Advanced Nonlinear Studies 15 (2015), 157–169

Resonant-Superlinear Elliptic Problems

Using Variational Methods

Edcarlos Domingos da Silva∗

Departamento de Matemática
Universidade Federal de Goiás, Caixa Postal 131, CEP: 74001-970 , Goiânia - GO, Brazil
e-mail: edcarlos@ufg.br
Bruno Ribeiro
Departamento de Matemática
Universidade Federal da Paraı́ba, CEP: 58059-900, Paraı́ba - PB, Brazil
e-mail: brunohcr@gmail.com

Received in revised form 02 April 2014

Communicated by Zhi-Qiang Wang

Abstract
In this work we establish existence and multiplicity of solutions for resonant-superlinear
elliptic problems using appropriate variational methods. The nonlinearity is resonant at −∞
and superlinear at +∞ and the resonance phenomena occurs precisely in the first eigenvalue
of the corresponding linear problem. Our main theorems are stated without the well known
Ambrosetti-Rabinowitz condition.
2010 Mathematics Subject Classification. Primary 35J20, Secondary 35J65.
Key words. Superlinear elliptic problems, Resonance Problems, Nonquadraticity Condition, Variational Methods

1 Introduction
In this paper we establish existence and multiplicity of solutions for the semilinear elliptic problem
⎧
⎪
⎨ −Δu = λ1 u + f (x, u) in Ω
⎪
⎩ u=0 (1.1)
on ∂Ω,

where Ω ⊆ RN , N ≥ 3, is a bounded domain with regular boundary and λ1 denotes the first positive
eigenvalue of (−Δ, H01 (Ω)). The nonlinearity f satisfies a unilateral nonquadratic growth condition,
which will be detailed below.
∗ Corresponding author. The author acknowledges the support CNPq grants 211623/2013-0

157
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158 E. D. da Silva, B. Ribeiro

Throughout this paper we assume that f is continuous and subcritical, i.e, f satisfies

f ∈ C(Ω × R, R)
( f0 )
| f (x, t)| ≤ C(1 + |t| p−1 ), t ∈ R, x ∈ Ω,

2N
where 2 < p < 2 = .
N−2
From a standard variational point of view, finding solutions of (1.1) in H01 (Ω) is equivalent to
finding critical points of the C 1 functional J : H01 (Ω) → R given by
  
1 λ1
J(u) = |∇u| dx −
2
u dx −
2
F(x, u)dx (1.2)
2 Ω 2 Ω Ω

where  t
F(x, t) = f (x, s)ds, x ∈ Ω, t ∈ R.
0

In order to control the behavior of f at +∞, we assume that

2F(x, t) − t f (x, t)
lim sup ≤ −k, (NQ+ )
t→+∞ tσ
holds uniformly for some k > 0 and x ∈ Ω, where we put
⎧
⎪
⎪
⎪
2
⎪
⎪ 1 if p < 2 +
⎨ N
σ= ⎪
⎪
⎪
⎪
⎪ 2
⎩ σ0 if p ≥ 2 + ,
N
and
N
(p − 2) < σ0 < p.
2
On the other hand, the behavior of f at −∞ gives the idea of unilateral nonquadratic growth. We
consider the following hypothesis:

lim 2F(x, t) − t f (x, t) = 0, (NQ− )

t→−∞

uniformly for x ∈ Ω. Moreover, our variational setting requires superlinear unilateral condition: we
also assume
f (x, t) f (x, t)
lim = +∞, lim = 0, ( f1 )
t→∞ t t→−∞ t
where the limits above are uniformly for x ∈ Ω.
Under these assumptions the problem (1.1) becomes resonant at −∞ and superlinear at +∞. The
novelty in this work is dealing with a resonant superlinear problem at the first eigenvalue and without
the Ambrosetti-Rabinowitz condition, which is replaced by the nonquadraticity condition stated in
(NQ+ ).
Let us introduce this problem by showing some related results that can be found in the literature.
Consider the following problem:
⎧
⎪
⎪ + q−1
⎨ −Δu = λ1 u + (u ) + h(x) in Ω
⎪
⎪
⎩ u=0
(1.3)
on ∂Ω,
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which was explored by Cuesta; de Figueiredo & Srikanth [8] with 2 < q < 2N/(N − 1) < 2 and
h ∈ L∞ (Ω) such that 
hφ1 dx < 0.
Ω
Here, we emphasize that problem (1.3) is not considered in [8] in case of
 
