Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 17, pp. 1–30.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2025.17

EIGENVALUE PROBLEMS FOR KIRCHHOFF-TYPE

EQUATIONS IN VARIABLE EXPONENT SOBOLEV SPACES

JUNICHI ARAMAKI

Abstract. In this article, we consider an eigenvalue problem for the Kirchhoff-

type equation containing p(·)-Laplacian and the mean curvature operator with
mixed boundary conditions. More precisely, we are concerned with the problem

with the Dirichlet condition on a part of the boundary and the Steklov bound-

ary condition on an another part of the boundary. We show that the eigenvalue
problem has infinitely many eigenpairs by using the celebrated Ljusternik-

Schnirelmann principle in the calculus of variation. Moreover, we derive that
in a variable exponent Sobolev space, there are two cases where the infimum

of all eigenvalues is equal to zero and is positive.

1. Introduction

In this article, we consider the following eigenvalue problem with mixed boundary
conditions

−M
(∫

Ω

A(x,∇u(x)) dx
)
div[a(x,∇u(x))] = 0 in Ω,

u(x) = 0 on Γ1,

M
(∫

Ω

A(x,∇u(x)) dx
)
n(x) · a(x,∇u(x)) = λg(x, u(x)) on Γ2.

(1.1)

Here Ω is a bounded domain of RN (N ≥ 2) with a Lipschitz-continuous (C0,1 for
short) boundary Γ satisfying that

Γ1 and Γ2 are disjoint non-empty open subsets of Γ such that Γ1 ∪ Γ2 = Γ, (1.2)

and the vector field n denotes the unit, outer, normal vector to Γ. Furthermore,
a(x, ξ) is a Carathéodory function on Ω × RN satisfying some structure condi-
tions associated with an anisotropic exponent function p(x) and A(x, ξ) is a func-
tion satisfying ∇ξA(x, ξ) = a(x, ξ). Here we say that a(x, ξ) is a Carathéodory
function on Ω × RN , if for a.e. x ∈ Ω, the map RN ∋ ξ 7→ a(x, ξ) is con-
tinuous and for every ξ ∈ RN , the map Ω ∋ x 7→ a(x, ξ) is measurable on
Ω. The operator u 7→ div[a(x,∇u(x))] is more general than the p(·)-Laplacian
∆p(x)u(x) := div[|∇u(x)|p(x)−2∇u(x)] and the mean curvature operator div[(1 +

|∇u(x)|2)(p(x)−2)/2∇u(x)]. This generality brings about difficulties and requires

2020 Mathematics Subject Classification. 49R50, 35A01, 35J62, 35J57.
Key words and phrases. Eigenvalue problem; Kirchhoff-type operator; p(·)-Laplacian;
mean curvature operator; mixed boundary value problem; variable exponent Sobolev space.
©2025. This work is licensed under a CC BY 4.0 license.

Submitted April 21, 2024. Published February 25, 2025.
1



2 J. ARAMAKI EJDE-2025/17

some conditions. The function M = M(s) defined in [0,∞) satisfies the following
condition

(A1) M : [0,∞) → [0,∞) is continuous and monotone non-decreasing, and there
exist 0 < m0 ≤ m1 < ∞ and k ≥ l ≥ 1 such that

m0s
l−1 ≤ M(s) ≤ m1(1 + sk−1) for s ≥ 0. (1.3)

We impose the mixed boundary conditions, that is, the Dirichlet condition on Γ1

and the Steklov condition on Γ2. The given data g : Γ2×R → R is a Carathéodory
function of special type and λ is a real number.

The study of differential equations with p(·)-growth conditions is a very interest-
ing topic recently. Studying such problem stimulated its application in mathemat-
ical physics, in particular, in elastic mechanics (Zhikov [36]), in electrorheological
fluids (Diening [12], Halsey [21], Mihăilescu and Rădulescu [28], Růz̆ic̆ka [31]). As
recent works, we can find some interesting related articles. See Alves et al. [2],
Alves and Tavares [3].

However, in even the case M ≡ 1, as we only find a few papers associate with the
problem with the mixed boundary condition in variable exponent Sobolev space as
in (1.1) (for example, Aramaki [5, 6]), we are convinced of the reason for existence
of this article.

When p(x) ≡ p (a constant), there are many articles for the p-Laplacian. For
example, see Lê [24], Anane [4], Friedlander [20]. For the p-Laplacian Dirichlet
eigenvalue problem:

−∆pu(x) = λ|u(x)|p−2u(x) in Ω,

u(x) = 0 on Γ,

we can see that the following properties hold.

(1) There exists a nondecreasing sequence of positive eigenvalues {λn} tending
to ∞ as n → ∞.

(2) The first eigenvalue λ1 is simple and only eigenfunctions associated with
λ1 do not change sign.

(3) The set of eigenvalues is closed.
(4) The first eigenvalue λ1 is isolated.

On the contrary, recently many authors study the p(·)-Laplacian. In particular,
Fan [15] has studied the eigenvalue problem for the p(·)-Laplacian with zero Neu-
mann boundary condition in a bounded domain, and Fan et al. [19] has studied
the eigenvalue problem for the p(·)-Laplacian Dirichlet problem. Mihăilescu and
Rădulescu [29] have studied nonhomogeneous quasilinear eigenvalue problem with
variable exponent. In Deng [11], the author treats only the p(·)-Laplacian in the
case Γ1 = ∅, that is,

−∆p(x)u(x) + |u(x)|p(x)−2u(x) = 0 in Ω,

|∇u(x)|p(x)−2 ∂u(x)

∂n
= λ|u(x)|p(x)−2u(x) on Γ.

(1.4)

As the author takes the variable exponent Sobolev space W 1,p(·)(Ω) as the base
space, the second term in the left-hand side of the first equation of (1.4) takes
the essential role. However, if we assume that Γ1 ̸= ∅, we can delete such a term
according to the Poincaré type inequality due to Ciarlet and Dinca [10].



EJDE-2025/17 EIGENVALUE PROBLEMS FOR KIRCHHOFF-TYPE EQUATIONS 3

For physical motivation to the problem (1.1), we consider the case where Γ = Γ1

and p(x) = 2. Then the equation

M(∥∇u∥2L2(Ω))∆u(x) = f(x, u(x)) (1.5)

is the Kirchhoff equation which arises in nonlinear vibration, namely

utt −M(∥∇u∥2L2(Ω))∆u = f(x, u) in Ω× (0, T ),

u = 0 on Γ× (0, T ),

u(x, 0) = u0(x), ut(x, 0) = u1(x) in Ω.

(1.6)

Equation (1.5) is the stationary counterpart of (1.6). Such a hyperbolic equation
is a general version of the Kirchhoff equation

ρutt −
(ρ0
h

+
E

2L

∫ L

0

∣∣∂u
∂x

∣∣2dx)∂2u

∂x2
= 0

presented by Kirchhoff [22]. This equation extends the classical d’Alembert wave
equation by considering the effect of the changes in the length of the strings during
the vibrations, where L, h,E, ρ and ρ0 are constants. In Afrouzu and Mirzapour
[1], the authors studied the p(·)-Kirchhoff type eigenvalue problem

−M
(∫

Ω

1

p(x)
|∇u(x)|p(x) dx

)
∆p(x)u(x) = λ|u(x)|q(x)−2u(x) in Ω,

u(x) = 0 on Γ.

(1.7)

They derived the existence of a nontrivial weak solution under some conditions on
the functions M,p(·), q(·) and a real number λ. Mendéz [27] considered the problem

−M
(∫

Ω

|∇u(x)|p(x) dx
)
div[p(x)|∇u(x)|p(x)−2∇u(x)]

= λp(x)|u(x)|p(x)−2u(x) in Ω,

u(x) = 0 on Γ.

(1.8)

The author showed that for any r > 0, there exists a eigenpair (u, λ) ∈ W
1,p(·)
0 (Ω)×

R of (1.8) satisfying M̂
( ∫

Ω
|∇u(x)|p(x) dx

)
= r, where M̂(t) =

∫ t

0
M(s) ds.

In this article, we extend these results to a class of operators containing p(·)-
Laplacian and the mean curvature operator. The purpose of this article is to solve
eigenvalue problem (1.1). According to some assumptions on the given function g,
we use the Ljusternik-Schnirelmann principle in the constrained variational method.
See Ljusternik and Schnirelmann [25] and Szulkin [32]. We will deal with the mixed
boundary value eigenvalue problem (1.1) for a class of operators involving the p(·)-
Laplacian and the mean curvature operator which seems to be a new topic. We
will show that there exist infinitely many eigenvalues {λ(n,α)} tending to ∞ as
n → ∞ for any fixed α > 0. Moreover, we will derive that under some condition,
the infimum λ∗ of the set of all eigenvalues of (1.1) is equal to zero, so there does
not exist a principal eigenvalue and the set of eigenvalues is not closed. We also
show that under some condition on the function g and variable exponent function
p in (1.1), there is a case where λ∗ is positive.

This article is organized as follows. In Section 2, we recall some results on
variable exponent Lebesgue-Sobolev spaces. In Section 3, we give the setting of
problem (1.1) rigorously and a main theorem (Theorems 3.20) on the eigenvalue
problem (1.1) in which we show the existence of infinitely many eigenpairs of (1.1).



4 J. ARAMAKI EJDE-2025/17

In Section 4, we present some sufficient conditions for the cases λ∗ = 0 and λ∗ > 0,
respectively.

2. Preliminaries

Throughout this article, Ω is a bounded domain in RN (N ≥ 2) with a C0,1-
boundary Γ and Ω is locally on the same side of Γ. Moreover, we assume that Γ
satisfies (1.2).

We only consider real vector spaces of real valued functions over R. For any space
B, we denote BN by the boldface character B. Hereafter, we use this character
to denote vectors and vector-valued functions, and we denote the standard inner

product of vectors a = (a1, . . . , aN ) and b = (b1, . . . , bN ) in RN by a·b =
∑N

i=1 aibi
and |a| = (a · a)1/2. Furthermore, we denote the dual space of B by B∗ and the
duality bracket by ⟨·, ·⟩B∗,B .

We recall some well-known results on variable exponent Lebesgue and Sobolev
spaces. See Fan and Zhang [17], Kovác̆ik and Rácosńık [23], Diening et al. [13] and
references therein for more details. We consider some new properties on variable
exponent Lebesgue space. We define C(Ω) = {p is a continuous function on Ω},
and for any p ∈ C(Ω), put

p+ = p+(Ω) = sup
x∈Ω

p(x) = max
x∈Ω

p(x), p− = p−(Ω) = inf
x∈Ω

p(x) = min
x∈Ω

p(x).

For any p ∈ C(Ω) with p− ≥ 1 and for any measurable function u on Ω, a modular
(for this notation, see [13, Definition 2.1.1]) ρp(·) = ρp(·),Ω is defined by

ρp(·)(u) =

∫
Ω

|u(x)|p(x) dx.

The variable exponent Lebesgue space is defined by

Lp(·)(Ω) = {u;u : Ω → R is a measurable function satisfying ρp(·)(u) < ∞}
equipped with the (Luxemburg) norm

∥u∥Lp(·)(Ω) = inf
{
τ > 0; ρp(·)

(u
τ

)
≤ 1

}
.

Then Lp(·)(Ω) is a Banach space. We also define the Sobolev space

W 1,p(·)(Ω) = {u ∈ Lp(·)(Ω); |∇u| ∈ Lp(·)(Ω)},
where ∇ is a gradient operator, that is, ∇u = (∂1u, . . . , ∂Nu), ∂i = ∂/∂xi, endowed
with the norm

∥u∥W 1,p(·)(Ω) = ∥u∥Lp(·)(Ω) + ∥∇u∥Lp(·)(Ω),

and ∥∇u∥Lp(·)(Ω) = ∥|∇u|∥Lp(·)(Ω).

The following three propositions are well known (see [19], Fan and Zhao [18],
Zhao et al. [35]).

Proposition 2.1. Let p ∈ C(Ω) with p− ≥ 1, and let u, un ∈ Lp(·)(Ω) (n =
1, 2, . . .). Then we have the following properties.

(i) ∥u∥Lp(·)(Ω) < 1(= 1, > 1) ⇔ ρp(·)(u) < 1(= 1, > 1).

(ii) ∥u∥Lp(·)(Ω) > 1 ⇒ ∥u∥p
−

Lp(·)(Ω)
≤ ρp(·)(u) ≤ ∥u∥p

+

Lp(·)(Ω)
.

(iii) ∥u∥Lp(·)(Ω) < 1 ⇒ ∥u∥p
+

Lp(·)(Ω)
≤ ρp(·)(u) ≤ ∥u∥p

−

Lp(·)(Ω)
.

(iv) limn→∞ ∥un − u∥Lp(·)(Ω) = 0 ⇔ limn→∞ ρp(·)(un − u) = 0.



