The existence of positive solution for singular Kirchhoff equation with two parameters

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R E S E A R C H

Open Access

The existence of positive solution for

singular Kirchhoff equation with two

parameters

Ke Di

1

_{and Baoqiang Yan}

1*

*_{Correspondence:}

[email protected]

1_{School of Mathematical Sciences,} Shandong Normal University, Jinan, P.R. China

Abstract

In this paper, we consider the singular Kirchhoﬀ equation with two parameters ⎧

⎪ ⎨ ⎪ ⎩

–a(_Ω|∇u(x)|2dx)u(x) +K(x)g(u) =

λ

f(x,u) +

μ

h(x) in

Ω

, u> 0 in

Ω

,

u= 0 on

∂Ω

.

By using the sub-supersolution method together with the comparison principle for elliptic equations, we obtain several existence and nonexistence theorems. Our works improve the results in the previous literature.

Keywords: Elliptic problems of Kirchhoﬀ type; Positive solution; Subsolution; Supersolution

1 Introduction

In this paper, we study the existence and nonexistence of solutions to the following singular nonlocal elliptic problem with two parameters:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

–a(_Ω|∇u(x)|2dx)u(x) +K(x)g(u) =λf(x,u) +μh(x) inΩ,

u> 0 inΩ,

u= 0 on∂Ω,

(1.1)

whereΩis a smooth bounded domain inRN (N≥2),a: [0, +∞)→(0, +∞) is a continu-ous and nondecreasing function with

inf

t∈[0,+∞)a(t) =a(0) =a0> 0,

K,h∈C0,γ₍_Ω¯_{) with}_K_{> 0 and}_h_{> 0 on}_Ω_{, and}_λ_and_μ_{are positive real numbers.}

This problem is related to the general form of the stationary counterpart of the hyper-bolic Kirchhoﬀ equation

ρ∂

2_u

∂t2 –

P0

h + E 2L

L

0

∂u

∂x 2dx

∂2u

∂t2 =g(x,u)

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for free vibrations of elastic strings [7]. Kirchhoﬀ ’s model takes into account the changes in length of the string produced by transverse vibrations, in whichLis the length of the string,

ρ is the mass density, andP0 is the initial tension. Many researchers began an extensive

research in equations of this type after Lions [10] put forward a basic framework. For example, Perera and Zhang [17,30] considered the following Kirchhof-type problems with 4-sublinear case, asymptotically 4-linear case, and 4-superlinear case terms:

⎧ ⎨ ⎩

–(a+b_Ω|∇u(x)|2_dx₎_u₍_x_{) =}_f₍_x_,_u₎ _in_Ω_,

u= 0 on∂Ω. (1.2)

They obtained nontrivial solutions via the Yang index and the invariant sets of descent flow. Sun and Tang [22] obtained the existence and multiplicity of weak solutions of (1.2) by using the mountain pass theorem, the local linking theorem, the fountain theorem, and the symmetric mountain pass lemma in critical point theory. Utilizing the local link-ing theory, Yang and Zhang [29] obtained the nontrivial solutions of (1.2) with quasilinear terms. Yang and Han [28] obtained infinitely many solutions whenf(x,u) is odd by using the fountain theorem. In recent years, many researchers studied the fractional Kirchhoff equation. In [9] and [24] the authors deal with fractional Schrödinger–Kirchhoff equa-tions. In [23] the authors obtained the multiplicity of solutions of a fractionalp-Laplacian Kirchhoff system. In [16] authors deal with nonlocal fractional problems. In addition, in [25] the authors obtained the local existence and blowup of solutions of nonlocal Kirch-hoff diffusion problems. Some other results can be found in [3–5,11–15,20,26], and their references.

