Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 65, pp. 1–16.

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

DOI: 10.58997/ejde.2024.65

QUASILINEAR BIHARMONIC EQUATIONS ON R4 WITH

EXPONENTIAL SUBCRITICAL GROWTH

ANTÔNIO DE PÁDUA FARIAS DE SOUZA FILHO

Abstract. This article studies the fourth-order equation

∆2u−∆u+ V (x)u−
1

2
u∆(u2) = f(x, u) in R4,

u ∈ H2(R4),

where ∆2 := ∆(∆) is the biharmonic operator, V ∈ C(R4,R) and f ∈ C(R4 ×
R,R) are allowed to be sign-changing. With some assumptions on V and f we

prove existence and multiplicity of nontrivial solutions in H2(R4), obtained via
variational methods. Three main theorems are proved, the first two assuming

that V is coercive to obtain compactness, and the third one requires only that
V be bounded. We work carefully with the sub-criticality of f to get a (PS)

condition for a related equation.

1. Introduction

In this article, we consider the fourth-order equation

∆2u−∆u+ V (x)u− 1

2
u∆(u2) = f(x, u) in R4,

u ∈ H2(R4),
(1.1)

where ∆2 := ∆(∆) is the biharmonic operator, V and f are continuous functions
that are allowed to be sign-changing.

In recent years, bi-harmonic and nonlocal operators arise in the description of
various phenomena in the pure mathematical research and real-world applications,
for example, for studying the traveling waves in suspension bridges [7, 10]. Recently
in [8], the authors studied the existence and multiplicity results for fourth-order el-
liptic equations on RN involving u∆(u2) and sign-changing potentials. The results
generalize some recent results on this kind of problems. To study this type of
problem, first consider the case where the potential V is coercive so that the work-
ing space can be compactly embedded into Lebesgue spaces. Next, we study the
case where the potential V is bounded so that the workspace is exactly H2(RN ),
which can not be compactly embedded into Lebesgue spaces. In [8], for sub-critical

2020 Mathematics Subject Classification. 35J62, 31B30, 35A15.
Key words and phrases. Biharmonic operator; exponential growth; variational methods; criti-

cal groups.
©2024. This work is licensed under a CC BY 4.0 license.

Submitted August 14, 2024. Published October 29, 2024.

1



2 A. P. F. SOUZA FILHO EJDE-2024/65

nonlinearity in the Sobolev sense, the authors defined

W (x) = V (x) +W0 ≥ 1, x ∈ RN , N ∈ N,

to deal with the potential allowed to be sign-changing. They then treated of the
following equivalent problem with the potential W > 0:

∆2u−∆u+W (x)u− 1

2
u∆(u2) = f(x, u) in RN ,

u ∈ H2(RN ),

Here, among other requirements, our nonlinearity f(x, t) satisfies subcritical expo-
nential growth in the sense of Adams’ Inequality, which is a Trudinger-Moser type

inequality for high dimensions, i.e., f(x, s) behaves like ±eαs
2

as t → ±∞ uniformly
in x ∈ R4, but slower than that.

2. Preliminaries

We now formulate assumptions for V and f :

(A1) V ∈ C(R4) is bounded from below, |V −1(−∞,M ]| < ∞ for all M > 0,
where | · | is the Lebesgue measure on R4.

(A2) V ∈ C(R4) is a bounded function such that the quadratic form B : X → R,

B(u) =
1

2

∫
R4

(|∆u|2 + |∇u|2 + V (x)u2) dx (2.1)

is non-degenerate and the negative space of B is finite-dimensional.
(A3) f : R → R is continuous and f(s) = o(s) near origin;
(A4) for (x, s) ∈ R4 × R we have 0 ≤ 4F (x, s) ≤ sf(x, s), moreover, for almost

all x ∈ R4;

lim
|s|→∞

F (x, s)

s4
= +∞, where F (x, s) =

∫ s

0

f(x, µ) dµ; (2.2)

(A5) f has subcritical exponential growth, that is,

lim
|s|→+∞

|f(s)|
eαs2

= 0 ∀α > 0;

(A6) For any r > 0, we have

lim
|x|→∞

sup
0<|t|≤r

|f(x, t)
t

| = 0.

Let H2(R4) be the standard Sobolev space. If V ∈ C(R4) is bounded from below,

we can choose a constant λ > 0 such that Ṽ (x) = V (x)+λ ≥ 1 for x ∈ R4. On the
linear subspace

X := {u ∈ H2(R4) :

∫
R4

V (x)|u|2 dx < ∞}

which is equip with the inner product

(u, v) =

∫
R4

(∆u∆v +∇u · ∇v + Ṽ (x)uv) dx

and the corresponding norm ∥ · ∥x. Note that if V ∈ C(R4) is bounded, then X is
precisely the standard Sobolev space H2(R4).



EJDE-2024/65 QUASILINEAR BIHARMONIC EQUATIONS 3

By the spectral theory of self-adjoint compact operators we have that the eigen-
value problem

∆2u−∆u+ V (x)u = λu, u ∈ X. (2.3)

possesses a complete sequence of eigenvectors and eigenvectors, such that

−∞ < λ1 ≤ λ2 ≤ . . . , λk → +∞,

where λk has been repeated according to its finite multiplicity. We denote by ϕk

the eigenfunction of λk with |ϕk|2 = 1, where | · |s is the Ls(R4)-norm. The main
results in this article can be stated as follows.

Theorem 2.1. Suppose (A1), (A4)–(A6) are satisfied. If 0 is not an eigenvalue of
(2.3), then (1.1) has a nontrivial solution u ∈ X.

