Three nontrivial solutions for some elliptic equation involving the N -Laplacian

Three nontrivial solutions for some elliptic equation involving the N -Laplacian

Sami Aouaoui

Institut Supérieur des Mathématiques Appliquées et de l’Informatique de Kairouan, Université de Kairouan, Avenue Assad Iben Fourat, 3100 Kairouan, Tunisie

Received 26 June 2014, appeared 12 February 2015 Communicated by Petru Jebelean

Abstract. A new variational approach is used in order to establish the existence of at least three nontrivial solutions to some elliptic equation involving the N-Laplacian and whose nonlinearity term enjoys a critical exponential growth. The well known Ambrosetti–Rabinowitz condition is not needed.

Keywords: N-Laplacian, Trudinger–Moser inequality, critical exponential growth, vari- ational method, local minima, Palais–Smale condition.

2010 Mathematics Subject Classification: 35D30, 35J20, 35J61, 58E05.

1 Introduction

In this paper, we deal with the following problem

−div

|∇u|^N−2∇u

+V(x)|u|^N−2u= λ

exp

a|u|^NN−1+ f(x,u), inR^N, (Pλ) where N≥2, a is some positive constant andλis some positive parameter. We assume (V₁) V: R^N →]0,+_∞[is a continuous function such that

V(x)≥ V₀, ∀ x∈_R^N, whereV₀ is a positive constant.

(F₁) f: R^N ×_R → _R is a Carathéodory function. We assume that f(x,s) ≥ 0, ∀ (x,s) ∈ R^N×[_0,+_∞[and there existC₀>_0, p>_0, α≥_{0 and}β≥0 such that

|f(x,s)| ≤C₀

|s|^α+|s|^βexp

p|s|^NN−1−S_N−2(p, s), ∀ (x,s)∈_R^N×_R, whereS_N−2(p, s) =_∑k^N₌⁻₀^{2 p}_k!^k|s|^NkN−1.

BEmail: sami_aouaoui@yahoo.fr

Elliptic problems similar to (P_λ), i.e. containing the N-Laplacian and a nonlinear term which behaves like exp α|s|^N/(N−1), as|s| → +_∞have been treated by many authors. We can for example cite [2–5,8–12,14–18,20,21,23]. This interest on that type of nonlinear equations is motivated by the Trudinger–Moser inequality (see [1,13,16,19,22]) which allows a variational analysis of these equations. Our work is a contribution in this direction. Here, we have to highlight the fact that in our paper we do not assume that the famous Ambrosetti–Rabinowitz condition (AR), that is

(AR) there are constantsθ > Nands₀ >0 such that 0<θ

Z _s

0

f(x,t)dt≤s f(x,s)_, ∀ |s| ≥s₀, ∀ x ∈_R^N_, or its weaker form,

(ARR) there existss₀>_{0 and} M>0 such that 0<

Z _s

0 f(x,t)dt≤ M|f(x,s)|, ∀ |s| ≥s₀, ∀ x∈_R^N,

holds. Knowing the important role of this condition in the establishment of existence and multiplicity results, we see that proving the existence of at least three nontrivial solutions could be considered as interesting. Some works dealing with exponential nonlinearities and where the (AR) condition is dropped were published (see, for example, [10–12]). In [12], the authors treated the case N = 2 and they used an appropriate version of the mountain pass theorem introduced by G. Cerami. In order to get the boundedness of some Palais–Smale sequence, they assumed that there existC∗ ≥0, θ≥1 such that

(H) H(x,t)≤θH(x,s) +C∗ ∀0< t<s, ∀x ∈_Ω, where H(x,u) =u f(x,u)−2F(x,u). This work is extended toN-dimensional space in [11]. In [10, Section 7], the authors assumed that(H)holds true withθ = 1 andC∗ =0. In addition, they assumed that there exists c> 0 such that for all(x,s)∈_R^N×[_0,+_∞[_, F(x,s)≤ c(|s|^N+ f(x,s))_.

Using a new variational approach, we establish the existence of at least three nontrivial solutions to the problem (P_λ). For this purpose, we will adapt some arguments developed in [7]. In fact, we will make use of a new Palais–Smale condition introduced by G. Bonanno in [7]

to prove the existence of at least two local minima of the energy functional which corresponds to the problem (P_λ). A third solution is obtained by a suitable version of the mountain pass theorem.

