Antisymmetric solutions for a class of quasilinear defocusing Schrödinger equations

Antisymmetric solutions for a class of quasilinear defocusing Schrödinger equations

Janete Soares Gamboa and Jiazheng Zhou

Universidade de Brasília, Departamento de Matemática, 70910-900, Brasília - DF - Brazil Received 15 October 2019, appeared 5 March 2020

Communicated by Dimitri Mugnai

Abstract. In this paper we consider the existence of antisymmetric solutions for the quasilinear defocusing Schrödinger equation inH¹(_R^N)_:

−_∆u+^k

2u∆u²+V(x)u=g(u),

where N ≥ 3, V(x) is a positive continuous potential, g(u) is of subcritical growth and k is a non-negative parameter. By considering a minimizing problem restricted on a partial Nehari manifold, we prove the existence of antisymmetric solutions via a deformation lemma.

Keywords: quasilinear Schrödinger equation, antisymmetric solutions, Nehari mani- fold.

2020 Mathematics Subject Classification: 35J20, 35J60, 35D05.

1 Introduction and main results

In this paper we are interested in the existence of antisymmetric solutions in H¹(_R^N)_{for the} modified quasilinear Schrödinger equation

−_∆u+^k

2u∆u²+V(x)u= g(u) inR^N, (1.1) where V : R^N → (0,∞) is a continuous and positive potential function, g : R → _R is a continuous and subcritical function,k ≥ 0 is a parameter. The existence of solutions for (1.1) is closely related to study of standing waves ω(x,t) = u(x)e^−(iEt)/¯h for the superfluid film equation arising in the plasma physics (see [9]),

i¯h∂tω =−_∆ω+W(x)ω−eh(|ω|²)ω+ ^k

2ω∆ω², (1.2)

where W(x) is a given potential and eh(u²)u = g(u) is a real function. So, ω(x,t) will be a such solution of (1.2) if and only ifu(x)solves equation (1.1) withV(x) =W(x)−E.

BCorresponding author. Email: zhou@mat.unb.br

For the case k = 0, equation (1.1) becomes a semilinear Schrödinger equation. The exis- tence of positive ground states or least action nodal solutions for the semilinear Schrödinger equation has been studied widely, we refer the readers to [3,8,24,26] and the references therein for the literature on nodal solutions of the semilinear Schrödinger equation.

Fork =−1, the modified quasilinear Schrödinger equation has received a lot of attention.

The appearance of the quasilinear partu∆u²makes the problem much more complicated, it is quite difficult to study the associated energy functional directly in the Sobolev spaceH¹(_R^N) and requires one to develop new techniques to apply variational methods. The existence of a positive ground state solution of equation (1.1) has been proved in [16] and [25] by introducing a parameter λ in front of the nonlinear term. In [17], by a change of variables, the authors studied the quasilinear problem was transformed to a semilinear one and the existence of a positive solution was proved using the Mountain-Pass Lemma in an Orlicz space. Different from the change of variable methods, in [20] the authors introduced new perturbation techniques and also proved the existence of solutions for a new kind of critical problems for the modified quasilinear Schrödinger equation in [21].

The existence of sign-changing solution is an interesting topic i.e. looking for solutions u with u⁺,u⁻ 6= 0, where u⁺(x) = max{u(x), 0} ≥ 0, and u⁻(x) = min{u(x), 0} ≤ 0, x ∈ _R^N. In [18] the authors proved the existence of sign-changing ground state solution for (1.1) withk =−1 andg(s) =|s|^p−2s,s∈ _Rwith 3≤ p <22^∗−1, that is,g having subcritical growth (22^∗plays the role of critical exponent here), andVis a continuous function such that 0<V₀ =inf_RNV(x)≤ lim_|x|→∞V(x) = V_∞ withV(x)≤V_∞−A/(1+|x|^m), for|x| ≥ M, for some real constantsA,M,m>0. The perturbation arguments in [21] was successfully applied to study the existence of multiple nodal solutions for a general class of sub-critical quasilinear Schrödinger equation in [19].

