedding results related to the fractional Sobolev space E, for more details, see [27].

Lemma 2.1

([27, Lemma 1]). Let (V1) hold. Then the embeddings E ↪ Hs(ℝN) ↪ Lϑ(ℝN) are continuous with

min{1,V0}||u||2,s2≤||u||E2,(6)

for all u ∈ E and ϑ∈[2,2s∗]. Moreover, for any R > 0 and ϑ∈[2,2s∗), the embedding E ↪ Lϑ(BR(0)) is compact.

Lemma 2.2

Suppose that (V1) and (V2) are fulfilled. If {υj} is a bounded sequence in E, then there exists υ ∈ E ∩ Lϑ(ℝN) such that up to a subsequence,

υj→υstronglyinLϑ(RN)

as j → ∞, for any ϑ∈[2,2s∗).

Proof

Since υj is bounded in E, by lemma 2.1 we have υj is bounded in Lϑ(ℝN). Then by the reflexivity of E, up to a subsequence, we get that υj ⇀ υ weakly in E ∩ Lϑ(ℝN) as j → ∞. Next we prove that

υj→υstronglyinLϑ(RN).

Now, for any ɛ > 0, there exists R1 > 0 such that

∫RN∖BR(0)1V(x)dx<ε

for all R ≥ R1, since 1V(x)∈L1(RN) by assumption (V2). Then, by Hölder inequality, we can get that

∫RN∖BR(0)V(x)12V(x)12−12s∗|υj−υ|2dx≤(∫RN∖BR(0)V(x)|υj−υ|2dx)12(∫RN∖BR(0)|υj−υ|2s∗dx)12s∗(∫RN∖BR(0)1V(x)dx)2s∗−222s∗≤ε,(7)

for all R ≥ R1.

Fix R1 > 0, we have

υj→υweaklyinL2s∗(BR1(0))∩Hs(BR1(0)),

by Theorem 6.7 of [17]. Since 2≤ϑ<2s∗, by Corollary 7.2 of [17], we obtain υj → υ strongly in Lϑ(BR1(0)), i.e. for above ɛ > 0, there exists N1 > 0 such that

∫BR1(0)|υj−υ|ϑdx<ε(8)

for all j ≥ N1. Combining (7) and (8), for all j ≥ N1, by interpolation inequality we have

∫RN|υj−υ|ϑdx=∫BR1(0)|υj−υ|ϑdx+∫RN∖BR1(0)|υj−υ|ϑdx≤ε+(∫RN∖BR1(0)|υj−υ|2dx)θϑ2(∫RN∖BR1(0)|υj−υ|2s∗dx)(1−θ)ϑ2s∗≤ε+(∫RN∖BR1(0)V(x)12V012s∗V(x)12−12s∗|υj−υ|2dx)θϑ2(∫RN∖BR1(0)|υj−υ|2s∗dx)(1−θ)ϑ2s∗≤ε+1V0θϑ22s∗(∫RN∖BR1(0)V(x)|υj−υ|2dx)θϑ4(∫RN∖BR1(0)|υj−υ|2s∗dx)(1−θ)θϑ22s∗2s∗×(∫RN∖BR1(0)1V(x)dx)θϑ(2s∗−2)42s∗≤Cε,

where C denotes various positive constants, and θ ∈ (0, 1) such that

1ϑ=θ2+1−θ2s∗.

Therefore, υj → υ strongly in Lϑ(ℝN ). □

Proposition 2.3

Let (E, ||·||E) be a Banach space and its dual space (E*, ||·||E*) and I ∈ C1(E, ℝ1)

  1. For c ∈ ℝ1, we say that I satisfies the Cc condition if for any sequence {xn} ⊂ E with

    I(xn)→c,||I′(xn)||E∗(1+||xn||E)→0.

  2. For c ∈ ℝ1, we say that I satisfies the (PS) condition if for any sequence {xn} ⊂ E with

    I(xn)→c,I′(xn)→0inE∗.
The following critical points theorem was established in [1, 2].

Lemma 2.4

([1, 2]). Let (E, ||·||) be a separable and reflexive real Banach space, and let Φ, ψ : E → ℝ be two continuously Gâteaux differentiable functionals. Assume that there exists z0 ∈ E such that Φ(z0) = ψ(z0) = 0 and infz∈EΦ(z)≥0 and that there exist z1 ∈ E, ϱ > 0 such that

  1. ϱ < Φ(z1);

  2. supΦ(z)<ϱψ(z)<ϱψ(z1)Φ(z1). Further, put

    a¯=ζϱϱψ(z1)Φ(z1)−supΦ(Z)<ϱψ(z),

    with ς > 1, assume that the functional

    J(z)=Φ(z)−λψ(z),(∀z∈E)

    is sequentially weakly lower semicontinuous, satisfies the (PS) condition, and

  3. lim||z||→+∞J(z)=+∞, for every λ ∈ [0, ā].