2N
q∈ , 2 .
N−1
The main difficult in their case is that there is no known a priori bounds of Brezis-Turner type for
q > 2N/(N − 1). The authors work on nonvariational arguments and such a priori bound must occur
in their settings. Our variational settings do not depend on these kinds of estimates because we
exploit the nonquadraticity conditions at infinity, proving that the functional for the problem (1.3)
satisfies the Cerami condition for any nonzero levels of energy. With this compactness property in
hand, classical critical point theorems can be applied.
We point out that problem (1.1) has also been considered by several authors. We refer the reader
to the works of Ambrosetti & Mancini [2], Calanchi & Ruf [7], de Figueiredo & Yang [9], Dancer &
Yan [10], Ortega & Kannan [12, 13], Ruf & Srikanth [14] and references therein.
In short, these works considered the following problem
⎧
⎪
⎨ −Δu = λu + g(x, u) in Ω
⎪
⎩ u=0 (1.4)
on ∂Ω,
where λ is a positive parameter and g satisfies ( f0 ). In that case, assuming λ < λ1 or λ > λ1 , λ 
λk , k ≥ 2, many results were obtained with respect to the existence and multiplicity of solutions for
(1.4). For instance, under the condition λ ∈ (λk , λk+1 ), k ≥ 2, where the sequence (λk ) stands for
the eigenvalues of (−Δ, H01 (Ω)), Ruf & Srikanth [14] showed the existence of nontrivial solutions for
(1.4) using the Generalized Mountain Pass Theorem. We also seize the opportunity to mention the
work of Arcoya & Villegas [1], which deals with a Neumann problem, with superlinear behavior at
+∞ and linear behavior at −∞ as well.
However, to the best our knowledge, there are few results for this problem when the parameter
λ is an eigenvalue of (−Δ, H01 (Ω)), i. e., when the problem (1.4) becomes resonant at −∞. In
this case, the problem (1.1) is more delicate because the associated functional J may not satisfy
the well known Palais-Smale condition at some levels of energy. That means, when the resonance
phenomena is sufficiently strong, J does not satisfy (PS )c for some c ∈ R.
As an example of our settings, consider the problem
⎧ + +
⎪
⎨ −Δu = λ1 u + u ln(1 + u ) + r(u) in Ω
⎪
⎩ u=0 (1.5)
on ∂Ω,
where we define ⎧
⎪
⎪ − μt if |t| ≤ 1
⎪
⎪
⎪
⎪
⎪
⎨ − μ(2 − t) if 1 ≤ t ≤ 2
r(t) = ⎪
⎪
⎪
⎪
⎪ μ(t + 2) if − 2 ≤ t ≤ −1
⎪
⎪
⎩
0 if |t| ≥ 2,
for any μ > 0. This problem is clearly superlinear at +∞ and resonant at −∞ in the first eigenvalue.
Moreover, it does not satisfy the well known Ambrosetti-Rabinowitz condition, used since the pi-
oneer work of Ambrosetti & Rabinowitz [3], which reads: there are θ ∈ (2, +∞) and M > 0 such
that
0 < θG(x, t) ≤ tg(x, t), x ∈ Ω, t ≥ M. (AR)
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160 E. D. da Silva, B. Ribeiro

It is very easy to see that (AR) implies that

G(x, t) ≥ Ctθ , x ∈ Ω, t ≥ M
for some C > 0. Observe that the nonlinearity g in (1.5) does not satisfy the (AR) condition. In fact,
the function
g(t) = t+ ln(1 + t+ ), x ∈ Ω, t ∈ R
provides us
G(x, t) = (t+ )2 ln(1 + t+ ) + o(tθ ), x ∈ Ω, t ≥ M.
Therefore, this function verifies the conditions (NQ+ ) and (NQ− ) mentioned above .
One of the main problems in this work is proving that the Cerami condition holds at some appro-
priate levels of energy. After that, using some additional conditions to ensure geometric properties
for J (so that convenient critical point theorems may be used), we shall prove existence and multi-
plicity of solutions for our problem. Next, we introduce a basic hypothesis to ensure that problem
(1.1) has at least one solution.
There is a real number t− < 0 such that

