CUBO, A Mathematical Journal

Vol. 24, no. 03, pp. 501–519, December 2022

DOI: 10.56754/0719-0646.2403.0501

Infinitely many solutions for a nonlinear Navier
problem involving the p-biharmonic operator

Filippo Cammaroto1, B

1 Department of Mathematical and

Computer Sciences, Physical Sciences

and Earth Sciences, University of

Messina, Viale F. Stagno d’Alcontres, 31,

98166 Messina, Italy.

fdcammaroto@unime.it B

ABSTRACT

In this paper we establish some results of existence of in-

finitely many solutions for an elliptic equation involving the

p-biharmonic and the p-Laplacian operators coupled with

Navier boundary conditions where the nonlinearities depend

on two real parameters and do not satisfy any symmetric

condition. The nature of the approach is variational and the

main tool is an abstract result of Ricceri. The novelty in the

application of this abstract tool is the use of a class of test

functions which makes the assumptions on the data easier to

verify.

RESUMEN

En este art́ıculo establecemos algunos resultados sobre la ex-

istencia de infinitas soluciones para una ecuación eĺıptica que

involucra los operadores p-biarmónico y p-Laplaciano acopla-

dos con condiciones de borde de Navier, donde las nolinea-

lidades dependen de dos parámetros reales y no satisfacen

ninguna condición simétrica. La naturaleza del enfoque es

variacional y la herramienta principal es un resultado abs-

tracto de Ricceri. La novedad de la aplicación de esta he-

rramienta abstracta es el uso de una clase de funciones test

que hacen que las hipótesis sobre la data sean más fáciles de

verificar.

Keywords and Phrases: p-biharmonic operator, p-Laplacian operator, Navier problem, multiplicity.

2020 AMS Mathematics Subject Classification: 35J35, 35J60.

Accepted: 28 November, 2022

Received: 21 October, 2022

©2022 F. Cammaroto. This open access article is licensed under a Creative Commons

Attribution-NonCommercial 4.0 International License.

http://cubo.ufro.cl/
http://dx.doi.org/10.56754/0719-0646.2403.0501
https://orcid.org/0000-0001-8229-8848
mailto:fdcammaroto@unime.it


502 F. Cammaroto CUBO
24, 3 (2022)

1 Introduction

In this paper we investigate the existence of infinitely many solutions to the following p-biharmonic

elliptic equation with Navier conditions,


 ∆

2
pu − ∆pu + V (x)|u|p−2u = λf(x, u) + µg(x, u) in Ω

u = ∆u = 0 on Ω
(Pλ,µ)

where Ω ⊂ Rn (n ⩾ 1) is a bounded domain with smooth boundary ∂Ω, p > max
{
1, n

2

}
, ∆2pu =

∆(|∆u|p−2∆u) is the p-biharmonic operator, ∆pu = ∇(|∇u|p−2∇u) is the p-Laplacian operator,
V ∈ C(Ω) satisfying infΩ V > 0, f, g : Ω × R → R are two Carathéodory functions with suitable
behaviors, λ ∈ R and µ > 0.

In the last years several authors have showed their interest in fourth-order differential problems

involving biharmonic and p-biharmonic operators, motivated by the fact that this type of equations

finds applications in fields such as the elasticity theory, or more in general, in continuous mechanics.

In particular, the fourth-order elliptic equations can describe the static form change of beam or

the motion of rigid body, so they are widely applied in physics and engineering. In 1990 Lazer and

Mckenna, in a large paper in which they investigated the oscillatory phenomena that led to the

collapse of the Tacoma Narrows bridge, considered fourth-order problems with the nonlinearity

(u + 1)+ − 1; this nonlinearity is useful to study traveling waves in suspension bridges. Anyway
the same authors observed that this kind of problems are interesting also when this particular

nonlinearity is replaced by a somewhat more general function F(·, u) (see [24, 31, 32]).

As regards fourth-order differential problems involving biharmonic and p-biharmonic operators,

a non-negligible part of the literature is devoted to the study of the existence of infinitely many

solutions to problems involving only the biharmonic or p-biharmonic operator (see, for instance,

[2, 4, 5, 6, 9, 10, 17, 18, 19, 29, 30, 40]) or considering also the presence of Laplacian or p-Laplacian

operator ([22, 26, 38, 42, 43]) and/or a term with a potential function ([11, 12, 13, 25, 28]); some

authors have also recently considered the case in which a nonlocal term is present ([16, 41]).

Unlike some papers concerning problems set in an unbounded domain (see [2, 4, 11, 12, 13, 18,

19, 30] and above all [25] which inspired us in the choice of this type of problem), most of the

literature is devoted to the bounded case. In this case, different approaches have been adopted for

obtaining infinitely many solutions. In a lot of papers symmetry conditions on the nonlinearities are

assumed together with the use of the symmetric mountain pass theorem of Ambrosetti Rabinowitz

(see [26, 40]) or with the use of the fountain theorem ([38, 42, 43]).

