Existence of solution for p-Laplacian boundary value problems with two singular and subcritical nonlinearities

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R E S E A R C H

Open Access

Existence of solution for

p

-Laplacian

boundary value problems with two singular

and subcritical nonlinearities

Q-Heung Choi

1

_{and Tacksun Jung}

2*

*_{Correspondence:}

[email protected] 2_{Department of Mathematics,}

Kunsan National University, Kunsan, Korea

Full list of author information is available at the end of the article

Abstract

We consider a boundary value problem forp-Laplacian systems with two singular and

subcritical nonlinearities. We obtain one theorem which shows that there exists at least one nontrivial weak solution for these problems under some conditions. We obtain this result by variational method and critical point theory.

MSC: 35D30; 35J35; 35J58; 35J75

Keywords: Boundary value problems forp-Laplacian systems; Singular potential;

Variational method; Critical point theory; (P.S.)ccondition

1 Introduction

LetΩbe a bounded domain ofRn_{with smooth boundary}_∂Ω_,_n_≥_{2. Let}_G_{be an open} sub-set inR2with compact complementC1∪C2=Rn\Gcontainingθ= (0, 0) ande= (e1,e2), whereθ = (0, 0)∈C1 ande= (e1,e2)∈C2,n≥2. In this paper we investigate existence and multiplicity of the solutions (u,v)∈W1,p₍_Ω_,_G_{) for the}_p_{-Laplacian system with two} singular and subcritical nonlinearities under the Dirichlet boundary condition:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

–div(|∇u|p–2_∇_u_{) =}_a_|_u_|p–2_u_–_grad

u(|u|2₊1_|_v_|2₎q–gradu(|u–e1|2+1|v–e2|2)r

+ 2α α+β|u|

α–1_|_v_|β _in_Ω_,

–div(|∇v|p–2∇v) =b|v|p–2v–grad_v 1

(|u|2₊_|_v_|2₎q–gradv₍_|_u_–_e₁_|2₊1_|_v_–_e 2|2)r

+_α2₊β_β|u|α_|_v_|β–1 _in_Ω_,

(1.1)

wherea,b,p,q,r,α, andβare real constants, and 1 <p<∞,q,r> 1, andp<α+β<p∗, wherep∗is a critical exponent such that

p∗=

⎧ ⎨ ⎩

np

n–p ifn>p, ∞ ifn≤p.

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{(u,v) =e}. We recommend the book [12] for the singular elliptic problems. We also rec-ommend Rˇadulescu’s paper [21] establishing the recent contributions in singular phenom-ena in nonlinear elliptic problems from blow-up boundary solutions to equations with singular nonlinearities in two types of stationary singular problems: the logistic equation

u=Φ(x,u,∇u) inΩ,

u> 0 inΩ,

u= +∞ on∂Ω

(1.2)

and the Lane–Emden–Fowler equation

–u=Ψ(x,u,∇u) inΩ,

u> 0 inΩ,

u= 0 on∂Ω,

(1.3)

whereΦis a smooth nonlinear function, whileΨ has one or more singularities. The solu-tions of (1.2) are called large (or blow-up) solutions. More studies on blow-up boundary solutions of logistic type equation like (1.2) can be found in [2–5,7,8,11,13,14,16–18,

20,22]. Singular Dirichlet boundary value problems for the Lane–Emden–Fowler equa-tion like (1.3) involving singular nonlinearities have been intensively studied in the last decades. The ﬁrst study in this direction is due to Fulks and Maybee [10], who proved existence and uniqueness of the problem

–u+u–α₌_u _in_Ω_,

u> 0 inΩ,

u= 0 on∂Ω

(1.4)

by using ﬁxed point arguments and no solution of (1.4), provided that 0 <α< 1 andλ1≥1 (that is, ifΩis small), whereλ1denotes the ﬁrst eigenvalue of –inH01(Ω). Shi and Yau studied in [23] the existence of radial symmetric solutions of the problem

u+λup–u–α= 0 inB1,

u> 0 inB1,

u= 0 on∂B1,

(1.5)

whereα> 0, 0 <p< 1,λ> 0 andB1is the unit ball inRN. They showed in [23] that there existsλ1>λ0> 0 such that (1.5) has no solution forλ<λ0, exactly one solution forλ=

λ0 orλ>λ1, and two solutions forλ0 <λ≤λ1. Dupaigne, Ghergu, and Rˇadulescu [9] proved existence and multiplicity of the Lane–Emden–Fowler equation with convection and singular potential

–u±d(x)g(u) =λf(x,u) +ν|∇u|α inΩ,

u> 0 inΩ,

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M. Trabelsi and N. Trabelsi [24] considered the semilinear elliptic system and proved ex-istence of the singular limit solutions for a two-dimensional semilinear elliptic system of Liouville type.