F(x, t− φ1 )dx > 0. (H1)
Ω
Making use of Ekeland Variational Principle, our first theorem give us a solution for the nonho-
mogeneous case which can be read as
Theorem 1.1 (Nonhomogeneous case) Suppose ( f0 ), ( f1 ) and (NQ− ) hold. Assume also that (H1)
holds. If f (x, 0 ≤ 0, then the problem (1.1) has at least one weak solution u ∈ H01 (Ω).
It is important to stress that the nonhomogenous case have been considered in many works (see
[8]). However, to the best our knowledge, there is no existence of solutions in our setting taking into
account one side resonant problems and superlinear elliptic problems.
Next we shall consider the homogeneous case, i.e, we assume that
f (x, 0) ≡ 0 in Ω. ( f2 )
Thus, the problem (1.1) admits u ≡ 0 as trivial solution. In order to ensure the existence of nontrivial
solutions we shall consider some additional hypothesis.
There are 0 < α < λ1 and a nonnegative function C ∈ L∞ (Ω) such that
f (x, t)
−C(x) ≤ lim ≤ α − λ1 , (H2)
t→0 t
for any x ∈ Ω. This condition avoids resonance at 0, but maintains resonance at −∞.
Now, applying the Mountain Pass Theorem, we can also prove the following result.
Theorem 1.2 (Homogenous case) Suppose ( f0 ), ( f1 ), ( f2 ), (NQ+ ) and (NQ− ) hold. Assume also
that (H2) holds. Then problem (1.1) has at least one nontrivial solution.
Combining our assumptions on f we have the following multiplicity result.
Corollary 1.1 (Multiplicity of solutions for homogenous case) Suppose ( f0 ), ( f1 ), ( f2 ), (NQ− ),
(H1) hold. Then the solution obtained in Theorem 1.1 is nontrivial. Also, assuming (NQ+ ) and
(H2), problem (1.1) has at least three nontrivial solutions.
This paper is organized as follows: In Section 2 we give some basic results on our problem (1.1),
such as the Cerami condition. Section 3 is devoted to prove some results concerning the geometry
of J (such as the mountain pass geometry). Finally, in Section 4 we prove our main theorems.
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Resonant-Superlinear elliptic problems using variational methods 161

2 Compactness results
In this section we prove that J satisfies the well known Cerami condition at some levels of energy.
Firstly, let us prove a technical result for our problem.

Lemma 2.1 Suppose (NQ− ) and ( f1 ) hold. Then

lim F(x, t) = 0. (2.1)

t→−∞

In particular there exists H ∈ L∞ (Ω) such that

−H(x) ≤ F(x, t) ≤ H(x), t ≤ 0, x ∈ Ω.

Proof. Fix  > 0. From (NQ− ), it follows that

− ≤ 2F(x, s) − s f (x, s) ≤ , x ∈ Ω, s ≤ −M,

for a suitable M > 0. So, we can prove that

 
 ∂ F(x, s) 
≤ ≤ − 3 , x ∈ Ω, s ≤ −M.
s3 ∂s s2 s
In this way, integrating the last expression on [T, t] where T < t ≤ −M, we obtain
   
 1 1 F(x, t) F(x, T )  1 1
− − ≤ − ≤ − .
2 t2 T 2 t2 T2 2 t2 T 2
Thus, by ( f1 ) and letting T → −∞, the last inequalities provide us
 
− ≤ F(x, t) ≤ , x ∈ Ω, t ≤ −M.
2 2
This finishes the proof.