Theorem 2.2. Suppose ((A1), (A4)–(A6) are satisfied. If f(x, ·) is odd for all
x ∈ R4, then (1.1) has a sequence of solutions {un} such that J(un) → +∞.

Similarly to [8], when that V satisfies (A2) and X is the standard Sobolev space
H2, we do not have the compact embedding X ↪→ Ls(R4) for s ∈ [2,∞) any more.
But we still have the following result.

Theorem 2.3. Suppose (A2)–(A6) are satisfied. Then (1.1) has a nontrivial solu-
tion u ∈ X.

3. Proof of theorem 2.1

In this section and the next section, we assume that (A1) holds. Now, let us
present some preliminary results necessary to demonstration of Theorem 2.1 and
that can be similarly used to the others main theorems.

The negative space of B is given by

X− = span{ϕ1, . . . , ϕℓ}
and X+ is the orthogonal complement of X− in X, such that X = X− ⊕X+. It is
well known that for u ∈ X±, there is a constant ã > 0 such that

B(u) ≥ ã∥u∥2X . (3.1)

Let us apply the linking theorem to find critical points of the functionals with
indefinite quadratic part, like J , with

∂Bρ ∩W = {u ∈ X+ : ∥u∥X = ρ}, Q = {u ∈ X− ⊕ R+ϕ : ∥u∥X ≤ R},
where ϕ ∈ X+\0. To prove Theorem 2.1 we need the following definition.

Definition 3.1. Let X be a Banach space, we say that functional J ∈ C1(X,R)
satisfies Palais-Smale condition at the level c ∈ R, ((PS)c for short notation) if any
sequence {un} ⊂ X satisfying J(un) → c, J ′(un) → 0 as n → ∞, has a convergent
subsequence. J satisfies (PS) condition if J satisfies (PS)c condition at all c ∈ R.

Having established the (PS) condition for the functional J , now we present some
concepts and results from infinite-dimensional Morse theory [14]. Let X be a Ba-
nach space, J : X → R be a C1-functional, u is an isolated critical point of J and
J(u) = c. Then

Cm(J, u) := Hm(Jc, Jc\{0}), m ∈ N = {0, 1, 2, . . . }
is called the m-th critical group of J at u, where Jc := J−1(−∞, c] and H∗ stands
for the singular homology with coefficients in Z.



4 A. P. F. SOUZA FILHO EJDE-2024/65

If J satisfies the (PS) condition and the critical values of J are bounded from
below by κ, then following Bartsch-Li [3], we define the m-th critical group of J at
infinity by

Cm(J,∞) := Hm(X, Jκ), m ∈ N.
It is well known that the homology on the right hand-side does not depend on the
choice of κ.

Proposition 3.2 ([11, Theorem 5.3]). Let E be a real Banach space with E =
V ⊕W , where V is finite dimensional. Suppose J ∈ C1(E,R), satisfies (PS), and

(i) there are constants ρ, d > 0 such that J |∂Br1
∩W ≥ d, and

(ii) there is an e ∈ ∂B1 ∩W and R > ρ such that if Q ≡ (B̄R ∩ V )⊕ {re : 0 <
r < R}, then J |∂Q ≤ 0. Then J possesses a critical value c̃ ≥ d which can
be characterized as

c̃ ≡ inf
h∈Γ

max
u∈Q

J(h(u)),

where

Γ = {h ∈ C(Q̄, E) : h = id on ∂Q}.

Proposition 3.3 ([3, Proposition 3.6]). If J ∈ C1(X,R) satisfies the condition
(PS) and Cm(J, 0) ̸= Cm(J,∞) for some m ∈ N, then J has a nonzero critical
point.

Proposition 3.4 ([9, Theorem 2.1]). Suppose J ∈ C1(X,R) has a local linking at
0 with respect to the decomposition X = X− ⊕

X+, i.e., for some ε > 0,

J(u) ≤ 0 foru ∈ X− ∩Bε,

J(u) > 0 foru ∈ (X+ \ {0}) ∩Bε,

where Bε = {u ∈ X : ∥u∥X ≤ ε}. If m = dimX− < ∞, then Cm(J, 0) ̸= 0.

Lemma 3.5. Assume that (A1), (A4), (A5) are satisfied, 0 is not an eigenvalue
of (2.3). Then J has a local linking at 0 with respect to the decomposition X =
X− ⊕X+.

Proof. For u ∈ X, we see that∫
R4

u2|∇u|2udx ≤ Big(

∫
R4

|u|6udx
)1/3(∫

R4

|∇u|3udx
)2/3

.

It follows from [1, Thm. 4.12 (Sobolev Imbedding Theorem)] that H2(R4) =
W 2,2(R4) ↪→ W 1,s(R4) for 2 ≤ s ≤ 2∗ = 8/(4 − 2) and H2(R4) ↪→ Lq(R4) for
2 ≤ q < ∞, we have ∫

R4

u2|∇u|2udx ≤ |u|26∥u∥2W 1,3 ≤ S∥u∥4X . (3.2)

By (A3) and (A4), we see that as ∥u∥X → 0,∫
R4

u2|∇u|2 = o(∥u∥2X),

∫
R4

F (x, u) = o(∥u∥2X).

Thus, as ∥u∥X → 0,

J(u) =
1

2
(∥u+∥2V − ∥u−∥2V ) +

1

2

∫
R4

u2|∇u|2udx−
∫
R4

F (x, u)udx



EJDE-2024/65 QUASILINEAR BIHARMONIC EQUATIONS 5

= B(u) +
1

2

∫
R4

u2|∇u|2udx−
∫
R4

F (x, u)udx

= B(u) + o(∥u∥2X).