Since the method of sub-supersolutions is an important tool to solve the existence of so-lutions for boundary value problems, naturally, some authors hope to use the theorem of sub-supersolutions to discuss the elliptic problems of Kirchhoff type. However, the nonlo-cal term brings some difficulties. To overcome the difficulty from the nonlononlo-cal term, Alves and Corrêa [1] considered following problem:

⎧ ⎨ ⎩

–a(_Ω|∇u(x)|2_dx₎_u₍_x_{) =}_f₍_x_,_u₎ _in_Ω_,

u= 0 on∂Ω, (1.3)

and obtained the existence of positive solutions by using the sub-supersolution method under the condition thatf :Ω×R→Ris an increasing function. Moreover, Alves and Corrêa [2] dealed with the quasilinear stationary Kirchhoﬀ equation

⎧ ⎨ ⎩

–a(_Ω|∇u(x)|2_dx₎_u₍_x_{) =}_f₍_x_,_u_,_∇_u₎ _in_Ω_,

u= 0 on∂Ω, (1.4)

and using the pseudomonotone operator theory, they established the existence of weak solutions by constructing a supersolution and a family of subsolutions. Recently, Yan, O’Regan, and Agarwal [27] discussed the following problem:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

–a(_Ω|∇u(x)|2dx)u(x) =F(x,u) inΩ,

u> 0 inΩ,

u= 0 on∂Ω.

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They presented some new deﬁnitions of a sub-supersolution to problem (1.5) and obtained the existence of classical solution to problem (1.5) whenF(x,u) =λuq_–_up+1 _{(0 <}_q_{< 1,}

p> 0).

Motivated by the ideas introduced in [6], we suppose thatf :Ω×[0,∞)→[0,∞) is a Hölder-continuous function that is positive onΩ×(0,∞). We also assume thatf is nondecreasing with respect to the second variable and is sublinear, that is,

(f1) the mapping(0,∞)s →f(x_s,s)is nonincreasing for allx∈Ω; (f2) lims→0f(x_s,s)= +∞andlims→+∞f(x_s,s)= 0uniformly inx∈Ω;

(f3) there existsn0> 0such thatλn0f(x,s)≥f(x,λn0+1s)for allλ≥1.

We assume thatg∈C0,γ_(0,_∞_{) is a nonnegative and nonincreasing function satisfying:}

(g1) lims→0g(s) = +∞;

(g2) there existC,δ0> 0andτ∈(0, 1)such thatg(s)≤Cs–τ for alls∈(0,δ0).

Example The functionf(s) =sα_{(0 <}_α_{< 1) fulﬁlls (f1)–(f3), whereas}_g₍_s_{) = (}_sα₊_sβ₎–1_{(0 <}

α< 1,β> 1) satisﬁes assumptions (g1)–(g2).

DenoteD={u∈C2₍_Ω₎_∩_C₍_Ω_);_g₍_u₎_∈_L1₍_Ω₎_}_.

We show in this paper that (1.1) has at least one solution inDforλbelonging to a certain range and anyμ> 0. We also prove that in some cases, (1.1) has no solutions inD, provided thatλandμare suﬃciently small.

Remark1.1

(i) Ifu∈D,v∈C2₍_Ω₎_∩_C₍_Ω_{), and}_{0 <}_u_<_v_in_Ω_{, then}_v_∈_D_.

(ii) Letu∈C2₍_Ω₎_∩_C₍_Ω₎_{be a solution of (}_1.1_{). Then}_u_∈_D_{if and only if}_u_∈_L1₍_Ω_).

Our main results are the following.

Theorem 1.1 Assume that f satisﬁes(f1)–(f3)and g satisﬁes(g1)–(g2).

(1) Iflimλ→+∞_a₍_λ2+2λn0)= +∞,then there existsλ¯such that(1.1)has at least one solution

inDfor allλ>λ¯ andμ> 0.

(2) There existλ∗andμ∗small enough such that(1.1)has no solution inDfor allλ<λ∗ andμ<μ_∗.

Theorem 1.2 Assume that f satisﬁes(f1)–(f2).If₀1g(s)ds= +∞,then(1.1)has no solution in D for anyλ,μ> 0.