In our investigation the approach is variational. More precisely we will apply the following critical

point theorem that Ricceri established in 2000 ([34, Theorem 2.5]), recalled below for the reader’s

convenience.



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Infinitely many solutions for a nonlinear Navier problem involving... 503

Theorem 1.1. Let X be a reflexive real Banach space, and let Φ, Ψ : X → R be two sequen-
tially weakly lower semicontinuous and Gâteaux differentiable functionals. Assume also that Ψ is

(strongly) continuous and coercive. For each r > infX Ψ, we put

φ(r) := inf
x∈Ψ−1(]−∞,r[)

Φ(x) − inf
Ψ−1(]−∞,r[)

ω
Φ

r − Ψ(x)

where Ψ−1(] − ∞, r[)w is the closure of Ψ
−1(] − ∞, r[) in the weak topology. Fixed λ ∈ R, then

a) if {rk} is a real sequence such that lim
k→∞

rk = +∞ and φ(rk) < λ, for each k ∈ N, the
following alternative holds: either Φ + λΨ has a global minimum or there exists a sequence

{xk} of critical points of Φ + λΨ such that lim
k→∞

Ψ(xk) = +∞;

b) if {sk} is a real sequence such that lim
k→∞

sk = (inf
x

Ψ)+ and φ(sk) < λ for each k ∈ N,
the following alternative holds: either there exists a global minimum of Ψ which is a local

minimum of Φ + λΨ or there exists a sequence {xk} of pairwise distinct critical points of
Φ + λΨ with lim

k→∞
Ψ(xk) = inf

X
Ψ, which weakly converges to a global minimum of Ψ.

Since its appearance in 2000 until our days, it has been a powerful tool to get multiplicity results

for different kinds of problems. In particular, it has been widely applied to obtain theorems

of existence of infinitely many solutions to problems associated with a vast range of differential

equations. In each of these applications, in order to guarantee that φ(rk) < λ (or φ(sk) < λ),

for each k ∈ N, and that the functional Φ + λΨ has no global minimum, it is necessary to use
some sequences of functions defined ad hoc. Generally, in these functions the norm of the variable

is raised to a suitable power which depends on the nature of the problem and that gives them

the requested regularity properties: in some applications the norm is used without power (see, for

instance, [3, 7, 14, 15, 23, 27, 39]), in some others it is raised to the second ([9, 10, 29, 33, 35, 36])

or to the third ([22, 28]) or to the fourth power ([1]); in [20, 21] the authors combined the norm

with trigonometric functions.

The choice of a particular sequence of functions inside the proof reflects heavily on the assumptions

and while there are some cases in which probably the choice is optimal, in some other cases it could

happen that a different choice of the sequence would make the result applicable in a greater number

of cases. This is the reason we have introduced an abstract class of test functions serving our

purpose. We will clarify this fact in Section 3, showing some examples. A similar line of reasoning

can be found in [8] and above all in [37] where the author does not choose the test functions

arbitrarily during the proof but he uses two generic functions whose properties are described in

the statement of his result.



504 F. Cammaroto CUBO
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2 Preliminaries

In this section we describe the variational framework in which we will work in our investigations.

To begin with, we denote by ω := π
n
2 /Γ

(
n
2
+ 1
)
the measure of the unit ball in Rn. If X is a

Banach space, the symbol B(x, r) stands for the open ball centered at x ∈ X and of radius r > 0.

Let Ω be a bounded smooth domain of Rn, n ≥ 1, p > max
{
1, n

2

}
and let V ∈ C(Ω) satisfy

infΩ V > 0. Put E = W
2,p(Ω) ∩ W 1,p0 (Ω); it is a reflexive Banach space when endowed with the

standard norm

∥u∥ =
(∫

Ω

|∆u|pdx
)1

p

.

Moreover, the assumptions on V assure that the position

∥u∥V =
(∫

Ω

(|∆u|p + |∇u|p + V (x)|u|p) dx
)1

p

for any u ∈ E, defines a norm equivalent to the standard one. Being p > n
2
, the Rellich-Kondrachov

theorem assures that E is compactly embedded in C0(Ω); in particular, there exists a constant

c∞ > 0 such that

∥u∥∞ ≤ c∞ ∥u∥ ≤ c∞ ∥u∥V (2.1)

for every u ∈ E. Now, motivated by the reasons that we have illustrated in the Introduction, let
us introduce the following class of functions. If {ak}, {bk}, {σk} are three real sequences with
0 < ak < bk and σk > 0, for each k ∈ N, let us denote by H({ak} , {bk} , {σk}) the space of all
sequences {χk} ⊂ W 2,p(]ak, bk[) satisfying

i) 0 ≤ χk(x) ≤ σk for a.e. x ∈]ak, bk[;

ii) lim
x→a+

k

χk(x) = σk, lim
x→b−

k

χk(x) = 0;

iii) lim
x→a+

k

χ′k(x) = lim
x→b−

k

χ′k(x) = 0;

iv) for all j ∈ {1, 2} there exists cj > 0, independent of k, such that

|χ(j)k (x)| ≤ cj
σk

(bk − ak)j
(2.2)

for a.e. x ∈]ak, bk[ and for all k ∈ N.

