Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 14, pp. 1–10.

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

EXISTENCE AND MULTIPLICITY RESULTS FOR

SUPERCRITICAL NONLOCAL KIRCHHOFF PROBLEMS

GIOVANNI ANELLO

Abstract. We study the existence and multiplicity of solutions for the non-

local perturbed Kirchhoff problem

−
(
a+ b

∫
Ω
|∇u|2 dx

)
∆u = λg(x, u) + f(x, u), in Ω,

u = 0, on ∂Ω,

where Ω is a bounded smooth domain in RN , N > 4, a, b, λ > 0, and f, g :

Ω× R→ R are Carathéodory functions, with f subcritical, and g of arbitrary
growth. This paper is motivated by a recent results by Faraci and Silva [4]

where existence and multiplicity results were obtained when g is subcritical

and f is a power-type function with critical exponent.

1. Introduction

Let Ω be a bounded smooth domain in RN , with N > 4, and let a, b, λ > 0. In
this article, we study the nonlocal Kirchhoff problem

−
(
a+ b

∫
Ω

|∇u|2 dx
)

∆u = λg(x, u) + f(x, u), in Ω,

u = 0, on ∂Ω,
(1.1)

where f, g : Ω× R→ R are Carathéodory functions satisfying

σf := ess sup(x,t)∈Ω×R
|f(x, t)|

1 + |t|p−1
< +∞, (1.2)

for some p ∈ (2, 2∗), where 2∗ = 2N
N−2 ;

ρg(C) := ess sup(x,t)∈Ω×[−C,C] |g(x, t)| < +∞, for each C > 0. (1.3)

Recall that 2∗ is the critical Sobolev exponent for the embedding Lm(Ω) ↪→W 1,2
0 (Ω).

Since we are assuming N > 4, one has 2∗ < 4.
Our aim is to establish some existence and multiplicity results for problem (1.1)

without assuming any other conditions on g, except the summability condition
(1.3). This paper is motivated by the results recently obtained by Faraci and Silva

2020 Mathematics Subject Classification. 35J20, 35J25.
Key words and phrases. Nonlocal problem; Kirchhoff equation; weak solution;

supercritical growth; variational methods.
©2023. This work is licensed under a CC BY 4.0 license.

Submitted March 20, 2022. Published February 15, 2023.

1



2 G. ANELLO EJDE-2023/14

[4] on the existence and multiplicity of solutions to the problem

−
(
a+ b

∫
Ω

|∇u|2 dx
)

∆u = λg(x, u) + |u|2
∗−2u, in Ω,

u = 0, on ∂Ω,
(1.4)

where a, b, λ > 0 and g satisfy the following conditions:

(A1) ess sup(x,t)∈Ω×R
|g(x,t)|

1+|t|p−1 < +∞, for some p ∈ (2, 2∗);

(A2) limt→0
g(x,t)
t = 0, uniformly for a.a. x ∈ Ω;

(A3) g(x, t)t > 0, for all t ∈ R \ {0} and for a.a. x ∈ Ω;
(A4) ess inf(x,t)∈Ω×A g(x, t) > 0, for some nonempty open set A ⊂ (0,∞).

In particular, under assumptions (A1)–(A4), Faraci and Silva proved in [4] that
problem (1.4) admits at least a nonzero solution if one of the following conditions
holds

• aN−4
2 b > C1(N) := 4(N−4)

N−4
2

N
N−2

2 cN
2∗

and λ is large,

• aN−4
2 b = C1(N).

Here, c2∗ is the best constant for the embedding L2∗
(Ω) ↪→ W 1,2

0 (Ω). Moreover,
a second solution is proved to exist for λ large, under the following more strict
condition on a, b

a
N−4

2 b ≥
( N

N − 2

)N−2
N

C1(N). (1.5)

Problem (1.1) is associated with the stationary version of the well known equation
proposed by Kirchhoff to describe the transversal oscillations of a stretched string.
For more details, we refer the reader to [4] or [7] and references therein. To the
best of our knowledge, the case in which problem (1.1) involves nonlinearities of
arbitrary growth has been addressed in few papers. Among them, we can cite
[1, 2, 3, 6]. However, in these papers only existence results were established.