It follows from the above estimate and (3.1) that the proof of our lemma is complete.
□

Setting g(x, s) = f(x, s) + γs, by (A4) we can see that

G(x, t) :=

∫ t

0

g(x, s) ds = F (x, t) +
γ

2
t2 ≤ t

4
g(x, t) +

τ

4
t2, (3.3)

where τ = b+ γ.
The functional J is equivalent to

J(u) =
1

2
∥u∥2X +

1

2

∫
R4

u2|∇u|2udx−
∫
R4

G(x, u)udx, (3.4)

with derivative given by

J ′(u)v = (u, v) +

∫
R4

(uv|∇u|2 + u2∇u · ∇v)udx−
∫
R4

g(x, u)vudx.

Lemma 3.6. Under the conditions of Theorem 2.1, there exist ρ > 0, ξ ∈ X with
∥ξ∥X > ρ such that J(ξ) < 0.

Proof. Combining 3.5 with (2.2), there exists a large K > 0 such that for any
e ∈ X, with ∥e∥X = 1, we have

lim
t→∞

J(te) ≤ lim
t→∞

[
t2B(e) +

t4

2
S∥e∥4X −Kt4|e|44

]
= −∞.

So, for some t0 > 0 there exists ρ > 0 such that J(t0e) < 0 with ∥t0e∥X > ρ. □

Lemma 3.7. Under (A1), the embedding of X into Lp(R4), for any p ∈ [2,+∞),
is compact.

Proof. Firstly, we may see that∫
R4

u2udx ≤
∫
R4

Ṽ (x)u2udx =

∫
R4

V (x)u2udx+

∫
R4

γu2udx < ∞.

Then, X is continuously embedded into H2(R4). Now let us to show that the
embedding of X into Lp(R4), with 2 ≤ p < ∞, is compact. Let un ⇀ 0 in X.
Hence, ∥un∥ is bounded and by the embedding continuous of X into Lp(R4) there
exists a constant C > 0 such that

|un|2 ≤ C ∀n ≥ 1.

Notice that

|un|22 =

∫
R4\B(0,R)

u2
nudx+

∫
B(0,R)

u2
nudx ∀n ≥ 1,

where R is large positive constant to be determined during the proof. We know
that un → 0 in L2(B(0, R)), for any R > 0. So, for any ε > 0 there exists N0(ε) ∈ N
such that

|un|22 ≤ ε

2
+

∫
R4\B(0,R)

u2
nudx ∀n ≥ N0(ε),



6 A. P. F. SOUZA FILHO EJDE-2024/65

Hence, we need to show that for any ε > 0 there exist R = R(ε) > 0 and N(ε) ∈ N
such that ∫

R4\B(0,R)

u2
nudx ≤ ε

2
∀n ≥ N(ε).

From (A1), it follows that there exists R > 0 such that

V (x) ≤ 2C

ε
∀x ∈ R4 \B(0, R).

Thus, ∫
R4\B(0,R)

u2
nudx ≤ ε

2C

∫
R4\B(0,R)

V (x)u2
nudx

≤ ε

2C

∫
R4\B(0,R)

Z(x)u2
nudx

≤ ε

2C
C =

ε

2
.

Therefore, un → 0 in L2(R4). By interpolation, un → 0 in Lt(R4) for any t ∈
[2,+∞). □

Lemma 3.8. Suppose that (A1), (A3)–(A5) hold. Then J satisfies the (PS) con-
dition.

Proof. Its clear that J satisfies the mountain pass geometry, that is, there exist α̃,
R̃ > 0 and e ∈ X such that J(u) ≤ α̃ with ∥u∥ = R̃ and J(e) < 0 for e ∈ X with

∥e∥ ≥ R̃. Observe that there is a sequence {un} ∈ X such that

∞ > C := sup
n

|J(un)|, J ′(un) → 0 as n → ∞. (3.5)

Firstly, we show that {un} is bounded in X. Otherwise, we have, up to a subse-
quence, ∥un∥ → ∞. Then, using the inequalities (3.3) and (3.5), we obtain

4C + ∥un∥X ≥ 4J(un)− J ′(un)un

= ∥un∥2X −
∫
R4

(4G(x, un)− g(x, un)un)udx

≥ ∥un∥2X − τ

∫
R4

u2
nudx.

(3.6)

Let ωn = un/∥un∥. Then there exists ω ∈ X, going if necessary to a subsequence,
by the Lemma 3.7 such that

ωn ⇀ ω in X, ωn → ω in L2(R4)

ωn → ω a.e. in R4 as n → ∞.

Multiplying by 1/∥un∥2X on both sides of (3.6) we have

τ

∫
R4

ω2
nudx ≥ 1 + on(1)

and then

τ

∫
R4

ω2udx ≥ 1 (3.7)



EJDE-2024/65 QUASILINEAR BIHARMONIC EQUATIONS 7

as n → ∞. So, ω ̸= 0. By (2.2) and (3.3), since |un(x)| → ∞ on {x ∈ R4 : ω(x) ̸= 0}
we see that

lim
n→∞

G(x, un(x))

∥un∥4X
= lim

n→∞

G(x, un(x))

u4
n(x)

ω4
n(x) = +∞. (3.8)

From (3.7), we have |{x ∈ R4 : ω(x) ̸= 0}| > 0. So, by Fatou’s lemma and (3.8),
we obtain that

lim inf
n→+∞

∫
R4

G(x, un)udx

∥un∥4X
≥ lim inf

n→+∞

∫
ω ̸=0

G(x, un)udx

∥un∥4X
= +∞. (3.9)

Thus, by (3.4), (3.5) and (3.2), we have

o(1) =
J(un)

∥un∥4X
=

1

∥un∥4X
(
1

2
∥un∥2X +

1

2

∫
R4

u2
n|∇un|2X −

∫
R4

G(x, un)udx)

≤ 1

2

1

∥un∥2X
+

S

2
−
∫
R4

G(x, un)

∥un∥4X
udx → −∞ as n → +∞.