Now, we show how the space H({ak} , {bk} , {σk}) help us to build some sequences in E that play
a crucial role in the proof of the main result.

If x0 ∈ Ω, {bk} ⊂]0, +∞[ such that B(x0, bk) ⊂ Ω, for each k ∈ N, and {χk} ∈ H({ak} , {bk} , {σk}),
consider the function uk : Ω → R defined by setting



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Infinitely many solutions for a nonlinear Navier problem involving... 505

uk(x) =



0 in Ω \ B(x0, bk),

σk in B(x0, ak),

χk(|x − x0|) in B(x0, bk) \ B(x0, ak)

for each k ∈ N.

Simple computations show that, fixed k ∈ N, for each i ∈ {1, . . . , n}, we have

∂uk
∂xi

(x) =



0 in Ω \ B(x0, bk),

0 in B(x0, ak),

χ′k(|x − x0|)
xi − x0i
|x − x0|

in B(x0, bk) \ B(xo, ak)

and

∂2uk
∂x2i

(x) =



0 in Ω \ B(x0, bk),

0 in B(x0, ak),

χ′′k(|x − x0|)
(xi − x0i )

2

|x − x0|2
+ χ′k(|x − x0|)

|x − x0|2 − (xi − x0i )
2

|x − x0|3
in B(x0, bk) \ B(x0, ak)

Using these computations together with (2.2), we get the following inequalities

|∇uk(x)| ⩽ |χ′k(|x − x0|)| ≤ c1
σk

(bk − ak)
,

and

|∆uk(x)| ⩽ |χ′′k(|x − x0|)| + |χ
′
k(|x − x0|)|

(n − 1)
|x − x0|

≤ c2
σk

(bk − ak)2
+ c1

σk
(bk − ak)

(n − 1)
ak

.

These inequalities allow us to estimate the norm of the functions uk as follows

∥uk∥
p
V =

∫
Ω

(|∆uk|p + |∇uk|p + V (x)|uk(x)|p) dx

=

∫
B(x0,bk)\B(x0,ak)

|∆uk(x)|pdx +
∫
B(x0,bk)\B(x0,ak)

|∇uk(x)|pdx +
∫
B(x0,bk)

V (x)|uk(x)|pdx

≤ ωσpk

{[
c2

(bk − ak)2
+

c1(n − 1)
ak(bk − ak)

]p
(bnk − a

n
k) +

[
c1

(bk − ak)

]p
(bnk − a

n
k) + b

n
k max

B(x0,bk)
V

}
.

Let us denote by C the class of all Carathéodory functions η : Ω×R → R satisfying sup|t|≤ξ |η(·, t)| ∈
L1(Ω) for all ξ > 0 and let f, g ∈ C.



506 F. Cammaroto CUBO
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We say that a function u ∈ E is a weak solution to (Pλ,µ) if∫
Ω

(
|∆u|p−2∆u∆v + |∇u|p−2∇u∇v + V (x)|u|p−2uv

)
dx = λ

∫
Ω

f(x, u(x))v(x)dx

+ µ

∫
Ω

g(x, u(x))v(x)dx

for each v ∈ E. Obviously the weak solutions to (Pλ,µ) are exactly the critical points in E of the
energy functional defined, for each u ∈ E, by

E(u) :=
1

p
Ψ(u) + λΦF (u) + µΦG(u),

where

Ψ(u) := ∥u∥pV , ΦF (u) := −
∫
Ω

F(x, u(x))dx, ΦG(u) := −
∫
Ω

G(x, u(x))dx,

where, for each (x, t) ∈ Ω × R,

F(x, t) :=

∫ t
0

f(x, s)ds, G(x, t) :=

∫ t
0

g(x, s)ds.

3 Results

The first multiplicity result deals with the case in which f has a global (m − 1)-sublinear growth,
with m < p, while different cases are considered for the behaviour of function g.

Theorem 3.1. Let V ∈ C(Ω) satisfy infΩ V > 0 and let f, g ∈ C such that:

(i1) there exist 1 < m < p and h ∈ L1(Ω) such that |f(x, t)| ≤ h(x)
(
1 + |t|m−1

)
for a.e. x ∈ Ω

and for all t ∈ R,

(i2) G(x, t) ≥ 0 for a.e. x ∈ Ω and for all t ≥ 0,

(i3) there exists x0 ∈ Ω and ρ > 0, p1, p2 > 1 such that B(x0, ρ) ⊆ Ω and

lim inf
t→+∞

∫
Ω
max|ξ|≤t G(x, ξ)dx

tp1
:= a < +∞, lim sup

t→+∞

∫
B(x0,ρ)

G(x, t)dx

tp2
:= b > 0.