Then there is an open interval Λ ⊂ [0, ā] and a number κ > 0 such that for each λ ∈ Λ, the equation J′ (z) = 0 admits at least three solutions in E having norm less than κ.

3 The main results and its proofs

In this section, we are ready to prove the Theorem 1.1. In the sequel, for the sake of clarity, we divide the proof of the theorem into several steps. We write the functional J as follows:

J(u)=Φ(u)−λψ(u),

where

Φ(u)=12∫R2N(u(x)−u(y))2|x−y|N+2sdxdy+12∫RNV(x)u2dx,

and

ψ(u)=∫RNF(x,u)dx.

We first give two preliminary lemmas.

Lemma 3.1

If (f1), (f2) and (V1) hold, then we have that

limϱ→0+supu∈Sϱ∫RNF(x,u)dxϱ=0

where Sϱ = Φ-1(−∞,ϱ).

Proof

By (f1) and (f2), there is a positive constants δ such that

|f(x,t)|≤min(1,V0)ε|t|2C∗2+cmin(1,V0)ϑ2|t|ϑ−1C∗ϑ|F(x,t)|≤min(1,V0)ε|t|24C∗2+cmin(1,V0)ϑ2|t|ϑC∗ϑ

for a fixed ϑ∈[2,2s∗) and for all t ∈ ℝN.

Moreover, by using lemma 2.1, we have

supu∈Sϱ⁡∫RNF^(x,u)dx≤∫RNmin(1,V0)ε|u|24C∗2+cmin(1,V0)ϑ2|u|ϑC∗ϑdx=min(1,V0)ε||u||224C∗2+cmin(1,V0)ϑ2|u|ϑϑC∗ϑ≤ε2ϱ+c2ϑ2ϱϑ2.

Thus, there exists ϱ(ɛ) > 0 such that, for every 0 < ϱ < ϱ(ɛ), we have

0≤supu∈Sϱ∫RNF^(x,u)dxϱ≤ε2+c2ϑ2ϱϑ−22≤ε.

The proof is complete. □

Lemma 3.2

Let σ ∈ [0, 1], t0 ∈ ℝ, and

Γ(t)=∫0+∞zt−1e−zdz,(∀t>0)

be the usual Gamma function, then we have

||uσt0||2<t02(1−σ)2πN2τN−2(1−σN)Γ(1+N2)S0(9)

where

uσt0=0,ifx∈RN∖B(x0,τ),t0(1−σ)τ(τ−|x−x0|),ifx∈B(x0,τ)∖B(x0,στ),t0,ifx∈B(x0,στ),

and

S0=max{(2π)N(1+1λ1),maxx∈B(x0,τ)V(x)λ1}.

Proof

Computing the standard seminorm of the function uσt0 in H1 (ℝN), we have

[uσt0]H1(RN)2=∫RN|∇uσt0|2dx=∫B(x0,τ)∖B(x0,στ)t02(1−σ)2τ2dx=t02(1−σ)2τ2[meas(B(x0,τ))−meas(B(x0,στ))]=t02(1−σ)2πN2τN−2(1−σN)Γ(1+N2),

where meas (B(x0, τ)) and meas (B(x0, στ)) denote the Lebesgue measure of B(x0, τ) and B(x0, στ).

Now, by standard arguments on the Fourier transform, we have

uσt0∈L2(RN)ifandonlyifℑuσt0∈L2(RN),

and

||uσt0||L2(RN)2=(2π)−N||ℑuσt0||L2(RN)2,

as well as

|∇uσt0|∈L2(RN)ifandonlyif|ξ|ℑuσt0∈L2(RN),

and

||∇uσt0||L2(RN)2=(2π)−N|||ξ|ℑuσt0||L2(RN)2.

Then, we have

||uσt0||2=∫RN|ξ|2s|ℑuσt0|2dξ+∫RNV(x)|uσt0|2dx<∫RN(1+|ξ|2)|ℑuσt0|2dξ+∫RNV(x)|uσt0|2dx=(2π)N(||∇uσt0||L2(RN)2+||uσt0||L2(RN)2)+∫RNV(x)|uσt0|2dx≤(2π)N(1+1λ1)||∇uσt0||L2(RN)2+maxx∈B(x0,τ)⁡V(x)||uσt0||L2(RN)2<max{(2π)N(1+1λ1),maxx∈B(x0,τ)V(x)λ1}[uσt0]H1(RN)2,

where

λ1=infu∈H1(B(x0,τ))∖0||∇u||L2(RN)2||u||L2(RN)2.