This paper is organized as follows. Some preliminary lemmas are given in Sect.2, and Sect.3is devoted to proofs of the results. Some ideas also come from [19] and [18].

2 Preliminaries

In this section, we consider the general problem ⎧

⎨ ⎩

–a(_Ω|∇u(x)|2dx)u(x) =F1(x,u) +F2(x,u) inΩ,

u= 0 on∂Ω. (2.1)

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Deﬁnition 2.1 The pair of functionsα,β∈C1₍_Ω₎_∩_C2₍_Ω_{) are a subsolution and}

super-solution of (2.1) ifα(x)≤β(x) forx∈Ωand ⎧

⎨ ⎩

–α(x)≤_b1

0F1(x,α(x)) +

1

a0F2(x,α(x)), x∈Ω,

α|∂Ω≤0,

⎧ ⎨ ⎩

–β(x)≥_a1

0F1(x,β(x)) +

1

b0F2(x,β(x)), x∈Ω,

β|∂Ω≥0,

wherea0=a(0) andb0=a(

ΩH(x)

2_dx_),_E_∈_Lp₍_Ω_{) (}_p_>_N_{); here}

E(x) = sup

u∈[α(x),β(x)]

F(x,u) , x∈Ω,

H(x) = 1 a0

Ω

Gx(x,y) E(y)dy, x∈Ω,

G(x,y) is the Green function for –u(x) =h, andu|∂Ω= 0.

Lemma 2.2(see [27]) LetΩ⊆RN₍_N_≥₁₎_{be a smooth bounded domain}_._{Let F}_:_Ω_×_R_→

Rbe a continuous function. Letα andβ be the subsolution and supersolution of (2.1), respectively.If

F1(x,u)≥0, F2(x,u)≤0, ∀x∈Ω,α(x)≤u≤β(x),

then problem(2.1)has at least one solution u such that

α(x)≤u(x)≤β(x) for all x∈Ω.

Letϕ1be the normalized positive eigenfunction corresponding to the ﬁrst eigenvalueλ1

of the problem ⎧ ⎨ ⎩

–u(x) =λu inΩ,

u= 0 on∂Ω.

(2.2)

Lemma 2.3(see [8]) _Ωϕ–₁s<∞if and only if s< 1.

Lemma 2.4(see [21]) Let F:Ω¯ ×(0,∞)→Rbe a Hölder-continuous function with

ex-ponentγ∈(0, 1)on each compact subset ofΩ×(0,∞)satisfying: (F1) lims→∞sup(s–1maxx∈ ¯ΩF(x,s)) <λ1;

(F2) for eacht> 0,there exists a constantD(t) > 0such that

F(x,r) –F(x,s)≥–D(t)(r–s) forx∈Ωandr≥s≥t;

(F3) there existη0> 0and an open subsetΩ0⊂Ωsuch that

min

x∈ ¯Ω

(5)

and

lim

s→0

F(x,s)

s = +∞ uniformly inx∈Ω0.

Then for any nonnegative functionϕ0∈C2,γ(∂Ω),the problem

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

–u(x) =F(x,u) inΩ,

u> 0 inΩ,

u=ϕ0 on∂Ω

(2.3)

has at least one positive solution u∈C2,γ₍_G₎_∩_C₍_Ω₎_{for any compact set G}_⊂_Ω_{∪ {}_x_∈ ∂Ω;ϕ0(x) > 0}.

Lemma 2.5(see [21]) Let F:Ω×(0,∞)→Rbe a continuous function such that the

mapping(0,∞)s → F(x_s,s) is strictly decreasing at each x∈Ω.Assume that there exists

ν,ω∈C2₍_Ω₎_∩_C₍_Ω₎_{such that}_:

(a) ω+F(x,ω)≤0≤ ν+F(x,ν)inΩ; (b) ν,ω> 0inΩandν≤ωon∂Ω;

(c) ν∈L1₍_Ω_).