Then the following facts hold:

(r1) if p1 < p < p2, for all λ ∈ R and for all µ > 0, the problem (Pλ,µ) admits a sequence of
non-zero weak solutions;

(r2) if p1 < p = p2, there exists µ1 > 0 such that for all λ ∈ R and for all µ > µ1, the problem
(Pλ,µ) admits a sequence of non-zero weak solutions;



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Infinitely many solutions for a nonlinear Navier problem involving... 507

(r3) if p1 = p < p2, there exists µ2 > 0 such that for all λ ∈ R and for all µ ∈]0, µ2[, the problem
(Pλ,µ) admits a sequence of non-zero weak solutions;

(r4) if p1 = p2 = p, there exists γ > 1 and CV,γ,ρ > 0 such that, if

CV,γ,ρ <
b

ωc
p
∞a

, (3.1)

(the previous inequality always being satisfied whether a = 0 or b = +∞) then µ1 < µ2 and
for all λ ∈ R and for all µ ∈]µ1, µ2[, the problem (Pλ,µ) admits a sequence of non-zero weak
solutions.

Proof. To prove (r1), let us apply part a) of Theorem 1.1 choosing X = E, Ψ defined as in

the Preliminaries and Φ = λΦF + µΦG. As we have already observed the critical points of the

functional Φ + 1
p
Ψ are precisely the weak solution of problem (Pλ,µ). The functionals Φ and Ψ are

sequentially weak lower semicontinuous and moreover Ψ is strongly continuous and coercive. In

our case the function φ is defined by setting

φ(r) = inf
∥u∥p

V
<r

Φ(u) + sup∥w∥p
V
≤r (−Φ)

r − ∥u∥pV

for each r > 0. Now, we wish to find a sequence {rk}k∈N such that lim
k→∞

rk = +∞ and φ(rk) < 1p
for each k ∈ N. To this aim it suffices to prove that for each k ∈ N there exists a function uk ∈ X,
with ∥uk∥

p
V < rk, such that

sup
∥w∥p

V
≤rk

{
λ

∫
Ω

F(x, w(x))dx + µ

∫
Ω

G(x, w(x))dx

}
− λ

∫
Ω

F(x, uk(x))dx+

−µ
∫
Ω

G(x, uk(x))dx <
1

p
(rk − ∥uk∥

p
V ) .

Thanks to (i3), fixed a > a, for each k ∈ N there exists αk ≥ k such that∫
Ω

max
|ξ|≤αk

G(x, ξ)dx ≤ aαp1k .

Now we choose uk = θE and

rk =
1

c
p
∞
α
p
k.

Obviously we have lim
k→∞

rk = +∞. Before proving (3), observe that, for each w ∈ X with ∥w∥
p
V ≤

rk, one has

∥w∥∞ ≤ c∞ ∥w∥V ≤ c∞r
1
p

k = αk

for each k ∈ N. Therefore, we obtain



508 F. Cammaroto CUBO
24, 3 (2022)

λ

∫
Ω

F(x, w(x))dx + µ

∫
Ω

G(x, w(x))dx ≤ |λ|
∫
Ω

|h(x)|
(
|w(x)| +

|w(x)|m

m

)
dx

+ µ

∫
Ω

max
|ξ|≤αk

G(x, ξ)dx ≤ |λ|∥h∥1
(
αk +

αmk
m

)
+ µaα

p1
k

≤ |λ|∥h∥1c∞r
1
p

k +
|λ|
m

∥h∥1cm∞r
m
p

k + µac
p1
∞r

p1
p

k <
1

p
rk

for k large enough, being 1 < m < p and p1 < p. So, thanks to part a) of Theorem 1.1, the

functional Φ + 1
p
Ψ has a global minimum, or there exists a sequence of weak solutions {uk} ⊂ E

such that lim
k→∞

∥uk∥ = +∞. This part of the proof will end if we show that the functional Φ + 1pΨ
has no global minimum. To this aim, using (i3), fixed 0 < b < b, we get βk ∈]0, +∞[ with βk ≥ k,
such that ∫

B(x0,ρ)

G(x, βk)dx ≥ bβ
p2
k

for each k ∈ N. After choosing γ > 1 such that B(x0, γρ) ⊆ Ω and a sequence {χk} ∈
H(ρ, γρ, {αk}), we consider

wk(x) =



0, in Ω \ B(x0, γρ),

βk, in B(x0, ρ),

χk(|x − x0|) in B(x0, γρ) \ B(x0, ρ).