The functional space in which the problem (P_λ) will be studied is E=

u∈W^1,N(_R^N)_,

Z

R^NV(x)|u|^Ndx<+_∞

, which is a reflexive Banach space equipped with the norm

kuk= _Z

R^N

|∇u|^N+V(x)|u|^Ndx _N¹

.

First, we recall the Trudinger–Moser inequality for the whole space R^N, N ≥ 2. In fact, we have the following result (forN=_{2, see [8,}19], and forN≥_{2, see [1,15])}

Z

R^N

h exp

α|u|^NN−1−S_N−2(α,u)ⁱdx<+_∞ foru∈W^1,N(_R^N)andα>0,

where SN−2(α, u) =_∑k^N₌⁻₀^{2 α}_k!^k|u|^NkN−1. Moreover, if |∇u|_LN(_R^N) ≤ 1, |u|_LN(_R^N) ≤ M < +_∞ and α < α_N, then there exists a positive constant C = C(N,M,α), which depends only on N,M

andαsuch that _Z

R^N

h exp

α|u|^NN−1−SN−2(α,u)ⁱdx≤C, (1.1) where α_N = NW

1 N−1

N−1 andW_N−1 is the measure of the unit sphere in R^N. Furthermore, using the above results together with Hölder’s inequality, ifα>_{0 and}q>0 then we have

Z

R^N|u|^qhexp

α|u|^NN−1−S_N−2(α,u)ⁱdx< +_∞, ∀ u∈W^1,N(_R^N).

More precisely, if kuk_W1,N(_R^N) ≤ M with αM^NN−1 < α_N, then there exists a positive constant C= C(α, M, q, N)such that

Z

R^N|u|^qhexp

α|u|^NN−1−S_N−2(α,u)ⁱdx≤Ckuk^q_W1,N(RN), (1.2) where

kuk_W1,N(_R^N)= Z

R^N

|∇u|^N+|u|^Ndx _N¹

is the norm in the Sobolev space W^1,N(_R^N). Observe that since V is positive and bounded from below, then clearly

E,→W^1,N(_R^N),→ L^q(_R^N), ∀ N≤ q<+_∞,

with continuous embeddings. Thus, there exists a positive constantχ₀ such that kuk_W1,N(_R^N)≤χ₀kuk, ∀u∈ E.

This last inequality together with (1.2) implies that there exists a constantC⁰ = C⁰(α,M,q)>0

such that _Z

R^N|u|^qhexp

α|u|^NN−1−SN−2(α,u)ⁱdx≤C⁰kuk^q, (1.3) provided thatkuk ≤ Mwith M < ¹

χ₀(^αN

α )^NN−1_. Assume that

(V₂) the function(V(x))⁻¹belongs to L^N1−1(_R^N).

Then, it is not difficult to show that E,→ L^q(_R^N), ∀ 1 ≤q< +∞, with compact embedding.

Letu∈ E. We have Z

R^N

_{Z u(x)}

0 exp

a|s|^NN−1ds

dx≤

Z

R^Nexp

a|u|^NN−1|u|dx

=

Z

R^N

exp

a|u|^NN−1−S_N−2(a,u)|u|dx +

Z

R^NS_N−2(a,u)|u|dx.

Thus, using (1.3), one can easily find a positive constant Ca >0 such that Z

R^N

_{Z u(x)}

0 exp

a|s|^NN−1ds

dx≤C_akuk, u∈ E, kuk ≤inf

1, 1 2χ₀

α_N a

^N_N⁻¹ . (1.4)

On the other hand, since α+1 ≥ 1, then the continuous (and also compact) embedding E,→ L^α+1(_R^N)holds and by consequence there exists a positive constantCα >0 such that

Z

R^N|u|^α+1dx ≤C_αkuk^α+1, ∀u∈ E. (1.5) Next, foru ∈Ewithkuk ≤ _2χ¹

0(^α_p^N)^NN−1, by (1.3) there exists a constantC_β,p >0 such that Z

R^N|u|^β+1exp

p|u|^NN−1−S_N−2(p,u)dx≤C_β,pkuk^β+1. (1.6) Definition 1.1. A pointu∈ Eis said to be a weak solution of the problem (P_λ) if it satisfies

Z

R^N|∇u|^N−2∇u∇vdx+

Z

R^NV(x)|u|^N−2uv dx

=

Z

R^Nexp

a|u|^NN−1v dx+

Z

R^N f(x,u)v dx, ∀ v∈E.