We introduce the space

Lp(Ω,R) =

u u:Ω→Ris measurable,

Ω

|u|pdx<∞

endowed with the norm

up_Lp₍_Ω₎=inf

λ> 0

Ω

u(_λx) p≤1

and the Sobolev space

W1,p(Ω,R) =u∈Lp(Ω,R)| ∇u(x)∈Lp(Ω,R) endowed with the norm

uW1,p₍_Ω_,_R₎=

Ω

_∇u(x) pdx+

Ω

u(x) pdx

1

p .

LetLp₍_Ω_,_R2_{) =}_Lp₍_Ω_,_R₎_×_Lp₍_Ω_,_R_{) and}_H₌_W1,p₍_Ω_,_R2_{) =}_W1,p₍_Ω_,_R₎_×_W1,p₍_Ω_,_R_{). Then}

Lp(Ω,R2) andHare Hilbert spaces with the norm

(u,v)_Lp₍_Ω_,_R2₎=uLp₍_Ω_,_R₎+v_Lp₍_Ω_,_R₎

and the norm

(u,v)_H=u_W1,p₍_Ω_,_R₎+v_W1,p₍_Ω_,_R₎,

respectively. It was proved in [15] that, for 1 <p<∞, the eigenvalue problem –div|∇u|p–2∇u=λ|u|p–2u inΩ,

u= 0 on∂Ω

has a nondecreasing sequence of nonnegative eigenvaluesλ(_jp)obtained by the Ljusternik– Schnirelman principle tending to∞asj→ ∞, where the first eigenvalueλ(₁p) is simple and only eigenfunctions associated withλ(₁p)do not change sign, the set of eigenvalues is closed, the first eigenvalueλ(₁p)is isolated. Thus there is a sequence of eigenfunctions (φ_j(p))j corresponding to the eigenvaluesλ(_jp)such that the first eigenfunctionφ₁(p) is positive or negative depending onp.

Let us set –pu= –div(|∇u|p–2∇u). Let us deﬁne an open subset of the Hilbert space

H=W1,p(Ω,R2):

ΛG=(u,v)∈H| ∇(u,v)∈LpΩ,R2,u(x),v(x)∈G=R2\(C1∪C2),

θ∈C!, (e1,e2)∈C2,∀

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In this paper, we are looking for weak solutions (u,v) of (1.1) inΛGsatisfying

Ω

–pu·z–pv·w–a|u|p–2u·z–b|v|p–2v·w

+grad_u 1

(|u|2₊_|_v_|2₎q·z+gradv 1

(|u|2₊_|_v_|2₎q ·w

+grad_u 1

(|u–e1|2₊_|_v_–_e_2|2₎r ·z+gradv

1

(|u–e1|2₊_|_v_–_e_2|2₎r·w – 2α

α+β|u| α–1_|_v_|β_·

z– 2β

α+β|u| α–1_|_v_|β_·

w

dx= 0 ∀(z,w)∈ΛG.

LetAbea₀0_b∈M2×2(R). Let us set

q_λ(p)

j

(a,b) =Detλ_j(p)I–A=λ(_jp)–aλ(_jp)–b.

Letμ1

λ(_ip)andμ

2

λ(_ip)be the eigenvalues of the matrix

λ(_ip)–a 0 0 λ(_ip)–b

∈M2×2(R), i.e.,

μ1 λ(_ip)=

1 2

2λ(_ip)–a–b–2λ_i(p)–a–b2– 4q_λ(p)

i (a,b)

,

μ2 λ(_ip)=

1 2

2λ(_ip)–a–b+2λ_i(p)–a–b2– 4q_λ(p)

i (a,b).

We note that weak solutions of (1.1) correspond to critical points of the continuous and Fréchet diﬀerentiable functionalf(u,v)∈C1₍_Λ_G_,_R_),

f(u,v) =1

p

Ω

|∇u|p+|∇v|p–a|u|p–b|v|pdx+

Ω

1

(|u|2₊|_v|2₎qdx +

Ω

1

(|u–e1|2₊|_v_–_e2|2₎rdx– 2

α+β

Ω

|u|α_|_v_|β_dx

=Ra,b(u,v) +

Ω

1

(|u|2₊_|_v_|2₎qdx+

Ω

1

(|u–e1|2₊_|_v_–_e_2|2₎rdx

– 2

α+β

Ω

|u|α|v|βdx,

whereRa,b(u,v) =1_p

Ω[|∇u|

p₊_|∇_v_|p_–_a_|_u_|p_–_b_|_v_|p_]_dx_{, which will be proved in Sect.}₃_. Whenp<α+β<p∗, the embeddingW₀1,p(Ω,G)→Lα+β₍_Ω_,_G_{) is continuous and}

com-pact, so we can assure that the functionalf(u,v) satisﬁes the (P.S.) condition, which will be also proved in Sect.3.