This is a contradiction. Therefore, the sequence {un} is bounded in X. Next, we
proof the existence of a subsequence of {un} which converges strongly in X. By
(A1) and Lemma 3.7, we have that un → u in Lp(R4) for any p ∈ [2,+∞). For a
fixed n ∈ N,

∥u− un∥2X = [J ′(un)− J ′(u)](u− un) +

∫
R4

[g(x, un)− g(x, u)] (u− un) dx

and

[J ′(un)− J ′(u)] (u− un) → 0,

because {un} is a bounded Palais-Smale sequence.
Let α > 0 and q > 0 to be determined during the proof. Hence, for some C(α, q),

applying [13, Lemma 2.3] for ∥un∥X ≤ M and α < 64π2/(5M2), yields

|
∫
R4

[g(x, un)− g(x, u)] (u− un) dx|

≤
∫
R4

[|g(x, un)|+ |g(x, u)|] |u− un|dx

≤ C(α, q)

∫
R4

[(|un|+ |u|) + (|un|q(eαu
2
n − 1) + |u|q(eαu

2

− 1))]|u− un|dx

≤ C1(|un|2 + |u|2)|un − u|2 + C2(

∫
R4

(|un|q(eαu
2
n − 1))5/4dx)4/5|un − u|5

+ C3(

∫
R4

(|u|q(eαu
2

− 1))5/4dx)4/5|un − u|5) → 0, as n → ∞. □

To study the functional J let us rewrite B in a more convenient representation.
Let us note that, if (A1) holds and 0 is not an eigenvalue of (2.3), or if (A2)
holds, then the quadratic form B is nondegenerate and the negative space of B is
finite-dimensional, and so we may choose an equivalent norm ∥ · ∥V on X such that

B(u) =
1

2

(
∥u+∥2V − ∥u−∥2V

)
,



8 A. P. F. SOUZA FILHO EJDE-2024/65

where u± is the orthogonal projection of u on X± being X± the positive/negative
space of B. Let us use the equivalent norm on X and rewrite J as

J(u) =
1

2
(∥u+∥2V − ∥u−∥2V ) +

1

2

∫
R4

u2|∇u|2udx−
∫
R4

F (x, u)udx. (3.10)

It follows from (3.2) that there exists some constant A0 > 0 such that∫
R4

u2|∇u|2 ≤ A0∥u∥4V . (3.11)

Lemma 3.9. If (A1) or (A2), and (A3)–(A5) hold, 0 is not an eigenvalue of (2.3),
then there exists L > 0 such that if J(u) ≤ −L, we have

d

dt
|t=1J(tu) < 0.

Proof. We argue by contradiction. Suppose that for any n ∈ N there exists un such
that J(un) ≤ −n but

J ′(un)un =
d

dt
|t=1J(tun) ≥ 0.

Then, arguing by contradiction, we can see that ∥un∥V → +∞ and

−4n ≥ 4J(un)− J ′(un)un

= (∥u+
n ∥2V − ∥u−

n ∥2V )−
∫
R4

4F (x, un)− f(x, un)unudx

≥ ∥u+
n ∥2V − ∥u−

n ∥2V

(3.12)

Let ωn = un

∥un∥v
and ω±

n be the orthogonal projection onX±. Since {ωn} is bounded
and dim X− < ∞, we have that X− is closed and

ω−
n → ω− in X−,

with ω− ∈ X−. If ω− = 0, then ∥ω+
n ∥V → 1, because ∥ωn∥2V = ∥ω+

n ∥2V + ∥ω−
n ∥2V =

1. By the definition of ω±
n , for large n we have

∥u+
n ∥2V = ∥un∥2V ∥ω+

n ∥2V = ∥un∥2V (1 + on(1))
2 ≥ ∥un∥2V ∥ω−

n ∥2V = ∥u−
n ∥2V .

Hence, using (3.12) we obtain

0 ≤ ∥u+
n ∥2V − ∥u−

n ∥2V ≤ −4n

which cannot happen. If ω− ̸= 0, then

ωn ⇀ ω in X,

where ω ̸= 0. As a direct consequence of (A4) and

+∞ = lim
l→+∞

4F (x, l)

l4
≤ lim

l→+∞

f(x, l)l

l4
,

we have

∞ ≤ lim inf
n→∞

∫
ω− ̸=0

4F (x, un)

∥un∥4V
dx

≤ lim inf
n→∞

∫
R4

4F (x, un)

∥un∥4V
dx ≤ lim inf

n→∞

∫
R4

f(x, un)un

∥un∥4V
dx.