Using the estimation of the norm made in the previous section, we get

∥wk∥
p
V ≤ ωβ

p
k

[
2p−1(γn − 1)
ρ2p−n(γ − 1)2p

c
p
2 +

(2p−1(n − 1)p + ρp)(γn − 1)
ρ2p−n(γ − 1)p

c
p
1 + γ

nρn max
B(x0,γρ)

V

]
.

If we put

CV,γ,ρ =
2p−1(γn − 1)
ρ2p−n(γ − 1)2p

c
p
2 +

(2p−1(n − 1)p + ρp)(γn − 1)
ρ2p−n(γ − 1)p

c
p
1 + γ

nρn max
B(x0,γρ)

V

we have

Φ(wk) +
1

p
Ψ(wk) = −λ

∫
Ω

F(x, wk(x))dx − µ
∫
Ω

G(x, wk(x))dx +
1

p
∥wk∥

p
V

≤ |λ|
∫
Ω

|h(x)|
(
|wk(x)| +

|wk(x)|m

m

)
dx − µ

∫
B(x0,ρ)

G(x, βk)dx +
ωCV,γ,ρ

p
β
p
k

≤ |λ|∥h∥1βk + |λ|∥h∥1
βmk
m

− µbβp2k +
ωCV,γ,ρ

p
β
p
k

and, since 1 < m < p < d2 and lim
k→∞

βk = +∞, the functional Φ + 1pΨ has no global minimum,

being lim
k→∞

Φ(wk) +
1

p
Ψ(wk) = −∞. This concludes the proof of (r1).



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Infinitely many solutions for a nonlinear Navier problem involving... 509

The proof of (r2) is similar. If p1 < p and p2 = p, we choose µ1 =
ωCV,γ,ρ

pb
(obviously if b = +∞ we

choose µ1 = 0). Therefore, if λ ∈ R and µ > µ1, choosing b such that
ωCV,γ,ρ

pµ
< b < b, in a similar

way we have

Φ(wk) +
1

p
Ψ(wk) ≤ |λ|∥h∥1βk + |λ|∥h∥1

βmk
m

−
(
µb −

ωCV,ρ,γ
p

)
β
p
k

and, thanks to the choice of b, also in this case the functional Φ + 1
p
Ψ has no global minimum.

This concludes (r2).

As for the proof of (r3), if p1 = p and p2 > p, we choose µ2 =
1

pc
p
∞a

(obviously if a = 0 we choose

µ2 = +∞). Then, fixing λ ∈ R and 0 < µ < µ2, we can choose a such that a < a < 1pcp∞µ. Similar
computations give

λ

∫
Ω

F(x, w(x))dx + µ

∫
Ω

G(x, w(x))dx ≤ |λ|∥h∥1c∞r
1
p

k +
|λ|
m

∥h∥1cm∞r
m
p

k + µac
p
∞rk <

1

p
rk

for k large enough, being 1 < m < p and µacp∞ <
1
p
.

Finally, the proof of (r4) relies on the considerations made in the previous two cases. We have

only to prove that µ1 < µ2, but this is guaranteed by the assumption (3.1).

Now, we are interested in the existence of infinitely many weak solutions in the case that the

nonlinearities f and g have a particular form.

Theorem 3.2. Let V ∈ C(Ω) satisfy infΩ V > 0, m < p, h ∈ L1(Ω), and r ∈ L1(Ω) \ {0} with
r ≥ 0 a.e. in Ω. Let s : R → R be a continuous function with

∫ t
0
s(ξ)dξ ≥ 0, for all t ≥ 0. Moreover

assume that there exists p1, p2 > 1, α, β > 0 and {αk}, {βk} satisfying lim
k→∞

αk = lim
k→∞

βk = +∞,
such that

max
|ξ|≤αk

∫ ξ
0

s(t)dt ≤ ααp1k ,
∫ βk
0

s(t)dt ≥ ββp2k

for each k ∈ N. Then, for the problem

 ∆

2
pu − ∆pu + V (x)|u|p−2u = λh(x)|u|m−2u + µr(x)s(u) in Ω

u = ∆u = 0 on Ω
(P λ,µ)

the following facts hold:

(r1) if p1 < p < p2, for all λ ∈ R and for all µ > 0, the problem (P λ,µ) admits a sequence of
non-zero weak solutions;

(r2) if p1 < p = p2, there exists µ1 > 0 such that for all λ ∈ R and for all µ > µ1, the problem
(P λ,µ) admits a sequence of non-zero weak solutions;



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(r3) if p1 = p < p2, there exists µ2 > 0 such that for all λ ∈ R and for all µ ∈]0, µ2[, the problem
(P λ,µ) admits a sequence of non-zero weak solutions;

(r4) if p1 = p2 = p, there exist x0 ∈ Ω, ρ > 0, γ > 1 and CV,γ,ρ > 0, such that, if

CV,γ,ρ <
β∥r∥L1(B(x0,ρ))
αωc

p
∞∥r∥L1(Ω)

, (3.2)

then µ1 < µ2 and for all λ ∈ R and for all µ ∈]µ1, µ2[, the problem (P λ,µ) admits a sequence
of non-zero weak solutions.