Now, we are ready to state our main results in the present paper. It consists of the following theorem.

Theorem 1.2. Assume that(V₁), (V₂),and(F₁)hold true. If there exists R>0such that

W_N−1

(R+1)^N−R^N

N +

Z

|x|<RV(x)dx+

Z

R≤|x|≤R+1V(x) (R+1− |x|)^Ndx

< N

"

W_N−1R^N

2N(4N)^N1 C_a+C₀ C_α+C_β,p

#N

,

(1.7)

then there exist0 < λ∗ < λ^∗ < +_∞such that (P_λ) admits at least three nontrivial weak solutions provided thatλ∗ < λ< λ^∗.

Example 1.3. We can takeV(x) =1+σ|x|^α withN(N−1)<αandσsmall enough. In this case, (1.7) holds forRchosen large enough.

2 Proof of Theorem 1.2

Foru∈ Eandλ>_{0, define} Φ(u) =

Z

R^N

|∇u|^N+V(x)|u|^N

N dx= kuk^N N Ψ(u) =

Z

R^N

_{Z u(x)}

0 exp

a|s|^NN−1ds+F(x,u)

dx, Iλ(u) =_Φ(u)−λΨ(u),

where F(x,s) = Rs

0 f(x,t)dt, (x,s)∈ _R^N×R. Clearly, the functional I_λ is well defined on E and by classical arguments (see [6]) it is of class C¹ and the critical points ofI_λ are nontrivial weak solutions of the problem (P_λ).

In order to prove our multiplicity results, we make use of a recent critical points results established by G. Bonanno in [7] by using a new Palais–Smale condition.

Definition 2.1. LetΦ0andΨ0be two continuously Gâteaux differentiable functionals defined on a real Banach space X and fixr₁,r2 ∈ [−_∞,+_∞], withr₁ < r2; we say that the functional I₀ = _Φ0−_Ψ0 verifies the Palais–Smale condition cut off lower atr₁ and upper atr₂ (in short

[r₁](PS)^[r2]) if any sequence(u_n)⊂X such that (i) (I₀(u_n))is bounded,

(ii) I₀⁰(un)→0 inX^∗ (whereX^∗ denotes the topological dual ofX,) (iii) r₁<_Φ0(un)<r2, ∀ n∈_N,

has a convergent subsequence. Clearly, ifr₁=−_∞andr₂ = +_∞it coincides with the classical (PS)condition. Moreover, ifr₁=−_∞andr₂∈ _Rit is denoted by(PS)^[r2].

The main tool to prove the existence of the two first weak solutions of (P_λ) is the following theorem.

Theorem 2.2([7, Theorem 5.1]). Let X be a real Banach space and let Φ,Ψ: X →_Rbe two contin- uously Gâteaux differentiable functions. Assume that there are r₁,r₂ ∈_Rwith r₁< r₂,such that

ς(r₁,r₂) = _inf

r1<_Φ(v)<r2

sup_r1<Φ(u)<r2Ψ(u)−_Ψ(v)

r2−_Φ(v) <ρ(r₁,r₂) = _sup

r₁<_Φ(v)<r₂

Ψ(v)−sup_Φ(u)≤r1Ψ(u) Φ(v)−r₁ , and for each λ ∈] ¹

ρ(r1,r2), _ς(r¹

1,r2)[the functional I_λ = _Φ−_λΨ satisfies ^[r1](PS)^[r2] condition. Then, for each λ ∈] ¹

ρ(r1,r2), _ς(r¹

1,r2)[ there is u_λ ∈ _Φ⁻¹(]r₁,r₂[) such that I_λ(u_λ) ≤ I_λ(u) for all u ∈ Φ⁻¹(]r₁,r₂[)and I_λ⁰(u_λ) =0.

Remark 2.3. Obviously, the critical pointuλof Iλgiven by Theorem2.2is a local minimum of the functional I_λ.

Lemma 2.4. Assume that the hypotheses of Theorem1.2hold true. For each

0<r< ¹

Ninf 1, 1

2χ0

N α_N N⁰p

N−1!

with N⁰ = ^N N−₁ andλ>_0,the functional I_λ satisfies(PS)^[r]_.