Our main result is as follows.

Theorem 1.1 Assume that a,b,p,q,r,α,andβare real constants,and1 <p<∞,q,r> 1,

and p<α+β<p∗,n≥2,

(i) 2λ(_ip)>a+b, (ii) q_λ(p)

i

(a,b) =detλ

(p)

i –a 0 0 λ(_ip)–b

> 0fori≥1.

Then(1.1)has at least one nontrivial weak solution(u(x),v(x))such that(u(x),v(x))= (0, 0)

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For the proof of Theorem1.1, we approach variational method and use critical point theory on eigenspaces. In Sect.2, we introduce the eigenspaces spanned by the

eigenfunc-tions corresponding to the eigenvalues of the matrixλ

(p)

i –a 0 0 λ(_ip)–b

∈M2×2(R) and obtain

some variational results on the eigenspaces. In Sect.3, we prove that the corresponding functional of (1.1) satisﬁes the (P.S.) condition and prove Theorem1.1.

2 Variational results on eigenspaces Letq_λ(p)

i

(a,b),μ1

λ(_ip), andμ

2

λ(_ip)be the numbers introduced in Sect.1. We note that

ifq_λ(p)

i

(a,b) < 0, thenμ1

λ(_ip)< 0 <μ

2

λ(_ip),

if 2λ(_ip)>a+bandq_λ(p)

i (a,b) > 0, then 0 <

μ1 λ(_ip)<μ

2

λ(_ip),

if 2λ(_ip)<a+bandq_λ(p)

i

(a,b) > 0, thenμ1 λ(_ip)<μ

2

λ(_ip)< 0,

if 2λ(_ip)=a+bandq_λ(p)

i

(a,b) > 0, thenμ1 λ(_ip)=μ

2

λ(_ip)= 0.

Let (c1

λ(_ip),d

1

λ(_ip)) and (c

2

λ(_ip),d

2

λ(_ip)) be the eigenvectors of

λ(_ip)–a 0 0 λ(_ip)–b

corresponding toμ1 λ(_ip)

andμ2

λ(_ip), respectively. Let us set

D_λ(p)

i

=(a,b)∈R2|q_λ(p)

i

(a,b) > 0 fori≥1,

D

λ(_ip)=Dλ(_ip)∩

2λ(_ip)≤a+b,

D

λ(_ip)=Dλ(_ip)∩

2λ(_ip)>a+b,

W_λ(p)

i =

φ_i(p)|–pφi(p)=λ (p) i φ

(p) i

H_λ(p)

i

=cφ(p),dφ(p)∈H|(c,d)∈R2,φ(p)∈W_λ(p)

i

,

H1

λ(_ip)=

c1

λ(_ip)φ

(p)_,_d1

λ(_ip)φ

(p)_∈_H_|_φ(p)_∈_W

λ(_ip)

,

H2

λ(_ip)=

c2

λ(_ip)φ

(p)_,_d2

λ(_ip)φ

(p)_∈_H_|_φ(p)_∈_W

λ(_ip)

,

H+(a,b) =

μ1

λ(_ip) >0

H1

λ(_ip)

⊕

μ2

λ(_ip) >0

E2

λ(_ip)

,

H–(a,b) =

μ1

λ(_ip) <0

H1

λ(_ip)

⊕

μ2

λ(_ip) <0

H2

λ(_ip)

,

H0(a,b) =

μ1

λ(_ip)=0

H1

λ(_ip)

⊕

μ2

λ(_ip)=0

H2

λ(_ip)

.

ThenH+₍_a_,_b_),_H–₍_a_,_b_{), and}_H0₍_a_,_b_{) are the positive, negative, and null spaces relative to} the quadratic formRa,b(u,v) inHand

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From now on we shall assume that 2λ(_ip)>a+bandq_λ(p)

i (a,b) > 0. Because

μ1

λ(_ip)> 0 and μ2

λ(_ip)> 0,∀i≥1,

H0=∅, H–=∅

and

H=H+.

We note thatHcan be split by two subspacesY1andY2such that

Y1=span

eigenfunctions corresponding to eigenvaluesμ1

λ(_ip)andμ

1

λ(_ip),

with 1≤i≤m,

Y2=span

eigenfunctions corresponding to eigenvaluesμ1

λ(_ip)andμ

2

λ(_ip),

withi≥m+ 1,

dimY1<∞and

H=Y1⊕Y2.

Let us set

X1=Y1∩ΛG,

X2=Y2∩ΛG.

Then

ΛG=X1⊕X2.