Thus (9) holds. □

Now, we prove Theorem 1.1.

Proof of Theorem 1.1

Step 1. We prove for every λ ∈ ℝ, the functional J is coercive and satisfies the compactness (PS) condition.

Let us fix λ ∈ ℝ. By (f2), there is a positive constant δ such that

|f(t)|≤min(1,V0)|t|C∗2(1+|λ|),

for every |t| ≥ δ.

So, we get

|F(t)|≤min(1,V0)|t|22C∗2(1+|λ|)+max|t|≤δ|f(t)||t|,

for every t ∈ ℝ.

Thus

J(u)=12||u||E2−λ∫RNF(x,u)dx≥12||u||E2−|λ||∫RNF(x,u)dx|≥12||u||E2−|λ|∫RN|F(x,u)|dx≥12(1+|λ|)||u||E2−|λ|∫RNmax|t|≤δ|f(t)||u|dx=12(1+|λ|)||u||E2−|λ|∫RNmax|t|≤δV(x)12V(x)12|f(t)||u|dx≥12(1+|λ|)||u||E2−|λ|max|t|≤δ|f(t)|(∫RNV(x)|u|2dx)12(∫RN1V(x)dx)12≥12(1+|λ|)||u||E2−C|λ|max|t|≤δ|f(t)|||u||E,

for every u ∈ E.

Then the functional J is bounded from below and J(u) → +∞ when ||u||E → +∞. Hence J is coercive.

Now we prove that J satisfies the (PS) condition. Let {un} ⊂ E be a (PS) sequence for J(u), that is

J(un)→c,J′(un)→0inE∗.

Taking into account the coercivity of J, the sequence {un} is necessarily bounded in E. Assume without loss of generality that {un} converges to u weakly in E, and by Lemma 2.2, we may assume that

{un(x)→u(x)a.e.inRun→uinLϑ(RN) (10)

where ϑ∈[2,2s∗).

To prove that {un} converges strongly to u in E, we first introduce a simple notation. Let ϕ ∈ E be fixed and denote by Bϕ the linear functional on E defined by

Bφ(υ)=∫R2N(φ(x)−φ(y))(υ(x)−υ(y))|x−y|N+2sdxdy

for all υ ∈ E.

Due to (f1) and (f2), there exists Cɛ > 0 such that

|f(t)|≤ε|t|+Cε|t|ϑ−1.

Then, by (10), we get

∫ R N ( f ( x , u n ) − f ( x , u ) ) ( u n − u ) d x ≤ ∫ R N ( ε | u n | + C ε | u n | ϑ − 1 + ε | u | + C ε | u | ϑ − 1 ) ( u n − u ) d x ≤ ε | | u n | | 2 | | u n − u | | 2 + ε | | u | | 2 | | u n − u | | 2 + C ε | | u n | | ϑ ϑ − 1 | | u n − u | | ϑ + C ε | | u | | ϑ ϑ − 1 | | u n − u | | ϑ → 0. (11)

Obviously, 〈J′(un) − J′(u), un − u) → 0 as n → ∞, since un ⇀ u in E and J′(un) → 0 in E*. Hence, (10) and (11) give as n → ∞

o(1)=〈J′(un)−J′(u),un−u〉=Bun(un−u)−Bu(un−u)+∫RNV(x)(un−u)2dx−∫RN(f(x,un)−f(x,u))(un−u)dx=Bun(un−u)−Bu(un−u)+∫RNV(x)(un−u)2dx+o(1).

That is

||un−u||E2=Bun(un−u)−Bu(un−u)+∫RNV(x)(un−u)2dx→0

as n → ∞.

Therefore, J satisfies the (PS) condition.

Step 2. We claim that the functional J(u) is weakly lower semicontinuous on E.

The functional

u↦∫R2N(u(x)−u(y))2|x−y|N+2sdxdy+∫RNV(x)u2dx,

is sequentially weakly lower semicontinuous on E.

Thus it is enough to prove that the map

u↦∫RNF(x,u)dx,

is sequentially weakly continuous on E. To this aim fix un ∈ E and u ∈ E such that un ⇀ u in E as n → ∞. Then, by Lemma 2.2, without loss of generality, we can assume that un → u strongly in Lϑ(ℝN) for 2 ≤ ϑ < 2 s ∗ and a.e. in ℝN. It is dominated by some function hϑ ∈ Lϑ(ℝN), that is,

|un|≤hϑ(x)a.e.x∈RN

for any n ∈ N and for any ϑ∈[2,2s∗).

By (f1) and (f2), there exists Cɛ such that

|f(t)|≤ε|t|+Cε|t|ϑ−1.