Thenν≤ωinΩ.

We observe that the hypotheses of Lemmas2.4and2.5are fulﬁlled for

Φλ(x,s) =

1 a0

λf(x,s) +h(x).

Lemma 2.6 Let f satisfy(f1)–(f2).Then for anyλ,μ> 0,according to Lemmas2.4and2.5,

the boundary value problem

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

–U(x) =λf(x,U) +μh(x) inΩ,

U> 0 inΩ,

U= 0 on∂Ω

(2.4)

has a unique solution Uλ,μ∈C2,γ(Ω)∩C(Ω).

3 Proofs of main theorems

DenoteK∗=max_x_∈_ΩK(x),K∗=min_x_∈_ΩK(x).

Proof of Theorem1.1 According to Lemma2.6, the boundary value problem

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

–U(x) =_a1

0(f(x,U) +h(x)) inΩ, U> 0 inΩ,

U= 0 on∂Ω

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has a unique solutionU∈C2,γ₍_Ω₎_∩_C₍_Ω_{). Let}_H_{: [0,}_∞₎_→_[0,_∞_{) be such that}

⎧ ⎨ ⎩

H(t) =g(H(t)) for allt> 0,

H(0) =H(0) = 0. (3.2)

Obviously,H∈C2(0, +∞)∩C1[0, +∞) exists by our assumption (g2). From (3.2) it fol-lows that His nonincreasing, whereasH andH are nondecreasing on (0,∞). Using this fact and applying the mean value theorem, we deduce that, for allt> 0, there exist

ξ1

t,ξt2∈(0,t) such that

H(t) t =

H(t) –H(0) t– 0 =H

_ξ1

t

≤H(t)

and

H(t) t =

H(t) –H(0) t– 0 =H

_ξ2

t

≥H(t).

These inequalities imply

H(t)≤tH(t)≤2H(t) for allt> 0.

Hence

1≤tH ₍_t₎

H(t) ≤2 for allt> 0. (3.3)

On the other hand, by (g2) and (3.2) there existsη> 0 such that

⎧ ⎨ ⎩

H(t)≤δ0 for allt∈(0,η),

H(t)≤CH–τ₍_t₎ _{for all}_t_∈_(0,_η_), (3.4)

which yields

H(t)≤ct1+2τ _{for all}_t∈_(0,_η_), _(3.5)

wherec> 0 is a constant. Set

αλ,μ=MH(ϕ1),

whereMis a positive constant. Then we can get

–αλ,μ+

1 a0

K(x)g(αλ,μ)

=λ1MH(ϕ1)ϕ1+

1 a0

K(x)gMH(ϕ1)

–MgH(ϕ1)

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FixM≥1. The monotonicity ofgleads to

gMH(ϕ1)

≤gH(ϕ1)

inΩ,

and by (3.6)

–αλ,μ+

1 a0

K(x)g(αλ,μ)

≤λ1MH(ϕ1)ϕ1+

1 a0

K∗gH(ϕ1)

–MgH(ϕ1)

|∇ϕ1|2 inΩ. (3.7)

By Hopf ’s maximum principle there existδ0andΣ⊂Ωsuch that

|∇ϕ1| ≥δ0 inΩ\Σ, |ϕ1| ≥δ0 inΣ.

OnΩ\Σ, we chooseM≥M1=max{1, K ∗

a0δ2}. Then we have

1 a0

K∗gH(ϕ1)

≤MgH(ϕ1)

|∇ϕ1|2 inΩ\Σ. (3.8)

ChoosingM≥max{M1, K ∗_g₍_H₍_δ

0))

a0λ1H(δ0)δ0}, we have

1 a0

K∗gH(ϕ1)

≤λ1MH(ϕ1)ϕ1 inΣ. (3.9)

It follows from (3.7)–(3.9) that

–αλ,μ+

1 a0

K(x)g(αλ,μ)≤2λ1MH(ϕ1)ϕ1 inΩ. (3.10)