Proof. Let (u_n) ⊂ E be such that (I_λ(u_n)) is bounded, I_λ⁰(u_n) → 0 and Φ(u_n) < r, ∀ n ∈ _N.

Sinceku_nk< (Nr)^N1, ∀n ∈N, then there existsu ∈Esuch thatu_n* uweakly inE. By(F₁) we have

Z

R^N|f(x,un)|^N0dx

≤ c_{1 Z}

R^N|u_n|^N0αdx+

Z

R^N|u_n|^N0βexp

N⁰p|u|^NN−1−S_N−2(N⁰p,u)dx

. Sincekunk ≤(Nr)^N1 < _2χ¹

0

αN

N⁰p

^N_N⁻¹

and by (1.3) we deduce that sup

n∈_N

_Z

R^N|f(x,u_n)|^N0dx

<+_∞.

This fact together with the compactness of the embeddingE,→L^N(_R^N)implies

n→+lim_∞ Z

R^N f(x,un)(un−u)dx=0. (2.1) On the other hand, we have

Z

R^Nexp

a|u_n|^NN−1(u_n−u)dx =

Z

R^Nexp

a|u_n|^NN−1 −S_N−2(a,u_n)(u_n−u)dx +

Z

R^NS_N−2(a,un)(un−u)dx.

Using the compact embeddings E ,→ L¹(_R^N)_and E ,→ L²(_R^N)together with (1.1), it is not difficult to prove that

n→+lim_∞ Z

R^Nexp

a|u_n|^NN−1(u_n−u)dx =0. (2.2) Combining (2.1) and (2.2) with the fact

I_λ⁰(un),un−u

→0 asn→+_∞,

we conclude that(u_n)is strongly convergent touinE. This ends the proof of Lemma2.4.

Fix a positive real numberrsuch that r ≤ ¹

4Ninf 1, 1

χ₀ N

α_N N⁰p

N−1

, 1

χ₀

Nα_N a

N−1! .

Lemma 2.5. Assume that the hypotheses of Theorem1.2hold true. Then, there isλ^∗ >0such that: if 0<λ<λ^∗,then the functional I_λadmits a nontrivial critical point u_λ satisfying

0<_Φ(u_λ)<r and I_λ(u_λ)≤ I_λ(w) for all w∈_Φ⁻¹(]_0,r[)_. Proof. Forλ>0 andR>0 as in (1.7), define the function

ϑ_λ =











δ_λ if |x|< R,

δ_λ(R+1− |x|) ifR≤ |x| ≤ R+1, 0 if |x|> R+1, withδ_λ is a real number satisfying

0<δ_λ <inf

λB A

_N¹₋₁ , r

A _N¹

!

, (2.3)

where A= ¹

N

WN−1

(R+1)^N−R^N

N +

Z

|x|<RV(x)dx+

Z

R≤|x|≤R+1V(x) (R+1− |x|)^Ndx

, (2.4) and

B=W_N−1

R^N

N . (2.5)

It is clear thatϑ_λ ∈Eand we have Z

R^N|∇ϑ_λ|^Ndx=

Z

R≤|x|≤R+1δ_λ^Ndx= δ_λ^NW_N−1

(R+1)^N−R^N

N .

On the other hand, we have Z

R^NV(x)ϑ_λ^Ndx= δ_λ^N _Z

|x|<RV(x)dx+

Z

R≤|x|≤R+1V(x) (R+1− |x|)^Ndx

. Thus,

Φ(ϑ_λ) =Aδ_λ^N. (2.6)

Next, since F(x,ϑ_λ)≥0, and exp a|s|^N/N−1 ≥1, we get Ψ(ϑ_λ)≥

Z

|x|<R

_{Z ϑ}

λ(x)

0 exp

a|s|^NN−1ds

dx≥

Z

|x|<Rϑ_λ(x)dx≥Bδ_λ. (2.7) By (2.6) and (2.7), it yields

Ψ(ϑ_λ)

Φ(ϑ_λ) ≥ ^B Aδ_λ^N−1. This inequality with (2.3) leads to

Ψ(ϑ_λ) Φ(ϑ_λ) > ¹

λ. (2.8)