EJDE-2025/17 EIGENVALUE PROBLEMS FOR KIRCHHOFF-TYPE EQUATIONS 5

(v) ∥un∥Lp(·)(Ω) → ∞ as n → ∞ ⇔ ρp(·)(un) → ∞ as n → ∞.

The following proposition is a generalized Hölder inequality.

Proposition 2.2. Let p ∈ C+(Ω), where C+(Ω) := {p ∈ C(Ω); p− > 1}. For any

u ∈ Lp(·)(Ω) and v ∈ Lp′(·)(Ω), we have∫
Ω

|u(x)v(x)|dx ≤
( 1

p−
+

1

(p′)−

)
∥u∥Lp(·)(Ω)∥v∥Lp′(·)(Ω) ≤ 2∥u∥Lp(·)(Ω)∥v∥Lp′(·)(Ω).

Here and from now on, for any p ∈ C+(Ω), p
′(·) denotes the conjugate exponent of

p(·), that is, p′(x) = p(x)/(p(x)− 1) for x ∈ Ω.

For p ∈ C+(Ω), we define, for x ∈ Ω,

p∗(x) =

{
Np(x)
N−p(x) if p(x) < N,

∞ if p(x) ≥ N.

Proposition 2.3. Let Ω be a bounded domain of RN with C0,1-boundary and let
p ∈ C+(Ω). Then we have the following properties.

(i) The spaces Lp(·)(Ω) and W 1,p(·)(Ω) are separable, reflexive and uniformly
convex Banach spaces.

(ii) If q(·) ∈ C(Ω) with q− ≥ 1 satisfies q(x) ≤ p(x) for all x ∈ Ω, then
W 1,p(·)(Ω) ↪→ W 1,q(·)(Ω), where ↪→ means that the embedding map is con-
tinuous.

(iii) If q(x) ∈ C(Ω) with q− ≥ 1 satisfies that q(x) < p∗(x) for all x ∈ Ω, then
the embedding map W 1,p(·)(Ω) ↪→ Lq(·)(Ω) is compact.

Next we consider the trace (cf. Fan [16]). Let Ω be a bounded domain of RN

with a C0,1-boundary Γ and p ∈ C(Ω) with p− ≥ 1. Since W 1,p(·)(Ω) ⊂ W 1,1(Ω),
the trace u

∣∣
Γ
to Γ of any function u in W 1,p(·)(Ω) is well defined as a function in

L1(Γ). We define

Tr(W 1,p(·)(Ω)) = (TrW 1,p(·))(Γ)

= {f ; f is the trace to Γ of a function F ∈ W 1,p(·)(Ω)}
equipped with the norm

∥f∥(TrW 1,p(·))(Γ) = inf{∥F∥W 1,p(·)(Ω);F ∈ W 1,p(·)(Ω) satisfying F
∣∣
Γ
= f}

for f ∈ (TrW 1,p(·))(Γ), where the infimum can be achieved. Then we can see that
(TrW 1,p(·))(Γ) is a Banach space. In the later, we also write F

∣∣
Γ
= g by F = g on

Γ. Moreover, for i = 1, 2, we denote

(TrW 1,p(·))(Γi) = {f
∣∣
Γi
; f ∈ (TrW 1,p(·))(Γ)}

equipped with the norm

∥g∥(TrW 1,p(·))(Γi) = inf{∥f∥(TrW 1,p(·))(Γ); f ∈ (TrW 1,p(·))(Γ) satisfying f
∣∣
Γi
= g},

where the infimum can also be achieved, so for any g ∈ (TrW 1,p(·))(Γi), there exists
F ∈ W 1,p(·)(Ω) such that F

∣∣
Γi
= g and ∥F∥W 1,p(·)(Ω) = ∥g∥(TrW 1,p(·))(Γi).

Let q ∈ C+(Γ) := {q ∈ C(Γ); q− > 1} and denote the surface measure on Γ
induced from the Lebesgue measure dx on Ω by dσx. We define

Lq(·)(Γ) =
{
u : Γ → R is a measurable function with respect to dσx



6 J. ARAMAKI EJDE-2025/17

satisfying

∫
Γ

|u(x)|q(x) dσx < ∞
}

and the norm is defined by

∥u∥Lq(·)(Γ) = inf
{
τ > 0;

∫
Γ

∣∣u(x)
τ

∣∣q(x) dσx ≤ 1
}
,

and we also define a modular on Lq(·)(Γ) by

ρq(·),Γ(u) =

∫
Γ

|u(x)|q(x) dσx.

Similarly as Proposition 2.1, we have the following proposition.

Proposition 2.4. Let q ∈ C(Γ) with q− ≥ 1, and let u, un ∈ Lq(·)(Γ). Then we
have the following properties.

(i) ∥u∥Lq(·)(Γ) < 1(= 1, > 1) ⇔ ρq(·),Γ(u) < 1(= 1, > 1).

(ii) ∥u∥Lq(·)(Γ) > 1 ⇒ ∥u∥q
−

Lq(·)(Γ)
≤ ρq(·),Γ(u) ≤ ∥u∥q

+

Lq(·)(Γ)
.

(iii) ∥u∥Lq(·)(Γ) < 1 ⇒ ∥u∥q
+

Lq(·)(Γ)
≤ ρq(·),Γ(u) ≤ ∥u∥q

−

Lq(·)(Γ)
.

(iv) ∥un∥Lq(·)(Γ) → 0 ⇔ ρq(·),Γ(un) → 0.

(v) ∥un∥Lq(·)(Γ) → ∞ ⇔ ρq(·),Γ(un) → ∞.

The Hölder inequality also holds for functions on Γ.

Proposition 2.5. Let q ∈ C(Γ) with q− > 1. Then the following inequality holds.∫
Γ

|f(x)g(x)|dσx ≤ 2∥f∥Lq(·)(Γ)∥g∥Lq′(·)(Γ) for all f ∈ Lq(·)(Γ), g ∈ Lq′(·)(Γ).

Proposition 2.6. Let Ω be a bounded domain of RN with a C0,1-boundary Γ and
let p ∈ C+(Ω). If f ∈ (TrW 1,p(·))(Γ), then f ∈ Lp(·)(Γ) and there exists a constant
C > 0 such that

∥f∥Lp(·)(Γ) ≤ C∥f∥(TrW 1,p(·))(Γ).

In particular, If f ∈ (TrW 1,p(·))(Γ), then f ∈ Lp(·)(Γi) and

∥f∥Lp(·)(Γi) ≤ C∥f∥(TrW 1,p(·))(Γ)

for i = 1, 2.

For p ∈ C+(Ω), we define, for x ∈ Ω,

p∂(x) =

{
(N−1)p(x)
N−p(x) if p(x) < N,

∞ if p(x) ≥ N.

The next proposition follows from Yao [33, Proposition 2.6].

Proposition 2.7. Let p ∈ C+(Ω). Then if q ∈ C+(Γ) satisfies q(x) < p∂(x) for all
x ∈ Γ, then the trace mapping W 1,p(·)(Ω) → Lq(·)(Γ) is well-defined and compact.
In particular, the trace mapping W 1,p(·)(Ω) → Lp(·)(Γ) is compact and there exists
a constant C > 0 such that

∥u∥Lp(·)(Γ) ≤ C∥u∥W 1,p(·)(Ω) for u ∈ W 1,p(·)(Ω).



EJDE-2025/17 EIGENVALUE PROBLEMS FOR KIRCHHOFF-TYPE EQUATIONS 7

Now we consider the weighted variable exponent Lebesgue space. Let p ∈ C(Ω)
with p− ≥ 1 and let a(x) be a measurable function on Ω with a(x) > 0 a.e. x ∈ Ω.
We define a modular

ρ(p(·),a(·))(u) =

∫
Ω

a(x)|u(x)|p(x) dx

for any measurable function u in Ω. Then the weighted Lebesgue space is defined
by

L
p(·)
a(·)(Ω) =

{
u is a measurable function on Ω satisfying ρ(p(·),a(·))(u) < ∞

}
equipped with the norm

∥u∥
L

p(·)
a(·)(Ω)

= inf
{
τ > 0;

∫
Ω

a(x)
∣∣u(x)

τ

∣∣p(x) dx ≤ 1
}
.

Then L
p(·)
a(·)(Ω) is a Banach space.

We have the following proposition (cf. Fan [14, Proposition 2.5]).

Proposition 2.8. Let p ∈ C(Ω) with p− ≥ 1. For u, un ∈ L
p(·)
a(·)(Ω), we have the

following.

(i) For u ̸= 0, ∥u∥
L

p(·)
a(·)(Ω)

= τ ⇔ ρ(p(·),a(·))
(
u
τ

)
= 1.

(ii) ∥u∥
L

p(·)
a(·)(Ω)

< 1(= 1, > 1) ⇔ ρ(p(·),a(·))(u) < 1(= 1, > 1).

(iii) ∥u∥
L

p(·)
a(·)(Ω)

> 1 ⇒ ∥u∥p
−

L
p(·)
a(·)(Ω)

≤ ρ(p(·),a(·))(u) ≤ ∥u∥p
+

L
p(·)
a(·)(Ω)

.

(iv) ∥u∥
L

p(·)
a(·)(Ω)

< 1 ⇒ ∥u∥p
+

L
p(·)
a(·)(Ω)

≤ ρ(p(·),a(·))(u) ≤ ∥u∥p
−

L
p(·)
a(·)(Ω)

.

(v) limn→∞ ∥un − u∥
L

p(·)
a(·)(Ω)

= 0 ⇔ limn→∞ ρ(p(·),a(·))(un − u) = 0.

(vi) ∥un∥Lp(·)
a(·)(Ω)

→ ∞ as n → ∞ ⇔ ρ(p(·),a(·))(un) → ∞ as n → ∞.

The author of [14] also derived the following proposition (cf. [14, Theorem 2.1]).

Proposition 2.9. Let Ω be a bounded domain of RN with a C0,1-boundary and
p ∈ C+(Ω). Moreover, let a ∈ Lα(·)(Ω) satisfy a(x) > 0 a.e. x ∈ Ω and α ∈ C+(Ω).
If q ∈ C(Ω) satisfies

1 ≤ q(x) <
α(x)− 1

α(x)
p∗(x) for all x ∈ Ω,

then the embedding map W 1,p(·)(Ω) ↪→ L
q(·)
a(·)(Ω) is compact.

Similarly, let q ∈ C(Γ) with q− ≥ 1 and let b(x) be a measurable function with
respect to dσx on Γ with b(x) > 0 σ-a.e. x ∈ Γ. We define a modular

ρ(q(·),b(·)),Γ(u) =

∫
Γ

b(x)|u(x)|q(x)dσx.

Then the weighted Lebesgue space on Γ is defined by

L
q(·)
b(·)(Γ) = {u is a σ-measurable function on Γ satisfying ρ(q(·),b(·)),Γ(u) < ∞}

equipped with the norm

∥u∥
L

q(·)
b(·)(Γ)

= inf
{
τ > 0;

∫
Γ

b(x)
∣∣u(x)

τ

∣∣q(x) dσx ≤ 1
}
.

Then L
q(·)
b(·)(Γ) is a Banach space.



8 J. ARAMAKI EJDE-2025/17

Proposition 2.10. Let q ∈ C(Γ) with q− ≥ 1. For u, un ∈ L
q(·)
b(·)(Γ), we have the

following.

(i) ∥u∥
L

q(·)
b(·)(Γ)

< 1(= 1, > 1) ⇔ ρ(q(·),b(·)),Γ(u) < 1(= 1, > 1).

(ii) ∥u∥
L

q(·)
b(·)(Γ)

> 1 ⇒ ∥u∥q
−

L
q(·)
b(·)(Γ)

≤ ρ(q(·),b(·)),Γ(u) ≤ ∥u∥q
+

L
q(·)
b(·)(Γ)

.

(iii) ∥u∥
L

q(·)
b(·)(Γ)

< 1 ⇒ ∥u∥q
+

L
q(·)
b(·)(Γ)

≤ ρ(q(·),b(·)),Γ(u) ≤ ∥u∥q
−

L
q(·)
b(·)(Γ)

.

(iv) limn→∞ ∥un − u∥
L

q(·)
b(·)(Γ)

= 0 ⇔ limn→∞ ρ(q(·),b(·)),Γ(un − u) = 0.

(v) ∥un∥Lq(·)
b(·)(Γ)

→ ∞ as n → ∞ ⇔ ρ(q(·),b(·)),Γ(un) → ∞ as n → ∞.

The following proposition plays an important role in the present paper.