Let us set

Bρ=

(u,v)∈ΛG|(u,v)_H≤ρ,

∂Bρ=

(u,v)∈ΛG|(u,v)_H=ρ,

Q=B¯R∩X1⊕

σ(z1,z2)|(z1,z2)∈∂B1∩

H1

μ

λ(_mp)₊₁⊕

H2

μ

λ(_mp)₊₁

⊂∂B1∩X2, 0 <σ<R

.

Let us deﬁne

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We note that weak solutions of (1.1) correspond to critical points of the continuous and Fréchet diﬀerentiable functionalf(u,v)∈C1₍_Λ_G_,_R_),

f(u,v) =1

p

Ω

1

(|u|2₊_|_v_|2₎qdx +

Ω

1

(|u–e1|2₊_|_v_–_e_2|2₎rdx–

Ω

2

α+β|u| α_|

v|β

dx.

Let us deﬁne

Cα(p,)β(Ω) = inf

(u,v)∈ΛG

τ> 0 (

Ω(|∇u|

p₊_|∇_v_|p₎_dx₎1p_{+ (}

Ω(|u|

p₊_|_v_|p₎_dx₎1p (_Ω|u|α|_v|β_dx₎α1+β

≤τ

. (2.1)

Lemma 2.1 Assume that a,b,p,q,r,α,andβare real constants,and1 <p<∞,q,r> 1,

p<α+β<p∗, 2λ_i(p)>a+b,and q_λ(p)

i

(a,b) > 0.Let i∈N and(a0,b0)∈∂D

λ(_ip)and(z1,z2)∈ ∂B1∩(Hμ1

λ(_mp)₊₁ ⊕H2

μ

λ(_mp₊₁)

)⊂∂B1∩X2.Then there exist a neighborhood W of(a0,b0),a

small numberσ > 0,and a large number R> 0such that,for any(a,b)∈W \D

λ(_ip), if

(u,v)∈∂Q=∂(B¯R∩X1⊕ {σ(z1z2)|0 <σ<R}),then sup

u∈∂Q

f(u,v) < 0 and sup u∈Q

f(u,v) <∞.

Proof Let (a,b)∈W\D

λ(_ip). Let us choose an element (z1,z2)∈∂B1∩X2and (u,v)∈X1⊕

{σ(z1,z2)|σ> 0}. Then, by (2.1), we have

f(u,v) = 1

p

Ω

1

(|u|2₊_|_v_|2₎qdx +

Ω

1

(|u–e1|2₊_|_v_–_e_2|2₎rdx– 2

α+β

Ω

|u|α|v|βdx

≤ 1

pμ

2

λ(mp)

(u,v)p_Lp₍_Ω₎+ 1

pσ

p_μ2

λ(mp)

+

Ω

1 (|u|2₊|_v|2₎q +

1

(|u–e1|2₊|_v_–_e2|2₎r

dx

– 2

α+β

Cα(p,β) –(α+β)

(Ω)(u,v)α_H+β.

Then there exist a large numberR> 0 and a small numberσ> 0 with 0 <σ<Rsuch that if (u,v)∈∂Q, then we have 0 <_Ω[₍_|_u_|2₊1_|_v_|2₎q +₍_|_u_–_e 1

1|2+|v–e2|2)r]dx≤C1for some constant

0 <C1< 1, and, byp<α+β<p∗, we have

f(u,v)≤ 1

pμ

2

λ(mp)

(u,v)p_Lp₍_Ω₎+ 1

pσ

p_μ2

λ(mp) +C1–

2

α+β

Ω

|u|α|v|βdx

≤ 1

pμ

2

λ(mp)

(u,v)p_Lp₍_Ω₎+ 1

pσ

p_μ2

λ(mp) +C1–

2

α+β

Cα(p,β) –(α+β)

(Ω)(u,v)α_H+β< 0.

Thus we have sup₍_u_,_v₎_∈_∂_Qf(u,v) < 0. Moreover, if (u,v) ∈ Q, then we have f(u,v) ≤ 1

pμ 2

λ(mp)

(u,v)p_Lp₍_Ω₎+1_pσpμ2

λ(mp)

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i (a,b) > 0.Let i∈N and(a0,b0)∈

∂D

λ(_ip)and(z1,z2)∈ ∂B1∩(Hμ1

λ(_mp₊₁) ⊕

H2

μ

λ(_mp)₊₁

)⊂∂B1∩X2.Then there exist a neighborhood W of(a0,b0)and a

small numberρ> 0such that,for any(a,b)∈W\D

λ(_ip),if(u,v)∈∂Bρ∩X2,then we have

inf u∈∂Bρ∩X2

f(u,v) > 0 and inf u∈Bρ∩X2

f(u,v) > –∞.