Now, letu∈ Ebe such thatΦ(u)<r. Clearly,kuk<(Nr)^N1. Since (Nr)^N1 < ¹

2χ₀ α_N

N⁰p ^N_N⁻¹

≤ ¹ 2χ₀

α_N p

^N_N⁻¹ , then by(F₁)we have

Z

R^NF(x,u)dx≤ C₀

C_αkuk^α+1+C_β,pkuk^β+1. (2.9) On the other hand, having in mind thatkuk<(Nr)^N1 ≤inf 1, _2χ¹

0(^α_a^N)^NN−1, and using (1.4) it yields

Z

R^N

_{Z u(x)}

0 exp

a|s|^NN−1ds

dx ≤C_akuk. (2.10)

By (2.9) and (2.10), we deduce

Ψ(u)≤ C_akuk+C₀

Cαkuk^α+1+C_β,pkuk^β+1. Since inf(α+1,β+1)≥1 andkuk<1, it follows that

Ψ(u)≤ C_a+C₀(Cα+C_β,p)kuk ≤ C_a+C₀(Cα+C_β,p)(Nr)^N1. (2.11) Set

λ^∗ = ^r

N−1 N

(_4N)^N1 _Ca+_C0(_Cα+_Cβ,p)^. By (2.11), we infer

sup_Φ(u)<rΨ(u)

r ≤ ¹

λ^∗. (2.12)

Thus,

sup_Φ(u)<rΨ(u) r < ¹

λ, provided thatλ< λ^∗. (2.13) From (2.13) and (2.8), we obtain

sup_Φ(u)<rΨ(u) r < ¹

λ < ^Ψ(ϑ_λ) Φ(ϑ_λ)^. Keeping in mind that 0<_Φ(ϑ_λ)<r, we easily deduce that

ς(0,r)<ρ(0,r), and

Φ(ϑ_λ) Ψ(ϑ_λ)^,

r

sup_Φ(u)<rΨ(u)

⊂ 1

ρ(_0,r)^, 1 ς(_0,r)

.

Finally, for 0 < λ < λ^∗, Theorem 2.2 guarantees the existence of a critical point u_λ of I_λ such that 0 < _Φ(u_λ) < r and I_λ(u_λ) ≤ I_λ(w), ∀ w ∈ _Φ⁻¹(]0,r[). This ends the proof of Lemma2.5.

Now, we will try to prove the existence of another critical point of I_λ as a second local minimum. This is made in the following lemma.

Lemma 2.6. Assume that the hypotheses of Theorem1.2hold true. Then, there existsλ∗ ∈]0,λ^∗[such that: ifλ∗ <λ<λ^∗,then the functional I_λadmits a critical pointuf_λwhich satisfies

r<_Φ(_fu_λ)<2r, and I_λ(_fu_λ)≤ I_λ(w), ∀w∈_Φ⁻¹(]r, 2r[). Proof. First set

λ∗ =^r A

^N_N⁻¹ 2A B , where AandBare given by (2.4) and (2.5). By (1.7), we get

λ∗ <λ^∗. Forλ∗ <λ<λ^∗, we keep using the function

ϑ_λ =











δ_λ if |x|< R,

δ_λ(R+1− |x|) ifR≤ |x| ≤ R+1, 0 if |x|> R+_1, with different conditions onδ_λ. In fact, here we chooseδ_λsuch that

r A

_N¹

<δ_λ<inf 2r

A _N¹

, Bλ

2A

_N¹₋₁!

. (2.14)

By (2.6) and (2.7), we get

Ψ(ϑ_λ)

2Φ(ϑ_λ) ≥ ^B 2Aδ_λ^N−1. Taking (2.14) into account, we infer

Ψ(ϑ_λ) 2Φ(ϑ_λ) > ¹

λ. (2.15)

Now, replacingrby(2r)in (2.11), it yields sup_Φ(u)<2rΨ(u)

2r ≤ N^N1 Ca+C0(Cα+C_β,p)(2r)^−NN−1

≤ N^N1 Ca+C0(Cα+C_β,p)r^−NN−1

≤ ¹ λ^∗

< ¹ λ.

