Ground-state solutions to a class of

Ground-state solutions to a class of

modified Kirchhoff-type transmissiom problems with critical perturbation

Ying Zhang, Zhanping Liang, Xiaoli Zhu and Fuyi Li

School of Mathematical Sciences, Shanxi University, Taiyuan 030006, Shanxi, P.R.China Received 20 February 2020, appeared 30 May 2020

Communicated by Dimitri Mugnai

Abstract. This paper discusses a class of modified Kirchhoff-type transmission prob- lems with critical perturbation. We establish an existence result of the ground-state solutions by using perturbation methods. Meanwhile, the limit properties of solution sequence are investigated.

Keywords: modified Kirchhoff-type transmission problem, critical perturbation, ground-state solution.

2010 Mathematics Subject Classification: 35J20, 35J60.

1 Introduction

LetΩbe a bounded domain inR³with a smooth boundaryΓ:=_∂Ω,Ω1⊂_R³be a subdomain ofΩwith a smooth boundaryΣ:=_∂Ω1andΩ1 ⊂Ω. Assume thatΩ2=_Ω\_Ω1is connected.

Obviously, Γ∩_Σ = _{∅ and} ∂Ω2 = _Γ∪Σ. In the present paper we study the existence of solutions for the following Kirchhoff-type transmission problem

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 α

_Z

Ω1

g²(u)|∇u|²

−div g²(u)∇u

+g(u)g⁰(u)|∇u|²= f(u) +λφ(u), inΩ1, β

_Z

Ω2

g²(v)|∇v|²

−div g²(v)∇v

+g(v)g⁰(v)|∇v|²=h(v) +λψ(v), inΩ2,

v=0, on Γ,

u=v, on Σ,

α Z

Ω1

g²(_u)|∇u|² ∂u

∂ν

= β Z

Ω2

g²(_v)|∇v|² ∂v

∂ν, on Σ,

(1.1)

where λ ∈ _R+ := [0,∞) and ν is the unit outward normal vector to ∂Ω1. This system is a modified version of Kirchhoff-type transmission problem because the appearance of nonlocal terms R

Ω1g²(u)|∇u|² andR

Ω2g²(v)|∇v|².

BCorresponding author. E-mail: fyli@sxu.edu.cn.

There are two motivations for studying equation (1.1). The first one is the generalized quasilinear Schrödinger equations. The second one is the classical Kirchhoff-type transmission problem.

In 2015, Deng, Peng, and Yan in [9] researched the generalized quasilinear Schrödinger equations

− div g²(u)∇u

+g(u)g⁰(u)|∇u|²+V(x)u= f(x,u), x∈_R^N, (1.2) where N > 3, the potential function V ∈ C(_R^N) and f ∈ C(_R^N×_R). If we take g²(t) = 1+(l(t²))⁰²_{/2 for}t ∈ _{R and}l being a suitable function defined onR+, then the equation (1.2) turns into

−_∆u+V(x)u−_∆[l(u²)]l⁰(u²)u= f(x,u), x ∈_R^N. (1.3) Solutions of (1.3) is related to the existence of solitary wave solutions for the following quasi- linear Schrödinger equation

i∂_tz= −_∆z+V(x)z− f(x,z)−_∆[l(|z|²)]l⁰(|z|²)z, x∈_R^N. (1.4) This quasilinear version of Schrödinger equations is derived from several models of various physical phenomena. The equation (1.4) is called the superfluid film equation in plasma physics when l(t) = t for t ∈ _R+, see [13] or [14,15]. If l(t) = (1+t)^1/2 for t ∈ _R+, the equation (1.4) was used for the self-channeling of a high-power ultrashort laser in matter, see [4,5,7,24]. In mathematics, many results about the equation (1.3) withl(t) =t^αfor someα>¹ have been obtained, see [1,2,6,8,10,18–20,22,23,29–31] and the references therein. Equation (1.3) with a general l was studied in the recent papers [9,25]. We can see that the equation (1.2) is more general and more practical than the equation (1.3).

If we choose g(t) = 1 for t ∈ _Rand λ = 0, then the equation (1.1) becomes the classical Kirchhoff-type transmission problem

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

−α Z

Ω1

|∇u|²

∆u= f(u), in Ω1,

−β _Z

Ω2

|∇v|²

∆v= h(v), in Ω2,

v =0, on Γ,

u= v, on Σ,

α _Z

Ω1

|∇u|² ∂u

∂ν =β _Z

Ω2

|∇v|² ∂v

∂ν, on Σ.