Sinceϕ1> 0 inΩ, from (3.3) we have

1≤H ₍_ϕ

1)ϕ1

H(ϕ1) ≤

2 inΩ. (3.11)

Then (3.10) and (3.11) yield

–αλ,μ+

1 a0

K(x)g(αλ,μ)≤4λ1MH(ϕ1) = 4λ1αλ,μ inΩ. (3.12)

TakeA0= 4a0λ1c–1|α(x)|∞, wherec=infx∈ ¯Ωf(x,|αλ,μ|∞) > 0. Ifλ>A0, then assumption

(f1) produces

λf(x,αλ,μ)

a0αλ,μ

≥A0f(x,|αλ,μ|∞)

a0|α(x)|∞ ≥4λ1 for allx∈Ω.

Combined with (3.12), this gives

–αλ,μ+

1 a0

K(x)g(αλ,μ)≤

1 a0

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DenoteΩ0={x∈Ω;ϕ1(x) <η}. By (3.4) and (3.5) it follows that

g(αλ,μ) =g

MH(ϕ1)

≤gH(ϕ1)

≤CH–τ(ϕ1)≤C0ϕ

–2τ

1+τ

1 inΩ0,

g(αλ,μ)≤g

MH(η) inΩ\Ω0.

These estimates combined with Lemma2.3yieldg(αλ,μ)∈L1(Ω), and soαλ,μ∈L1(Ω).

Set

βλ,μ=λn0+1U,

whereλ≥max{1,μ}.

We now prove that there existsA0such thatαλ,μ≤βλ,μfor allλ>A0.

Since

βλ,μ+

1 a0

λf(x,βλ,μ) +h(x)

=λn0+1_U₊ 1 a0

λfx,λn0+1_U₊_h₍_x₎

≤λn0+1

U+ 1 a0

f(x,U) +h(x)= 0,

we can get

βλ,μ+Φλ(x,βλ,μ)≤0≤ αλ,μ+Φλ(x,αλ,μ) inΩ,

αλ,μ,βλ,μ> 0 inΩ,

αλ,μ=βλ,μ on∂Ω,

αλ,μ∈L1(Ω).

By Lemma2.5it follows thatαλ,μ≤βλ,μonΩfor allλ>A0.

Deﬁne

E1(x) =λf(x,βλ,μ) +μh(x) +K(x)g(αλ,μ), x∈Ω,

H1(x) =

1 a0

Ω

Gx(x,y) E1(y)dy, x∈Ω,

b1=a

Ω

H1(x)2dx

,

that is,

E(x) = sup

u∈[αλ,μ(x),βλ,μ(x)]

F(x,u) ≤E1(x), x∈Ω,

H(x) = 1 a0

Ω

Gx(x,y) E(y)dy≤H1(x), x∈Ω,

b0=a

Ω

H(x)2dx

≤b1=a

Ω

H1(x)2dx

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We have

b1=a

Ω

H1(x)2dx

=a Ω 1 a0 Ω

Gx(x,y) λf

y,λn0+1_U₊_μ_h₍_y_{) +}_K₍_y₎_g_MH₍_ϕ

1) dy 2 dx ≤a Ω 1 a0 Ω

Gx(x,y) λf

y,λn0+1_U₊_λ_h₍_y_{) +}_K∗_g_H₍_ϕ

1) dy 2 dx ≤a λ a0 2 Ω Ω

Gx(x,y) f

y,λn0+1_U₊_h₍_y_{) +}_K∗_g_H₍_ϕ

1) dy 2 dx ≤a

λ2+2n0 a2 0 Ω Ω

Gx(x,y) f(y,U) +h(y) +K∗g

H(ϕ1)

dy 2 dx . Let C= 1 a0 2 Ω Ω

Gx(x,y) f(y,U) +h(y) +K∗g

H(ϕ1)

dy

2

dx,

that is,

b1≤a

Cλ2+2n0_,

whereCis a positive constant. Sincelimλ→+∞_a₍_λ2+2λn₀₎= +∞, there existsB0such that

λ

a(Cλ2+2n0₎≥4λ1c

–1_|_α_|

∞ for allλ≥B0.