Proposition 2.11. Let Ω be a bounded domain of RN with a C0,1-boundary Γ and
let p ∈ C+(Ω). Assume that 0 < b ∈ Lβ(·)(Γ), β ∈ C+(Γ). If r ∈ C(Γ) satisfies

1 ≤ r(x) <
β(x)− 1

β(x)
p∂(x) for all x ∈ Γ,

then the embedding map W 1,p(·)(Ω) ↪→ L
r(·)
b(·)(Γ) is compact.

Proof. Let u ∈ W 1,p(·)(Ω). Set h(x) = β′(x)r(x). From the hypothesis, we have
h(x) < p∂(x) for all x ∈ Γ. By Proposition 2.7, the embedding map W 1,p(·)(Ω) ↪→
Lh(·)(Γ) is compact. Since |u(x)|r(x) ∈ Lβ′(·)(Γ), it follows from the Hölder inequal-
ity (Proposition 2.5) that∫

Γ

b(x)|u(x)|r(x) dσx ≤ 2∥b∥Lβ(·)(Γ)∥|u|r(·)∥Lβ′(·)(Γ) < ∞.

Hence W 1,p(·)(Ω) ⊂ L
r(·)
b(·)(Γ). We show that the embedding W 1,p(·)(Ω) ↪→ L

r(·)
b(·)(Γ)

is compact. Let un → 0 weakly in W 1,p(·)(Ω). Then un → 0 strongly in Lh(·)(Γ).
Since

ρβ′(·),Γ(|un|r(·)) =
∫
Γ

|un(x)|r(x)β
′(x)dσx =

∫
Γ

|un(x)|h(x) dσx → 0,

we have ∥|un|r(·)∥Lβ′(·)(Γ) → 0 from Proposition 2.10 (iv). Therefore,∫
Γ

b(x)|un(x)|r(x) dσx ≤ 2∥b∥Lr(·)(Γ)∥|un|r(·)∥Lβ′(·)(Γ) → 0.

Thus it also follows from Proposition 2.10 (iv) that ∥un∥Lr(·)
b(·)(Γ)

→ 0, soW 1,p(·)(Ω) ↪→

L
r(·)
b(·)(Γ) is compact. □

Now we consider the Nemytskii operator.

Proposition 2.12. Let q ∈ C(Ω) with q− ≥ 1 and a be a measurable function with
a(x) > 0 for a.e. x ∈ Ω. Assume that

(1) A function F (x, t) is a Carathéodory function on Ω× R.
(2) The growth condition holds: there exist c ∈ Lq1(·)(Ω) with c(x) ≥ 0 a.e.

x ∈ Ω, q1 ∈ C(Ω) with q−1 ≥ 1 and a constant c1 > 0 such that

|F (x, t)| ≤ c(x) + c1a(x)
1/q1(x)|t|q(x)/q1(x) for a.e. x ∈ Ω and all t ∈ R.



EJDE-2025/17 EIGENVALUE PROBLEMS FOR KIRCHHOFF-TYPE EQUATIONS 9

Then the Nemytskii operator NF : L
q(·)
a(·)(Ω) ∋ u 7→ F (x, u(x)) ∈ Lq1(·)(Ω) is contin-

uous and there exists a constant C > 0 such that

ρq1(·)(NF (u)) ≤ C(ρq1(·)(c) + ρ(q(·),a(·))(u)) for all u ∈ L
q(·)
a(·)(Ω).

In particular, if q1(x) ≡ 1, then NF : L
q(·)
a(·)(Ω) → L1(Ω) is continuous.

For a proof of the above proposition, see Aramaki [9, Proposition 7]. The propo-
sition is an extension of [6, Proposition 2.12]. Similarly we have the following
proposition.

Proposition 2.13. Let r ∈ C(Γ2) with r− ≥ 1 and b be a σ-measurable function
with b(x) > 0 σ-a.e. x ∈ Γ2. Assume that

(1) The function H(x, t) is a Carathéodory function on Γ2 × R.
(2) The growth condition holds: there exist d ∈ Lr1(·)(Γ2) with d(x) ≥ 0 σ-a.e.

x ∈ Γ2, r1 ∈ C(Γ2) with r1 ≥ 1, and a constant d1 > 0 such that

|H(x, t)| ≤ d(x) + d1b(x)
1/r1(x)|t|r(x)/r1(x) for σ-a.e. x ∈ Γ2 and all t ∈ R.

Then the Nemytskii operator NH : L
r(·)
b(·)(Γ2) ∋ v 7→ H(x, v(x)) ∈ Lr1(·)(Γ2) is

continuous and there exists a constant C > 0 such that

ρr1(·),Γ2
(NH(v)) ≤ C(ρr1(·),Γ2

(d) + ρ(r(·),b(·)),Γ2
(v)) for all v ∈ L

r(·)
b(·)(Γ2).

In particular, if r1(x) ≡ 1, then NH : L
r(·)
b(·)(Γ2) → L1(Γ2) is continuous.

Now we define the space

X = {v ∈ W 1,p(·)(Ω); v = 0 on Γ1}. (2.1)

Then it is clear that X is a closed subspace of W 1,p(·)(Ω), so X is a reflexive and
separable Banach space. We can see the following Poincaré-type inequality (cf.
[10]).

Proposition 2.14. Let Ω be a bounded domain of RN with a C0,1-boundary and
let p ∈ C+(Ω). Then there exists a constant C = C(Ω, N, p) > 0 such that

∥u∥Lp(·)(Ω) ≤ C∥∇u∥Lp(·)(Ω) for all u ∈ X.

In particular, ∥∇u∥Lp(·)(Ω) is equivalent to ∥u∥W 1,p(·)(Ω) for u ∈ X.

For a proof of the above proposition see [5, Lemma 2.5].
Thus we can define the norm on X so that

∥v∥X = ∥∇v∥Lp(·)(Ω) for v ∈ X, (2.2)

which is equivalent to ∥v∥W 1,p(·)(Ω) from Proposition 2.14.

3. Assumptions and main theorem

Let p ∈ C+(Ω) be fixed. Assume that the following:

(A2) A : Ω × RN → R is a function satisfying that for a.e. x ∈ Ω, the function
A(x, ·) : RN ∋ ξ 7→ A(x, ξ) is of C1-class, and for all ξ ∈ RN , the function
A(·, ξ) : Ω ∋ x 7→ A(x, ξ) is measurable. Moreover, suppose that A(x,0) =
0 and put a(x, ξ) = ∇ξA(x, ξ). Then a(x, ξ) is a Carathéodory function.

For items (A3)–(A5), c, k0, k1 > 0 denote constants, h0 ∈ Lp′(·)(Ω) is a non-negative
function, and h1 ∈ L1

loc(Ω) with h1(x) ≥ 1 for a.e. x ∈ Ω.



10 J. ARAMAKI EJDE-2025/17

(A3) |a(x, ξ)| ≤ c(h0(x) + h1(x)|ξ|p(x)−1) for all ξ ∈ RN and a.e. x ∈ Ω.
(A4) A is p(·)-uniformly convex, that is,

A

(
x,

ξ + η

2

)
+ k1h1(x)|ξ − η|p(x) ≤ 1

2
A(x, ξ) +

1

2
A(x, η)

for all ξ, η ∈ RN and a.e. x ∈ Ω.
(A5) k0h1(x)|ξ|p(x) ≤ a(x, ξ) · ξ ≤ p(x)A(x, ξ) for all ξ ∈ RN and a.e. x ∈ Ω.
(A6) (a(x, ξ)− a(x, η)) · (ξ− η) > 0 for all ξ, η ∈ RN with ξ ̸= η and a.e. x ∈ Ω.
(A7) A(x,−ξ) = A(x, ξ) for all ξ ∈ RN and a.e. x ∈ Ω.

Remark 3.1. (i) The condition (A3) is more general than that of Mashiyev et al.
[26] who considered the case h1(x) ≡ 1. In our case, to overcome this we have to
consider the space Y defined by (3.2) later as a basic space rather than the space
X defined by (2.1).

(ii) (A5) implies that A is p(·)-sub-homogeneous, that is,

A(x, sξ) ≤ A(x, ξ)sp(x) for each ξ ∈ RN , a.e. x ∈ Ω and s ≥ 1. (3.1)

For a proof, see Aramaki [7, (4.14)].

Example 3.2. Let

(i) A(x, ξ) = h(x)
p(x) |ξ|

p(x) with p− ≥ 2, h ∈ L1
loc(Ω) satisfying h(x) ≥ 1 a.e.

x ∈ Ω.
(ii) A(x, ξ) = h(x)

p(x) ((1 + |ξ|2)p(x)/2 − 1) with p− ≥ 2, h ∈ Lp′(·)(Ω) satisfying

h(x) ≥ 1 a.e. x ∈ Ω.

Then A(x, ξ) and a(x, ξ) = ∇ξA(x, ξ) of (i) and (ii) satisfy (A2)–(A7).

Remark 3.3. In Example 3.2, when h(x) ≡ 1, (i) corresponds to the p(·)-Laplacian
and (ii) corresponds to the prescribed mean curvature operator for nonparametric
surface.

For the function h1 ∈ L1
loc(Ω) with h1(x) ≥ 1 for a.e. x ∈ Ω, we define a modular

on X by

ρ̃(p(·),h1(·))(v) =

∫
Ω

h1(x)|∇v(x)|p(x) dx for v ∈ X,

where the space X is defined by (2.1). We define our basic space

Y = Y (Ω) = {v ∈ X; ρ̃(p(·),h1(·))(v) < ∞} (3.2)

equipped with the norm

∥v∥Y = inf
{
τ > 0; ρ̃(p(·),h1(·))

(v
τ

)
≤ 1

}
.

Proposition 3.4. The space (Y, ∥ · ∥Y ) is a separable and reflexive Banach space.

For a proof of the above propositon see Aramaki [8, Proposition 3.4]. We note
that C∞

0 (Ω) ⊂ Y . Since h1(x) ≥ 1 a.e. x ∈ Ω, it follows that

ρ̃(p(·),h1(·))(v) = ρp(·)(h
1/p(·)
1 |∇v|) ≥ ρp(·)(|∇v|) for v ∈ Y

and

∥v∥Y = ∥h1/p(·)
1 ∇v∥Lp(·)(Ω) ≥ ∥∇v∥Lp(·)(Ω) = ∥v∥X for v ∈ Y. (3.3)

From (3.3) and Proposition 2.1, we have the following proposition.



EJDE-2025/17 EIGENVALUE PROBLEMS FOR KIRCHHOFF-TYPE EQUATIONS 11

Proposition 3.5. Let p ∈ C+(Ω) and let u, un ∈ Y (n = 1, 2, . . .). Then the
following properties hold:

(i) Y ↪→ X and ∥u∥X ≤ ∥u∥Y .
(ii) ∥u∥Y > 1(= 1, < 1) ⇔ ρ̃(p(·),h1(·))(u) > 1(= 1, < 1).

(iii) ∥u∥Y > 1 ⇒ ∥u∥p
−

Y ≤ ρ̃(p(·),h1(·))(u) ≤ ∥u∥p
+

Y .

(iv) ∥u∥Y < 1 ⇒ ∥u∥p
+

Y ≤ ρ̃(p(·),h1(·))(u) ≤ ∥u∥p
−

Y .
(v) limn→∞ ∥un − u∥Y = 0 ⇔ limn→∞ ρ̃(p(·),h1(·))(un − u) = 0.
(vi) ∥un∥Y → ∞ as n → ∞ ⇔ ρ̃(p(·),h1(·))(un) → ∞ as n → ∞.

We assume that the function g in (1.1) satisfies

(A8) The function g(x, t) is of the form g(x, t) = b(x)|t|r(x)−2t, where b satisfies
0 < b ∈ Lβ(·)(Γ2) with β ∈ C+(Γ2), and r ∈ C+(Γ2) satisfies

r(x) <
β(x)− 1

β(x)
p∂(x) for all x ∈ Γ2.

If we define

G(x, t) =

∫ t

0

g(x, s) ds,

then G(x, t) = b(x)
r(x) |t|

r(x), so we have

r(x)G(x, t) = b(x)|t|r(x) = g(x, t)t > 0 (3.4)

for σ-a.e. x ∈ Γ2 and all 0 ̸= t ∈ R.
Now we introduce the notion of a weak solution and an eigenfunction for the

problem (1.1).

Definition 3.6. (i) We say that a pair (u, λ) ∈ Y × R is a weak solution of
(1.1), if

M
(∫

Ω

A(x,∇u(x)) dx
)∫

Ω

a(x,∇u(x)) ·∇v(x) dx = λ

∫
Γ2

g(x, u(x))v(x)dσx (3.5)

for all v ∈ Y .
(ii) Such a pair (u, λ) ∈ Y ×R with u ̸= 0 is called an eigenpair, λ is called an

eigenvalue and u is called an associated eigenfunction.