(1.5)

It is well known that this problem is related to the stationary analogue of the problem

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 u_tt−α

_Z

Ω1

|∇u|²

∆u= f(u)_, x∈_Ω1,t>_0, vtt−β

Z

Ω2

|∇v|²

∆v =g(v), x∈_Ω2,t>0,

v=0, on Γ,

u=v, on Σ,

α _Z

Ω1

|∇u|² ∂u

∂ν = β _Z

Ω2

|∇v|² ∂v

∂ν, on Σ, u(₀) =u₀,u_t(₀) =u₁, x∈_Ω1, v(0) =v₀,v_t(0) =v₁, x∈_Ω2,

(1.6)

which models the transverse vibrations of a membrane composed of two different materials inΩ1andΩ2. According to [21], we call the problem (1.6) a transmission problem because the boundary conditions u = v and α R

Ω1|∇u|^2∂u

∂ν = β R

Ω2|∇v|^2∂v

∂ν on Σ. This transmission problem (1.6) arises in physics and biology phenomena, such as in the study of electromag- netic processes in ferromagnetic media with different dielectric constants [3], and in thinking about the population distribution of subjects living in an environment composed of different ecological media. In 2003, Ma and Muñoz Rivera [21] discussed the existence and nonexistence of positive solution to the Kirchhoff-type transmission problem (1.5) by using minimization arguments with f andghaving subcritical growth. In [16], Li, Zhang, Zhu, and Liang investi- gated the existence of the ground-state solutions to the following Kirchhoff-type transmission problem with critical perturbation

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

−α _Z

Ω1

|∇u|²

∆u= f(u) +λu⁵, inΩ1,

−β _Z

Ω2

|∇v|²

∆v= g(v) +λv⁵, inΩ2,

v=0, onΓ,

u=v, onΣ,

α Z

Ω1

|∇u|² ∂u

∂ν = β Z

Ω2

|∇v|² ∂v

∂ν, onΣ.

(1.7)

Here, we will establish the existence of ground-state solutions to Kirchhoff-type transmis- sion problem with more generalgand more general perturbation termsφandψ. To obtain the existence of ground-state solutions to the more general Kirchhoff-type transmission problem (1.1), we assume that four pairs of functions(α,g,f),(β,g,h),(α,g,φ), and(β,g,ψ)belong to the set A, where a pair of functions (α,g,f) is said to belongs to A, if (α,g,f) satisfies the following assumptions

(A₀) α∈C¹(_R+)is an increasing function andα(0)>0;

(A₁) there existsγ∈(0, 2)such that[α(s)−α(0)]/s^γ is decreasing on(0,∞); (G) g∈C¹(_R,R+)is even withg⁰(s)>^{0 fors} ∈_R+andg(0) =1;

(F₀) f ∈C¹(_R,R)and lims→0 f(s)/s=0;

(F₁) there existsl_f ∈_Rsuch that

|slim|→_∞

f(s)

g(s)G⁵(s) =l_f, where G(s) = Rs

0 g(t)dt for s ∈ _{R. And if} l_f = 0, we call that f has a quasicritical growth; if l_f 6=0, we call that f has a critical growth;

(F2) f(s)/(g(s)|G(s)|^2γG(s))is nondecreasing on (0,∞)and nonincreasing on (−_{∞, 0}), and lim_|s|→∞ F(s)/|G(s)|^2γ+2= _{∞, where}F(s) =Rs

0 f(t)dtfors∈_Randγis as in (A₁).

Remark 1.1. Assuming thatg satisfies (G) andγ∈(0, 2), let f(s) =g(s)|G(s)|^2γG(s)ln|G(s)|

andφ(s) =g(s)(G(s))⁵fors∈_{R. Then} f andφsatisfy (F₀), (F₁), and (F₂).

Example 1.2. Let α(s) = ₁+s² for s ∈ _{R+, and for} γ ∈ (_{0, 2})_{, define} g(s) = s²+_1, f(s) = (s²+1)s³/3+s

2γ s³/3+s ln

s³/3+s

, φ(s) = (s² +1) s³/3+s5

for s ∈ _R. Then (_α,g,f)_and(_α,g,φ)_{belong to}A_.

Example 1.3. For a,b > 0, let α(s) = a+bs for s ∈ _R+, and for γ ∈ (0, 2), define g(s) =

√

2s²+1, f(s) =

√2 4

p2s²+1

√ 2sp

2s²+1+ln√

2s+^p2s²+1

2γ√ 2sp

2s²+1 +ln√

2s+^p2s²+1

×ln

√ 2sp

2s²+1+ln√

2s+^p2s²+1 , φ(s) =

√2 4

√

2s²+1h√ 2s√

2s²+1+ln√ 2s+√

2s²+1i5

for s ∈ _{R. Then} (α,g,f) and (α,g,φ)belong toA.

Remark 1.4. We know that the critical exponent of equation (1.7) is 6 which has a significant influence on the properties of the solution. The critical exponent of equation (1.1) is different for differentg and the critical exponent depends on G⁶. This is an interesting phenomenon.