Choosingλ¯=max{1,μ,A0,B0}, we easily to see that

–αλ,μ≤

1 b0

λf(x,αλ,μ) +μh(x)

– 1

a0

K(x)g(αλ,μ) inΩfor allλ>λ¯.

Since

–λn0_U₌ 1 a0

λn0_f₍_x_,_U_{) +}_h₍_x₎≥ 1 a0

fx,λn0+1_U₊_h₍_x₎ _in_Ω_,

we get

–λn0+1_U≥ λ a0

fx,λn0+1_U₊_h₍_x₎ _in_Ω_.

Hence

–βλ,μ≥

1 a0

λf(x,βλ,μ) +μh(x)

– 1

b0

K(x)g(βλ,μ) inΩfor allλ>λ¯.

By Lemma2.2(1.1) has at least one solutionuλ,μsuch thatαλ,μ≤uλ,μ≤βλ,μinΩ. Since

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Nonexistence forλ,μsmall. Letλ,μ> 0. Set

Θλ,μ(x.s) =

1 a0

λf(x,s) +μh(x) –K(x)g(s) inΩ.

SinceK∗> 0, assumption (g1) implieslims→0Θλ,μ(x.s) = –∞uniformly inx∈Ω. Then

there existsc> 0 such that

Θλ,μ(x.s) < 0 for all (x,s)∈Ω×(0,c). (3.13)

Lets≥c. From (f1) we deduce

Θλ,μ(x.s)

s ≤

1 a0

λf(x,s)

s +μ h(x)

s

≤ 1

a0

λf(x,c)

c +μ

|h|∞ s

for allx∈ ¯Ω. Fixμ≤cλ1a0

2|h|∞ and letM=supx∈Ω

f(x,c)

a0c > 0. From the above inequality we have

Θλ,μ(x.s)

s ≤λM+

λ1

2 for all (x,s)∈Ω×(c, +∞). (3.14)

SetA(λ) =λM. Inequalities (3.13) and (3.14) yield

Θλ,μ(x.s)≤A(λ)s+ λ1

2s for all (x,s)∈Ω×(0, +∞). (3.15)

Moreover,A(λ)→0 asλ→0. If (1.1) has a solutionuλ,u, then

λ1

Ω

u2_λ_,_μ(x)dx≤

Ω

_∇uλ,μ(x) 2

dx

= –

Ω

uλ,μ(x)uλ,μ(x)dx,

and using (3.15), we get

λ1

Ω

u2λ,μ(x)dx≤

A(λ) +λ1

2 Ω

u2λ,μ(x)dx.

SinceA(λ)→0 asλ→0, this relation leads to a contradiction forλ,μ> 0 suﬃciently

small. The proof of Theorem1.1is now complete.

Proof of Theorem 1.2 Suppose to the contrary that existλandμ such that (1.1) has a solutionuλ,μ∈D, and by Lemma2.6the boundary value problem

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

–V(x) =_a₍ 1

Ω|∇uλ,μ|2dx)(λf(x,V) +μh(x)) inΩ,

V> 0 inΩ,

V= 0 on∂Ω

(3.16)

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Set

Ψλ,μ(x,s) =

1 a(_Ω|∇uλ,μ|2dx)

λf(x,s) +μh(x).

Since

Vλ,μ+Ψλ,μ(x,Vλ,μ)≤0≤ uλ,μ+Ψλ,μ(x,uλ,μ) inΩ,

by Lemma2.5we getuλ,μ≤Vλ,μinΩ.

Consider the perturbed problem ⎧

⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

–u+K∗g(u+) =_a₍ 1

Ω|∇uλ,μ|2dx)(λf(x,u) +μh(x)) inΩ,

u> 0 inΩ,

u= 0 on∂Ω.