Proof Let (a,b)∈W\D

λ(_ip)and (u,v)∈X2. Then, by (2.1), we have

f(u,v) = 1

p

Ω

1

(|u|2₊|_v|2₎qdx +

Ω

1

(|u–e1|2₊_|_v_–_e_2|2₎rdx–

Ω

2

α+β|u| α_|

v|β

dx

≥ 1

pμ

1

λ(_mp)₊₁(u,v)

p Lp₍_Ω₎+

Ω

1 (|u|2₊|_v|2₎q +

1

(|u–e1|2₊|_v_–_e2|2₎r

dx

– 2

α+β

C_α(p_,_β)–(α+β)(Ω)(u,v)α_H+β ≥ 1

pμ

1

λ(_mp)₊₁(u,v)

p Lp(Ω)–

2

α+β

C(αp,β) –(α+β)

(Ω)(u,v)α_H+β.

Since p<α+β <p∗, there exists a small number ρ> 0 such that if (u,v)∈∂Br∩X2, thenf(u,v) > 0. Thusinf₍_u_,_v₎_∈∂Br∩X2f(u,v) > 0. Moreover, if (u,v)∈Br∩X2, thenf(u,v)≥

–_α₊2_β(C_α(p_,_β))–(α+β)₍_Ω₎₍_u_,_v₎α+β

H > –∞. Thusinf(u,v)∈Br∩X2f(u,v) > –∞. So the lemma is

proved.

Let us deﬁne

γ =inf h∈Γsup_u_∈_Qf

h(u,v).

i

(a,b) > 0.Let i∈N and(a0,b0)∈∂D

λ(_ip).Then there

exist a neighborhood W of(a0,b0),a small numberσ,a small numberρ> 0,and a large

number R> 0such that,for any(a,b)∈W\D

λ(_ip),

0 < inf u∈∂Bρ∩X2

f(u,v)≤γ =inf h∈Γsup_u_∈_Qf

h(u,v)≤sup u∈Q

f(u,v) <∞.

Proof By Lemma2.1, we have

γ =inf h∈Γsup_u_∈_Qf

h(u,v)≤sup u∈Q

f(u,v) <∞.

By Lemma2.2, we have

0 < inf u∈∂Bρ∩X2

f(u,v)≤inf h∈Γsup_u_∈_Qf

h(u,v)=γ.

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3 (P.S.)condition and proof of Theorem1.1

We need some lemma for the proof thatf(u,v) satisﬁes the (P.S.) condition.

Lemma 3.1([1,6]) Let1 <p<∞.Let1 <τ≤p∗.Then the embedding

H=W1,pΩ,R2→LτΩ,R2

is continuous and compact and,for every(u,v)∈C₀∞(Ω,R2_),_{we have}

(u,v)_Lτ₍_Ω¯_,_R2₎≤C(u,v)_H

for a positive constant C independent of u. By Lemma3.1, we obtain the following.

Lemma 3.2 Assume that1≤p<∞,a,b,p,q,r,α,andβ are real constants and q,r> 1

and p<α+β<p∗.Then all the solutions of(1.1)belong toΛG⊂H.

Proof

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

u= ––1_p (a|u|p–2_u_–_grad

u(|u|2₊1_|_v_|2₎q –gradu(|u–e1|2+1|v–e2|2)r +

2α α+β|u|

α–1_|_v_|β₎

inΩ,

v= ––1_p (b|v|p–2v–grad_v 1

(|u|2₊_|_v_|2₎q –gradv₍_|_u_–_d_|2₊1_|_v_–_d_|2₎r + 2β α+β|u|

α_|_v_|β–1₎ inΩ.

(3.1)

Since the right-hand side of (1.1) belongs to Lα+β₍_Ω_,_G_{), where}_p_<_α₊_β _<_p∗_{, and by}

Lemma3.1, the embeddingW1,p(Ω,G)→Lα+β₍_Ω_,_G_{) is continuous and compact, it}

fol-lows that ––1_p is a compact operator and the solutions of (3.1) are inW1,p₍_Ω_,_G_{) =}_Λ_G_.

Lemma 3.3 Assume that1≤p<∞,a,b,p,q,r,α,andβ are real constants and q,r> 1

and p<α+β<p∗.Then the functional f(u,v)is continuous,Fréchet diﬀerentiable with Frechet derivative in´ ΛG,

Df(u,v)·(z,w) =

Ω

–pu·z–pv·w–a|u|p–2u·z–b|v|p–2v·w

+grad_u 1

(|u|2₊|_v|2₎q ·z+gradv 1

(|u|2₊|_v|2₎q ·w

+grad_u 1

(|u–e1|2₊_|_v_–_e_2|2₎r·z+gradv

1

(|u–e1|2₊_|_v_–_e_2|2₎r ·w – 2α

α+β|u| α–1_|_v_|β_·

z– 2β

α+β|u| α_|

v|β–1·w

dx ∀(z,w)∈ΛG.