We also mention that, by the previous lemma, we have

lim t f (x, t) = lim f (x, t) = 0, x ∈ Ω.

t→−∞ t→−∞

In particular, it follows that the problem (1.1) becomes strong resonant at −∞. These problems have
been considered in several works since the paper of Benci; Bartolo & Fortunato [4]. We also refer
the reader to the articles [5, 6].
Let H be a Hilbert space. We recall that a functional J : H → R, of class C 1 , satisfies the Cerami
condtion at level c ∈ R, in short (Ce)c , if any sequence (un )n∈N ∈ H such that

J(un ) → c, J  (un )(1 + un ) → 0, as n → ∞ (2.2)

has a convergent subsequence. When J satisfies the (Ce)c property for any c ∈ R we simply say that
J satisfies the (Ce) property.
Next we will prove the Cerami property in order to ensure the compactness required in the proof
of our main theorems.

Proposition 2.1 Assume (NQ+ ), (NQ− ) and ( f0 ), ( f1 ). Then the functional J defined in (1.2) satisfies
the (Ce)c condition for any c ∈ R\{0}.
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162 E. D. da Silva, B. Ribeiro

Proof. Let (un )n∈N ∈ H01 (Ω) be a (Ce)c unbounded sequence at some level c ∈ R. We will prove that
c = 0.
Define vn = un /un . It follows that vn is bounded in H01 (Ω) and there exists v ∈ H01 (Ω) satisfying
the following properties:
• vn  v in H01 (Ω), vn → v in Lq (Ω), q ∈ [1, 2 ),
• vn (x) → v(x) a.e. in Ω,
• |vn (x)| ≤ h(x), for some h ∈ Lq (Ω), q ∈ [1, 2 ).
We begin by proving that v ≤ 0 a. e. in Ω.
By (NQ+ ) and (NQ− ) it is readily seen that
2F(x, t) − t f (x, t) ≤ M, x ∈ Ω, t ∈ R, (2.3)
for some M > 0.
Now, suppose that v > 0 for some Ω 5 ⊂ Ω, |Ω|5 > 0. So that, using (2.3), Fatou’s Lemma and
+
(NQ ), it follows that


−2c = lim sup{J (un )un − 2J(un )} = lim sup 2F(x, un ) − un f (x, un )dx
n→∞ n→∞ Ω

≤ lim sup{2F(x, un ) − un f (x, un )}dx
Ω n→∞
  
≤ + lim sup{2F(x, un )−un f (x, un )}dx (2.4)
v>0 v≤0 n→∞

≤ lim sup{2F(x, un ) − un f (x, un )}dx + M|Ω| → −∞, as n → ∞,
v>0 n→∞

which is a contradiction. Consequently, v ≤ 0 a. e in Ω, that is, v ≡ v− in Ω.

Now, we observe that
   
−2 − 2 f (x, un ) − 2 J (un )v−n
|∇vn | dx − λ1 (vn ) dx − |vn | = →0
Ω Ω Ω un un 
as n → ∞. Moreover, the Lebesgue Convergence Theorem together with ( f1 ) ensure that

f (x, un ) − 2
|vn | → 0.
Ω un
This can be verified because of ( f0 ) holds and the fact that v ≤ 0 a. e. ∈ Ω. As a consequence, since
v−n  v− in H01 (Ω) and using the fact that the norm | | is a s.c.i function in the weak topology, we
have     
−2 − 2
0≤ |∇v| dx − λ1
2
v dx ≤ lim inf
2
|∇vn | dx − λ1 (vn ) dx → 0.
Ω Ω n→∞ Ω Ω
Therefore,  
|∇v|2 dx − λ1 v2 dx = 0
Ω Ω
and v−n → v strongly in H01 (Ω). Hence, v is a weak solution to the eigenvalue problem

−Δu = λ1 u in Ω,
u = 0 on ∂Ω.
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Resonant-Superlinear elliptic problems using variational methods 163

In this way, v = tφ1 , t ≤ 0, where φ1 denotes the first positive eigenfunction associated to λ1 . Later
on we shall prove that t is negative, i.e, the function v is negative in Ω.
We notice that 
lim 2F(x, un ) − un f (x, un )dx = −2c. (2.5)
n→∞ Ω
Moreover, by (NQ ) and (NQ− ), we see also that
+