For example, when g(s) =√

2s²+1 fors ∈_R, the critical exponent is 12; when g(s) =s²+1 fors ∈R, the critical exponent is 18.

For any given subdomain D ofR³, the standard norm on L^p(D) is denoted by| · |_{p,D for} p ∈ [1,∞). Let H¹(_Ω1)and H¹(_Ω2)be the usual Sobolev spaces. Then H¹(_Ω1)×H¹(_Ω2)is also a Sobolev space with the norm

k(u,v)k= |∇u|²_2,Ω

1+|u|²_2,Ω

1+|∇v|²_2,Ω

2+|v|²_2,Ω

2

1/2

, (u,v)∈ H¹(_Ω1)×H¹(_Ω2). (1.8) Our analysis is based on the following Sobolev space

E={(u,v)∈ H¹(_Ω1)×H_Γ¹(_Ω2):u=von Σ}, where

H_Γ¹(_Ω2) ={v∈ H¹(_Ω2):v=0 on Γ}.

In [21] Ma and Muñoz Rivera established the following lemma which gave the definition of norm for the Sobolev spaceE.

Lemma 1.5([21, Lemma 1]). E is a closed subspace of H¹(_Ω1)×H¹(_Ω2), and k(u,v)k_E = |∇u|²_2,Ω

1+|∇v|²_2,Ω

2

1/2

, (u,v)∈E, defines also a norm on E, which is equivalent to the standard norm(1.8).

Remark 1.6. From Lemma1.5, we know that the space Eis embedded into L^p(_Ω1)×L^q(_Ω2) for all p,q ∈ [1, 6], and these embeddings are compact for all p,q ∈ [1, 6). In particular, for eachp= q∈[1, 6], there existsν_p>0 such that

|(u,v)|_p:=|u|^p_p,Ω

1+|v|^p_p,Ω

2

1/p

6^νpk(u,v)k_E, (u,v)∈E. (1.9) In order to solve the transmission problem (1.1), due to the appearance of nonlocal terms R

Ω1 g²(u)|∇u|²andR

Ω2 g²(v)|∇v|², the potential working space seems to be E₀=

(u,v)∈ E: Z

Ω1

g²(u)|∇u|²< _∞,

Z

Ω2

g²(v)|∇v|²< _∞

.

Obviously, E₀ may not be a linear space under the assumed condition of (G). To avoid this drawback, we gave a change of variables,

(u,v) =G⁻¹(u₁),G⁻¹(v₁), (u₁,v₁)∈ E,

which is motivated by [9,25]. According to the properties of g,G, and G⁻¹ which will be given in Section 2, if (u₁,v₁) ∈ E, then (u,v) = (G⁻¹(u₁),G⁻¹(v₁)) ∈ E (see Remark 2.3), R

Ω1g²(u)|∇u|² = R

Ω1g²(G⁻¹(u₁))|∇G⁻¹(u₁)|² = |∇u₁|²₂ < _{∞, and} R

Ω2 g²(v)|∇v|² = R

Ω2g²(G⁻¹(v₁))|∇G⁻¹(v₁)|²=|∇v₁|²₂ <∞. Thus, it follows from the change of variables that Ecan be used as the working space and the transmission problem (1.1) turns into

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−α Z

Ω1

|∇u₁|²

g(G⁻¹(u₁))_∆u₁ = f(G⁻¹(u₁)) +λφ(G⁻¹(u₁))_{, inΩ1,}

−β Z

Ω2

|∇v₁|²

g(G⁻¹(v₁))_∆v1 =h(G⁻¹(v₁)) +λψ(G⁻¹(v₁)), inΩ2,

v₁=0, onΓ,

u₁= v₁, onΣ,

α _Z

Ω1

|∇u₁|² ∂u₁

∂ν = β _Z

Ω2

|∇v₁|² ∂v₁

∂ν, onΣ.

(1.10)

Furthermore, we can prove that if(u₁,v₁)∈ E∩ H_loc² (_Ω1)×H²_loc(_Ω2)is a strong solution to the equation (1.10), then(u,v) = (G⁻¹(u₁),G⁻¹(v₁))∈ E∩ H_loc² (_Ω1)×H_loc² (_Ω2)is a strong solution to the equation (1.1). Here, we call that(u,v)∈ E∩ H_loc² (_Ω1)×H_loc² (_Ω2)is a strong solution to the transmission problem (1.10) or (1.1) if the first two equations in (1.10) or (1.1) hold in the sense of almost everywhere. Actually, we only need to verify that for any an open bounded setD ⊂_R³if u₁ ∈ H²(D)_{, then} G⁻¹(u₁)∈ H²(D)_{(see Lemma}4.2). Moreover, because of the continuity of g,G, and G⁻¹, to obtain a strong solution to the transmission problem (1.10), it suffices to seek for the weak solution to the following transmission problem