We stress out that variational methods are not directly applicable when su-
percritical nonlinearities are involved. Usually, in this case, an auxiliary problem
involving a suitable truncation of the supercritical nonlinearity is introduced. After
that, one shows, by using L∞-norm estimates, that the solutions of the auxiliary
problem are also solutions of the original problem. We will make use of this tech-
nique to prove our main results.

Now we recall some basic concepts of variational methods. Let h : Ω × R → R
be a Carathéodory function and let H : Ω × R → R be the primitive of h, defined
by

H(x, ξ) =

∫ ξ

0

h(x, t)dt, for all (x, ξ) ∈ Ω× R. (1.6)

Consider the set Xh ⊆W 1,2
0 (Ω) given by

Xh =
{
u ∈W 1,2

0 (Ω) : x ∈ Ω→ H(x, u(x)) is summable in Ω
}

By Sobolev embeddings, the set Xh is the whole W 1,2
0 (Ω) whenever

ess sup(x,t)∈Ω×R
|H(x, t)|
1 + |t|2∗ < +∞.



EJDE-2023/14 SUPERCRITICAL NONLOCAL KIRCHHOFF PROBLEMS 3

Throughout the paper, if h : Ω × R → R and Xh are as above, we denote by
Ih : Xh → R the energy functional associated with the problem

−
(
a+ b

∫
Ω

|∇u|2 dx
)

∆u = h(x, u), in Ω,

u = 0, on ∂Ω,
(1.7)

which is defined by

Ih(u) =
a

2

∫
Ω

|∇u(x)|2 dx+
b

4

(∫
Ω

|∇u(x)|2 dx
)2

−
∫

Ω

H(x, u(x)) dx,

for all u ∈ Xh. By a solution of problem (1.7) we mean any function u ∈ W 1,2
0 (Ω)

satisfying, for each v ∈ W 1,2
0 (Ω), the following conditions: the function x ∈ Ω →

h(x, u(x))v(x) belongs to L1(Ω), and(
a+ b

∫
Ω

|∇u(x)|2 dx
)∫

Ω

∇u(x)∇v(x) =

∫
Ω

h(x, u(x))v(x) dx.

When Xh = W 1,2
0 (Ω) and Ih is differentiable in W 1,2

0 (Ω), the solutions of (1.7)

are exactly the critical points of Ih. We denote by Ĩλ the energy functional associ-
ated with problem (1.4), that is

Ĩλ := Ih, where h(x, t) = λg(x, t) + |t|2
∗−2t.

A key ingredient in the proofs of the results in [4] is the sequential weak lower
semicontinuity of the functional

Φ(u) =
a

2

∫
Ω

|∇u(x)|2 dx+
b

4

(∫
Ω

|∇u(x)|2 dx
)2

− 1

q

∫
Ω

|u(x)|q dx, u ∈W 1,2
0 (Ω),

when q = 2∗. It is well known that Φ is sequentially weakly lower semicontinuous
for 0 < q < 2∗, but this is not true, in general, if q = 2∗. In [4] the condition

a
N−4

2 b ≥ C1(N) assumes a key role since it just ensures the sequential weak lower

semicontinuity of Φ in the case q = 2∗. Thus, if one assumes a
N−4

2 b ≥ C1(N) and
the subcritical growth condition i) on g, one gets the sequential weak lower semicon-

tinuity of Ĩλ. When q > 2∗, the set Xh corresponding to h(x, t) = λg(x, t) + |t|q−2t

is strictly contained in W 1,2
0 (Ω) and, moreover, the functional Φ (and therefore also

the functional Ĩλ) is never sequentially weakly lower semicontinuous in Xh. Thus,
the arguments used in [4] cannot be applied when q > 2∗ and, in general, when a
nonlinearity of arbitrary growth is involved.

As said above, in the present paper, we address the question of the existence
and multiplicity of solutions to problem (1.1) in the case g has an arbitrary growth.
We will establish existence and multiplicity results by assuming only condition
αg) on g, and imposing (as in [4]) some constrains on a, b. However, differently
to the problem (1.4) considered in [4], where the parameter λ is multiplied by the
subcritical nonlinearity, in our case the parameter λ is multiplied by the nonlinearity
of arbitrary growth. This allows to deduce that the solutions of the auxiliary
truncated problem are also solutions of the original problem, for λ small enough.