Then, by the continuity of F, it follows that

|F(x,un)|≤h(x),

a.e. x ∈ ℝN, for some h ∈ L1(ℝN), and

F(x,un)→F(x,u),

a.e. in ℝN.

Hence, by Lebesgue dominated convergence theorem, we have

limn→+∞∫RNF(x,un)dx=∫RNF(x,u)dx,

that is the map

u↦∫RNF(x,u)dx,

is weakly continuous in E. Thus, J is weakly semicontinuous in E.

Step 3. We show that there exists a u0 ∈ E, ϱ > 0 such that ϱ < Φ(u0) and supΦ(u)<ϱΨ(u)<ϱΨ(u0)Φ(u0).

By (f3), there exists t0 ∈ ℝN such that F(t0) > 0. Further, let σ0 [0, 1] be such that

F(t0)σ0N−(1−σ0N)max|t|≤|t0||F(t)|>0.

Indeed, since

||uσt0||∞=maxx∈B(x0,τ)|uσt0|≤|t0|,

(recalled the definition of uσt0 ), it follows that

∫B(x0,τ)∖B(x0,σ0τ)F(uσt0)dx≥−max|t|≤|t0|⁡|F(t0)|∫B(x0,τ)∖B(x0,σ0τ)dx=−(1−σ0N)max|t|≤|t0|⁡|F(t)|τNωN,

where ωN denotes the volume of the unit ball in ℝN.

Then, we have that

Ψ(uσt0)=∫B(x0,τ)F(uσt0)dx=∫B(x0,σ0τ)F(uσt0)dx+∫B(x0,τ)∖B(x0,σ0τ)F(uσt0)dx≥F(t0)σ0NτNωN+∫B(x0,τ)∖B(x0,σ0τ)F(uσt0)dx≥[F(t0)σ0N−(1−σ0N)max|t|≤|t0||F(t)|]τNωN. (12)

By Lemma 3.2, we have

Φ(uσt0)≤C, (13)

where

C=t022(1−σ)2πN2τN−2(1−σN)Γ(1+N2)S0.

By Lemma 3.1, there exists ϱ > 0 such that the function uσt0∈E verifies the following conditions:

uσt0∈R∖Sϱ, (14)

and

supu∈SϱΦ(u)ϱ<[F(t0)σ0N−(1−σ0N)max|t|≤|t0||F(t)|]τNωNC. (15)

Then by (14), we have

ϱ<Φ(uσt0),

as well as, by (12) and (13), it follows that:

[F(t0)σ0N−(1−σ0N)max|t|≤|t0||F(t)|]τNωNC≤Ψ(uσt0)Φ(uσt0). (16)

Hence, (15) and (16) give

supu∈SϱΦ(u)≤Ψ(uσt0)Φ(uσt0). (17)

By choosing u0=uσt0, we get that ϱ < Φ(u0), and sup Φ ( u ) ∈ ϱ Ψ ( u ) < ϱ Ψ ( u 0 ) Φ ( u 0 ) .

Set

a¯=(1+ϱ)ϱϱΨ(u0)Φ(u0)−supΦ(u)<ϱΨ(u),

Clearly, 1 + ϱ > 1, and ā is a positive constant.

It’s easy to see that infx∈EΦ(u)=0, and by choosing u1 = 0, we have

Φ(u1)=Ψ(u1)=0.

Then, by Step 1, Step 2, and Step 3, all the assumptions of Lemma 2.4 are verified. Thus there is an open interval Λ ⊆ [0, ā] and a number κ > 0 such for all λ ∈ Λ, the functional J admits three solutions in E having norm less than κ. Since one of them may be a trivial one, we have at least two distinct, nontrivial weak solutions of problem (2). ☐

Remark 3.3

Our hypotheses are similar to those employed by Bisci-Rădulescu [24] of (2) on bounded subset Ω of ℝN . Moreover, (f2) is weaker than the condition (h). So Theorem 1.1 extends Theorem 1 of [24] and Theorem 2 of [25].

Acknowledgement

Supported by the National Natural Science Foundation of China (Nos. 11201095, 71373059), Postdoctoral research startup foundation of Heilongjiang (No. LBH-Q14044) and the Science Research Funds for Overseas Returned Chinese Scholars of Heilongjiang Province (No. LC201502), Research Fund for the Doctoral Program of Higher Education of China (No. 20122304110018).

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Received: 2016-11-8
Accepted: 2017-6-19
Published Online: 2017-8-8

© 2017 Jia et al.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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https://doi.org/10.1515/math-2017-0084
Keywords for this article
Fractional Laplacian; Variational methods; Nontrivial solution
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