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−α _Z

Ω1

|∇u₁|²

∆u₁ = ^f(G⁻¹(u₁))

g(G⁻¹(u₁)) +λφ(G⁻¹(u₁))

g(G⁻¹(u₁))^{, inΩ1,}

−β _Z

Ω2

|∇v₁|²

∆v₁= ^h(G⁻¹(v₁))

g(G⁻¹(v₁))+λψ(G⁻¹(v₁))

g(G⁻¹(v₁))^{, inΩ2,}

v₁ =0, on Γ,

u₁ =v₁, on Σ,

α _Z

Ω1

|∇u₁|² ∂u₁

∂ν =β _Z

Ω2

|∇v₁|² ∂v₁

∂ν, on Σ.

(1.11)

In fact, if (u₁,v₁) ∈ E is a weak solution to the transmission problem (1.11), then it should satisfy, for all(w₁,z₁)∈ E,

α |∇u₁|²_2,Ω

1

Z

Ω1

∇u₁· ∇w₁+β |∇v₁|²_2,Ω

2

Z

Ω2

∇v₁· ∇z₁

=

Z

Ω1

f(G⁻¹(u₁)) g(G⁻¹(u₁))^w1+

Z

Ω2

h(G⁻¹(v₁)) g(G⁻¹(v₁))^z1+λ

Z

Ω1

φ(G⁻¹(u₁)) g(G⁻¹(u₁))^w1+λ

Z

Ω2

ψ(G⁻¹(v₁)) g(G⁻¹(v₁))^z1. Hence,u₁∈ H¹(_Ω1)weakly solves the equation

−α |∇u₁|²_2,Ω

1

∆u₁ =a(x)(₁+u₁)_{, inΩ1,}

with

a(x) = ¹ 1+u₁(x)

f(G⁻¹(u₁))

g(G⁻¹(u₁))+λφ(G⁻¹(u₁)) g(G⁻¹(u₁))

=: 1 1+u₁(x)

fe(u₁) +λeφ(u₁), where fe(s) := ^f(G−1(s))

g(G⁻¹(s)) and φe(s) := ^φ(G−1(s))

g(G⁻¹(s)) for s ∈ R. The condition (F2) implies that a∈ L^3/2_loc (_Ω1). By the Brézis–Kato theorem, see also [26, Lemma B.3, p. 244] , we know thatu₁∈ L^q_loc(_Ω1)for any q ∈ [1,∞). Theorem 8.8 in [11, p. 183] shows that u₁ ∈ H¹(_Ω1)∩H_loc² (_Ω1) and

−α |∇u₁|²_2,Ω

1

∆u₁ = ef(u₁) +λφe(u₁), a.e.x ∈_Ω1. Similarly, we can prove thatv₁ ∈ H_Γ¹(_Ω2)∩H_loc² (_Ω1)such that

−β |∇v₁|²_2,Ω2∆v1 =eh(v₁) +λψe(v₁), a.e.x∈ _Ω2, whereeh(s) = ^h(G−1(s))

g(G⁻¹(s)) andψe(s) = ^ψ(G−1(s))

g(G⁻¹(s)) fors∈R. So the problem (1.11) holds in the sense of almost everywhere and(u₁,v₁)∈ H¹(_Ω1)∩H²_loc(_Ω1)× H_Γ¹(_Ω2)∩H_loc² (_Ω2) is a strong solution to the equation. Here, let(u,v) = (G⁻¹(u₁)_,G⁻¹(v₁))_{. Then}(u,v)is a strong solution to the transmission problem (1.1). For the convenience, removing the subscripts ofu₁,v₁ , we rewrite (1.11) as the following transmission problem

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−α Z

Ω1

|∇u|²

∆u= ^f(G⁻¹(u))

g(G⁻¹(u))+λφ(G⁻¹(u))

g(G⁻¹(u))^{, inΩ1,}

−β Z

Ω2

|∇v|²

∆v= ^h(G⁻¹(v))

g(G⁻¹(v))+λψ(G⁻¹(v))

g(G⁻¹(v))^{, inΩ2,}

v=0, onΓ,

u=v, onΣ,

α _Z

Ω1

|∇u|² ∂u

∂ν =β _Z

Ω2

|∇v|² ∂v

∂ν, onΣ.