Besides (A1), we assume on the nonlinearity f the following two additional
conditions:

(A5) lim supξ→0
1
ξ2

∫ ξ
0
f(x, t)dt < aλ1/2, uniformly for a.a. x ∈ Ω;

(A6) lim inf |ξ|→+∞
1
ξ2

∫ ξ
0
f(x, t)dt > aλ1/2, uniformly for a.a. x ∈ Ω.



4 G. ANELLO EJDE-2023/14

Here,

λ1 := inf
u∈W 1,2

0 (Ω)\{0}

∫
Ω
|∇u(x)|2 dx∫

Ω
|u(x)|2 dx

is the first eigenvalue of the Laplacian on Ω.
Under (A1), (A2), (A5), and (A6), we will be able to prove a multiplicity result

for problem (1.1) for all a, b > 0, with b ≤ β(a), where β(a) is a suitable number
depending on a), and for all λ small enough. We will also show that, if conditions
(A5) and (A6) are replaced by

(A7) lim infξ→0
1
ξ2

∫ ξ
0
f(x, t)dt > aλ1

2 , uniformly for a.a. x ∈ Ω,

an existence result can be proved, again for λ small, without imposing any constrain
on a, b.

Note that the constrain 0 < b ≤ β(a) is a sort of opposite condition to (1.5) as-

sumed in [4] (indeed, observe that (1.5) can be rewritten b ≥ β(a) := a−
N−2

4 C1(N)).
Our main results read as follows:

Theorem 1.1. Let a > 0. Assume f satisfying (A1), (A5), (A6), and g satisfying
(A2). Then, there exists β(a) > 0 with the following property: for each b ∈ (0, β(a)],
there exists λ(a, b) > 0, such that, for each λ ∈ [0, λ(a, b)], problem (1.1) admits at
least three distinct solutions.

Theorem 1.2. Let a, b > 0. Assume f satisfying (A1) and (A7) and g satisfying
(A2). Then, there exists λ(a, b) > 0 such that, for all λ ∈ [0, λ(a, b)], problem (1.1)
admits at least a nonzero solution.

2. Notation and preliminary lemmas

Throughout this paper, we use of the following notation:

(1) for each u ∈ W 1,2
0 (Ω), ‖u‖ :=

( ∫
Ω
|∇u(x)|2 dx

)1/2
denotes the Poincaré

norm of u;

(2) for each m ∈ [1,+∞[ and u ∈ Lm(Ω), ‖u‖m :=
( ∫

Ω
|u(x)|m dx

)1/m
denotes

the norm of u in the space Lm(Ω);
(3) for each u ∈ L∞(Ω), ‖u‖∞ := ess supx∈Ω |u(x)| denotes the norm of u in

the space L∞(Ω);
(4) for each m ∈ [1, 2∗],

cm := sup
u∈W 1,2

0 (Ω)\{0}

‖u‖m
‖u‖

denotes the best constant for the Sobolev embedding Lm(Ω) ↪→ W 1,2
0 (Ω).

Note that λ1 := c−2
2 .

(5) for each λ ∈ R and C > 0, gC : Ω× R→ R and hλ,C : Ω× R→ R are the
functions defined by

gC(x, t) =


g(x, t) if (x, t) ∈ Ω× [−C,C],

g(x,C) if (x, t) ∈ Ω× (C,+∞),

g(x,−C) if (x, t) ∈ Ω× (−∞,−C).

hλ,C(x, t) = λgC(x, t) + f(x, t), for each (x, t) ∈ Ω× R. (2.1)



EJDE-2023/14 SUPERCRITICAL NONLOCAL KIRCHHOFF PROBLEMS 5

The next lemmas provide regularity estimates for the solutions of the problem

−
(
a+ b

∫
Ω

|∇u|2 dx
)

∆u = hλ,C(x, u), in Ω,

u = 0, on ∂Ω,
(2.2)

In particular, by these estimates we will infer that, for certain values of C and λ,
every solution of (2.2) is also a solution of (1.1).