(3.13)

But, it follows from (3.11) that

J ′(un)un

∥un∥4V



EJDE-2024/65 QUASILINEAR BIHARMONIC EQUATIONS 9

=
1

∥un∥4V
(∥u+

n ∥2V − ∥u−
n ∥2V ) + 2

1

∥un∥4V

∫
R4

u2
n|∇un|2 −

1

∥un∥4V

∫
R4

f(x, un)un

≤ on(1) + 2A0 −
1

∥un∥4V

∫
R4

f(x, un)un → −∞ as n → +∞,

which is a contradiction, because we are assuming that J ′(un)un ≥ 0.
Now, since the proof does not depend of the compact embedding X ↪→ L2(R4),

the result is true if we assume (A2) instead of (A1). □

Lemma 3.10. Let B := B(0, 1) the unit ball in X. Then, there exists u ∈ ∂B such
that tuu ∈ X \B and J(tuu) < 0, for some tu > 0.

Proof. For t > 0 and u ∈ ∂B, we have

J(tu)

t4
=

1

t2
(∥u+

n ∥2V − ∥u−
n ∥2V ) +

1

2

∫
R4

u2|∇u|2udx−
∫
R4

F (x, tu)

t4
udx. (3.14)

By (3.14), (2.2) and (3.2), we obtain

J(tu) → −∞ as t → +∞.

Hence, there exists tu > 0 such that J(tuu) < 0. □

Now, let us introduce some concepts and results from infinite-dimensional Morse
theory [4]. Let X be a real Banach space, J ∈ C1(X,R), u be an isolated critical
point of J and J(u) = c. Then the ith critical group of J at u is defined by

Ci(J, u) := Hi(Jc, Jc\{0}), i = 0, 1, 2, . . . ,

where Jc = {u ∈ X : J(u) ≤ c}, and Hi(·, ·) denotes a singular relative homology
group of pair (·, ·) with integer coefficients.

If J satisfies the (PS) condition and the critical values of J are bounded from
below by some a, then, following Bartsch and Li [3], the critical groups of J at
infinity

Ci(J,∞) := Hi(X, Jc), i = 0, 1, 2, . . . ,

do not depend on the choice of a, because the homology on the right satisfies this.

Lemma 3.11. Assume that the conditions (A1) or (A2), and (A3)–(A5) are sat-
isfied. Then Ci(J,∞) = 0 fori = 0, 1, 2, . . . .

Proof. By Lemma 3.10, for à ≥ A > 0 large enough, for any v ∈ S there exists a
unique tv > 0 such that J(tvv) = −Ã. So, letting u = tvv we have J ′(u)u < 0.
By the Implicit Function Theorem, there is a unique continuous T : W0 ⊂ S →
W1 ⊂ R, for some W0 and W1 open neighborhoods such that for F (s, v) = J(sv),

we have F (T (v), v) = J(T (v)v) = −Ã. Then, T (v) = tv and we obtain a unique

application φ ∈ C(S,R) such that J(φ(v)v) = −Ã. Moreover, if J(u) = −Ã, then
φ(u) = 1. Using the function φ we can construct a strong deformation retract
η : X\B → J−Ã

η(u) =

{
u, if J(u) ≤ −Ã,

φ( u
∥u∥X

) u
∥u∥X

, if J(u) > −Ã

and we obtain

Ci(J,∞) = Hi(X, J−Ã)
∼= Hi(X,X\B) = 0, i = 0, 1, 2, . . . . □



10 A. P. F. SOUZA FILHO EJDE-2024/65

Proof ofTheorem2.1. By Lemma 3.8, J satisfies the Palais-Smale condition and by
Lemma 3.5, J has a local linking at 0 with respect to the decomposition X− ⊕

X+.
Hence, since m = dimX− < ∞, we have Cm(J, 0) ̸= 0 = Cm(J,∞). Then, it
follows from Proposition 3.3 that J has a critical point u, which is a nontrivial
solution of (1.1). □

4. Proof of theorem 2.2

In the proof of Theorem 2.2 we apply the following symmetric mountain pass
theorem.

Proposition 4.1 ([2, Theorem 9.12]). Let X be an infinite dimensional Banach
space, J ∈ C1(X,R) be even, satisfies (PS) condition and J(0) = 0. If X = Y ⊕Z
with dim Y < ∞, and J satisfies

(1) there are constants ρ, α > 0 such that J |∂Bρ∩Z ≥ α,
(2) for any finite dimensional subspace W ⊂ X, there is an R = R(W ) such

that J ≤ 0 on W\BR(W ) then J has a sequence of critical values cj → +∞.

Lemma 4.2. For m ∈ N, let Zm = span{ϕm, ϕm+1, . . . } and set

σm = sup
u∈Zm,∥u∥=1

|u|2

Then σm → 0 as m → ∞.

Proof. For m ∈ N large, let us take u ∈ Zm with ∥u∥ = 1. Thus, we obtain

λm

∫
R4

u2udx ≤
∫
R4

(|∆u|2 + |∇u|2 + V (x)u2)udx

or equivalently

(λm + γ)

∫
R4

u2udx ≤
∫
R4

(|∆u|2 + |∇u|2 + Ṽ (x)u2)udx = ∥u∥2X = 1.

Therefore, as m → ∞,

|σm| ≤ 1√
λm + γ

→ 0. □

Proof of Theorem 2.2. Note that here we are considering that the functional J is
even and satisfies the (PS) condition. Then, it suffices to show that the Proposition
4.1 is applicable to J .

(1) It follows from (A3) and (A4) that for fixed α > 32π and q > 2, the existence
of two constants c1, c2 > 0 such that

|G(x, s)| ≤ c1|s|2 + c2|s|q(eαs
2

− 1) ∀s ∈ R. (4.1)

We have that the embedding X ↪→ H2(R4) is continuous, namely there exists a
constant L such that

∥u∥H2 ≤ L∥u∥X ∀u ∈ X.