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Proof Let us set H(x,u,v) = 1_p(a|u|p ₊_b_|_v_|p_{) –} 1

(|u|2₊_|_v_|2₎q – ₍_|_u_–_e 1

1|2+|v–e2|2)r +

2

α+β|u| α_|_v_|β_.

Then Hu(x,u,v) = a|u|p–2u–gradu(|u|2₊1_|_v_|2₎q –gradu(|u–e1|2+1|v–e2|2)r +

2α α+β|u|

α–1_|_v_|β _and

Hv(x,u,v) =b|v|p–2v–gradv(|u|2₊1_|_v_|2₎q –gradv(|u–e1|2+1|v–e2|2)r +

2β α+β|u|

α_|_v_|β–1_{. First we shall} prove thatf(u,v) is continuous. Foru,v∈ΛG,

f(u+z,v+w) –f(u,v)

= 1

p

Ω

–p(u+z), –p(v+w)

·(u+z,v+w)dx–

Ω

H(x,u+z,v+w)dx

–1

p

Ω

(–pu, –pv)·(u,v)dx+

Ω

H(x,u,v)dx

= 1 2

Ω

–p(u+z), –p(v+w)

·(u+z,v+w) – (–pu, –pv)·(u,v)

dx

–

Ω

H(x,u+z,v+w) –H(x,u,v)dx .

We have

1₂_Ω–p(u+z), –p(v+w)

·(u+z,v+w) – (–pu, –pv)·(u,v)

dx

≤O(z,w)_H (3.2)

and

_ΩH(x,u+z,v+w) –H(x,u,v)dx

≤

Ω

Hu(x,u,v),Hv(x,u,v)

·(z,w) +O(z,w)_Hdx =O(z,w)_H. (3.3)

Thus we have

f(u+z,v+w) –f(u,v) =O(z,w)_H.

Next we shall prove thatf(u,v) is Fréchet diﬀerentiable. Foru,v∈ΛG,

f(u+z,v+w) –f(u,v) –Df(u,v)·(z,w)

= 1

p

Ω

–p(u+z), –p(v+w)

·(u+z,v+w)dx–

Ω

H(x,u+z,v+w)dx

–1

p

Ω

(–pu, –pv)·(u,v)dx+

Ω

H(x,u,v)dx

–

Ω

–pu–Hu(x,u,v), –pv–Hv(x,u,v)

·(z,w)dx

= 1

p

Ω

–p(u+z), –p(v+w)

·(u+z,v+w)

– (–pu, –pv)·(u,v) – (–pu, –pv)·(z,w)

dx

–

Ω

H(x,u+z,v+w) –H(x,u,v) –Hu(x,u,v),Hv(x,u,v)

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By (3.2) and (3.3),

f(u+z,v+w) –f(u,v) –Df(u,v)·(z,w) =O(z,w)2_H.

Thusf ∈C1.

Lemma 3.4(A priori estimate) Assume that 1≤p<∞, a,b,p,q,r,α,andβ are real constants,q,r> 1,p<α+β<p∗, 2λ(_ip)>a+b,and q_λ(p)

i

(a,b) > 0.Let(un,vn)nbe any

se-quence inΛG andγ ∈R be any positive real number.Then there exist constants Ci=Ci(γ),

i= 1, 2, 3,such that if(un,vn)n∈ΛG satisﬁes that f(un,vn)→γ and Df(un,vn)→θ,then

lim

n→∞(un,vn)Lα+β₍_Ω₎≤C1, lim

n→∞

Ω

1 (|un|2+|vn|2)q

dx≤C2,

Ω

1

(|un–e1|2+|vn–e2|2)r

dx≤C3.

Proof Letγ ∈Rbe any positive real number. Let (un,vn)nbe any sequence inΛGsuch thatf(un,vn)→γ andDf(un,vn)→θ. Then there exists a small number> 0 such that

γ +

≥ lim

n→∞f(un,vn) –nlim→∞ 1

pDf(un,vn)·(un,vn)

= lim n→∞

Ω

1 (|un|2+|vn|2)q

dx+

Ω

1

(|un–e1|2+|vn–e2|2)r

dx

– 2

α+β

Ω

|un|α|vn|βdx

– lim n→∞

1

p

Ω

grad_u 1 (|un|2+|vn|2)q·

un+gradv

1 (|un|2+|vn|2)q ·

vn

+grad_u 1

(|un–e1|2+|vn–e2|2)r ·

un+gradv

1

(|un–e1|2+|vn–e2|2)r ·

vn

– 2α

α+β|un| α–1_|_v

n|β·un– 2β α+β|un|

α_|_v

n|β–1·vn

dx ∀(un,vn)∈ΛG.