Proof. We want to apply Theorem 3.1 choosing f(x, t) = h(x)|t|m−2t and g(x, t) = r(x)s(t) for all
(x, t) ∈ Ω × R. The hypotheses (i1), (i2) are obviously verified. Since r ̸≡ 0 we can choose x0 ∈ Ω
and ρ > 0 such that B(x0, ρ) ⊂ Ω and r > 0 in B(x0, ρ). Then we have:

∫
Ω

max
|ξ|≤αk

G(x, ξ)dx =

∫
Ω

max
|ξ|≤αk

(∫ ξ
0

r(x)s(t)dt

)
dx = ∥r∥L1(Ω) max

|ξ|≤αk

∫ ξ
0

s(t)dt ≤ ∥r∥L1(Ω)αα
p1
k

and

∫
B(x0,ρ)

G(x, βk)dx =

∫
B(x0,ρ)

(∫ βk
0

r(x)s(t)dt

)
dx = ∥r∥L1(B(x0,ρ))

∫ βk
0

s(t)dt ≥ ∥r∥L1(B(x0,ρ))ββ
p2
k .

Therefore

lim inf
t→+∞

∫
Ω
max|ξ|≤t G(x, ξ)dx

tp1
≤ ∥r∥L1(Ω)α < +∞

and

lim sup
t→+∞

∫
B(x0,ρ)

G(x, t)dx

tp2
≥ ∥r∥L1(B(x0,ρ))β > 0.

So, (i3) is also verified with a = α∥r∥L1(Ω) and b = β∥r∥L1(B(x0,ρ)). Therefore we can apply the
Theorem 3.1 and obtain the conclusions (r1)–(r4).

Now, we want to exhibit two examples. In the first one we present a function s verifying the

hypotheses of Theorem 3.2.

Example 3.3. Let p > 1, δ > 1 and let s : R → R be the function such that

S(t) =

∫ t
0

s(ξ)dξ =




0, in ] − ∞, 0],

−2δt3 + 3δt2, in ]0, 1],

2p(k−1)δk in
]
2k−1δ

k−1
p , 2k−1δ

k
p

]
k ≥ 1,

Akt
3 + Bkt

2 + Ckt + Dk in
]
2k−1δ

k
p , 2kδ

k
p

]
k ≥ 1



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Infinitely many solutions for a nonlinear Navier problem involving... 511

where

Ak := −2(p−3)k+4δ
(p−3)k

p
(
δ − 2−p

)
, Bk := 9 · 2(p−2)k+2δ

(p−2)k
p

(
δ − 2−p

)
,

Ck := −3 · 2(p−1)k+3δ
(p−1)k

p
(
δ − 2−p

)
, Dk := 2

pkδk
(
5δ − 22−p

)
.

Using MATLAB by MathWorks, we have plotted the graph of the function S (for δ = 2 and p = 2),

showed in the following image.

-2 0 2 4 6 8 10 12 14

0

20

40

60

80

100

120

140

160

180

200

The function s satisfies all the assumption of Theorem 3.2 with α = 1, β = δ, αk = 2
k−1δ

k
p and

βk = 2
kδ

k
p , for each k ∈ N. In particular

max
|ξ|≤αk

∫ ξ
0

s(t)dt =

∫ 2k−1δ kp
0

s(t)dt = 2p(k−1)δk = α
p
k

and ∫ βk
0

s(t)dt = 2pkδk+1 = δβ
p
k

for all k ∈ N.

In Theorems 3.1 and 3.2, inequalities (3.1) and (3.2) serve to assure that µ1 < µ2; moreover the

value of CV,γ,ρ depends heavily also on constants cj and then on the choice of the sequence {χk}.
Obviously, fixed the nonlinearity, the smaller the constant CV,γ,ρ the easier the inequalities (3.1)

and (3.2) will be verified. The next example is in this direction.

Example 3.4. Let p > 1, Ω = B(0, 1) in Rn, x0 = 0, r ∈ L1(Ω) \ 0, with r ≥ 0, V (x) = |x|2R2 + 1,
for all x ∈ B(0, 1), ρ = 1

2
, γ = 2 and {σk} ⊂]0, +∞[ with limk→∞ σk = +∞. Let

{
χ1k
}
,
{
χ2k
}
∈

H(1
2
, 1, {σk}) the sequences defined by

χ1k(x) = 4σk(4x
3 − 9x2 + 6x − 1)



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and

χ2k(x) =
σk
2

cos(π(2x − 1) + 1)

for all x ∈]1
2
, 1[ and for each k ∈ N. We observe that, for each x ∈]1

2
, 1[,

|χ1k
′
(x)| ≤ 3σk, |χ1k

′′
(x)| ≤ 24σk

and then the constants cj(
{
χ1k
}
), defined in (2.2), are respectively c1(

{
χ1k
}
) = 3

2
and c2(

{
χ1k
}
) = 6.