(2.16)

By (2.15) and (2.16), we obtain Ψ(ϑ_λ) 2Φ(ϑ_λ) > ¹

λ > ^{supΦ(u)<2rΨ}(u)

2r , for eachλ∗ <λ< λ^∗. (2.17) On the other hand, since _λ¹ > _λ¹∗ andδ_λ <(^Bλ_2A)^N1−1 <(^Bλ_A)^N1−1, we infer

sup_Φ(u)≤rΨ(u) r < ¹

λ < ^Ψ(ϑ_λ)

Φ(ϑ_λ)^{. (2.18)}

One can easily show that (2.17) implies 1

λ > ^{supr<Φ(u)<2rΨ}(u)−_Ψ(ϑ_λ)

2r−_Φ(ϑ_λ) ^{. (2.19)}

Similarly, inequality (2.18) implies 1

λ < ^Ψ(ϑ_λ)−sup_Φ(u)≤rΨ(u)

Φ(ϑ_λ)−r . (2.20)

Next, by (2.14) and (2.6) it yields r<_Φ(ϑ_λ)<2r. Consequently, ς(r, 2r)≤ ^{supr<Φ(u)<2rΨ}(u)−_Ψ(ϑ_λ)

2r−_Φ(ϑ_λ) ^. This inequality together with (2.19) gives

ς(r, 2r)< ¹

λ. (2.21)

Similarly, we get

Ψ(ϑ_λ)−sup_Φ(u)≤rΨ(u)

Φ(ϑ_λ)−r ≤ρ(r, 2r). Taking (2.20) into account, it follows that

1

λ <ρ(r, 2r). (2.22)

Combining (2.21) and (2.22), we deduce that 1

ρ(r, 2r) <λ< ¹

ς(r, 2r)^, ∀ λ∗ <λ<λ^∗. Since

0<2r< ¹

Ninf 1, 1

χ²₀ N

α_N N⁰p

N−1! ,

then by Lemma2.4, the functional I_λ satisfies^[r](PS)^[2r]. Hence, all the conditions of Theorem 2.2 are fulfilled. We conclude that, for eachλ∗ < λ < λ^∗, the functional Iλ admits a critical pointfu_λ satisfying

r< _Φ(_fuλ)<_{2r, and Iλ}(_ufλ)≤ Iλ(_w)_, ∀ w∈_Φ⁻¹(]_{r, 2r}[)_. SinceΦ(u_λ)<r, thenu_λ 6=_fu_λ.

In order to prove the existence of a third critical point of I_λ, the following result is needed.

Theorem 2.7 ([7, Theorem 6.2]). Let X be a real Banach space and let Φ0,Ψ0: X → _R be two continuously Gâteaux differentiable functions withΦ0 convex. Put I₀ = _Φ0−_Ψ0 and assume that x₀,x₁∈ X are two local minima of I₀. Put m₀=min_t∈[0,1]Ψ0(tx₁+ (1−t)x₀)and assume that there are r0 >max{_Φ0(x0),Φ0(x₁)}and s0 ≥0such that

sup

Φ0(x)<r0+s0

Ψ0(x)<s₀+m₀,

and I₀ satisfies the (PS)^[r0+s0] condition. Then, I₀ admits at least a third critical point x₃ such that Φ0(x₃)<r₀+s₀.

Forλ∗ <λ<λ^∗, take x₁ =u_λ, x₂= _fu_λ, r₀ =3r ands₀=r. Since r₀+s₀=4r< ¹

Ninf 1, 1

2χ₀ N

α_N N⁰p

N−1! , then I_λ satisfies(PS)^[r0+s0]. Arguing as in (2.11), we can easily obtain

sup

Φ(u)<4r

Ψ(u)≤(4N)^N1 Ca+C0 Cα+C_β,p r^N1. Thus,

sup_Φ(u)<4rΨ(u)

r ≤(4N)^N1 C_a+C₀ C_α+C_β,p

r^−NN−1 = ¹ λ^∗. Sinceλ< λ^∗, then

sup

Φ(u)<4r

Ψ(u)< ^r λ.

By the virtue of Theorem 2.7, the functional I_λ admits at least a third critical point ffu_λ such thatΦ(u^f_fλ)<r₀+s₀=4r. This ends the proof of Theorem1.2.

References

[1] S. Adachi, K. Tanaka, Trudinger type inequalities inR^N and their best constants,Proc.