If we define a function associated with the function M by

M̂(t) =

∫ t

0

M(s) ds for t ≥ 0,

then we see that M̂ ∈ C1([0,∞)) and satisfies

m0

l
tl ≤ M̂(t) ≤ m1

(
t+

1

k
tk
)

for t ≥ 0. (3.6)

Moreover, since M̂ ′(t) = M(t) is monotone non-decreasing and satisfies (1.3), M̂(t)
is convex and strictly monotone increasing on [0,∞).

We define functionals on Y by

Φ(u) =

∫
Ω

A(x,∇u(x)) dx, Ψ(u) = M̂(Φ(u)), K(u) =

∫
Γ2

G(x, u(x))dσx

(3.7)
for u ∈ Y . It follows from (A7) and (A8) that Φ, Ψ and K are even functionals,
that is, Φ(−u) = Φ(u), Ψ(−u) = Ψ(u) and K(−u) = K(u) for all u ∈ Y .



12 J. ARAMAKI EJDE-2025/17

Lemma 3.7. (i) We have

k0
p+

ρ̃(p(·),h1(·))(u) ≤ Φ(u) ≤ c(2∥h0∥Lp′(·)(Ω)∥∇u∥Lp(·)(Ω) + ρ̃(p(·),h1(·))(u))

for u ∈ Y , where c and k0 are the constants in (A.1) and (A5).
(ii) We have

Φ
(u+ v

2

)
+ k1ρ̃(p(·),h1(·))(u− v) ≤ 1

2
Φ(u) +

1

2
Φ(v)

for all u, v ∈ Y , where k1 is the constant in (A4). In particular, Φ is convex, that
is, Φ((1− τ)u+ τv) ≤ (1− τ)Φ(u) + τΦ(v) for all u, v ∈ Y and τ ∈ [0, 1].

Proof. (i) easily follows from (A5) and the Hölder inequality (Proposition 2.2). (ii)
easily follows from (A4) and the continuity of A(x, ξ) with respect to ξ. □

The functional Ψ defined by (3.7) is a continuous modular on a real Banach space
Y in the sense of [13, Definition 2.1.11], that is, Ψ has the following properties:

(a) Ψ(0) = 0. This easily follows from A(x,0) = 0 and the definition of M̂ .
(b) Ψ(−u) = Ψ(u) for every u ∈ Y . This follows from (A7).

(c) Ψ is convex. Indeed, since M̂ is convex and strictly monotone increasing
and Φ is convex, for any u, v ∈ Y and τ ∈ [0, 1] we have

Ψ((1− τ)u+ τv) = M̂(Φ((1− τ)u+ τv))

≤ M̂((1− τ)Φ(u) + τΦ(v))

≤ (1− τ)Ψ(u) + τΨ(v).

(d) The function [0,∞) ∋ λ 7→ Ψ(λu) is continuous for every u ∈ Y . Indeed,
let [0,∞) ∋ λn → λ0 as n → ∞. Here we can assume that 0 ≤ λn ≤ λ0 +1
for large n ∈ N. From (A.0) and (A5), we have

|A(x, λn∇u(x))| ≤ c(λ0 + 1)h0(x)|∇u(x)|+ c(λ0 + 1)p
+

h1(x)|∇u(x)|p(x).

Since h0 ∈ Lp′(·)(Ω) and |∇u(·)| ∈ Lp(·)(Ω) and u ∈ Y , the right-hand side
in the above inequality is an integrable function independent of n. Clearly,
we see that A(x, λn∇u(x)) → A(x, λ0∇u(x)) as n → ∞ for a.e. x ∈ Ω. By
the Lebesgue dominated convergent theorem, we see that Φ(λnu) → Φ(λ0u)
as n → ∞, so Ψ(λnu) → Ψ(λ0u).

(e) Ψ(u) = 0 implies u = 0. Indeed, if Ψ(u) = 0, then Φ(u) = 0. Hence it
follows from (A5) and the Poincaré-type inequality (Proposition 2.14) that
u = 0. .

Thus we can define a modular space

YΨ = {u ∈ Y ; lim
τ→0

Ψ(τu) = 0} = {u ∈ Y ; Ψ(τu) < ∞ for some τ > 0}

with the Luxemburg norm

∥u∥Ψ = inf
{
τ > 0;Ψ

(u
τ

)
≤ 1

}
for u ∈ YΦ.

Then (YΨ, ∥ · ∥Ψ) is a normed linear space over R from [13, Theorem 2.1.7]. Clearly
we see that YΨ = Y , and the norms ∥ · ∥Ψ and ∥ · ∥Y are equivalent (cf. [8, Lemma
4.3]).



EJDE-2025/17 EIGENVALUE PROBLEMS FOR KIRCHHOFF-TYPE EQUATIONS 13

From now on, we denote a ∨ b = max{a, b} and a ∧ b = min{a, b} for any real
numbers a and b. Since

Φ(u) ≥ k0
p+

∫
Ω

h1(x)|∇u(x)|p(x) dx ≥ k0
p+

(∥u∥p
+

Y ∧ ∥u∥p
−

Y ),

it follows from (3.6) that

Ψ(u) = M̂(Φ(u)) ≥ m0

l

( k0
p+

(∥u∥p
+

Y ∧ ∥u∥p
−

Y )
)l

, (3.8)

Lemma 3.8. If un → u weakly in Y and Ψ(un) → Ψ(u) as n → ∞, then we have
Ψ
(
un−u

2

)
→ 0 as n → ∞. In particular, un → u strongly in Y as n → ∞.

Proof. Let un → u weakly in Y and Ψ(un) → Ψ(u) as n → ∞. Then, if we use [13,
Lemma 2.4.17] (cf. Aramaki [9, Lemma 20]), then we can show that Ψ

(
un−u

2

)
→ 0

as n → ∞, so un → u strongly in Y using (3.8). □

First we list the properties of Ψ.

Proposition 3.9. (i) Ψ is coercive, that is, Ψ(u) → ∞ as ∥u∥Y → ∞.
(ii) Ψ is sequentially weakly lower-semicontinuous on Y .
(iii) Ψ ∈ C1(Y,R) and the Fréchet derivative Ψ′ of Ψ satisfies

⟨Ψ′(u), v⟩Y ∗,Y = M(Φ(u))

∫
Ω

a(x,∇u(x)) ·∇v(x) dx for u, v ∈ Y. (3.9)

(iv) Ψ ∈ WY , that is, if un → u weakly in Y and lim infn→∞ Ψ(un) ≤ Ψ(u),
then the sequence {un} has a strongly convergent subsequence.

(v) Ψ is bounded on every bounded subset of Y .

Proof. (i) follows from (3.8). (ii) follows from Aramaki [7, Proposition 4.4] and the

fact that M̂ is monotone increasing and continuous. (iii) follows from [7, Proposition

4.1] and M̂ ∈ C1([0,∞)).
(iv) Let un → u weakly in Y and lim infn→∞ Ψ(un) ≤ Ψ(u). Since Ψ is sequen-

tially weakly lower semi-continuous, Ψ(u) ≤ lim infn→∞ Ψ(un), so that
lim infn→∞ Ψ(un) = Ψ(u). Hence there exists a subsequence {un′} of {un} such
that limn′→∞ Ψ(un′) = Ψ(u). By Lemma 3.8, we see that un′ → u strongly in Y .

(v) follows from Lemma 3.7 (i) and (3.6). □

Next we derive the properties of Ψ′.

Proposition 3.10. (i) Ψ′ is strictly monotone in Y , that is,

⟨Ψ′(u)−Ψ′(v), u− v⟩Y ∗,Y > 0 for all u, v ∈ Y with u ̸= v.

Moreover, Ψ′ is bounded on every bounded subset of Y and coercive in the
sense that

lim
∥u∥Y →∞

⟨Ψ′(u), u⟩Y ∗,Y

∥u∥Y
= ∞.

(ii) Ψ′ is of (S+)-type, that is, if un → u weakly in Y and

lim sup
n→∞

⟨Ψ′(un), un − u⟩Y ∗.Y ≤ 0,

then un → u strongly in Y .
(iii) The mapping Ψ′ : Y → Y ∗ is a homeomorphism.



14 J. ARAMAKI EJDE-2025/17

Proof. (i) In general, when a functional f : Y → R is of C1-class, f is strictly
convex if and only if f ′ : Y → Y ∗ is strictly monotone (cf. Zeidler [34, Proposition
25.10]), that is,

⟨f ′(u)− f ′(v), u− v⟩Y ∗,Y > 0 for all u, v ∈ Y with u ̸= v.

From (A6),

⟨Φ′(u)−Φ′(v), u−v⟩Y ∗,Y =

∫
Ω

(a(x,∇u(x)−a(x,∇v(x)))·(∇u(x)−∇v(x)) dx > 0

for all u, v ∈ Y with u ̸= v, so Φ′ is strictly monotone in Y , so Φ is strictly convex.

The function M̂ is strictly monotone increasing and convex. Hence for u, v ∈ Y
with u ̸= v and τ ∈ (0, 1), since Φ((1− τ)u+ τv) < (1− τ)Φ(u) + τΦ(v), we have

M̂(Φ(1− τ)u+ τv)) < M̂((1− τ)Φ(u) + τΦ(v)) ≤ (1− τ)M̂(Φ(u)) + τM̂(Φ(v)),

so Ψ((1− τ)u+ τv) < (1− τ)Ψ(u) + τΨ(v). Thus Ψ is strictly convex, so Ψ′(·) =
M(Φ(·))Φ′(·) is strictly monotone in Y .

It follows from the Hölder inequality (Proposition 2.2) and Proposition 3.5 (i)
that

|⟨Ψ′(u), v⟩Y ∗,Y |

= M(Φ(u))
∣∣ ∫

Ω

a(x,∇u(x)) ·∇v(x) dx
∣∣

≤ cM(Φ(u))

∫
Ω

(h0(x)|∇v(x)|+ h1(x)|∇u(x)|p(x)−1|∇v(x)|) dx

= cM(Φ(u))

∫
Ω

(h0(x)|∇v(x)|+ h1(x)
1/p′(x)|∇u(x)|p(x)−1h1(x)

1/p(x)|∇v(x)|) dx

≤ 2cm1(1 + Φ(u)k−1)(∥h0∥Lp′(·)(Ω)∥v∥Y

+ ∥h1/p′(·)
1 |∇u|p(·)−1∥Lp′(·)(Ω)∥h

1/p(·)
1 |∇v|∥Lp(·)(Ω)

= 2cm1(1 + Φ(u)k−1)(∥h0∥Lp′(·)(Ω) + ∥h1/p′(·)
1 |∇u|p(·)−1∥Lp′(·)(Ω))∥v∥Y

for all v ∈ Y . Hence we have

∥Ψ′(u)∥Y ∗ ≤ 2cm1(1 + Φ(u)k−1)(∥h0∥Lp′(·)(Ω) + ∥h1/p′(·)
1 |∇u|p(·)−1∥Lp′(·)(Ω)).

Here we note that

Φ(u)k−1 ≤ ck−1(2∥h0∥Lp′(·)(Ω)∥u∥Y + ∥u∥p
+

Y ∨ ∥u∥p
−

Y )k−1,

ρp′(·)(h
1/p′(·)
1 |∇u|p(·)−1) =

∫
Ω

h1(x)|∇u(x)|p(x) dx ≤ ∥u∥p
+

Y ∨ ∥u∥p−Y .

If ∥u∥ ≤ M , then it is clear that there exists a constant C(A1) > 0 such that
∥Ψ′(u)∥Y ∗ ≤ C(A1), so Ψ′ is bounded on every bounded subset of Y .

Let ∥u∥Y > 1. Then from (A1) and (A5),

⟨Ψ′(u), u⟩Y ∗,Y = M(Φ(u))

∫
Ω

a(x,∇u(x) ·∇u(x) dx

≥ k0M(Φ(u))

∫
Ω

h1(x)|∇u(x)|p(x) dx

≥ kl0
(p+)l−1

m0∥u∥(l−1)p−

Y ∥u∥p
−

Y



EJDE-2025/17 EIGENVALUE PROBLEMS FOR KIRCHHOFF-TYPE EQUATIONS 15

=
m0k

l
0

(p+)l−1
∥u∥lp

−

Y .

Since lp− > 1, this implies the coerciveness of Ψ′.
(ii) Let un → u weakly in Y and lim supn→∞⟨Ψ′(un), un − u⟩Y ∗,Y ≤ 0. Since Ψ′

is monotone from (i), ⟨Ψ′(un)−Ψ′(u), un − u⟩Y ∗,Y ≥ 0. Hence

0 ≤ lim inf
n→∞

⟨Ψ′(un)−Ψ′(u), un − u⟩Y ∗,Y

= lim inf
n→∞

⟨Ψ′(un), un − u⟩Y ∗,Y

≤ lim sup
n→∞

⟨Ψ′(un), un − u⟩Y ∗,Y ≤ 0.