Also, we would also like to mention [10,11,13,15,18] and references therein for some recent progress of the study of the quasilinear Schrödinger equation fork < 0. However, in [12,14], the nonlinearity gis permitted to behave in a critical way, under the more restrictive assumption thatVis symmetric radially positive and differentiable continuous function with V⁰(r) ≥ 0 for r ≥ 0. Their approach was based on Mountain Pass Theorem on Nehari manifolds.

But, for the case k > 0, it seems that there are few work about this type of problems.

The existence results of solutions, we like to mention [1] and the existence of sign-changing solutions, we like to mention [2].

The existence of τ-antisymmetric solutions, in [5] and [6], the authors proved existence of τ-antisymmetric solutions for the problem

−_∆u+V(x)u= g(u) inR^N, by considering the limit problem

−_∆u+V_∞u=g(u) inR^N.

In [7], the authors showed the existence ofτ-antisymmetric solutions for the system (−_∆u+u=|u|^2p−2u+β(x)|v|^p|u|^p−2u, inR^N,

−_∆v+ω²v=|v|^2p−2v+β(x)|u|^p|v|^p−2v, inR^N under suitable assumptions by considering the limit problem

(−_∆u+u=|u|^2p−2u+β_∞|v|^p|u|^p−2u, inR^N,

−_∆v+ω²v =|v|^2p−2v+β_∞|u|^p|v|^p−2v, inR^N,

and other additional conditions.

However, for the casek6=0, it seems that the existence results of solutions ofτ-antisymmetric solutions to equation (1.1) has not been considered yet. Thus the aim of the present paper is to study the existence of τ-antisymmetric solution for a quasilinear defocusing Schrödinger equation.

To state the main results, we may assume that the potential functionVis continuous such thatV(x)≥V₀ >0 for allx∈ _R^N, and:

(V₁) V(τx) =V(x), whereτ:R^N →_R^N is a nontrivial orthogonal involution that is a linear orthogonal transformation onR^N such thatτ6=Id andτ²=Id;

(V₂) V is 1-periodic inx_i, 1≤i≤ N;

(V₃) V is radially symmetric, i.e.V(x) =V(|x|)andV ∈L^∞(_R^N); (V₄) lim_|x|→∞V(x) =_∞.

The nonlinearitygis supposed to satisfy:

(G₁) g∈C(_R,R)is such thatg(0) =0 and odd;

(G2) lim_|t|→0^g(_t^t) =0 and lim sup_{|t|→∞ |}^g_t|⁽q^t−⁾1 <_∞for someq∈(2, 2^∗); (G₃) 0<θG(s)≤ sg(s),s6=0 for some 2<θ <2^∗, whereG(u) =Ru

0 g(t)dt;

(G₄) t 7−→ ^g_t⁽_ρ^t),t >0 is non-decreasing for someρ>1.

Our principal result shows the existence of aτ-antisymmetric solution, that is u satisfies (1.1) andu(τx) =−u(x)_.

Theorem 1.1. Suppose that(V₁)holds and one of(V₂),(V₃)and(V₄)is satisfied and the conditions (G₁)–(G₄)hold. Then there exists k0>0such that for each k∈(0,k0)equation(1.1)has at least one τ-antisymmetric solution u∈ H¹(_R^N)∩L^∞(_R^N)with

max

x∈_R^N|u(x)| ≤ √^σ

k, whereσ=

"

4− ¹ ρ−

s 1 ρ² +⁸

ρ

! /8

#1/2

. (1.3)

The antisymmetric solution found in Theorem1.1minimizes the energy functional among all possible solutions for (1.1), and so we can call it the least action antisymmetric solution.

This work contributes to the literature of modified quasilinear defocusing Schrödinger equation in the two senses: on the hand, we found anτ-antisymmetric solution instead of a limit problem, we used several different conditions of the functionV; on the other hand, we just need the function g to be continuous, so we can not use directly Ekeland’s variational principle.

The paper is organized as follows. In Section 2, we introduce the variational framework for the quasilinear defocusing Schrödinger equation. In Section 3, establishing some auxil- iary lemmas and build a homeomorphism between sphere and Nehari manifold. Finally in Section 4, we prove the existence ofτ-antisymmetric solution for (1.1) with subcritical growth and obtaining aL^∞-estimate.