(1.12)

In the following, we make our efforts to find the weak solution to the transmission problem (1.12). To this end, we define the energy functionalI :E→_Rassociated with the transmission problem (1.12)

I_λ(u,v) = ¹

2A |∇u|²_2,Ω

1

+¹

2B |∇v|²_2,Ω

2

−

Z

Ω1

F(G⁻¹(u))−

Z

Ω2

H(G⁻¹(v))

−λ Z

Ω1

Φ(G⁻¹(u))−λ Z

Ω2

Ψ(G⁻¹(v)), (u,v)∈E, where A(s) =Rs

0 α(t)dt,B(s) =Rs

0 β(t)dt fors∈_R+, and H(s) =Rs

0 h(t)dt,Φ(s) =Rs

0 φ(t)dt, Ψ(s) =Rs

0 ψ(t)dtfors∈R. It can be verified that I_λis of classC¹. And for all(u,v),(w,z)∈ E, hI_λ⁰(u,v),(w,z)i= α(|∇u|²_2,Ω

1)

Z

Ω1

∇u· ∇w+β(|∇v|²_2,Ω

2)

Z

Ω2

∇v· ∇z−

Z

Ω1

f(G⁻¹(u)) g(G⁻¹(u))^w

−

Z

Ω2

h(G⁻¹(v)) g(G⁻¹(v))^z−λ

Z

Ω1

φ(G⁻¹(u)) g(G⁻¹(u))^w−λ

Z

Ω2

ψ(G⁻¹(v)) g(G⁻¹(v))^z.

Let Fe(s) = F(G⁻¹(s)),He(s) = H(G⁻¹(s)),Φe(s) = _Φ(G⁻¹(s)), and Ψe(s) = _Ψ(G⁻¹(s))for s∈_R. Then, for all (u,v),(w,z)∈E, we have that

I_λ(u,v) = ¹

2A |∇u|²_2,Ω

1

+ ¹

2B |∇v|²_2,Ω

2

−

Z

Ω1

Fe(u)−

Z

Ω2

He(v)−λ Z

Ω1

Φe(u)−λ Z

Ω2

Ψe(v)_,

and

hI_λ⁰(u,v),(w,z)i=α |∇u|²_2,Ω

1

Z

Ω1

∇u· ∇w+β |∇v|²_2,Ω

2

Z

Ω2

∇v· ∇z

−

Z

Ω1

ef(u)w−

Z

Ω2

eh(v)z−λ Z

Ω1

φe(u)w−λ Z

Ω2

ψe(v)z. (1.13) Then we say that (u,v) ∈ Eis a weak solution to the transmission problem (1.12) if and only if (u,v)is a critical point of the functional I_λ in E, i.e., I_λ⁰(u,v) =0. To sum up, it suffices to seek a critical point of the functional I_λ in E to achieve a strong solution to the transmission problem (1.1).

Now, we state our main results through the following theorems.

Theorem 1.7. Assume that(α,g,f),(β,g,h) ∈ A with l_f = l_h = 0, (α,g,φ),(β,g,ψ) ∈ A with l_φ,l_ψ 6=0, andφ(s)s > 0,ψ(s)s >0for s 6= 0. Then there existsλ₀ > 0such that both the problem (1.12) and(1.1) have a ground-state solution (u_λ,v_λ)for allλ ∈ [0,λ₀). Furthermore, it holds that (u_λ,v_λ)→(u₀,v₀)in E asλ→0, where(u₀,v₀)is a ground-state solution to the problem(1.1)with λ=0.

Corollary 1.8. LetΩ2= _∅,α(s) =1,g(s) =√

1+2s², f(s) =|s|^q−2, andφ(s) =|s|¹⁰s for s∈_R.

Then the following equation has a ground-state solution u_λ for allλ∈[0,λ₀), (−_∆u−_∆(u²)u=|u|^q−2u+λ|u|¹⁰u, inΩ,

u=0, on∂Ω, (1.14)

where q∈(4, 12). Furthermore, it holds that u_λ →u₀in H₀¹(_Ω)asλ→0, where u₀is a ground-state solution to the above problem with λ=0.

Remark 1.9. According to [8], for a single quasilinear Schrödinger equation (1.14) in a bounded domain inR³, there exists a suitable energy levelc^∗such that ifc(λ)<c^∗, then the associated energy functional satisfies the (PS)_c(λ) condition, where c^∗ = S³/6 and S is the best Sobolev constant forD^1,2(_R³),→ L⁶(_R³). However, a large amount of calculations is required to prove that c(λ)<c^∗ by verifying

sup

t∈_R+

I_λ(tu_e)<c^∗,

where u_e is a modification of U and U attains the best Sobolev constant S. In this paper to avoid this difficulty, we adopt the perturbation method from [12,32].

Remark 1.10. Let g(s) = 1 andφ(s) = ψ(s) = s⁵ fors ∈ R. Then by Theorem1.7, we have that the transmission problem (1.1) also has a ground-state solution, which has been achieved in [16]. Thus, Theorem1.7could be regarded as a generalization of Theorem 1.1 in [16].