(3.17)

SinceK∗> 0, it follows thatuλ,μandVλ,μare subsolution and supersolution for (3.17),

respectively. Then there exists a solutionu∈C2,γ(Ω) of (3.17) such that

uλ,μ≤u≤Vλ,μ.

Integrating in (3.17), we deduce

–

Ω

udx+K∗

Ω

g(u+)dx=

1 a(_Ω|∇uλ,μ|2dx)

Ω

λf(x,u) +μh(x)

dx.

Hence

–

∂Ω ∂u

∂n +K∗

Ω

g(u+)dx≤M, (3.18)

whereM> 0 is a constant. Since∂u

∂n ≤0 on∂Ω, relation (3.18) yieldsK∗

Ωg(u+)dx≤

M, and soK∗Ωg(Vλ,μ+)dx≤M. Thus, for any compact subsetω⊂Ω, we have

K∗

ω

g(Vλ,μ+)dx≤M.

Letting→0, this relation leads toK∗ωg(Vλ,μ)dx≤M. Therefore

K∗

Ω

g(Vλ,μ)dx≤M. (3.19)

Chooseδ> 0 suﬃciently small and deﬁneΩδ :={x∈Ω;dist(x,∂Ω)≤δ}. Taking into

ac-count the regularity of the domain, we get that there existsk> 0 such that

Vλ,μ≤kdist(x,∂Ω) for allx∈Ωδ.

Then

Ω

g(Vλ,μ)dx≥

Ωδ

g(Vλ,μ)dx≥

Ωδ

(12)

which contradicts (3.19). It follows that problem (1.1) has no solution inD, and the proof

of Theorem1.2is now complete.

Acknowledgements

The authors thank the referees for their comments.

Funding

This work is supported by the National Natural Science Foundation of China (61603226) and the Fund of Natural Science of Shandong Province (ZR2018MA022).

Availability of data and materials Not applicable.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally to the writing of this paper. Both authors read and approved the ﬁnal manuscript.

Authors’ information Not applicable.

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Received: 6 November 2018 Accepted: 11 February 2019 References

1. Alves, C.O., Corrêa, F.J.S.A.: On existence of solutions for a class of problem involving a nonlinear operator. Commun. Appl. Nonlinear Anal.8, 43–56 (2014)

2. Alves, C.O., Corrêa, F.J.S.A.: A sub-supersolution approach for a quasilinear Kirchhoﬀ equation. J. Math. Phys.56, 591–608 (2015)

3. Alves, C.O., Corrêa, F.J.S.A., Ma, T.M.: Positive solutions for a quasilinear elliptic equation of Kirchhoﬀ type. Comput. Math. Appl.49, 85–93 (2005)

4. Bensedik, A., Bouchekif, M.: On an elliptic equation of Kirchhoﬀ type with a potential asymptotically linear at inﬁnity. Math. Comput. Model.49, 1089–1096 (2009)

5. Cheng, B., Wu, X.: Existence results of positive solutions of Kirchhoﬀ type problems. Nonlinear Anal.71, 4883–4892 (2009)

6. Ghergu, M., R ˘adulescu, V.: Sublinear singular elliptic problems with two parameters. J. Diﬀer. Equ.195, 520–536 (2003) 7. Kirchhoﬀ, G.: Mechanik. Teubner, Leipzig (1883)

8. Lazer, A.C., McKenna, P.J.: On a singular nonlinear elliptic boundary value problem. Proc. Am. Math. Soc.111, 721–730 (1991)

9. Liang, S., Repovš, D., Zhang, B.: On the fractional Schrödinger–Kirchhoﬀ equations with electromagnetic ﬁelds and critical nonlinearity. Comput. Math. Appl.75, 1778–1794 (2018)

10. Lions, J.L.: On some questions in boundary value problems of mathematical physics. In: Contemporary Developments in Continuum Mechanics and Partial Diﬀerential Equations. North-Holland Mathematics Studies, vol. 36, pp. 284–346 (1977)