Therefore, limn→∞ M(Φ(un))⟨Φ′(un), un−u⟩Y ∗,Y = 0. Since un → u weakly in Y ,
the sequence {∥un∥Y } is bounded. Hence since M(Φ(un)) is bounded from Lemma
3.7 (i), we have limn→∞ M(Φ(un))⟨Φ′(u), un − u⟩Y ∗,Y = 0. Therefore,

lim
n→∞

M(Φ(un))⟨Φ′(un)− Φ′(u), un − u⟩Y ∗,Y = 0.

Thereby, since M(Φ(un)) ≥ 0 and ⟨Φ′(un) − Φ′(u), un − u⟩Y ∗,Y ≥ 0, we obtain
that limn→∞ M(Φ(un)) = 0 or limn→∞⟨Φ′(un) − Φ′(u), un − u⟩Y ∗,Y = 0. Indeed,
if we put an = M(Φ(un)) and bn = ⟨Φ′(un) − Φ′(u), un − u⟩Y ∗,Y , then it suffices
to derive that an ≥ 0, bn ≥ 0 and limn→∞ anbn = 0 implies that limn→∞ an = 0
or limn→∞ bn = 0. For any subsequence {n′} of N, we have limn′→∞ an′bn′ = 0.
If limn′→∞ an′ does not exist or exists and is equal to a positive number, then
there exist ε0 > 0 and a subsequence {an′′} of {an′} such that an′′ ≥ ε0 for any
an′′ . Hence we have an′′bn′′ ≥ ε0bn′′ ≥ 0. Since limn′′→∞ an′′bn′′ = 0, we see that
limn′′→∞ bn′′ = 0, so according to the convergent principal we have limn→∞ bn = 0.
If limn′→∞ an′ = 0 for any subsequence {an′}, then we clearly have limn→∞ an = 0.

When M(Φ(un)) → 0 as n → ∞, we have Φ(un) → 0 = Φ(0). By Lemma 3.8
with M ≡ 1, un → 0 strongly in Y (in this case we necessarily have u = 0). When

lim
n→∞

⟨Φ′(un)− Φ′(u), un − u⟩Y ∗,Y = lim
n→∞

⟨Φ′(un), un − u⟩Y ∗,Y = 0,

since Φ′ is of (S+)-type (cf. [9, Proposition 21 (ii)]), we have un → u strongly in Y .
(iii) Since Ψ′ is strictly monotone from (i), Ψ′ is injective. We show that Ψ′ :

Y → Y ∗ is surjective. Let w ∈ Y ∗. Define a functional on Y by

φ(u) := Ψ(u)− ⟨w, u⟩Y ∗,Y for u ∈ Y.

From (A1) and Lemma 3.7 (i), for ∥u∥Y > 1, we see that

φ(u) ≥ M̂(Φ(u))− ⟨w, u⟩Y ∗,Y ≥
( k0
p+

)l

∥u∥lp
−

Y − ∥w∥Y ∗∥u∥Y .

Since lp− > 1, φ is coercive. Since Ψ is sequentially weakly lower semi-continuous,
φ is so. If we put γ = infu∈Y φ(u)(< ∞), then there exists a sequence {un} ⊂ Y
such that γ = limn→∞ φ(un). Since φ is coercive, the sequence {un} is bounded.
Since Y is a reflexive Banach space, there exist a subsequence {un′} of {un} and
u0 ∈ Y such that un′ → u0 weakly in Y , so φ(u0) ≤ lim infn′→∞ φ(un′) = γ. This
implies that γ > −∞ and u0 is a minimizer of φ, so φ′(u0) = 0, i.e., Ψ′(u0) = w.
Therefore, Ψ′ has an inverse operator (Ψ′)−1 : Y ∗ → Y . We show that (Ψ′)−1

is continuous. Let fn → f in Y ∗ as n → ∞. Then there exist un, u ∈ Y such
that Ψ′(un) = fn and Ψ′(u) = f . Then {un} is bounded in Y . Indeed, if {un} is



16 J. ARAMAKI EJDE-2025/17

unbounded, then there exists a subsequence {un′} of {un} such that ∥un′∥Y → ∞
as n′ → ∞. Hence

⟨Ψ′(un′), un′⟩Y ∗,Y = ⟨fn′ , un′⟩Y ∗,Y ≤ ∥fn′∥Y ∗∥un′∥Y ≤ C∥un′∥Y
for some constant C > 0. This contradict the coerciveness of Ψ′.

Since Y is a reflexive Banach space, there exist a subsequence (still denoted by
{un′}) and u0 ∈ Y such that un′ → u0 weakly in Y . Hence

lim
n′→∞

⟨Ψ′(un′), un′ − u0⟩Y ∗,Y = lim
n′→∞

⟨Ψ′(un′)−Ψ′(u), un′ − u0⟩Y ∗,Y

= lim
n′→∞

⟨fn′ − f, un′ − u0⟩Y ∗,Y = 0.

Since Ψ′ is of (S+)-type, we see that un′ → u0 strongly in Y . According to the
continuity of Ψ′, Ψ′(un′) = fn′ → f = Ψ′(u0) = Ψ′(u), so we have u0 = u from
the injectiveness of Ψ′. By the convergent principle (cf. [34, Theorem 10.13 (i)]),
for full sequence {un}, un → u strongly in Y , that is, (Ψ′)−1(fn) → (Ψ′)−1(f) as
n → ∞. □

For the functional K defined by (3.7), we have the following proposition.

Proposition 3.11. Under hypotheses (A8), we have the following.

(i) K ∈ C1(Y,R) and

⟨K ′(u), v⟩Y ∗,Y =

∫
Γ2

g(x, u(x))v(x) dσx for u, v ∈ Y. (3.10)

(ii) K is sequentially weakly continuous in Y .
(iii) K ′ : Y → Y ∗ is weakly-strongly continuous, that is, if un → u weakly in Y

as n → ∞, then K ′(un) → K ′(u) strongly in Y ∗ as n → ∞.

Proof. (i) and (ii) follows from Aramaki [7, Proposition 4.2, Proposition 4.4]. So
we only verify (iii). Let un → u weakly in Y . Then

⟨K ′(un)−K ′(u), v⟩Y ∗,Y =

∫
Γ2

(g(x, un(x))− g(x, u(x)))v(x)dσx for v ∈ Y.

From Proposition 2.11 and (A8), the embeddingW 1,p(·)(Ω) ↪→ L
r(·)
b(·)(Γ2) is compact.

Since Y ↪→ X ↪→ W 1,p(·)(Ω), there exists a constant C > 0 such that

∥v∥
L

r(·)
b(·)(Γ2)

≤ C∥v∥Y for all v ∈ Y.

By the Hölder inequality (Proposition 2.5), for any v ∈ Y , we have

|⟨K ′(un)−K ′(u), v⟩Y ∗,Y |

≤
∫
Γ2

b(x)−1/r(x)|g(x, un(x))− g(x, u(x))|b(x)1/r(x)|v(x)|dσx

≤ 2∥b(·)−1/r(·)|g(·, un(·))− g(·, u(·))|∥Lr′(·)(Γ2)
∥b(·)1/r(·)|v(·)|∥Lr(·)(Γ2).

Since

∥b(·)1/r(·)v(·)∥Lr(·)(Γ2) = ∥v∥
L

r(·)
b(·)(Γ2)

≤ C∥v∥Y ,

we have

∥K ′(un)−K ′(u)∥Y ∗ ≤ 2C∥b(·)−1/r(·)|g(·, un(·))− g(·, u(·))|∥Lr′(·)(Γ2)
.



EJDE-2025/17 EIGENVALUE PROBLEMS FOR KIRCHHOFF-TYPE EQUATIONS 17

We want to show that ∥K ′(un) − K ′(u)∥Y ∗ → 0 as n → ∞. By Proposition 2.4
(iv), it suffices to show that

ρr′(·),Γ2

(
b(·)−1/r(·)g(·, un(·))− b(·)−1/r(·)g(·, u(·))

)
→ 0 as n → ∞. (3.11)

We can see that

ρr′(·),Γ2

(
b(·)−1/r(·)g(·, un(·))− b(·)−1/r(·)g(·, u(·))

)
=

∫
Γ2

b(x)−r′(x)/r(x)|g(x, un(x))− g(x, u(x))|r
′(x)dσx.

Since un → u weakly in Y and the embedding map Y ↪→ L
r(·)
b(·)(Γ2) is compact,

we can see that un → u strongly in L
r(·)
b(·)(Γ2). From [6, Theorem A.1], there

exist a subsequence {un′} of {un} and f ∈ Lr(·)(Γ2) such that b(x)1/r(x)un′(x) →
b(x)1/r(x)u(x) σ-a.e. x ∈ Γ2 and |b(x)1/r(x)un′(x)| ≤ f(x) for σ-a.e. x ∈ Γ2. Since
b(x) > 0 σ-a.e. x ∈ Γ2, un′(x) → u(x) σ-a.e. x ∈ Γ2, so we see that g(x, un′(x)) →
g(x, u(x)) σ-a.e. x ∈ Γ2. From (A8), we have

b(x)−r′(x)/r(x)|g(x, un′(x)− g(x, u(x))|r
′(x)

≤ b(x)−r′(x)/r(x)(b(x)|un′(x)|r(x)−1 + b(x)|u(x)|r(x)−1)r
′(x)

≤ b(x)r
′(x)−r′(x)/r(x)(|un′(x)|r(x) + |u(x)|r(x))

≤ b(x)(|un′(x)|r(x) + |u(x)|r(x))

≤ 2f(x)r(x).

The last term is an integrable function in Ω independent of n′. Thus by the Lebesgue
dominated convergence theorem, we have

ρr′(·),Γ2

(
b(·)−1/r(·)g(·, un′(·))− b(·)−1/r(·)g(·, u(·))

)
→ 0 as n′ → ∞.

From the convergent principle [34, Proposition 10.13], we see that (3.11) holds, so
∥K ′(un)−K ′(u)∥Y ∗ → 0 as n → ∞. □

Remark 3.12. From (3.9), (3.10) and Definition 3.6, we can see that (u, λ) ∈ Y ×R
is a weak solution of (1.1) if and only if

Ψ′(u) = λK ′(u). (3.12)

In particular, we have ⟨Ψ′(u), u⟩Y ∗,Y = λ⟨K ′(u), u⟩Y ∗,Y . If (u, λ) is an eigenpair of
(1.1), then from (A5), (A1) and (A8)it follows that

⟨Ψ′(u), u⟩Y ∗,Y

= M(Φ(u))

∫
Ω

a(x,∇u(x)) ·∇u(x) dx

≥ m0

(∫
Ω

A(x,∇u(x)) dx
)l−1

∫
Ω

a(x,∇u(x)) ·∇u(x) dx

≥ m0

(∫
Ω

1

p(x)
a(x,∇u(x)) ·∇u(x) dx

)l−1
∫
Ω

a(x,∇u(x)) ·∇u(x) dx

≥ m0

(p+)l−1

(∫
Ω

a(x,∇u(x)) ·∇u(x) dx
)l



18 J. ARAMAKI EJDE-2025/17

≥ m0k
l
0

(p+)l−1

(∫
Ω

h1(x)|∇u(x)|p(x) dx
)l

≥ m0k
l
0

(p+)l−1
(∥u∥p

+

Y ∧ ∥u∥p
−

Y )l > 0

and from (3.12) and (3.4),

⟨K ′(u), u⟩Y ∗,Y =

∫
Γ2

g(x, u(x))u(x)dσx > 0,

so we have

λ =
⟨Ψ′(u), u⟩Y ∗,Y

⟨K ′(u), u⟩Y ∗,Y
> 0. (3.13)

This means that any eigenvalue of problem (1.1) is positive.

To solve the eigenvalue problem (3.12), we apply the constrained variational
method. We take Ψ as an objective functional and K as a constraint functional.
For any fixed α > 0, put

Mα = {u ∈ Y ;K(u) = α}. (3.14)

If u ∈ Mα, then from (A8),

⟨K ′(u), u⟩Y ∗,Y =

∫
Γ2

g(x, u(x))u(x)dσx

≥ r−
∫
Γ2

G(x, u(x))dσx

= r−K(u) = r− α > 0,

(3.15)

soK ′(u) ̸= 0. HenceMα is a C1-submanifold of Y with codimension one. Moreover,
Mα is weakly closed subset of Y . Indeed, let uj ∈ Mα and uj → u weakly in Y
as j → ∞. Since K is sequentially weakly continuous from Proposition 3.11 (ii),
α = K(uj) → K(u), so u ∈ Mα.