2F(x, t) − t f (x, t) ≤ −kt+σ + C, x ∈ Ω, t ∈ R, (2.6)

for any k > 0 and for some C > 0. Thus, it follows from (2.5) and (2.6) that
 
−M ≤ 2F(x, un ) − un f (x, un )dx ≤ −k (u+n )σ dx + C|Ω| (2.7)
Ω Ω

for some M > 0. From (2.7) we obtain that

(u+n )n∈N ∈ Lσ (Ω) is a bounded sequence. (2.8)

Now, using the interpolation inequality, we also prove that

u+n  p ≤ u+n 1−t + t

σ un 2 (2.9)
1 1−t t
where = +  . Hence, we see that
p σ 2
u+n  pp ≤ Cu+n  pt . (2.10)

We claim that pt < 2. Indeed, supposing p ≥ 2 + 2/N and using σ > (N/2)(p − 2), we can prove
that pt < 2 . Furthermore, assuming that 2 < p < 2 + 2/N, we put σ = 1 showing once again that
pt < 2.
We notice that (2.10) and ( f0 ) give us the following convergence:

f (x, un ) +
un → 0,
Ω un 
2

as n → 0.
As a consequence, using that J  (un )u+n /un 2 → 0, we have

lim v+n 2 = λ1 v+ 22 = 0. (2.11)

In that case, using (2.11) and since v−n → v in H01 (Ω), it follows that vn → v in H01 (Ω), which means
that v = 1 and this implies that v < 0. So

un (x) → −∞, n → ∞, a. e. in Ω. (2.12)

Next, from (2.2) we see that J  (un )u+n → 0, and using ( f0 ) we have the following estimates

u+n 2 ≤ C + λ1 u+n 22 + f (x, un )u+n dx
Ω
≤ C(1 + u+n 22 + u+n  pp ) (2.13)

for some C > 0. If σ ≥ 2 it follows from (2.8) that u+n 2 is bounded as well. Therefore, by (2.10)
and (2.13) we get
u+n 2 ≤ C(1 + u+n  pt ).
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164 E. D. da Silva, B. Ribeiro

and since pt < 2, we prove that (u+n ) is bounded in H01 (Ω). On the other hand, supposing σ < 2 and
arguing as in (2.9), we see that
u+n 2 ≤ u+n 1−s + s
σ un 2

1 1−s s
where = +  . It is easily seen that 0 < s < 1. This fact together with (2.8) give
2 σ 2

u+n 22 ≤ Cu+n 2s . (2.14)

As a consequence it follows from (2.10), (2.13) and (2.14) that

u+n 2 ≤ C(1 + u+n 2s + u+n  pt ).

Then, (u+n ) ∈ H01 (Ω) is again a bounded sequence.

Take h ∈ H01 (Ω) such that u+n  h. This implies that u+n → h in L p (Ω) and then we have
+
un (x) → h(x) for almost every x ∈ Ω. But from (2.12) it follows that h = 0. In particular, we get

• u+n → 0 in L p (Ω),

• u+n (x) → 0 a. e. Ω,

• |u+n | ≤ g(x), g ∈ L p (Ω).

Hence, from (2.5) and (2.12), we see that

 
−2c = lim 2F(x, u−n ) − u−n f (x, u−n )dx + lim 2F(x, u+n ) − u+n f (x, u+n )dx
n→∞ Ω n→∞ Ω
= 0 (2.15)

The first integral above goes to zero since |2F(x, u−n ) − u−n f (x, u−n )| ≤ C, by (NQ− ) (and so, the
Lebesgue Dominated Convergence Theorem applies). The second integral also goes to zero by the
same theorem, since

|2F(x, u+n ) − u+n f (x, u+n )| ≤ C(1 + |u+n | p ) ≤ C(1 + g p (x)), x ∈ Ω.

Therefore, by (2.15), we evidently have c = 0. The proof of this lemma is now complete.