11. Lu, S.S.: Multiple solutions for a Kirchhoﬀ-type equation with general nonlinearity. Adv. Nonlinear Anal.7, 293–306 (2016)

12. Ma, T.: Remarks on an elliptic equation of Kirchhoﬀ type. Nonlinear Anal.63, 1967–1977 (2005)

13. Mao, A., Luan, S.: Sign-changing of a class nonlocal quasilinear elliptic boundary value problems. J. Math. Anal. Appl. 383, 239–243 (2011)

14. Mao, A., Zhang, Z.: Sign-changing and multiple solutions of Kirchhoﬀ type problems without the P.S. condition. Nonlinear Anal., Theory Methods Appl.70, 1275–1287 (2009)

15. Mao, A., Zhu, X.: Existence and multiplicity results for Kirchhoﬀ problems. Mediterr. J. Math.14, 58 (2017).

https://doi.org/10.1007/s00009-017-0875-0

16. Molica Bisci, G., R ˘adulescu, V., Servadei, R.: Variational Methods for Nonlocal Fractional Problems. Encyclopedia of Mathematics and Its Applications, vol. 162. Cambridge University Press, Cambridge (2016)

17. Perera, K., Zhang, Z.: Nontrivial solutions of Kirchhoﬀ-type problems via the Yang index. J. Diﬀer. Equ.221, 246–255 (2006)

18. R ˘adulescu, V.: Singular phenomena in nonlinear elliptic problems: from blow-up boundary solutions to equations with singular nonlinearities. In: Handbook of Diﬀerential Equations: Stationary Partial Diﬀerential Equations, vol. 4, Chap. 7, pp. 485–593 (2007)

19. R ˘adulescu, V.: Combined eﬀects in nonlinear singular elliptic problems with convection. Rev. Roum. Math. Pures Appl. 53, 543–553 (2008)

20. Ricceri, B.: On an elliptic Kirchhoﬀ-type problem depending on two parameters. J. Glob. Optim.46, 543–549 (2010) 21. Shi, J., Yao, M.: On a singular nonlinear semilinear elliptic problem. Proc. R. Soc. Edinb.128, 1389–1401 (1998) 22. Sun, J., Tang, C.: Existence and multiplicity of solutions for Kirchhoﬀ type equations. Nonlinear Anal., Theory Methods

(13)

23. Xiang, M., R ˘adulescu, V., Zhang, B.: Multiplicity of solutions for a class of quasilinear Kirchhoﬀ system involving the fractionalp-Laplacian. Nonlinearity29, 3186–3205 (2016)

24. Xiang, M., R ˘adulescu, V., Zhang, B.: Combined eﬀects for fractional Schrödinger–Kirchhoﬀ systems with critical nonlinearities. ESAIM Control Optim. Calc. Var.24, 1249–1273 (2018)

25. Xiang, M., R ˘adulescu, V., Zhang, B.: Nonlocal Kirchhoﬀ diﬀusion problems: local existence and blow-up of solutions. Nonlinearity31, 3228–3250 (2018)

26. Xu, L., Chen, H.: Ground state solutions for Kirchhoﬀ-type equations with a general nonlinearity in the critical growth. Adv. Nonlinear Anal.7, 535–546 (2018)

27. Yan, B., O’Regan, D., Agarwal, R.P.: The existence of positive solutions for Kirchhoﬀ-type problems via the sub-supersolution method. An. ¸Stiin¸t. Univ. ‘Ovidius’ Constan¸ta26, 5–41 (2018)

28. Yang, M., Han, Z.: Existence and multiplicity results for Kirchhoﬀ type problems with four-superlinear potentials. Appl. Anal.91, 2045–2055 (2012)

29. Yang, Y., Zhang, J.: Nontrivial solutions of a class of nonlocal problems via local linking theory. Appl. Math. Lett.23, 377–380 (2010)