Now, let us choose m ∈ N, Zm and σm such that by Lemma 4.2 we have

c̄ =
1

2
− c1σ

2
m > 0

and we set

Y = span{ϕ1, . . . , ϕm−1}, Z = span{ϕm, ϕm+1, . . . }.
Then, we have X = Y ⊕ Z.



EJDE-2024/65 QUASILINEAR BIHARMONIC EQUATIONS 11

If we consider ∥u∥X ≤ 1
L
√
α
and then ∥u∥H2 ≤ 1√

α
, we can apply [12, Lemma 1]

such that∫
R4

G(x, u)udx ≤ c1|u|22 + L(α, q, c1)∥u∥qX ∀u ∈ X, ∥u∥X ≤ 1

L
√
α
.

Therefore, using (4.1) for u ∈ Z = Zm, as ∥u∥X → 0 we obtain

J(u) =
1

2
∥u∥2X +

1

2

∫
R4

u2|∇u|2udx−
∫
R4

G(x, u)udx

≥ 1

2
∥u∥2X −

∫
R4

G(x, u)udx

≥ 1

2
∥u∥2X − c1|u|22 − L(α, q, c1)∥u∥qX

≥
(1
2
− c1σ

2
m

)
∥u∥2X − L(α, q, c1)∥u∥qX

= c̄∥u∥2X + o(∥u∥2X).

Thus (1) was verified.
(2) Its sufficient to show that J is anti-coercive, i.e. J(un) → −∞ as ∥un∥X →

+∞. We argue by contradiction: let us suppose that there exists {un} ⊂ W ⊂
X and L < 0 such that ∥un∥X → +∞ but J(un) ≥ L. Let vn = un

∥un∥X
be

the normalized sequence and, up to a sub-sequence, vn → v ∈ W \ {0}, because
dimW < ∞. Continuing as it was done in (3.9) we obtain

1

∥un∥4X

∫
R4

G(x, un)udx → +∞.

Hence, it follows from (3.2) that

J(un) =
1

2
∥un∥2X +

1

2

∫
R4

u2
n|∇un|2udx−

∫
R4

G(x, un)udx

≤ 1

2
∥un∥4X

( 1

∥un∥2X
+ S − 2

∥un∥4X

∫
R4

G(x, un)udx
)
→ −∞.

This contradicts J(un) ≥ L. □

5. Proof or theorem 2.3

In this section we consider the potential V satisfying the assumption (A2) instead
of (A1) and so X is equivalent to standard Sobolev space H2(RN ) and we do not
have the compact embedding X ↪→ L2(RN ).

Lemma 5.1. Under the assumptions of Theorem 2.3, {un} is a (PS) sequence of
J , that is, as n → +∞

sup
n

|J(un)| < ∞, J ′(un) → 0 .

Then {un} is bounded in X.

Proof. Otherwise, up to a subsequence. we assume that ∥un∥V → ∞. Let vn =
∥un∥−1

V un. Then vn = v+n + v−n → v = v+ + v− ∈ X, v±n , v
± ∈ X±. If v = 0, then

v−n → v− = 0 because dimX− < ∞ and X− ∩X+ = {0}. Since

∥v+n ∥2V + ∥v−n ∥2V = 1,



12 A. P. F. SOUZA FILHO EJDE-2024/65

for n large enough we have

∥v+n ∥2V − ∥v−n ∥2V ≥ 1

2
.

Now, using (A4) for n large enough, we obtain

1 + sup
n

|J(un)|+ ∥un∥V

= J(un)−
1

4
J ′(un)un

=
1

4

(
∥u+

n ∥2V − ∥u−
n ∥2V

)
−
∫
R4

(
F (x, un)−

1

4
f(x, un)un

)
udx

≥ 1

4
∥un∥2V (∥v+n ∥2V − ∥v−n ∥2V )−

∫
R4

(
F (x, un)−

1

4
f(x, un)un

)
udx

≥ 1

8
∥un∥2V ,

a contradiction to ∥un∥V → +∞. Thus, we obtain that the (PS) sequence {un} is
bounded in X.

If we suppose v ̸= 0, then there exists Ω = {x ∈ R4 : v(x) ̸= 0} with positive
Lebesgue measure such that for x ∈ Ω we have

F (x, un)

u4
n(x)

v4n(x) → +∞,

thanks to (2.2). On the other hand, using (3.11), we obtain∫
Ω

F (x, un)

u4
n(x)

v4n(x)udx =
1

∥un∥4V

∫
Ω

F (x, un(x))udx

≤ 1

∥un∥4V

∫
R4

F (x, un(x))udx

≤ ∥u+
n ∥2V − ∥u−

n ∥2V
2∥un∥4V

+
1

2∥un∥4V

∫
R4

u2
n|∇un|2udx− J(un)

∥un∥4V

≤ 1 +
A0

2
.

Thus, we have that {un} is bounded in X. □

Now, let us to investigate the C1-functional V : H2(R4) → R, defined by

V(u) = 1

2

∫
R4

u2|∇u|2udx,

with derivative

V ′(u)v =

∫
R4

(uv|∇u|2 + u2∇u · ∇v)udx, u, v ∈ H2(R4),

to obtain the (PS) condition for J .

Lemma 5.2. The functional V : H2(R4) → R is weakly lower semi-continuous;
V ′ : H2(R4) → H−2(R4) is weakly sequentially continuous.