In a similar way, for each x ∈]1
2
, 1[, we have

|χ2k
′
(x)| ≤ πσk, |χ2k

′′
(x)| ≤ 2π2σk

and, in this case, the constants cj(
{
χ2k
}
) are respectively c1(

{
χ2k
}
) = π

2
and c2(

{
χ2k
}
) = π

2

2
.

Now let us consider a sequence of functions that, in combination with the norm, raises it to the

second power; namely

χ3k(x) =



σk
(
−8x2 + 8x − 1

)
in ]1

2
, 3
4
[

αk(8x
2 − 16x + 8) in ]3

4
, 1[

(3.3)

for each k ∈ N. In this case

|χ3k
′
(x)| ≤ 4σk, |χ3k

′′
(x)| ≤ 16σk

and then c1(
{
χ3k
}
) = 2 and c2(

{
χ3k
}
) = 4. With respect to these three sequences of test functions

the smallest CV,γ,ρ (among the three) depends on the values of n and p. For instance, for n = 3

and p = 2 the smallest CV,γ,ρ is the one in correspondence with the sequence {χ3k}; in fact, using
MATLAB again to compute these constants, one has

CV,γ,ρ({χ1k}) ≈ 1270, CV,γ,ρ({χ
2
k}) ≈ 969, CV,γ,ρ({χ

3
k}) = 912.

But, for instance, for n = 4 and p = 3, the smallest CV,γ,ρ is the one in correspondence with the

sequence {χ2k} being

CV,γ,ρ({χ1k}) ≈ 73737, CV,γ,ρ({χ
2
k}) ≈ 53988, CV,γ,ρ({χ

3
k}) = 67262.

Obviously if we consider the function s of Example 3.3, taking a posteriori δ >
ωcp∞∥r∥L1(Ω)CV,γ,ρ

∥r∥
L1(B(0, 1

2
)

the corresponding problem admits a sequence of non-zero weak solutions; but if δ is fixed a priori,

Theorems 3.1 and 3.2 could be always applied as long as one manages to find an appropriate

sequence {χk} while it is not sure that a generic application of Theorem 1.1 can be applied because
the assumptions depends heavily by the particular sequence of test functions fixed during the proof.



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Infinitely many solutions for a nonlinear Navier problem involving... 513

The last theorem concerns the case in which the growth exponent of nonlinearity f(x, t) is exactly

p − 1. In this situation the existence of infinite weak solutions will be obtained not for each λ ∈ R
but in an appropriate interval.

Theorem 3.5. Let V ∈ C(Ω) satisfy infΩ V > 0 and let f, g ∈ C such that (i2) and (i3) are
verified. Moreover, suppose that:

(̃i1) there exist h ∈ L1(Ω) such that |f(x, t)| = h(x)
(
1 + |t|p−1

)
for a.e. x ∈ Ω and for all t ∈ R.

Then the following facts hold:

(r̃1) if p1 < p < p2, for all λ such that |λ| < 1∥h∥1cp∞ (for all λ if h = 0) and for all µ > 0, the
problem (Pλ,µ) admits a sequence of non-zero weak solutions;

(r̃2) if p1 < p = p2, there exists µ1 > 0 such that, for all µ > µ1, there exists λµ > 0 such that,

for all |λ| < λµ, the problem (Pλ,µ) admits a sequence of non-zero weak solutions;

(r̃3) if p1 = p < p2, there exists µ2 > 0 such that, for all µ ∈]0, µ2[, there exists λµ > 0 such that,
for all |λ| < λµ, the problem (Pλ,µ) admits a sequence of non-zero weak solutions;

(r̃4) if p1 = p2 = p, there exists γ > 1 and CV,γ,ρ > 0 such that, if

CV,γ,ρ <
b

ωc
p
∞a

(3.4)

then µ1 < µ2 and for all µ ∈]µ1, µ2[, there exists λµ > 0 such that, for all |λ| < λµ the
problem (Pλ,µ) admits a sequence of non-zero weak solutions.