It is well known that when u ∈ Mα, a pair (u, λ) ∈ Y × R solves (3.12) if and
only if u is a critical point of Ψ with respect to Mα, that is,

⟨Ψ′(u), h⟩Y ∗,Y = 0 for all h ∈ TuMα,

(see for example [34, Proposition 43.21]). Here TuMα is the tangent space of Mα

at u ∈ Mα and we can see that

TuMα = Ker(K ′(u)) = {v ∈ Y ; ⟨K ′(u), v⟩Y ∗,Y = 0}.
Let P : Y → TuMα be the natural projection. Note that the bounded linear
map K ′(u) : Y → R is surjective. We denote the restriction of Ψ to Mα by

Ψ̃ = Ψ
∣∣
Mα

and the derivative dΨ̃(u) ∈ Y ∗ of Ψ̃ at u ∈ Mα can be defined by

⟨dΨ̃(u), v⟩Y ∗,Y = ⟨Ψ′(u), Pv⟩Y ∗,Y for v ∈ Y .
For u ∈ Mα, put w = (Ψ′)−1(K ′(u)). Then since we have (3.15), we see that

K ′(u) ̸= 0. From (A7), the functional Ψ is even, so Ψ′ is odd and so Ψ′(0) = 0. Since
(Ψ′)−1 is injective, we have w ̸= 0. From strict monotonicity of Ψ′ (Proposition
3.11 (i)),

⟨K ′(u), w⟩Y ∗,Y = ⟨K ′(u), (Ψ′)−1(K ′(u))⟩Y ∗,Y = ⟨Ψ′(w), w⟩Y ∗,Y > 0. (3.16)

Hence since w = (Ψ′)−1(K ′(u)) ̸∈ TuMα, we can see that

Y = TuMα ⊕ {β(Ψ′)−1(K ′(u));β ∈ R}.



EJDE-2025/17 EIGENVALUE PROBLEMS FOR KIRCHHOFF-TYPE EQUATIONS 19

For every v ∈ Y , there exists a unique β ∈ R such that v = Pv + β(Ψ′)−1(K ′(u)).
Since Pv ∈ TuMα = Ker(K ′(u)), we have

⟨K ′(u), v⟩Y ∗,Y = β⟨K ′(u), (Ψ′)−1(K ′(u))⟩Y ∗,Y .

Thus from (3.14), we can write

β =
⟨K ′(u), v⟩Y ∗,Y

⟨K ′(u), (Ψ′)−1(K ′(u))⟩Y ∗,Y
.

Now we have

⟨dΨ̃(u), v⟩Y ∗,Y

= ⟨Ψ′(u), Pv⟩Y ∗,Y

= ⟨Ψ′(u), v⟩Y ∗,Y −
〈
Ψ′(u),

⟨K ′(u), v⟩Y ∗,Y

⟨K ′(u), (Ψ′)−1(K ′(u))⟩Y ∗,Y
(Ψ′)−1(K ′(u))

〉
Y ∗,Y

=
〈
Ψ′(u)− ⟨Ψ′(u), (Ψ′)−1(K ′(u))⟩Y ∗,Y

⟨K ′(u), (Ψ′)−1(K ′(u))⟩Y ∗,Y
K ′(u), v

〉
Y ∗,Y

for all v ∈ Y.

Thus we have

dΨ̃(u) = Ψ′(u)− λ(u)K ′(u),

where

λ(u) =
⟨Ψ′(u), (Ψ′)−1(K ′(u))⟩Y ∗,Y

⟨K ′(u), (Ψ′)−1(K ′(u))⟩Y ∗,Y
.

Proposition 3.13. For each α > 0, the functional Ψ̃ : Mα → R satisfies (PS)c-

condition for any c ∈ R, that is, if any sequence {un} ⊂ Mα such that Ψ̃(un) → c

and ∥dΨ̃(un)∥Y ∗ → 0 as n → ∞, then {un} contains a convergent subsequence.

Proof. Let {un} ⊂ Mα satisfy that Ψ̃(un) → c and dΨ̃(un) → 0 in Y ∗ as n → ∞.
Then since from (3.6) and (A5),

Ψ̃(un) = M̂(Φ(un))

≥ m0

l

( k0
p+

∫
Ω

h1(x)|∇un(x)|p(x) dx
)l

≥ m0

l

( k0
p+

∥un∥p
+

Y ∧ ∥un∥p
−

Y

)l

,

{un} is bounded in Y . Since Y is a reflexive Banach space from Proposition 3.4,
there exist a subsequence {un′} of {un} and u0 ∈ Y such that un′ → u0 weakly in Y .
By Proposition 3.11 (ii) and (iii), K ′(un′) → K ′(u0) in Y ∗ and K(un′) → K(u0)
as n → ∞. Thereby, u0 ∈ Mα. Put wn′ = (Ψ′)−1(K ′(un′)). Since K ′(un′) →
K ′(u0) ̸= 0 in Y ∗ from (3.15), we see that wn′ → w0 ̸= 0 in Y , where w0 =
(Ψ′)−1(K ′(u0)). Thus

⟨K ′(un′), (Ψ′)−1(K ′(un′))⟩Y ∗,Y

= ⟨Ψ′(wn′), wn′⟩Y ∗,Y → ⟨Ψ′(w0), w0⟩Y ∗,Y > 0.
(3.17)

On the other hand,

|⟨Ψ′(un′), (Ψ′)−1(K ′(un′))⟩Y ∗,Y | = |⟨Ψ′(un′), wn′⟩Y ∗,Y | ≤ ∥Ψ′(un′)∥Y ∗∥wn′∥Y .



20 J. ARAMAKI EJDE-2025/17

Since un′ → u0 weakly in Y , we see that {un′} is bounded in Y , so by Proposition
3.10 (i), ∥Ψ′(un′)∥Y ∗ is bounded. Hence, there exists a constant c2 > 0 such that

|⟨Ψ′(un′), (Ψ′)−1(K ′(un′))⟩Y ∗,Y | ≤ c2. (3.18)

From (3.17) and (3.18), {λ(un′)} is bounded in R. Passing to a subsequence, we

may assume that λ(un′) → λ0 for some λ0 ∈ R. Since dΨ̃(un′) → 0 in Y ∗, we see
that Φ′(un′) − λ(un′)K ′(un′) → 0 as n′ → ∞. Hence, since K ′(un′) → K ′(u0) in
Y ∗,

Ψ′(un′) = (Ψ′(un′)− λ(un′)K ′(un′)) + λ(un′)K ′(un′) → λ0K
′(u0)

in Y ∗ as n′ → ∞. Therefore, we see that un′ → (Ψ′)−1(λ0K
′(u0)) strongly in Y as

n′ → ∞. □

Here we recall the notion of “genus” which wass introduced in Rabinowitz [30,
Chapter 7] or [34, Section 44.3]. Let E be a real Banach space and let E denote
the family of subsets A ⊂ E \ {0} such that A is closed in E and symmetric with
respect to 0, that is, x ∈ A implies −x ∈ A. For ∅ ̸= A ∈ E , define the genus of A
to be n ≥ 1 (denoted by γ(A) = n) if there is a map φ ∈ C(A,Rn \ {0}) with φ
odd and n is the smallest integer with this property. When there does not exist a
finite such n, set γ(A) = ∞. Finally set γ(∅) = 0. The main properties of genus
will be listed in the next proposition.

Proposition 3.14. Let A,B ∈ E. Then the following properties hold.

(i) If there exits an odd map f ∈ C(A,B), then γ(A) ≤ γ(B).
(ii) If A ⊂ B, then γ(A) ≤ γ(B).
(iii) γ(A ∪B) ≤ γ(A) + γ(B).
(iv) If A is compact, then γ(A) < ∞ and there exists δ > 0 such that if we put

Nδ(A) = {x ∈ E; ∥x − A∥ := inf{∥x − y∥; y ∈ A} ≤ δ}, then Nδ(A) ∈ E
and γ(Nδ(A)) = γ(A).

(v) If Ω is a bounded neighborhood of 0 in Rn, and there exists a mapping
h : A → ∂Ω with h an odd homeomorphism, then γ(A) = n.

For a proof of the above proposition, see [30, Lemma 7.5 and Proposition 7.7] or
[32, Proposition 2.3]. We note that it can be easily seen that when A ∈ E , A ̸= ∅ if
and only if γ(A) ≥ 1.

We apply the notion with E = Y . Let Σα = {H ⊂ Mα : H is compact and
symmetric}, γ(H) be the genus of H ∈ Σα, and define

c(n,α) = inf
H∈Σα,γ(H)≥n

sup
u∈H

Ψ̃(u) (n = 1, 2, . . .). (3.19)

The following proposition is due to [32, Corollary 4.3].

Proposition 3.15 (Ljusternik-Schnirelmann principle). Assume that M is a closed
symmetric C1-submanifold of a real Banach space B and 0 ̸∈ M . Let f ∈ C1(M,R)
be an even functional and bounded from below. Define

cj = inf
H∈Γj

sup
u∈H

f(u) for j = 1, 2, . . . ,

where
Γj = {H ⊂ M : H is compact, symmetric and γ(H) ≥ j}.

If Γk ̸= ∅ for some k ≥ 1 and f satisfies (PS)c-condition for c := cm = cm+1 =
· · · = ck with 1 ≤ m ≤ k, then f has at least k − m + 1 distinct pairs of critical
points.



EJDE-2025/17 EIGENVALUE PROBLEMS FOR KIRCHHOFF-TYPE EQUATIONS 21

Since Y is a separable reflexive Banach space, it is well known that there exist
{en}∞n=1 ⊂ Y and {fn}∞n=1 ⊂ Y ∗ such that ⟨fn, em⟩Y ∗,Y = δnm, where δnm is the
Kronecker delta and

Y = span{e1, e2, . . .} and Y ∗ = span{f1, f2, . . .}.
We define the spaces

Yj = span{ej}, Zn = ⊕n
j=1Yj , Wn = ⊕∞

j=nYj .

If we apply Proposition 3.15 with B = Y , M = Mα and f = Ψ̃, then we obtain

the following lemma. We note that Ψ̃ is bounded from below on Mα and satisfies
(PS)c-condition with respect ot Mα for any c ∈ R by Proposition 3.13.

Lemma 3.16. For any m ∈ N, we have Γm ̸= ∅. Thus we see that all c(m,α)

defined by (3.19) are critical values of Ψ̃ with respect to Mα and

−∞ < c(m,α) ≤ c(m+1,α) < ∞ for every m ∈ N.

Proof. For each fixed m ∈ N, we claim that

c(m) := inf{K(u) : u ∈ Zm, ∥u∥Y = 1} > 0. (3.20)

Indeed, assume that c(m) = 0. Then there exists a sequence {uj} ⊂ Zm such that
∥uj∥Y = 1 and

0 ≤ K(uj) ≤
1

j
. (3.21)

Since the sequence {uj} is bounded in Y , there exist a subsequence {uj′} of {uj}
and u0 ∈ Y such that uj′ → u0 weakly in Y as j′ → ∞. Since ⟨fk, uj′⟩Y ∗,Y = 0 for
any k > m, we have ⟨fk, u0⟩Y ∗,Y = 0 for all k > m, so we see that u0 ∈ Zm. Since
dimZm = m < ∞, uj′ → u0 strongly in Zm, so in Y . Thereby ∥u0∥Y = 1, so we
can see that K(u0) > 0. On the other hand, letting j′ → ∞ in (3.21), we see that
K(u0) = 0. This is a contradiction.

For 0 ̸= u ∈ Zm, since ∥u/∥u∥Y ∥Y = 1, it follows from (3.20) and (3.4) that

c(m) ≤ K
( u

∥u∥Y

)
=

∫
Γ2

1

∥u∥r(x)Y

G(x, u(x))dσx ≤ 1

∥u∥r+Y ∧ ∥u∥r−Y
K(u).