Remark 2.1 Using the last result we see that our prototype problem (1.3) satisfies the (Ce)c for any
c ∈ R\{0}. In addition, the functional J does not satisfy (Ce)0 with h ≡ 0. In fact, the unbounded
sequence un = −nφ1 , n ∈ N, verifies

(u+n ) p
J(un ) = − dx = 0
Ω p

and 

J (un ), φ = − (u+n ) p−1 φdx = 0, φ ∈ H01 (Ω).
Ω

This shows that J cannot satisfy the (Ce)0 property for several nonlinearities f under conditions ( f1 )
and ( f0 ).
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At this point we consider some auxiliary functions. Define

g(x, t) = λ1 t + f (x, t), x ∈ Ω, t ∈ R.

Then we consider the following functions


− g(x, t), t ≤ 0
g (x, t) =
g(x, 0), t ≥ 0.

and 
g(x, t), t ≥ 0
g+ (x, t) =
g(x, 0), t ≤ 0.
Consider the associated functionals
 
1
J ± (u) = |∇u|2 dx − G± (x, u)dx, u ∈ H01 (Ω);
2 Ω Ω

where  t
±
G (x, t) = g± (x, s)ds, x ∈ Ω, t ∈ R.
0

It is well known that J ± is of class C 1 and its critical points give us negative or positive solutions for
problem (1.1). First of all, it is essential to prove the Cerami condition for J ± . So our next result can
be read as

Proposition 2.2 Suppose ( f1 ), ( f2 ) and (NQ)− hold. Then the functional J − satisfy (Ce)c condition
for any c  0.

Proof. It is worthwhile to mention that

g− (x, t) g− (x, t)
lim = λ1 , lim =0
t→−∞ t t→∞ t
hold uniformly in x ∈ Ω. Furthermore we see that

lim 2G− (x, t) − g− (x, t)t = 0

t→±∞

holds uniformly in x ∈ Ω. Thus the function

2G− (x, t) − g− (x, t)t is bounded in L∞ (Ω).

Once again, we shall prove the Cerami condition arguing by contradiction. Consider an un-
bounded sequence (un ) ∈ H01 (Ω) which is a Cerami sequence at level c  0.
un
Define vn = which is normalized. Given φ ∈ H01 (Ω) we also see that
un 
   −
J (un )φ g (x, un )φ
= ∇vn φdx − dx
un  Ω Ω un 
Taking the limit in the last identity and using ( f0 ) and Lebesgue Convergence Theorem we can prove
that  
∇vφdx = λ1 v− φdx.
Ω Ω
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In other words v is weak solution for the elliptic problem

⎧ −
⎪
⎨ −Δv = λ1 v in Ω
⎪
⎩ v=0 on ∂Ω,

Using the Maximun priciple it follows that v < 0 due to the fact that v  0. Actually, using φ = vn
as testing function, we conclude that v = 1. Thus v is an eigenfunction that can be rewritten as
v = tφ1 , t < 0. In this way, by Lebesgue Convergence Theorem, we obtain

2c = lim 2G− (x, un ) − g− (x, un )un dx = 0.
n→∞ Ω

Therefore c = 0 and which is a contradiction. So the proof of this proposition is now finished.

Proposition 2.3 Suppose ( f1 ), ( f2 ) and (NQ)+ . Then the functional J + satisfy (Ce)c condition for
any c  0.

Proof. The proof follows the same ideas discussed in the proof of Lemma 2.1. We will omit the
details.

3 Variational framework
In this section we shall exploit some results about the geometry for the functional J proving, under
specific hypotheses, that J has a mountain pass geometry or it is bounded from below.

Proposition 3.1 Suppose that ( f0 ) and ( f1 ) hold. Then J − is bounded from below, i.e, there exists
C > 0 such that
J −1 (u) ≥ −C, for any u ∈ H01 (Ω).

Proof. First of all, we recall that G− has the following behavior

λ1 2 λ1
G− (x, t) = t + F(x, t) ≤ t2 + H(x), t ≤ 0, x ∈ Ω.
2 2
Here, we used Lemma 2.1 for a suitable function H ∈ L∞ (Ω). In particular, the functional J − is
bounded from below on H01 (Ω). In fact, given u ∈ H01 (Ω), it follows that

1 2 1 λ1
J − (u) = u − G− (x, u)dx ≥ u2 − u22 − H∞ |Ω| ≥ −H∞ |Ω|
2 Ω 2 2
> −∞.