Proof. The proof is similar to that of Theorem 3.1. In fact, computing the two main evaluations

for m = p, we get:

λ

∫
Ω

F(x, w(x))dx + µ

∫
Ω

G(x, w(x))dx ≤ |λ|∥h∥1c∞r
1
p

k +
|λ|
p
∥h∥1cp∞rk + µac

p1
∞r

p1
p

k (3.5)

and

Φ(wk) +
1

p
Ψ(wk) ≤ |λ|∥h∥1βk + |λ|∥h∥1

β
p
k

p
− µbβp2k +

ωCV,γ,ρ
p

β
p
k. (3.6)

To prove (r̃1), fix λ such that |λ| ≤ 1∥h∥1cp∞ and µ > 0. Thanks to the choice of λ and to the fact
that p1 < p then, from (3.5) we get

λ

∫
Ω

F(x, w(x))dx + µ

∫
Ω

G(x, w(x))dx <
1

p
rk (3.7)

for k large enough (remember that lim
k→∞

rk = +∞); moreover, from (3.6) we obtain

lim
k→∞

Φ(wk) +
1

p
Ψ(wk) = −∞ (3.8)

because p < p2.



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To prove (r̃2), it is sufficient to choose µ1 =
ωCV,γ,ρ

pb
. Fixed µ > µ1 and b in a similar way as

done in Theorem 3.1, we define λµ = min
{

1
∥h∥1c

p
∞
,
µpb−ωCV,γ,ρ

∥h∥1

}
. Fixed λ such that |λ| < λµ,

obviously, from (3.5), we get (3.7) (for k large enough) because p1 < p and thanks to the choice of

λ. Moreover, using (3.6), the choice of λ and µ guarantees that (3.8) holds.

To prove (r̃3), it is sufficient to choose µ2 =
1

pc
p
∞a

. Fixed µ ∈]0, µ2[ and a in a similar way as done
in Theorem 3.1, we choose λµ =

1−µpcp∞a
∥h∥1c

p
∞

. Fixed λ such that |λ| < λµ, obviously, from (3.6), we
get (3.8) because p < p2. Moreover, using (3.5), the choice of λ and µ guarantees that (3.7) holds.

In the last case, to prove (r̃4), we observe that, thanks to (3.4), we have µ1 < µ2. So, fixed

µ ∈]µ1, µ2[, and choosing a and b in a similar way as done in Theorem 3.1, we define λµ =
min

{
1−µcp∞a
∥h∥1

,
µpb−ωCV,γ,ρ

∥h∥1

}
. Fixed λ such that |λ| < λµ, obviously, from (3.5), we get (3.7) (for

k large enough) because of the choice of λ and µ. Moreover, using (3.6), the choice of λ and µ

guarantees that (3.8) holds.

We conclude with an example related to case (r̃4) of Theorem 3.5. In this case we consider the

one-dimensional setting, providing an explicit estimate of the constant c∞ in (3.4).

Example 3.6. Let n = 1, Ω =] − 1, 1[, p1 = p2 = p = 2, V (x) = x2 + 1 for all x ∈] − 1, 1[,
h ∈ L1(]−1, 1[), r ∈ L1(]−1, 1[)\{0} with r ≥ 0 in ]−1, 1[ and

∫ 1/2
−1/2 r(x)dx > 0. It is well-known

that, for all u ∈ W 2,2(] − 1, 1[) ∩ W 1,20 (] − 1, 1[), one has

max
x∈]−1,1[

|u(x)| ≤
√
2

2
∥u′∥L2(]−1,1[)

and

∥u′∥L2(]−1,1[) ≤
2

π
∥u′′∥L2(]−1,1[) ,

so

max
x∈]−1,1[

|u(x)| ≤
√
2

π
∥u′′∥L2(]−1,1[) ≤

√
2

π
∥u∥V

and then c∞ =
√
2

π
. Now choosing x0 = 0, ρ =

1
2
, γ = 2, δ >

1064∥r∥
L1(]−1,1[)

π2∥r∥
L1(]− 12 ,

1
2 [)

, and g(t, x) =

r(x)s(t) (where the function s is that of Example 3.3), assumptions (i2) and (i3) are satisfied with

a = ∥r∥L1(]−1,1[) and b = δ∥r∥L1(]− 12 , 12 [). Using the sequence {χ
3
k} of Example 3.4 as test function,

we compute CV,γ,ρ = 266 (lower than those associated with the other two sequences). It is easy to

see that

b

ωc
p
∞a

=
δπ2∥r∥

L1(]− 12 ,
1
2 [)

8∥r∥L1(]−1,1[)
> 266

then (3.4) is satisfied and then the fact (r̃4) holds. In particular, for all µ ∈
]
266
δ
, π

2

4∥r∥
L1(]−1,1[)

[
,

there exists λµ > 0 (defined inside the proof of Theorem 3.5) such that, for all |λ| < λµ the problem
(Pλ,µ) admits a sequence of non-zero weak solutions.



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Infinitely many solutions for a nonlinear Navier problem involving... 515

Acknowledgment

The author is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro

Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).



516 F. Cammaroto CUBO
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	Introduction
	Preliminaries
	Results