Proof. Let {un} be a sequence in H2(R4) such that un ⇀ u in H2(R4). Then, by
the compact embedding H2(R4) ↪→ H1

loc(R4) we have un → u in H1
loc(R4). Hence,

going if necessary to a subsequence, we obtain

∇un → ∇u a.e. in R4, un → u a.e. in R4, (5.1)



EJDE-2024/65 QUASILINEAR BIHARMONIC EQUATIONS 13

and then by Fatou’s lemma,∫
R4

u2|∇u|2udx ≤ lim inf
n→+∞

∫
R4

u2
n|∇un|2udx,

that is

V(u) ≤ lim inf
n→+∞

V (un).

Thus, V is weakly lower semi-continuous. To investigate the weak lower semicon-
tinuity of V ′ we need to see that for u ∈ H2(R4), since H2(R4) ↪→ W 1,4(R4), we
have ∫

R4

(|∇un|2|un|)4/3udx ≤
(∫

R4

|∇un|2
4
3

3
2udx

)2/3(∫
R4

|un|
4
3

3
1udx

)1/3

≤ C∥un∥8/3H2(R4)∥un∥4/3H2(R4)

and then ∣∣ ∫
R4

|∇un|2unudx
∣∣ ≤ C∥un∥4H2(R4). (5.2)

Thus, {un} is bounded in L4/3(R4) and combining with (5.1) we may apply the
Brézis-Lieb lemma to obtain |∇un|2un ⇀ |∇u|2u in L4/3(R4). Hence, for any
φ ∈ H2(R4), we have φ ∈ L4(R4) and∫

R4

|∇un|2unφudx →
∫
R4

|∇u|2uφudx. (5.3)

Similarly, we have∫
R4

|u2∇u|4/3udx ≤
(∫

R4

|u|4udx
)2/3(∫

R4

|∇u|4udx
)1/3

≤ C∥un∥8/3H2(R4)∥un∥4/3H2(R4).

Thus, the sequence {u2
n∇un} is bounded in L4/3(R4) and converges point-wise to

u2∇u. Again, by Brézis-Lieb lemma we obtain

u2
n∇un → u2∇u, in [L4/3(R4)]4.

For each φ ∈ H2(R4) we have φ ∈ W 1,4(R4) and then φ ∈ L4(R4), which implies∫
R4

u2
n∇un · ∇φudx →

∫
R4

u2∇u · ∇φudx. (5.4)

Now, using (5.3) and (5.4) we have∫
R4

|∇un|2unφ+ u2
n∇un · ∇φudx →

∫
R4

|∇u|2uφ+ u2∇u · ∇φudx,

that is,

V ′(un)φ → V ′(u)φ ,

and then V ′ is weakly sequentially continuous. Moreover, if un ⇀ u in H2(R4), we
obtain

lim inf
n→+∞

∫
R4

(|∇un|2un(un − u) + u2
n∇un · ∇(un − u))udx

= lim inf
n→+∞

(4V(un)− V ′(un)u)

≥ 4V(u)− V ′(u)u = 0.

(5.5)

□



14 A. P. F. SOUZA FILHO EJDE-2024/65

Lemma 5.3. Operator J satisfies the (PS) condition.

Proof. Let {un} be a (PS) sequence. It follows from Lemma 5.1 that {un} is
bounded in X and so, up to a subsequence, we obtain un ⇀ u in X. We claim that

lim sup
n→+∞

∫
R4

f(x, un)(un − u)udx ≤ 0. (5.6)

Indeed, letting ε̄ > 0 and α > 0 such that 2 < 32π2

αM2 with ∥un∥V ≤ M , as a
consequence of Lemma 5.1. Then by (A5) and [13, Theorem 2.2], for r ≥ 1 large
enough, we have∫

R4∩{|un|≥r}
f(x, un)(un − u)udx

≤ ε̄

∫
R4∩{|un|≥r}

(eαu
2
n − 1)|un − u|udx

≤ ε̄
(∫

R4∩{|un|≥r}
(eαu

2
n − 1)2udx

)1/2(∫
R4∩{|un|≥r}

(un − u)2udx
)1/2

≤ ε̄C(
32π2

αM2
)
(∫

R4

(eα(
32π2

αM2 )u2
n − 1)udx

)1/2

× |un − u|2

≤ ε̄C
(∫

R4

(e32π
2(un

M )2 − 1)udx
)1/2

≤ ε

3
,

for small ε > 0. Moreover, by (A6) there exists R > 0 such that∫
R4∩{|x|≤R}∩{|un|≤r}

f(x, un)(un − u)udx

≤ sup
|t|<r,|x|≤R

|f(x, t)|
|t|

∫
R4∩{|x|≤R}∩{|un|≤r}

|un||un − u|udx ≤ ε

3
.

Finally, since un → u in L2(B(0, R)), by (f0) and (A5) we have∫
R4∩{|x|≤R}∩{|un|≤r}

f(x, un)(un − u)udx

≤ (

∫
R4∩{|x|≤R}∩{|un|≤r}

|f(x, un)|2udx)1/2

×
(∫

R4∩{|x|≤R}∩{|un|≤r}
|un − u|2udx

)1/2

≤ ε

3

and then we conclude that∫
R4

f(x, un)(un − u)udx ≤ ε,

for small ε > 0. Now, by weak convergence we have∫
R4

(∆un∆u+∇un · ∇u+ V (x)unu)udx

→
∫
R4

(|∆u|2 + |∇u|2 + V (x)u2)udx

= ∥u+∥2V − ∥u−∥2V .