This paper is organized as follows. We give some preliminaries in Section 2. Theorem1.7is proved in Section 3. Throughout this paper we denote C_i fori∈ _N:={1, 2, . . .}as constants which can be different from line to line.

2 Preliminaries

In this section we first give some properties of the functions α,g,ef, and A,G,G⁻¹,Fevia the following lemmas.

Lemma 2.1.

(i) Assume thatαsatisfies the condition (A0). Then A(s)>^α(0)s for s ∈_R+.

(ii) Assume that α satisfies the conditions (A₀) and (A₁). Then [A(s)−α(0)s]/s^γ+1, α(s)s− (γ+1)A(s) +γα(0)s, and A(s)/s^γ+1are decreasing on(0,∞). Furthermore, we have that

(γ+1)A(s)−α(s)s>^γα(0)s, s ∈_R+, (2.1) and

α⁰(s)s6^γ[α(s)−α(0)]< γα(s), s∈_R+. (2.2) Lemma 2.2. The functions g,G, and G⁻¹have the following properties under the assumption of (G):

(i) G and G⁻¹ are both odd, and

t6^G(t)6^g(t)t, t ∈_R+, s/g(G⁻¹(s))6^G−1(s)6^{s, s}∈_R+; (ii) lim_s→0G⁻¹(s)/s =1andlim_s→_∞G⁻¹(s)/s=1/g(_∞), where g(_∞) =lim_s→_∞g(s); (iii) G⁻¹(s)/

|s|^2γsg(G⁻¹(s))is nonincreasing on(0,∞)and nondecreasing on(−_{∞, 0}); (iv) [G⁻¹(s)]²−G⁻¹(s)s/g(G⁻¹(s))is nondecreasing on(_0,∞)and nonincreasing on(−_{∞, 0});

(v) if f is a continuous function and (F₂) holds, then f(G⁻¹(s))s/[(2γ+2)g(G⁻¹(s))]−F(G⁻¹(s)) is increasing on(_0,∞)and decreasing on(−_{∞, 0}).

Proof. (i), (ii), and (iv) can be derived from [17, (1), (2), and (4) of Lemma 2.2]. As for (iii), because g is even, we need only to prove that the conclusion holds on (0,∞). In fact, since [G(t)/t]^2γ+1g(t)is nondecreasing on(0,∞), [G(t)]^2γ+1g(t)/tis also nondecreasing on (0,∞), and thenG⁻¹(s)/[s^2γ+1g(G⁻¹(s))]is nonincreasing on(0,∞).

Finally, we prove that (v) holds. Indeed, since f(t)/[g(t)|G(t)|^2γG(t)]is nondecreasing on (0,∞)and nonincreasing on(−_{∞, 0}), according to [17, Lemma A.1], f(t)G(t)/[(2γ+2)g(t)]− F(t)is nondecreasing on (0,∞) and nonincreasing on(−_{∞, 0}), and then f(G⁻¹(s))s/[(2γ+ 2)g(G⁻¹(s))]−F(G⁻¹(s)) is nondecreasing on (0,∞) and nonincreasing on (−_∞, 0), that is, (v) holds. The proof is complete.

Remark 2.3. Let (u,v)∈ E. Then it follows from g(t) > ^{1 fort} ∈ _R+, and (i) of Lemma2.2 that(G⁻¹(u),G⁻¹(v))∈ E.

Lemma 2.4. Assume that g satisfies (G) and f satisfies (F0), (F₁), and (F2). Let ef(s) = _g^f(₍^G_G⁻₋¹₁⁽₍^s_s⁾⁾₎₎ for s∈_{R. Then} ef has the following properties:

(F₀⁰) ef ∈C¹(_R)andlims→0 ef(s)/s=0;

(F₁⁰)

|slim|→_∞

ef(s) s⁵ = l_f;

(F₂⁰) ef(s)_/(|s|^2γs)is nondecreasing on(_0,∞)and nonincreasing on(−_{∞, 0}), and

|slim|→_∞Fe(s)/|s|^2γ+2 =_∞.

From [16], the function ef possesses some other properties as mentioned in the following Remark2.5. With those properties, we know that Lemmas2.6–2.8hold.

Remark 2.5. It follows from (F⁰₂) and [17, Lemma A.1] that ef(s)s−2(γ+1)Fe(s)is nondecreas- ing onR+and nonincreasing(−_{∞, 0}]_{, and then}

ef(s)s−2(γ+1)Fe(s)>^{0, s}∈_R, (2.3) and

ef⁰(s)s−(2γ+1)ef(s)>^{0, s}∈_R+. (2.4) Lemma 2.6. Suppose that f satisfies the conditions (F0)and (F₁) and g satisfies the conditions (G).