Lemma 2.1. It holds

tx ≤ γx + γx−yty, for each t, γ, x, y > 0, with x < y, (2.3)

Proof. One has ( t
γ

)x ≤ 1 if t ≤ γ,
( t
γ

)x ≤ ( t
γ

)y
if t ≥ γ.

Hence, ( t
γ

)x ≤ 1 +
( t
γ

)y
from which (2.3) follows. �

The following two regularity lemmas are well known (see for instance [8], Ap-
pendix B, and Theorem 8.16 of [5]).

Lemma 2.2. Let p ∈ [2, 2∗), K > 0, and let l : Ω × R → R be a Carathéodory
function such that |l(x, t)| ≤ K(1 + |t|p−1), for each t ∈ R and for a.a. x ∈ Ω.

Moreover, let u ∈W 1,2
0 (Ω) satisfying∫

Ω

∇u(x)∇v(x) dx =

∫
Ω

l(x, u(x))v(x) dx, for each v ∈W 1,2
0 (Ω).

Then, u ∈ C1,α(Ω), for some α ∈ (0, 1).

Lemma 2.3. Let s > N/2 and l ∈ Ls(Ω). Assume that u ∈W 1,2
0 (Ω) satisfies∫

Ω

∇u(x)∇v(x) dx =

∫
Ω

l(x)v(x) dx, for each v ∈W 1,2
0 (Ω).

Then, u ∈ L∞(Ω), and there exists a constant Ks > 0, independent of u, l, such
that ‖u‖∞ ≤ Ks‖l‖s.

Lemma 2.4. Let a, b, C, λ > 0 and let f, g : Ω × R → R satisfying conditions
(A1) and (A2), respectively. Then, there exists a constant γ > 0, independent of
a, b, C, λ, such that, for every solution u of problem (2.2) one has

‖u‖2∗ ≤ γ
[
λb−1ρg(C) + b−1 + b

3
p−4
]1/3

(2.4)

Proof. Let u ∈W 1,2
0 (Ω) be a solution of (2.2). Then

0 = I ′hλ,C (u)(u) = (a+ b‖u‖2)‖u‖2 −
∫

Ω

hλ,C(x, u(x))u(x) dx. (2.5)

Moreover, one has

(a+ b‖u‖2)‖u‖2 ≥ b‖u‖4 ≥ bc−4
2∗ ‖u‖42∗

and, by (2.5) and conditions (A1) and (A2),

(a+ b‖u‖2)‖u‖2 =

∫
Ω

hλ,C(x, u(x))u(x) dx ≤ λρg(C)‖u‖1 + σf‖u‖1 + σf‖u‖pp.



6 G. ANELLO EJDE-2023/14

Consequently,

b‖u‖32∗ ≤ σ1(λρg(C) + 1 + ‖u‖p−1
2∗ ), (2.6)

for some constant σ1 > 0 independent of a, b, λ, C.
Recall that, since N > 4, one has 2∗ < 4. Therefore, one has p− 1 < 2∗− 1 < 3.

Then applying Lemma 2.1 with t = ‖u‖2∗ , γ = (2−1σ−1
1 b)

1
p−4 , x = p−1, and y = 3,

one obtains

‖u‖p−1
2∗ ≤

(
2−1σ−1

1 b
) p−1
p−4 + 2−1σ−1

1 b‖u‖32∗ . (2.7)

By (2.6) and (2.7), one has

‖u‖32∗ ≤ 2σ1b
−1
(
λρg(C) + 1 + (2−1σ−1

1 b)
p−1
p−4

)
from which (2.4) easily follows. �

Lemma 2.5. Let a, b > 0 and let f, g : Ω × R → R satisfying conditions (A1)
and (A2), respectively. Then, there exists C = C(a, b), such that for every λ ∈
(0, ρg(C)−1) and every solution u of (2.2), one has ‖u‖∞ ≤ C.

Proof. Since 2 < p < 2∗ and N
2 = 2∗

2∗−2 , we can fix s ∈ R such that

max
{N

2
,

2∗

p− 1

}
< s <

2∗

p− 2
.