EJDE-2024/65 QUASILINEAR BIHARMONIC EQUATIONS 15

Since X− is a finite-dimensional vector space, we obtain u−
n → u− and ∥u−

n ∥V →
∥u−∥V . Then, since J ′(un)(un − u) = o(1), we obtain∫

R4

f(x, un)(un − u)udx

=

∫
R4

(∆un∆(un − u) +∇un · ∇(un − u) + V (x)un(un − u))udx

+

∫
R4

(|∇un|2un(un − u) + u2
n∇un · ∇(un − u))udx+ o(1)

= (∥u+
n ∥2V − ∥u−

n ∥2V )− (∥u+∥2V − ∥u−∥2V ) + o(1)

+

∫
R4

(|∇un|2un(un − u) + u2
n∇un · ∇(un − u))udx

= (∥u+
n ∥2V − ∥u+∥2V ) +

∫
R4

(|∇un|2un(un − u) + u2
n∇un · ∇(un − u))udx+ o(1).

Hence, by (5.6)

0 ≥ lim sup
n→+∞

(∥u+
n ∥2V − ∥u+∥2V )

+ lim inf
n→+∞

∫
R4

(
|∇un|2un(un − u) + u2

n∇un · ∇(un − u)
)
udx

(5.7)

and so combining (5.5) with (5.7) we obtain

∥u+∥2V ≤ lim inf
n→+∞

∥u+
n ∥2V ≤ lim sup

n→+∞
(∥u+

n ∥2V − ∥u+∥2V ) + ∥u+∥2V ≤ ∥u+∥2V .

Therefore,
lim
n→∞

∥u+
n ∥2V = ∥u+∥2V

which implies
lim

n→∞
∥un∥2V = ∥u∥2V . (5.8)

Thus, combining (5.8) and un ⇀ u in X, it follows from Radon-Riesz theorem that
un → u in X. □

Proof of Theorem 2.3. Since the conclusion of Lemma 3.9 remains valid if instead
of (A1), V satisfies (A2), there exists L > 0 such that if J(u) ≤ −L, then

d

dt

∣∣
t=1

J(tu) < 0. (5.9)

Applying Lemma 3.11 we obtain Ci(J,∞) = 0. But analogously to Lemma 3.5
we can show that J has a local linking at 0 with respect to the decomposition
X = X− ⊕X+. Since m = dimX− < ∞, we have Cm(J, 0) ̸= 0 = Cm(J,∞). By
Proposition 3.3, J has a critical point u, which concludes the proof of Theorem
2.3. □

References

[1] R. A. Adams, J. J. Fournier; Sobolev spaces. Elsevier, 2003.
[2] A. Ambrosetti, P.H. Rabinowitz; Dual variational methods in critical point theory and ap-

plications, J. Funct. Anal., 14 (1973), 349–381
[3] T. Bartsch, S. Li; Critical point theory for asymptotically quadratic functionals and applica-

tions to problems with resonance, Nonlinear Anal., 28 (1997), 419–441.
[4] K. C. Chang; Infinite-dimensional Morse theory and multiple solution problems, Progress in

Nonlinear Differential Equations and their Applications 6 (Birkhäuser, Boston, 1993).



16 A. P. F. SOUZA FILHO EJDE-2024/65

[5] Y. Chen, P. McKenna; Traveling waves in a nonlinear suspension beam: theoretical results

and numerical observations, J. Differ. Equ. 137 (1997), 325-355.

[6] J. M. Do Ó, E. Medeiros, U. Severo; A Nonhomogeneous elliptic problem involving critical

growth in dimension two, J. Math. Anal. Appl. 345, No. 1 (2008), 286–304.

[7] A. Lazer, P. McKenna; Large-Amplitude Periodic Oscillations in Suspension Bridges: Some
New Connections with Nonlinear Analysis. SIAM Rev. 32 (1990), 537-578.

[8] S. B. Li, Z. H. Zhao; Solutions for fourth order elliptic equations on RN involving u∆(u2)

and sign-changing potentials, J. Differential Equations 267 (2019), No. 3, 1581–1599.
[9] J. Q. Liu; The Morse index of a saddle point, J. Systems Sci. Math. Sci. 2 (1989), 32–39.

[10] P. J. McKenna, W. Walter; Travelling waves in a suspension bridge. SIAM J. Appl. Math.
50 (1990), 703–715.

[11] P. Rabinowitz; Minimax methods in critical point theory with applications to differential

equations, in: CBMS Regional Conf. Ser. in. Math., vol. 65, American Mathematical Society,
Providence, RI, 1986.

[12] F. Sani; A biharmonic equation in R4 involving nonlinearities with critical exponential

growth, Commun. Pure Appl. Anal., 12 (2013), 405–428
[13] F. Sani; A biharmonic equation in R4 involving nonlinearities with subcritical exponential

growth, Adv. Nonlinear Stud., 11 (2011), no. 4, 889–904.

[14] Z. Wang, H.-S. Zhou; Positive solution for a nonlinear stationary Schrödinger-Poisson sys-
tem in R3, Discrete Contin. Dvn. Syst., 18 (2007). 809–816.

[15] M. Willem; Minimax theorems, Birkhäuser, Boston, 1996.

Antônio de Pádua Farias de Souza Filho
Departamento de Ciências Exatas e Naturais, Universidade Federal Rural do Semi-

Árido, 59900-000, Pau dos Ferros-RN, Brazil

Email address: padua.filho@ufersa.edu.br


	1. Introduction
	2. Preliminaries
	3. Proof of theorem ??
	4. Proof of theorem ??
	5. Proof or theorem ??
	References