Then for each u∈ H¹(_Ω), one has that limt→0

Z

Ω1

fe(tu)u t =0.

Lemma 2.7. Suppose that f satisfies the conditions (F₀)and (F₁)and g satisfies the conditions (G). If u_n*u6=₀in H¹(_Ω)and|t_n| →_∞, then

nlim→_∞ Z

Ω

ef(t_nu_n)u_n

|t_n|^2γt_n =_∞.

Lemma 2.8. Suppose that f satisfies the conditions (F0)and (F₁) and g satisfies the conditions (G).

Then for each u∈ H¹(_Ω)and u6=0, it holds that

|tlim|→_∞ Z

Ω1

ef(tu)u

|t|^2γt =_∞.

3 Existence and convergence of ground-state solutions

In this section, assuming that the all conditions of Theorem 1.7 hold, we will establish the existence of ground-state solutions to the problems (1.12) and complete the proof of Theo- rem 1.7. First, we verify that the functional I_λ has a mountain pass geometric structure and the functional I₀ satisfies the Palais–Smale (PS for short) condition.

For eachλ∈_R+, let

Γλ ={γ∈C([0, 1],E):γ(0) =0,I_λ(γ(1))<0} and define

c(λ) = inf

γ∈_Γmax

t∈[0,1]I_λ(γ(t)). Lemma 3.1. Γλ 6= _∅and c(λ)>0forλ∈ _R+.

Proof. For any given ε ∈ 0,

2(1+λ)ν₂²−1

min{α(0),β(0)} and p ∈ (2γ+2, 6], we obtain from (F⁰₀)_{and (F}⁰₁)that there existsC_ε,p,C_ε >0 such that

|ef(s)|,|eh(s)|6^ε|s|+|s|⁵+Cε,p|s|^p−1, s ∈_R, (3.1)

|Fe(s)|_,|He(s)|6^{ε s2}+s⁶

+C_ε,p|s|^p_, s∈_R,

|φe(s)|,|ψe(s)|6^ε|s|+C_ε|s|⁵, s∈_R, (3.2)

|Φe(s)|,|Ψe(s)|6^εs2+C_εs⁶, s∈_R,

where Fe(s) = Rs

0 ef(t)dt,He(s) = Rs

0eh(t)dt,Φe(s) = Rs

0 φe(t)dt, and Ψe(s) = Rs

0 ψe(t)dt fors ∈ _R.

Then it follows from the Sobolev inequality (1.11) that for(u,v)∈ E,

Z

Ω1

Fe(u) +

Z

Ω2

He(v)

6 ^εν22k(u,v)k²_E+εν₆⁶k(u,v)k⁶_E+ν_p^pC_ε,pk(u,v)k^p_E,

and

Z

Ω1

Φe(u) +

Z

Ω2

Ψe(v)

6^εν2₂k(u,v)k²_E+ν₆⁶C_εk(u,v)k⁶_E. Thus, combining this and (i) of Lemma2.1, we have that for(u,v)∈E,

I_λ(u,v)

= ¹

2A |∇u|²_2,Ω

1

+¹

2B |∇v|²_2,Ω

2

−

Z

Ω1

Fe(u)−

Z

Ω2

He(v)−λ _Z

Ω1

Φe(u) +

Z

Ω2

Ψe(v)

> ¹ 2

α(0)|∇u|²_2,Ω

1 +β(0)|∇v|²_2,Ω

2

−(1+λ)εν₂²k(u,v)k²_E

−ν_p^pCε,pk(u,v)k_E^p−(ε+λCε)ν₆⁶k(u,v)k⁶_E

>

1

2min{α(0),β(0)} −(1+λ)εν₂²

k(u,v)k²_E−ν^p_pCε,pk(u,v)k^p_E−(ε+λCε)ν₆⁶k(u,v)k⁶_E. Hence, lettingρ>0 small enough, it is easy to see that inf{I_λ(u,v):k(u,v)k_E =ρ}>0.

Next, for each(u,v)∈ E\ {0}, according to (ii) of Lemma2.1, the following limits exist

a_∞ := lim

t→_∞

A

t²|∇u|²_2,Ω

1

2t^2γ+2 ∈_R+, b_∞ := lim

t→_∞

B

t²|∇v|²_2,Ω

2

2t^2γ+2 ∈_R+. For any given M > (a_∞+b_∞)^h(1+λ)|u|^2γ_2γ⁺₊²_2,Ω

1+|v|^2γ_2γ⁺₊²_2,Ω

2

i−1

, it follows from (F⁰₂) and (F⁰₀) that there existsC>0 such that

Fe(s),He(s),Φe(s),Ψe(s)> ^M|s|^2γ+2−C, s∈ _R.