Then

0 < p− 1− 2∗

s
< 1, 0 < 2− p+

2∗

s
< 1, s >

N

2
. (2.8)

Now, let C > 0, λ ∈ (0, ρg(C)−1), and let u ∈ W 1,2
0 (Ω) be a solution of (2.2). By

Lemma 2.2 we known that u ∈ C1(Ω). Hence, the function

x ∈ Ω→ hλ,C(x, u(x))

belongs to L∞(Ω). By Lemma 2.3, Lemma 2.4, conditions (A1), (A2), and (2.8),
we infer, recalling λρg(C) < 1, that

a‖uλ‖∞ ≤ σs
[
λρg(C) + 1 +

(∫
Ω

|uλ(x)|s(p−1) dx
)1/s]

≤ σs
[
λρg(C) + 1 + ‖uλ‖

p−1− 2∗
s∞ ‖uλ‖

2∗
s

2∗

]
≤ σ′s

[
λρg(C) + 1 + ‖uλ‖

p−1− 2∗
s∞

(
λb−1ρg(C) + b−1 + b

3
p−4

)2∗/(3s)]
≤ σ′′s

[
1 + ‖uλ‖

p−1− 2∗
s∞

(
b−1 + b

3
p−4

)2∗/(3s)]
where the constants σs, σ

′
s, σ
′′
s > 0 are independent of a, b, λ, C. In particular, if

‖uλ‖∞ ≥ 1, one has (in view of (2.8))

a‖uλ‖
2−p+ 2∗

s∞ ≤ σ′′s
[
1 +

(
b−1 + b

3
p−4

)2∗/(3s)]
Thus, if C is the constant defined by

C2−p+ 2∗
s = σ′′s a

−1
[
1 +

(
b−1 + b

3
p−4

)2∗/(3s)]
+ 1 (2.9)

one has in any case ‖uλ‖∞ ≤ C. �



EJDE-2023/14 SUPERCRITICAL NONLOCAL KIRCHHOFF PROBLEMS 7

3. Proofs of main results

Proof of Theorem 1.1. Let a, b > 0. By conditions (A1) and (A6), we can find two
constants µ, τ > 0 (both depending on a) such that∫ ξ

0

f(x, t)dt > λ1

(a
2

+ µ
)
ξ2 − τ, for each ξ ∈ R and a.a. x ∈ Ω.

Let ψ be the positive eigenfunction associated with λ1 and normalized with respect

to the norm ‖ · ‖. Moreover, put θ =
√

2τ |Ω|
µ .

By the above inequality, for b < β(a) := µ2

τ |Ω| , one gets

a

2
‖θψ‖2 +

b

4
‖θψ‖4 −

∫
Ω

∫ θψ(x)

0

f(x, t)dt <
aθ2

2
+
bθ4

4
−
(a

2
+ µ

)
θ2 + τ |Ω|

= −µθ2 +
bθ4

4
+ τ |Ω|

= −τ |Ω|+ 4bτ2|Ω|2

4µ2
< 0.

Thus, if we consider the functional If : W 1,2
0 (Ω)→ R defined by

If (u) =
a

2
‖u‖2 +

b

4
‖u‖4 −

∫
Ω

∫ u(x)

0

f(x, t)dt, for all u ∈W 1,2
0 (Ω),

we realize that

inf
W 1,2

0 (Ω)
If < 0, if 0 < b ≤ β(a). (3.1)

Now, by (A1) and (A5), we can also find two constants δ, η > 0 such that∫ ξ

0

f(x, t)dt ≤ λ1

(a
2
− η
)
|ξ|2 + δ|ξ|p,

for each ξ ∈ R and a.a. x ∈ Ω. Consequently,

If (u) ≥ a

2
‖u‖2 −

(a
2
− η
)
‖u‖2 −

δcpp
p
‖u‖p = η‖u‖2 −

δcpp
p
‖u‖p,

for each u ∈W 1,2
0 (Ω). Therefore, since p > 2, if we fix

0 < ε <
( ηp
δcpp

) 1
p−2

and take (3.1) into account, we obtain, for all b ∈ (0, β(a)),

inf
‖u‖=ε

If (u) > 0 = If (0) = inf
‖u‖≤ε

If (u) > inf
u∈W 1,2

0 (Ω)
If (u). (3.2)

Now, let λ ∈ R, and let C = C(a, b) > 0 be the constant defined in (2.9). Moreover,
let hλ,C be the function defined in (2.1). Since p < 2∗ < 4, by assumptions (1.2)
and (1.3), it easily follows that

lim
‖u‖→+∞

Ihλ,C (u) = +∞. (3.3)