Bylimn→∞Df(un,vn) =θ, we have

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

limn→∞un=limn→∞(–p–agp)–1(–gradu₍_|_u_n_|2₊1_|_v_n_|2₎q –gradu₍_|_u_n–_e₁_|2₊1_|_v_n–_e₂_|2₎r + 2α

n|β),

limn→∞vn=limn→∞(–p–bgp)–1(–gradv(|un|2+1|vn|2)q –gradv(|un–e1|2+1|vn–e2|2)r

+_α2₊β_β|un|α|vn|β–1),

where gp(t) =|t|p–2tfort= 0 andgp(0) = 0. By 2λi(p)>a+bandqλ_i(p)(a,b) =Det(λ

(p) j I–

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(–p–bgp)–1are positive operators. Since

lim n→∞

–grad_u 1 (|un|2+|vn|2)q

–grad_u 1

(|un–e1|2+|vn–e2|2)r + 2α

n|β

> 0,

lim n→∞

–grad_v 1 (|un|2+|vn|2)q

–grad_v 1

(|un–e1|2+|vn–e2|2)r + 2β

α+β|un| α_|

vn|β–1

> 0,

and (–p–agp)–1and (–p–bgp)–1are positive operators, it follows thatlimn→∞un> 0 andlimn→∞vn> 0. Then we have

–grad_u 1 (|un|2+|vn|2)q

un–gradv

1 (|un|2+|vn|2)q

vn= 2q 1 (|un|2+|vn|2)q

> 0,

–grad_u 1

(|un–e1|2+|vn–e2|2)r

un> 0 and –gradv

1

(|un–e1|2+|vn–e2|2)r

vn> 0.

Therefore we have

γ + ≥ lim n→∞ Ω 1 (|un|2+|vn|2)q

dx

+

Ω

1

(|un–e1|2+|vn–e2|2)r

dx– 2

α+β

Ω

|un|α|vn|βdx

–1

pnlim→∞

Ω

grad_u 1 (|un|2+|vn|2)q·

un+gradv

1 (|un|2+|vn|2)q ·

vn

+grad_u 1

(|un–e1|2+|vn–e2|2)r ·

un+gradv

1

(|un–e1|2+|vn–e2|2)r ·

vn

– 2α

n|β·un– 2β α+β|un|

α_|_v

n|β–1·vn

dx

=

2q p + 1

lim n→∞ Ω 1 (|un|2+|vn|2)q

dx+ lim n→∞

Ω

1

(|un–e1|2+|vn–e2|2)r

dx

–1

pnlim→∞

Ω

grad_u 1

(|un–e1|2+|vn–e2|2)r ·

undx

–1

pnlim→∞

Ω

grad_v 1

(|un–e1|2+|vn–e2|2)r ·

vndx

+

2

p–

2

α+β lim n→∞ Ω

|un|α|vn|β

dx ∀(un,vn)∈ΛG.

Since 1 < p < ∞, q,r > 1, p < α + β < p∗, we have that 2_pq + 1 > 0 and 2_p – 2

α+β > 0. Since

Ω

1

(|un–e1|2+|vn–e2|2)r dx> 0, –

1

plimn→∞

Ωgradu₍_|_u_n–_e₁_|2₊1_|_v_n–_e

2|2)r ·undx> 0,

and –_p1limn→∞Ωgradv(|un–e1|2+1|vn–e2|2)r ·vndx> 0, it follows that there exist constants

Ci=Ci(γ),i= 1, 2, 3, such thatlimn→∞(un,vn)Lα+β₍_Ω₎≤C₁,lim_n_→∞

Ω(

1

|un|2+|vn|2)qdx≤

C2, and_Ω ₍_|_u_n–_e 1

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Lemma 3.5 If any sequence(un,vn)ninΛG satisﬁes (un,vn)→(z,w)∈∂ΛG,

then

f(un,vn)→ ∞.

Proof The proof can be checked easily. Now, we shall prove thatf(u,v) satisﬁes (P.S.)γ withγ> 0 as follows.

Lemma 3.6(Palais–Smale condition) Assume that1≤p<∞,a,b,p,q,r,α,andβare real constants,and q,r> 1and p<α+β<p∗.Letγ be any positive real number.Then f(u,v)

satisﬁes the Palais–Smale condition:if(un,vn)n∈ΛG is any sequence such that f(un,vn)→

γ and Df(un,vn)→θ,θ= (0, 0),then(un,vn)has a convergent subsequence(uni,vni)such

that(uni,vni)converges strongly to(u0,v0)∈ΛG.