This ends the proof.

In order to apply the Mountain Pass Theorem for the proof of our main results we consider the
following result.

Proposition 3.2 Suppose ( f0 ), (H1), (H2) hold. Then the functional J has the following mountain
pass geometry:

i) J(u) ≥ β, u = γ, u ∈ H01 (Ω);

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ii) There is a nonzero real number t such that

J(t φ1 ) < 0;

for suitable β, γ > 0.

Proof. Recall that for all t > 0 we can prove that
 
t2
J(tφ1 ) = φ1 2 − G(x, tφ1 )dx = − F(x, tφ1 )dx → −∞
2 Ω Ω

as t → ∞. This prove the item ii) for any t big enough.

In addition, by (H2), we see easily that
g(x, t)
lim sup ≤α
t→0 t
This implies that
α 2
G(x, t) ≤ t + C |t|q , t ∈ R, x ∈ Ω,
2
for some q ∈ (2, 2 ) and C > 0 where 0 < α < λ1 . In particular, using the estimate just above,
variational inequalities and Sobolev’s embedding, it follows that
  
1 α
J(u) ≥ |∇u| dx −
2
u + C
2
|u|q dx
2 Ω 2 Ω Ω
 
1 α
= 1− u2 − C uq
2 λ1
   
1 α
= 1− − C u q−2
u2 ,
2 λ1

for any u ∈ H01 (Ω). Putting u = γ, with γ > 0 small enough, the desired result follows. So we
finish the proof.

At this stage we stress that

g± (x, t)
lim sup ≤α
t→0 t
where α ∈ (0, λ1 ). As a consequence we obtain the following result

Proposition 3.3 Suppose ( f0 ), (H1), (H2) hold. Then the functionals J ± has also the mountain pass
geometry, i.e, we have the following conditions:
i) J ± (u) ≥ β, u = γ, u ∈ H01 (Ω);
ii) There is a nonzero real number t such that

J ± (t φ1 ) < 0;

for suitable β, γ > 0.

Proof. The proof follows from reasoning similar to that given in in the previous proof. We leave the
details for the reader.
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168 E. D. da Silva, B. Ribeiro

4 The proof of the main theorems

In this section we shall prove our main theorems stated in the Introduction.

4.1 Proof of Theorem 1.1

Recall that J − is bounded from below, see Proposition 3.1. On the other hand, by (H1), we see that

J − (t φ1 ) = − F(x, t φ1 )dx < 0.
Ω

As a consequence it follows that

 
inf J − (u), u ∈ H01 (Ω) ≤ J − (t φ1 ) < 0.

Now, by Proposition 2.2 and applying the Ekeland Variational Principle, we obtain a critical point
u for J − such that J − (u ) < 0. Using the test function given by the positive part of u and the fact
that f (x, 0) ≤ 0, by hypothesis, we have
 
|∇u+ |2 dx = g− (x, u )u+ dx ≤ 0.
Ω Ω

This shows that u is a nonpositive function in Ω. Therefore u is a critical point of J and the
problem (1.1) admits at least one solution. So we finish the proof.

4.2 Proof of Theorem 1.2

Using Propositions 3.2 and 2.1, we obtain at least one mountain pass critical point u for J; so that
problem (1.1) admits at least one nontrivial solution with positive energy. This completes the proof.

4.3 Proof of Corollary 1.1

Using the functionals J ± we know that problem (1.1) has two nontrivial solutions with positive
energy, see Propositions 2.2, 2.3 and 3.3. Besides that, using the maximum principle, one solution is
positive and another one is negative. Moreover, by Theorem 1.1 we guarantee one more nontrivial
solutions which has negative energy. So problem (1.1) admits at least three nontrivial solutions. This
ends the proof.

Acknowledgments: This work was finished while the first author was a pos-doctorate in Rome,
Italy that was partially supported by CNPq grants 211623/2013-0. We would like to thank the
Referee for carefully reading our manuscript and for giving constructive comments which helped
improving the results in this paper.

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