Since, by standard results, Ihλ,C is sequentially weakly lower semicontinuous in

W 1,2
0 (Ω), we infer that Ihλ,C is bounded below on W 1,2

0 (Ω) as well. Consequently,



8 G. ANELLO EJDE-2023/14

we can consider the functions ω, ω1 : R→ R defined by

ω(λ) = inf
‖u‖=ε

Ihλ,C (u)− inf
‖u‖≤ε

Ihλ,C (u),

ω1(λ) = inf
‖u‖≤ε

Ihλ,C (u)− inf
u∈W 1,2

0 (Ω)
Ihλ,C (u)

for each λ ∈ R. Since λ ∈ R→ Ihλ,C (u) is an affine function for each u ∈W 1,2
0 (Ω),

we have that the functions ω, ω1 are both the difference of two concave functions,
and so they are continuous in R. By (3.2), we also have

ω(0) = inf
‖u‖=ε

If (u)− inf
‖u‖≤ε

If (u) > 0

ω1(0) = inf
‖u‖≤ε

If (u)− inf
u∈W 1,2

0 (Ω)
If (u) > 0

Thus, by the continuity of ω and ω1, we can find λ(a, b) ∈ (0, ρg(C)−1) such that

ω(λ) = inf
‖u‖=ε

Ihλ,C (u)− inf
‖u‖≤ε

Ihλ,C (u) > 0

ω1(λ) = inf
‖u‖≤ε

Ihλ,C (u)− inf
u∈W 1,2

0 (Ω)
Ihλ,C (u) > 0

for each λ ∈ [0, λ(a, b)]. Fix λ ∈ [0, λ(a, b)]. By the above two inequalities and by
the sequential weak lower semicontinuity of Ihλ,C , one infers that

• Ihλ,C admits a local minimum point uλ ∈W 1,2
0 (Ω), such that ‖uλ‖ < ε;

• Ihλ,C admits a global minimum point vλ ∈W 1,2
0 (Ω),

with

Ihλ,C (vλ) < Ihλ,C (uλ) = inf
‖u‖≤ε

Ihλ,C (u) < inf
‖u‖=ε

Ihλ,C (u).

Of course, uλ, vλ are critical points of Ihλ,C . Observe also that the inequality
Ihλ,C (vλ) < inf‖u‖≤ε Ihλ,C (u) implies ‖vλ‖ > ε. Hence, in particular, the functional
Ihλ,C turns out to have the mountain pass geometry. In addiction, we know, again
by standard results, that:

• the functional

u ∈W 1,2
0 (Ω)→ a

2
‖u‖2 +

b

4
‖u‖4

is differentiable in W 1,2
0 (Ω) with continuously invertible derivative;

• the functional

u ∈W 1,2
0 (Ω)→

∫
Ω

(∫ u(x)

0

hλ,C(x, t)dt
)
dx

is differentiable in W 1,2
0 (Ω) with compact derivative.

Therefore, taking (3.3) into account, we infer that Ihλ,C satisfies the Palais-Smale
condition (see, for instance, Example [9, 38.25]). By applying the classical Mountain
Pass Theorem by Ambrosetti-Rabinowitz, we derive the existence of a third critical
point wλ for Ihλ,C , which is of mountain pass type. Finally, since λ ∈ (0, ρg(C))
and C is as in (2.9), by Lemma 2.5 we conclude that uλ, vλ, wλ are three distinct
solutions of (1.1). �



EJDE-2023/14 SUPERCRITICAL NONLOCAL KIRCHHOFF PROBLEMS 9

Proof of Theorem 1.2. Let a, b > 0. By conditions (A1) and (A7), we can find two
constants µ, τ (depending on a) such that∫ ξ

0

f(x, t)dt >
(a

2
+ µ

)
ξ2 − τ |ξ|p, for each ξ ∈ R and a.a. x ∈ Ω.