Amer. Math. Soc.128(2000), 2051–2057.MR1646323;url

[2] Adimurthi, Existence of positive solutions of the semilinear Dirichlet problem with crit- ical growth for the N-Laplacian, Ann. Sc. Norm. Super. Pisa Cl. Sci. 17(1990), 393–413.

MR1079983

[3] F. S. B. Albuquerque, C. O. Alves, E. S. Medeiros, Nonlinear Schrödinger equation with unbounded or decaying radial potentials involving exponential critical growth in R²,J. Math. Anal. Appl.409(2014), 1021–1031.MR3103216;url

[4] C. O. Alves, M. A. S. Souto, Multiplicity of positive solutions for a class of prob- lems with exponential critical growth inR²,J. Differential Equations244(2008), 1502–1520.

MR2396507;url

[5] C. O. Alves, J. M.doÓ, O. H. Miyagaki, On nonlinear perturbations of a periodic elliptic problem in R² involving critical growth, Nonlinear Anal. 56(2004), 781–791. MR2036791;

url

[6] H. Berestycki, P. L. Lions, Nonlinear scalar field equations, I. Existence of ground state, Arch. Rational Mech. Anal.82(1983) 313–346.MR695535;url

[7] G. Bonanno, A critical point theorem via the Ekeland variational principle, Nonlinear Anal.75(2012), 2992–3007.MR2878492;url

[8] D. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in R², Commun. Partial Differ. Equ.17(1992), 407–435.MR1163431;url

[9] L. R. deFreitas, Multiplicity of solutions for a class of quasilinear equations with expo- nential critical growth,Nonlinear Anal.95(2014), 607–624.MR3130547;url

[10] N. Lam, G. Lu, Existence and multiplicity of solutions to equations of N-Laplacian type with critical exponential growth inR^N,J. Funct. Anal. 262(2012), 1132–1165.MR2863858;

url

[11] N. Lam, G. Lu,N-Laplacian equations inR^N with subcritical and critical growth without the Ambrosetti–Rabinowitz condition,Adv. Nonlinear Stud.13(2013), 289–308.MR3076792 [12] N. Lam, G. Lu, Elliptic equations and systems with subcritical and critical exponential growth without the Ambrosetti–Rabinowitz condition, J. Geom. Anal.24(2014), 118–143.

MR3145918;url

[13] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J.

20(1970/71), 1077–1092.MR301504;url

[14] J. M.doÓ, Semilinear Dirichlet problems for the N-Laplacian in R^N with nonlinearities in the critical growth range,Differential Integral Equations9(1996), 967–979.MR1392090 [15] J. M.doÓ, N-Laplacian equations in R^N with critical growth,Abstr. Appl. Anal.2(1997),

301–315.MR1704875;url

[16] J. M.doÓ, E. Medeiros, U. Severo, On a quasilinear nonhomogeneous elliptic equation with critical growth inR^N,J. Differential Equations246(2009), 1363–1386.MR2488689;url [17] J. M. do Ó, E. Medeiros, U. Severo, A nonhomogeneous elliptic problem involving

critical growth in dimension two,J. Math. Anal. Appl.345(2008), 286–304.MR2422651;url [18] J. M. do Ó, M. de Souza, E. de Medeiros, U. Severo, An improvement of the Trudinger–Moser inequality and applications, J. Differential Equations 256(2014), 1317–

1349.MR3145759;url

[19] B. Ruf, A sharp Trudinger–Moser type inequality for unbounded domains inR²,J. Funct.

Anal.219(2005), 340–367.MR2109256;url

[20] M. de Souza, E. de Medeiros, U. Severo, On a class of quasilinear elliptic prob- lems involving Trudinger–Moser nonlinearities, J. Math. Anal. Appl. 403(2013), 357–364.

MR3037474;url

[21] E. Tonkes, Solutions to a perturbed critical semilinear equation concerning the N-Laplacian inR^N,Comment. Math. Univ. Carolin. 40(1999), 679–699.MR1756545

[22] N. S. Trudinger, On imbeddings into Orlicz spaces and some applications,J. Math. Mech.

17(1967), 473–483.MR216286

[23] Y. Wang, J. Yang, Y. Zhang, Quasilinear elliptic equations involving the N-Laplacian with critical exponential growth inR^N,Nonlinear Anal.71(2009), 6157–6169.MR2566522;

url