Thus we have K(u) ≥ c(m)∥u∥r+Y ∧ ∥u∥r−Y for all u ∈ Zm. Therefore, Zm ∩Mα is
a bounded and closed subset of Zm, so is compact by dimZm < ∞. Since K is an
even functional, Zm ∩Mα is clearly symmetric. Let G = {u = u1e1 + · · ·+umem ∈
Zm;K(u) < α}. Then G can be identified with an open neighborhood of 0 in Rm

by a trivial odd homeomorphism. Since the identity map: Zm ∩ Mα → ∂G is an
odd homeomorphism, using Proposition 3.14 (v), we have γ(Zm ∩ Mα) = m, so
Γm ̸= ∅. Since Γm+1 ⊂ Γm, we can see that −∞ < c(m,α) ≤ c(m+1,α) < ∞. □

Lemma 3.17. Assume that a functional χ : Y → R is sequentially weakly contin-
uous and satisfies χ(0) = 0. Then for any fixed r > 0,

lim
n→∞

sup
u∈Wn,∥u∥Y ≤r

|χ(u)| = 0. (3.22)

Proof. Put dn = supu∈Wn,∥u∥Y ≤r |χ(u)|. Then there exists uj ∈ Wn with ∥uj∥Y ≤ r

such that limj→∞ |χ(uj)| = dn. Since Y is a reflexive Banach space, there exist a

subsequence {uj′} of {uj} and u(n) ∈ Y such that uj′ → u(n) weakly in Y . Hence

∥u(n)∥Y ≤ lim infj′→∞ ∥uj′∥Y ≤ r. Since Wn is a closed subspace of Y , we see



22 J. ARAMAKI EJDE-2025/17

that Wn is weakly closed, so u(n) ∈ Wn. Since χ is sequentially weakly continuous,
|χ(uj′)| → |χ(u(n))| as j′ → ∞. Thereby |χ(u(n))| = dn. Since dn+1 ≤ dn for all

n ∈ N, limn→∞ dn = d0 ≥ 0 exists. Since {u(n)} satisfies ∥u(n)∥Y ≤ r, there exists

a subsequence {u(n′)} of {u(n)} and u0 ∈ Y such that u(n′) → u0 weakly in Y , so

∥u0∥Y ≤ r. Since again χ is sequentially weakly continuous, |χ(u(n′))| = dn′ →
|χ(u0)| = d0. Since Y is reflexive, we can look upon u0 ∈ Y ∗∗ = Y . Therefore, for

any fj ∈ Y ∗, since u(n′) ∈ Wn′ , we have

⟨u0, fj⟩Y ∗∗,Y ∗ = ⟨fj , u0⟩Y ∗,Y = lim
n′→∞

⟨fj , u(n′)⟩Y ∗,Y = 0.

Thus we have u0 = 0. Since χ(0) = 0, we have d0 = 0, that is, (3.22) holds. □

Proposition 3.18. We have limn→∞ infu∈Wn∩Mα
∥u∥Y = ∞.

Proof. Suppose that the conclusion is false. Then there exist c1 > 0 and un ⊂
Wn ∩Mα such that ∥un∥Y ≤ c1 for large n ∈ N. Then

sup
u∈Wn,∥u∥Y ≤c1

|K(u)| ≥ |K(un)| = α.

Therefore,

lim
n→∞

sup
u∈Wn,∥u∥Y ≤c1

|K(u)| ≥ lim
n→∞

|K(un)| = α > 0.

If we apply Lemma 3.17 with χ = K, this is a contradiction. □

Proposition 3.19. We have

lim
n→∞

c(n,α) = ∞. (3.23)

Proof. By Proposition 3.18, for any c > 1, there exists n0 ∈ N such that for
any n ≥ n0 and u ∈ Wn ∩ Mα, we have ∥u∥Y > c. For any H ∈ Σα, we have
γ(H ∩ Zn−1) ≤ n − 1. On the other hand, we have codimWn = n − 1. Hence
for any H ∈ Σα with γ(H) ≥ n, H ∩ Wn is non-empty. Indeed, since H =
(H ∩ Zn−1) ∪ (H ∩Wn), it follows from Proposition 3.14 (iii) that

n ≤ γ(H) ≤ γ(H ∩ Zn−1) + γ(H ∩Wn) ≤ n− 1 + γ(H ∩Wn),

so γ(H ∩Wn) ≥ 1. Hence H ∩Wn ̸= ∅. For n ≥ n0, using (3.19), we have

c(n,α) = inf
H∈Σα,γ(H)≥n

sup
u∈H

Ψ̃(u)

= inf
H∈Σα,γ(H)≥n

max
{

sup
u∈H∩(Y \Zn−1)

Ψ̃(u), sup
u∈H∩Zn−1

Ψ̃(u)
}

≥ inf
H∈Σα,γ(H)≥n

sup
u∈H∩(Y \Zn−1)

Ψ̃(u)

= inf
H∈Σα,γ(H)≥n

max
{

sup
u∈H∩((Y \Zn−1)\Wn)

Ψ̃(u), sup
u∈H∩Wn

Ψ̃(u)
}

≥ inf
H∈Σα,γ(H)≥n

sup
u∈H∩Wn

Ψ̃(u)

≥ inf
H∈Σα,γ(H)≥n

sup
u∈H∩Wn

m0

l

( k0
p+

∥u∥p
−

Y

)l

≥ m0

l

( k0
p+

cp
−
)l

.

Since c > 1 is arbitrary, we thus get (3.23). □



EJDE-2025/17 EIGENVALUE PROBLEMS FOR KIRCHHOFF-TYPE EQUATIONS 23

Theorem 3.20. Assume that (A2)–(A8) hold and fix α > 0. Then for every n ∈ N,
c(n,α) defined by (3.19) is a critical value of Ψ̃ with respect to the submanifold Mα

such that

0 < c(n,α) ≤ c(n+1,α) < ∞ and c(n,α) → ∞ as n → ∞.

Moreover, (1.1) has infinitely many eigenpair sequence {(u(n,α), λ(n,α))} such that

K(±u(n,α)) = α,Ψ(±u(n,α)) = c(n,α) and 0 < λ(n,α) → ∞ as n → ∞.

Proof. Taking Proposition 3.15, (3.12), (3.13), Lemma 3.16, and Proposition 3.19
into consideration, it suffices to show that λ(n,α) → ∞ as n → ∞. It follows from
(A8) that

⟨K ′(u(n,α)), u(n,α)⟩Y ∗,Y ≤ r+K(u(n,α)) = r+α.

Hence

λ(n,α) =
⟨Ψ′(u(n,α)), u(n,α)⟩Y ∗,Y

⟨K ′(u(n,α)), u(n,α)⟩Y ∗,Y
≥

⟨Ψ′(u(n,α)), u(n,α)⟩Y ∗,Y

r+α
. (3.24)

Assume that λ(n,α) ≤ M for all n ∈ N. Then by (3.24), ⟨Ψ′(u(n,α)), u(n,α)⟩Y ∗,Y ≤
Mr+α =: c2. On the other hand, from (A5), we have

⟨Ψ′(u(n,α)), u(n,α)⟩Y ∗,Y = M(Φ(u(n,α)))⟨Φ′(u(n,α)), u(n,α)⟩Y ∗,Y

≥ m0

(
Φ(u(n,α))

)l−1
∫
Ω

a(x,∇u(n,α)(x)) ·∇u(n,α)(x) dx

≥ m0

( 1

p+

∫
Ω

a(x,∇u(n,α)(x)) ·∇u(n,α)(x) dx
)l−1

×
∫
Ω

a(x,∇u(n,α)(x)) ·∇u(n,α)(x) dx

=
m0

(p+)l−1

(∫
Ω

a(x,∇u(n,α)(x)) ·∇u(n,α)(x) dx
)l

≥ m0

(p+)l−1
kl0

(∫
Ω

h1(x)|∇u(n,α)(x)|p(x)dx
)l

=
m0

(p+)l−1
kl0(ρ̃(p(·),h1(·))(u(n,α)))

l.

Therefore, we have ρ̃(p(·),h1(·))(u(n,α)) ≤ c3 for some constant c3. In particular,
∥u(n,α)∥Y ≤ c4 for all n ∈ N with some constant c4. Then from Lemma 3.7 (i),

Φ(u(n,α)) ≤ c(2∥h0∥Lp′(·)(Ω)∥u(n,α)∥Y + ρ̃(p(·),h1(·))(u(n,α))) ≤ c5

for some constant c5 > 0. Since M̂ is bounded for every bounded subset from

(A1), we see that c(n,α) = Ψ(u(n,α)) = M̂(Φ(u(n,α))) is bounded from above. This
contradicts Proposition 3.19. □

Remark 3.21. We do not know whether problem (1.1) only has eigenvalue se-
quences of the form {λ(n,α)}.

Remark 3.22. We assume the following more restrictive conditions instead of (A1)
and (A5):

(A1’) M : [0,∞) → [0,∞) is continuous and monotone non-decreasing, and there
exist 0 < m0 ≤ m1 < ∞ and l ≥ 1 such that

m0s
l−1 ≤ M(s) ≤ m1s

l−1 for s ≥ 0.



24 J. ARAMAKI EJDE-2025/17

(A5’) k0h1(x)|ξ|p(x) ≤ a(x, ξ) · ξ = p(x)A(x, ξ) for a.e. x ∈ Ω and all ξ ∈ RN .

We note that (i) in Example 3.2 satisfies (A5’), but (ii) does not satisfy this condi-
tion.

Under assumptions (A1)–(A4), (A6)–(A8), (A1’), and (A5’), we have

λ(n+1,α) ≥
p−r−m2

0

p+r+m2
1

λ(n,α). (3.25)

In particular, if p(x) = p (a constant), r(x) = r ( a constant) and m0 = m1, then
we have λ(n+1,α) ≥ λ(n,α).

Proof. Let un be the eigenfuntion associated with the eigenvalue λ(n,α) for n =
1, 2, . . .. From assumption(A5’), (A8) and Theorem 3.20, we have

λ(n+1,α) =
⟨Ψ′(un+1), un+1⟩Y ∗,Y

⟨K ′(un+1), un+1⟩Y ∗,Y
=

M(Φ(un+1))⟨Φ′(un+1), un+1⟩Y ∗,Y∫
Γ2

K ′(un+1), un+1⟩Y ∗,Y

=
M(Φ(un+1))

∫
Ω
a(x,∇un+1(x)) ·∇un+1(x) dx∫

Γ2
g(x, un+1(x))un+1(x)dσx

≥
m0Φ(un+1)

l−1
∫
Ω
p(x)A(x,∇un+1(x)) dx∫

Γ2
r(x)G(x, un+1(x))dσx

≥ m0p
−

r+α
Φ(un+1)

l

≥ m0p
−

r+α

l

m1
M̂(Φ(un+1))

=
m0p

−l

r+αm1
c(n+1,α) ≥

m0lp
−

r+αm1
c(n,α).

The last inequality follows from Theorem 3.20.
On the other hand, from (A1’), (A5’) and (A8), we have

c(n,α) = α
Ψ(un)

K(un)

= α
M̂(Φ(un))∫

Γ2
G(x, un(x))dσx

≥ α
m0

l Φ(un)
l∫

Γ2

1
r(x)g(x, un(x))un(x)dσx

≥
αm0

l Φ(un)
l−1Φ(un)

1
r− ⟨K ′(un), un⟩Y ∗,Y

≥
αm0

l
1

m1
M(Φ(un))

∫
Ω

1
p(x)a(x,∇un(x)) ·∇un(x) dx

1
r− ⟨K ′(un), un⟩Y ∗,Y

≥
αm0

l
1

m1

1
p+M(Φ(un))⟨Φ′(un), un⟩Y ∗,Y

1
r− ⟨K ′(un), un⟩Y ∗,Y

=
αm0r

−

lm1p+
⟨Ψ′(un), un⟩Y ∗,Y

⟨K ′(un), un⟩Y ∗,Y

=
αm0r

−

lm1p+
λ(n,α).



EJDE-2025/17 EIGENVALUE PROBLEMS FOR KIRCHHOFF-TYPE EQUATIONS 25

Thus we obtain the estimate (3.25). □

4. The infimum of all the eigenvalues

In this section, we consider the infimum of all the eigenvalues of the problem
(1.1). We show that there exist two cases where the infimum is equal to zero, and
positive according to the hypoetheses on the variable exponent.

Put Λ = {λ is an eigenvalue of problem (1.1)} and λ∗ = inf Λ. For a subset
A ⊂ Ω and δ > 0, put

B(A, δ) = {x ∈ RN ; dist(x,A) < δ}, BΩ(A, δ) = B(A, δ) ∩ Ω,

BΓ2
(A, δ) = B(A, δ) ∩ Γ2.

Here, for x0 ∈ Ω, if A = {x0}, then we simply write B({x0}, δ), BΩ({x0}, δ)
and BΓ2({x0}, δ) by B(x0, δ), BΩ(x0, δ) and BΓ2(x0, δ), respectively. Assume that
(A1)–(A8), hold.

Lemma 4.1. For δ, α > 0, if we define

βδ(u) =

∫
BΩ(Γ2,δ)

h1(x)|∇u(x)|p(x) dx for u ∈ Y,

then we have
β(δ,α) := inf

u∈Mα

βδ(u) > 0.

Proof. First we consider Y (BΩ(Γ2, δ)). We extend the function b(x) on Γ2 in (A8)

to a function b̃(x) on Γ̃2, where Γ̃2 := Γ2 ∪ (∂BΩ(Γ2, δ) \ Γ) by a positive constant

outside ∂BΩ(Γ2, δ) \ Γ, and define G̃ and K̃ as in (3.10) and (3.12), respectively.

Since δ > 0, we have Γ̃1 := ∂BΩ(Γ2, δ) ∩ Γ1 ̸= ∅, so Y (BΩ(Γ2, δ)) is the same

properties as Y , if we replace Γ1 in Y with Γ̃1. Thus Y ↪→ Y (BΩ(Γ2, δ)) and βδ is
a modular on Y (BΩ(Γ2, δ)).