Proof Let (un,vn)nbe any sequence inΛGsuch thatf(un,vn)→γ,γ > 0 andDf(un,vn)→

θ. By Lemma3.4,lim_n_→∞(un,vn)Lα+β₍_Ω₎is ﬁnite. Thus (u_n,v_n)_nis bounded inLα+β(Ω). Then, up to subsequence, (un,vn)nconverges weakly to some (u0,v0). SinceDf(un,vn)→θ, we have

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

limn→∞un= –limn→∞–1_p (a|un|p–2un–gradu(|un|2+1|vn|2)q –gradu

1 (|un–e1|2+|vn–e2|2)r

+_α2₊α_β|un|α–1|vn|β),

lim_n_→∞vn= –limn→∞–1p (b|vn|p–2vn–gradv(|un|2+1|vn|2)q–gradv

1 (|un–e1|2+|vn–e2|2)r

+_α2₊β_β|un|α|vn|β–1). By Lemma3.4, (un,vn)n,

a|un|p–2un–gradu 1 (|un|2+|vn|2)q

–grad_u 1

(|un–e1|2+|vn–e2|2)r + 2α

n|β

n

and

b|vn|p–2vn–gradv 1 (|un|2+|vn|2)q

–grad_v 1

(|un–e1|2+|vn–e2|2)r + 2β

n|β

n

are bounded inLα+β₍_Ω_{). Since the embedding}_Λ_G_into_Lα+β₍_Ω_),_p_<_α₊_β_<_p∗_{, is compact,}

––1_p is a compact operator, it follows that (un,vn)nhas a convergent subsequence (uni,vni) converging strongly to some (u0,v0) such that

DF(u0,v0) = lim

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We claim that (u0,v0)=θand (u0,v0)= (e1,e2). By contradiction, we suppose that (u0,v0) =

θ or (u0,v0) = (e1,e2). Thenf(u0,v0) =∞, which is absurd. Thus (u0,v0)=θ and (u0,v0)=

(e1,e2).

Proof of Theorem1.1 Assume thata,b,p,q,r,α, andβare real constants, and 1 <p<∞,

q,r> 1,p<α+β<p∗, 2λ_i(p)>a+b, andq_λ(p)

i

(a,b) > 0. By Lemma3.3,f(u,v) is continuous and Fréchet diﬀerentiable inΛGandDf ∈C. By Lemma3.6,f(u,v) satisﬁes the Palais– Smale condition. We claim thatγ > 0 is a critical value off(u,v), that is,f(u,v) has a critical point (u0,v0) such that

f(u0,v0) =γ,

Df(u0,v0) = 0.

In fact, by contradiction, we suppose thatγ > 0 is not a critical value off(u,v). Then by Theorem A.4 in [6], for any¯∈(0,γ) > 0, there exist a constant∈(0,¯) and a deformation

η∈C([0, 1]×ΛG,ΛG) such that

(i) η(0, (u,v)) = (u,v)for all(u,v)∈ΛG,

(ii) η(s, (u,v)) = (u,v)for alls∈[0, 1]iff(u,v) /∈[γ –¯,γ+¯], (iii) f(η(1, (u,v)))≤γ –iff(u,v)≤γ+.

We can chooseh∈Γ such that

sup u∈Q

fh(u,v)≤γ+

and

fh(u,v)<γ –¯ on∂Q.

This leads tof(h(u,v)) /∈[γ –¯,γ +¯]. Thus by (ii),

η1,h(u,v)=h(u,v) on∂Q.

Henceη(1,h(u,v))∈Γ. By (iii) and the deﬁnition ofγ,

γ ≤sup u∈Q

fη1,h(u,v)=sup u∈Q

fh(u,v)≤γ –,

which is a contradiction. Thusγis a critical value off(u,v). Thusf(u,v) has a critical point (u0,v0) with a critical value

γ =F(u0,v0)

such that

0 < inf u∈∂Bρ∩X2

f(u,v)≤γ ≤sup u∈Q

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By Lemma3.4,

(u0,v0)=θ= (0, 0), (u0,v0)= (e1,e2).

Thus (1.1) has at least one nontrivial solution (u0,v0) such that (u0,v0)=θ and (u0,v0)=

(e1,e2). Thus Theorem1.1is proved.

Acknowledgements

Not applicable.

Funding

Q-Heung Choi was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2017R1D1A1B03030024). Tacksun Jung was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (NRF-2017R1A2B4005883).

Abbreviations

Not applicable.

Availability of data and materials

Not applicable.

Competing interests

The authors declare that there is no conﬂict of interests regarding the publication of this paper.

Authors’ contributions

Q-HC introduced the main ideas of multiplicity study for this problem. TJ participated in applying the method for solving this problem and drafted the manuscript. All authors contributed equally to reading and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics Education, Inha University, Incheon, Korea.}2_{Department of Mathematics, Kunsan National} University, Kunsan, Korea.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

Received: 4 February 2019 Accepted: 17 May 2019

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