Let ψ be the positive eigenfunction associated with λ1 and normalized with respect
to the norm ‖ · ‖. Since p > 2, one has

−µθ2‖ψ‖2 +
bθ4

4
‖ψ‖4 + τθp‖ψ‖pp = θ2

(
− µ+

bθ2

4
+ τθp−2‖ψ‖pp

)
< 0.

for θ > 0 small enough.
Fix such a θ and let C = C(a, b) be the constant defined in (2.9). By the previous

inequality, we can find λ(a, b) ∈ (0, ρg(C)) such that, for λ ∈ [0, λ(a, b)] and hλ,C
as in (2.1), one has

Ihλ,C (θψ) =
aθ2

2
+
bθ4

4
−
∫

Ω

(∫ θψ

0

hλ,C(x, t)dt
)
dx

≤ aθ2

2
+
bθ4

4
−
(a

2
+ µ

)
θ2 + τθp‖ψ‖pp + λθρg(C)‖ψ‖1

≤ θ2
(
− µ+

bθ2

4
+ τθp−2‖ψ‖pp

)
+ λθρg(C)‖ψ‖1 < 0

This means that

inf
W 1,2

0 (Ω)
Ihλ,C < 0.

Since Ihλ,C also satisfies the coercivity condition (3.3), then Ihλ,C admits a nonzero
global minimum point uλ, which is a solution of (1.1) in view of the condition
λ < ρg(C)−1 and Lemma 2.5. �

4. Conclusion

In this paper, we have considered a supercritical non local problem of Kirchhoff
type and we have proved, via variational methods and truncation arguments, both
existence and multiplicity results. The main feature of these results is that the
presence of the nonlocal term allows to obtain the multiplicity of solutions even
in the supercritical case. We point out that we found very few results where the
multiplicity of solutions is established for critical or supercritical problems. Among
them, we have mentioned the interesting paper [4]. In [4], the right hand-side in
the problem considered there is a sum of a subcritical nonlinearity multiplied by
a parameter λ and a critical nonlinearity (of power-type). Therefore, the problem
considered in [4] is different from problem (1.1) considered here, where, instead, the
parameter λ multiplies the supercritical term. We think that an interesting question
is to investigate, by the approach used in present paper, the possible extension to
the supercritical case of the multiplicity result obtained in [4].

References

[1] N. Azzouz, A. Bensedik; Existence results for an elliptic equation of Kirchhoff-type with

changing sign data. Funkc. Ekvacioj, Ser. Int., 55 (1), 55-66 (2012).
[2] S. Chen, V. D. Radulescu, X. Tang; Normalized solutions of nonautonomous Kirchhoff equa-

tions: sub- and super-critical cases. Appl. Math. Optim. 84,(1), 773-806, (2021).



10 G. ANELLO EJDE-2023/14

[3] F. J. S. A. Corrêa, G. M. Figueiredo; On the existence of positive solution for an elliptic

equation of Kirchhoff type via Moser iteration method. Bound. Value Probl. 2006, Article

ID 79679, 10 p. (2006).
[4] F. Faraci, K. Silva; On the Brezis-Nirenberg problem for a Kirchhoff type equation in high

dimension, Calc. Var. 60 (1), Paper no. 22, (2021), 33 pp.

[5] D. Gilbarg, N. S. Trudinger; Elliptic Partial Differentil Equations of Second Order, Berlin
Heidelberg, Springer Verlag (2001).

[6] Q. Li, Quanqing; K. Teng, X. Wu; Existence of nontrivial solutions for Schrödinger-Kirchhoff

type equations with critical or supercritical growth. Math. Methods Appl. Sci. 41(3), 1136-
1144 (2018).

[7] P. Pucci, V. D. Radulescu; Progress in nonlinear Kirchhoff problems. Nonlinear Anal. 186,

1–5 (2019).
[8] M. Struwe; Variational Methods. Applications to nonlinear partial differential equations and

Hamiltonian systems, Berlin, Springer (1996).
[9] E. Zeidler; Nonlinear Functional Analysis and its Applications III: Variational Methods and

Optimization, New York, Springer, 1985.

Giovanni Anello

Department of Mathematics and Computer Science, Physical Science and Earth Science,
University of Messina, Viale F. Stagno d’Alcontres 31, Italy

Email address: ganello@unime.it


	1. Introduction
	2. Notation and preliminary lemmas
	3. Proofs of main results
	4. Conclusion
	References