Assume that β(δ,α) = 0. Then there exist {un} ⊂ Mα such that βδ(un) →
0 as n → ∞. Hence ∥un∥Y (BΩ(Γ2,δ))

→ 0 as n → ∞, where ∥u∥Y (BΩ(Γ2,δ))
=

inf
{
τ > 0;βδ

(
u
τ

)
≤ 1

}
.

On the other hand, we have

K̃(un) =

∫
∂BΩ(Γ2,δ))

G̃(x, un(x))dσx ≥
∫
Γ2

G(x, un(x))dσx = K(un) = α > 0.

Since K̃ is continuous on Y (BΩ(Γ2, δ)), we can see that K̃(un) → K̃(0) = 0. This
is a contradiction. □

Lemma 4.2. For α > 0, let u0 be an eigenfunction associated with λ(1,α). Then

Ψ(u0) = c(1,α) = inf{Ψ(u);u ∈ Mα}.

Proof. Put bα = inf{Ψ(u);u ∈ Mα}. Since c(1,α) = infH∈Σα,γ(H)≥1 supu∈H Ψ̃(u),

if u ∈ H and H ∈ Σα ⊂ Mα with γ(H) ≥ 1, then Ψ̃(u) = Ψ(u) ≥ bα. Thus
c(1,α) ≥ bα.

By the definition of bα, there exists a sequence {un} ⊂ Mα such that bα =

limn→∞ Ψ(un). For large n, bα + 1 ≥ Ψ(un) = M̂(Φ(un)) ≥ m0

l ( k0

p+ ∥un∥p
+

Y ∧
∥un∥p

−

Y )l. Thus {un} is bounded in Y . So there exist a subsequence {un′} of {un}
and u∗ ∈ Y such that un′ → u∗ weakly in Y . Then Ψ(u∗) ≤ lim infn′→∞ Ψ(un′) =



26 J. ARAMAKI EJDE-2025/17

bα. Since Mα is a weakly closed subset of Y , u∗ ∈ Mα, so Ψ(u∗) ≥ bα. Thus we
have Ψ(u∗) = bα. By (A7), Ψ(±u∗) = bα. LetH0 = {±u∗}, then clearly γ(H0) = 1.
Therefore, c(1,α) ≤ supu∈H0

Ψ(u) = bα. Thus we have c(1,α) = bα. □

From now on, we suppose that the following more restrictive assumption than
(A8) on the given function g hold.

(A8’) (A8) holds with r(x) = lp(x), where l is a constant in (A1), that is, g(x, t) =
b(x)|t|lp(x)−2t with a function b(x) satisfying the condition in (A8) with
r(x) = lp(x).

Theorem 4.3. Assume that (A1)–(A7), (A8’) hold, moreover, suppose that there
exists δ > 0 such that p(x) = p (a constant) for all x ∈ BΩ(Γ2, δ). Then we have
λ∗ > 0.

Proof. Let u be the eigenfunction of problem (1.1), associated with λ. ThenK(u) >
0. In fact, let K(u) = 0. Since ⟨Ψ′(u), u⟩Y ∗,Y = λ⟨K ′(u), u⟩Y ∗,Y , it follows from
(A8’) that

M(Φ(u))

∫
Ω

a(x,∇u(x)) ·∇u(x) dx

= λ

∫
Γ2

g(x, u(x))u(x)dσx

= λlp

∫
Γ2

G(x, u(x))dσx

= λlpK(u) = 0.

Hence from (A5) and M(Φ(u)) > 0, we have

0 =

∫
Ω

a(x,∇u(x)) ·∇u(x) dx ≥ k0

∫
Ω

h1(x)|∇u(x)|p(x) dx.

Thus we have ∇u(x) = 0 a.e. x ∈ Ω. From Proposition 2.14, we have u = 0 a.e. in
Ω. This is a contradiction.

We show that there exists t0 > 0 such that u1 := 1
t0
u ∈ M1. Indeed, since

g(x, t)t = lpG(x, t) for σ-a.e. x ∈ Γ2 and all t ∈ R, it follows from (3.4) that
K(ut ) = t−lpK(u) for t > 0. Here we can see that K(ut ) → 0 as t → ∞ and
K(ut ) → ∞ as t → +0. Since K(ut ) is continuous with respect to t ∈ (0,∞),
it follows from the intermediate value theorem that there exists t0 > 0 such that
K( u

t0
) = 1, so u1 := u

t0
∈ M1.

Now since K(u1) = 1, it follows from (A1), (A5), (A8’), and Lemma 4.1 that

λ =
⟨Ψ′(u), u⟩Y ∗,Y

⟨K ′(u), u⟩Y ∗,Y

=
M(Φ(u))

∫
Ω
a(x,∇u(x)) ·∇u(x) dx∫

Γ2
g(x, u(x))u(x)dσx

≥
m0

( ∫
Ω

1
p(x)a(x,∇u(x)) ·∇u(x) dx

)l−1 ∫
Ω
a(x,∇u(x)) ·∇u(x) dx∫

Γ2
g(x, u(x))u(x)dσx

≥ m0

(p+)l−1

( ∫
BΩ(Γ2,δ)

a(x,∇u(x)) ·∇u(x) dx
)l

lp
∫
Γ2

G(x, u(x))dσx



EJDE-2025/17 EIGENVALUE PROBLEMS FOR KIRCHHOFF-TYPE EQUATIONS 27

≥ m0k
l
0

lp(p+)l−1

( ∫
BΩ(Γ2,δ)

h1(x)|∇u(x)|pdx
)l∫

Γ2
G(x, u(x))dσx

=
m0k

l
0

lp(p+)l−1

( ∫
BΩ(Γ2,δ)

h1(x)|t0∇u1(x)|pdx
)l∫

Γ2
G(x, t0u1(x))dσx

=
m0k

l
0

lp(p+)l−1

tlp0
( ∫

BΩ(Γ2,δ)
h1(x)|∇u1(x)|pdx

)l
tlp0

∫
Γ2

G(x, u1(x))dσx

≥ m0k
l
0

lp(p+)l−1
βl
(δ,1) > 0.

Thus we have λ∗ = inf Λ ≥ m0k
l
0

lp(p+)l−1 β
l
(δ,1) > 0. □

Next we will treat the case λ∗ = 0. From the absolute continuity of integral, we
obtain the following lemma which is needed later.

Lemma 4.4. Let u ∈ Y be given. Then for any ε > 0, there exists δ0 > 0 such
that for any 0 < δ < δ0,

βu(δ) :=

∫
BΩ(Γ2,δ)

A(x,∇u(x)) dx < ε.

Theorem 4.5. Assume that (A2)–(A7), (A1’), (A8’) hold. Moreover, suppose that
there exist δ > 0 and x0 ∈ Γ2 such that the following hold:

(i) p(x) = p (a constant) for all x ∈ BΓ2
(x0, δ).

(ii) p(x) < p for all x ∈ BΩ(x0, δ).
(iii) h1 ∈ L1(BΩ(x0, δ)), where h1 is the function of (A3)–(A5).

Then we have limα→∞ λ(1,α) = 0, so λ∗ = 0.

Proof. Replacing δ > 0 with smaller one, if necessary, we may assume thatB(x0, δ)∩
Γ ⊂ Γ2. Choose 0 ≤ u ∈ C∞(Ω) such that u(x) = 1 for x ∈ BΩ(x0, δ/4) and
u(x) = 0 for x ∈ Ω \BΩ(x0, δ/2). We note that from (iii) it follows that u ∈ Y . By
Lemma 4.4, for any ε > 0, there exists δ0 ∈ (0, δ/4) such that for each δ1 ∈ (0, δ0),( ∫

BΩ(Γ2,δ1)
A(x,∇u(x)) dx

)l
K(u)

< ε/(2c), where c =
m2

1p
+

lm0p−
2l−1.

Since p ∈ C(Ω), it follows from (ii) that for any x ∈ BΩ(x0, δ/2) \ B(Γ2, δ0), we
have

p(x)− p ≤ p+(BΩ(x0, δ/2) \B(Γ2, δ0))− p := −ε0 < 0.

We note that p(x) = p on suppu ∩ Γ2. If we define h(t) = K(tu) = tlpK(u), then
h is differentiable in (0,∞) and h′(t) = lptlp−1K(u) > 0, so h is strictly monotone
increasing and clearly h(t) → 0 as t → +0 and h(t) → ∞ as t → ∞. Hence for
any α > 0, there exists unique t(α) > 0 such that t(α)u ∈ Mα. Clearly t(α) → 0
as α → +0 and t(α) → ∞ as α → ∞. So there exists α0 > 1 such that for

any α ∈ (α0,∞), max{1, (2ε−1cΦ(u)l

K(u) )
1/(lε0)} < t(α). Let u0 be the eigenfunction

associated with λ(1,α). Then from (A1’) we have

λ(1,α) =
M(Φ(u0))

∫
Ω
a(x,∇u0(x)) ·∇u0(x) dx∫

Γ2
g(x, u0(x))u0(x) dσx



28 J. ARAMAKI EJDE-2025/17

≤
m1Φ(u0)

l−1
∫
Ω
p(x)A(x,∇u0(x)) dx

lp−
∫
Γ2

G(x, u0(x))dσx

≤ m1p
+Φ(u0)

l

lp−α

≤ m1p
+Ψ(u0)

m0p−α
.

By Lemma 4.2, since Ψ(u0) = c(1,α) = inf{Ψ(u);u ∈ Mα}, we have Ψ(u0) ≤
Ψ(t(α)u). Hence

λ(1,α) ≤
m1p

+Ψ(t(α)u)

m0p−α
=

m1p
+

m0p−
Ψ(t(α)u)

K(t(α)u)
.

Thus using (3.1), we have

λ(1,α) ≤
m1p

+

m0p−
Ψ(t(α)u)

K(t(α)u)

≤ m1p
+

m0p−

m1

l

( ∫
Ω
A(x, t(α)∇u(x)) dx

)l∫
Γ2

G(x, t(α)u(x))dσx

≤ c

( ∫
BΩ(x0,δ/2)\BΩ(Γ2,δ1)

A(x, t(α)∇u(x)) dx
)l∫

BΓ2
(x0,δ/2)

G(x, t(α)u(x))dσx

+ c

( ∫
BΩ(x0,δ/2)∩BΩ(Γ2,δ1)

A(x, t(α)∇u(x)) dx
)l∫

BΓ2
(x0,δ/2)

G(x, t(α)u(x))dσx

≤ c

(∫
BΩ(x0,δ/2)\BΩ(Γ2,δ1)

t(α)p(x)A(x,∇u(x)) dx
)l

t(α)lp
∫
BΓ2

(x0,δ/2)
G(x, u(x))dσx

+ c

( ∫
BΩ(x0,δ/2)∩BΩ(Γ2,δ1)

t(α)p(x)A(x,∇u(x)) dx
)l

t(α)lp
∫
BΓ2

(x0,δ/2)
G(x, u(x))dσx

= c

( ∫
BΩ(x0,δ/2)\BΩ(Γ2,δ1)

t(α)p(x)−pA(x,∇u(x)) dx
)l∫

BΓ2
(x0,δ/2)

G(x, u(x))dσx

+ c

( ∫
BΩ(x0,δ/2)∩BΩ(Γ2,δ1)

t(α)p(x)−pA(x,∇u(x)) dx
)l∫

BΓ2
(x0,δ/2)

G(x, u(x))dσx

≤ ct(α)−lε0
Φ(u)l

K(u)
+ c

( ∫
BΩ(Γ2,δ1)

A(x,∇u(x)) dx
)l

K(u)

<
ε

2
+

ε

2
= ε.

Therefore, 0 < λ(1,α) < ε for all α > α0. Since ε > 0 is arbitrary, we have
limα→∞ λ(1,α) = 0. □

Remark 4.6. (1) If p(x) = p (a constant) in Ω, then it is well known that λ∗ =
λ(1,α) = λ1 and so λ∗ is a principal eigenvalue.



EJDE-2025/17 EIGENVALUE PROBLEMS FOR KIRCHHOFF-TYPE EQUATIONS 29

(2) For a variable exponent p(x), under some assumptions, λ∗ = 0. This means
that under some assumptions, there does not exist a principal eigenvalue and the
set of eigenvalues is not closed.

(3) For a variable exponent p(x), under some assumptions, we have λ∗ > 0.

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Junichi Aramaki

Division of Science, Faculty of Science and Engineering, Tokyo Denki University,
Hatoyama-machi, Saitama 350-0394, Japan

Email address: aramaki@hctv.ne.jp


	1. Introduction
	2. Preliminaries
	3. Assumptions and main theorem
	4. The infimum of all the eigenvalues
	References