Thus, we have that I_λ(t(u,v))6 ¹

2A t²|∇u|²_2,Ω

1

+ ¹

2B t²|∇v|²_2,Ω

2

−M(1+λ)t^2γ+2h

|u|^2γ_2γ⁺₊²_2,Ω

1+|v|^2γ_2γ⁺₊²_2,Ω

2

i

+C(1+λ)[|_Ω1|+|_Ω2|]

= t^2γ+2



 A

t²|∇u|²_2,Ω

1

2t^2γ+2 +

B

t²|∇v|²_2,Ω

2

2t^2γ+2 −M(1+λ)^h|u|^2γ_2γ⁺₊²_2,Ω

1 +|v|^2γ_2γ⁺₊²_2,Ω

2

i

+ ^C(1+λ)

t^2γ+2 [|_Ω1|+|_Ω2|]





→ −_∞, t →_∞.

The proof is complete.

Lemma 3.2. For eachλ∈_R+, any PS sequence of the functional I_λ is always bounded. Particularly, forλ=0, the functional I₀satisfies the PS condition.

Proof. As for the boundedness of PS sequence, one only needs to observe that (2.1) and (2.3) imply the AR condition. Here for the completeness, we sketch out the proof. Assume that λ∈_R+,c∈ _{R, and}{(u_n,v_n)}is a (PS)_c sequence of I_λ. Then according to (2.1) and (2.3), for sufficiently large nwe have that

c+1+k(u_n,v_n)k_E > ^Iλ(u_n,v_n)− ¹

2γ+2hI_λ⁰(u_n,v_n),(u_n,v_n)i

= ¹

2A(|∇u_n|²_2,Ω

1)− ¹

2γ+2α(|∇u_n|²_2,Ω

1)|∇u_n|²_2,Ω

1

+ ¹

2B(|∇vn|²_2,Ω

2)− ¹

2γ+2β(|∇vn|²_2,Ω

2)|∇vn|²_2,Ω

2

+

Z

Ω1

1

2γ+2ef(u_n)u_n−Fe(u_n)

+

Z

Ω2

1

2γ+2eh(v_n)v_n−He(v_n)

+λ Z

Ω1

1

2γ+2φe(un)un−Φe(un)

+λ Z

Ω2

1

2γ+2ψe(vn)vn−Ψe(vn)

> ^γα(0)

2γ+2|∇u_n|²_2,Ω

1 + ^γβ(0)

2γ+2|∇v_n|²_2,Ω

2

> ^γ

2γ+2min{α(0),β(0)}k(u_n,v_n)k²_E. (3.3) It follows that{(u_n,v_n)}is bounded inE.

Now, we illustrate that the functionalI0satisfies the PS condition. In fact, let{(un,vn)}be a PS sequence ofI₀. First, from the above conclusion we can get the boundedness of{(u_n,v_n)}

in E. Without loss of generality, there exists (u,v) ∈ Esuch that(u_n,v_n)*(u,v)asn →_∞. Owing to (3.1) and the compact embedding E ,→ L^p(_Ω1)×L^p(_Ω2) for p ∈ [1, 6), we can derive that

nlim→_∞ Z

Ω1

ef(u_n)(u_n−u) =0, lim

n→_∞ Z

Ω2

eh(v_n)(v_n−v) =0. (3.4) Thus, similarly to Lemma 3.2 in [16], we can prove thatk(u_n−u,v_n−v)k²_E →0. The proof is complete.

It follows from the mountain pass theorem that the following corollary holds.

Corollary 3.3.

K_c(0) :={(u,v)∈E: I₀⁰(u,v) =0,I₀(u,v) =c(0)} 6=_∅. (3.5) Define

N_λ =(u,v)∈E\ {0}:hI_λ⁰(u,v),(u,v)i=0 , d(λ) =inf

N_λ I_λ. (3.6) We now prove that N_λ 6= _∅and provide some properties of the mappingd(·).

Lemma 3.4. Let(u,v)∈E\{0}.

(i) For eachλ∈_R+, there exists a unique t(λ)>0such that t(λ)(u,v)∈ N_λ,hI_λ⁰(t(u,v)),t(u,v)i

> 0 for t ∈ (0,t(λ)),hI_λ⁰(t(u,v)),t(u,v)i < 0 for t ∈ (t(λ),∞), and I_λ(t(λ)(u,v)) = max_t∈_R+ I_λ(t(u,v)).

(ii) The function t(·):R+→(0,∞)is continuously differentiable and t⁰(λ) =

R

Ω1φe(t(λ)u)t(λ)u+R

Ω2ψe(t(λ)v)t(λ)v

W₁(t(λ)_,(u,v)) ^{, (3.7)}