Notation

We will use the following notations frequently:

•C,C₀,C₁,C₂, . . . denote positive (possibly different) constants.

•B_R denotes the open ball centered at the origin with radiusR>0.

•C₀^∞(_R^N)denotes functions infinitely differentiable with compact support inR^N.

•_{For 1}≤ s≤_∞, L^s(_R^N)denotes the usual Lebesgue space with the norms

|u|_s:=

Z

R^N|u|^s1/s_{, 1}≤s <_∞;

|u|_∞:=inf{C>0 :|u(x)| ≤Calmost everywhere in R^N}.

•H¹(_R^N)denotes the Sobolev spaces with usual norm kuk_1,2:= |∇u|²₂+|u|²₂^1/2.

•The weak convergence in H¹(_R^N)is denoted by*, and the strong convergence by→.

2 The modified problem

Formally, this equation has a variational structure, that is, by considering I(u) = ¹

2 Z

R^N(1−k|u|²)|∇u|²+¹ 2

Z

R^NV(x)|u|²−

Z

R^NG(u), a functionu∈ H¹(_R^N)is said to be a weak solution of equation (1.1) if it satisfies

Z

R^N(1−k|u|²)∇u∇ϕ−k Z

R^N|∇u|²uϕ+

Z

R^NV(x)uϕ=

Z

R^Ng(u)ϕ for all ϕ∈ H¹(_R^N), which meanshI⁰(u),ϕi=0 for all ϕ∈ H¹(_R^N).

First, we point out that, under the hypothesisV(x)≥V0 >0 for allx ∈_R^N, the subset E=

u∈ H¹(_R^N)

Z

R^NV(x)u²(x)<_∞

is a closed subspace ofH¹(_R^N). Moreover, kuk²_E =

Z

R^N|∇u|²+

Z

R^NV(x)u²(x)

defines a norm on E. However, the presence of the second order nonhomogeneous term u∆u² prevents us to work directly with the functionalI, because it is not even well defined in general inH¹(_R^N).

In order to prove the main results, we first establish the existence of nontrivial solution for a modified quasilinear Schrödinger equation. More precisely, we will show the existence of sign changing solutions for the following quasilinear Schrödinger equations

−div(l²(u)∇u) +l(u)l⁰(u)|∇u|²+V(x)u= g(u)_, x∈_R^N _(2.1) with l(t) = √

1−kt² for|t| < σ/√

k for k > 0, where V : R^N → _R is a continuous function andσ > 0 was chosen in (1.3). Clearly, whenl(t) = √

1−kt², we derive that (2.1) turns into

(1.1). Then, by using Morse typeL^∞-estimate, we will prove that there existk₀such that for all k∈[0,k0)the solution found verifies the estimate max_R^N|u|< σ/√

k. After that, we conclude that the solutions obtained are solutions of the original equation (1.1).

For the equation (2.1), we will considerl:R→_Rdefined by

l(t) =











p1−kt², if 0≤t< ^√σ

k, σ³

√k kt√

1−σ² +

s1

ρ, ift≥ ^√σ

k,

and l(t) = l(−t) _{for all} t ≤ 0. So, it follows from the choice of σ = σ(ρ) > _{0 for} ρ > _{1 in} (1.3) that l ∈ C¹(_R,(^p1/ρ, 1))is an even function and it increases in (−_{∞, 0})and decreases in [0,+_∞).

Note that (2.1) is the Euler–Lagrange equation associated to the energy functional I_k(u) = ¹

2 Z

R^Nl²(u)|∇u|²+ ¹ 2

Z

R^NV(x)|u|²−

Z

R^NG(u) (2.2)

for|u|< _σ/√ k.

In the sequel, we will prove the existence of nontrivial antisymmetric critical points u of (2.2) satisfying sup_x∈RN|u(x)| ≤ σ/√

k. This means that it is a nontrivial antisymmetric solution of (2.1) withl(u) =√

1−ku², and so, a nontrivial antisymmetric solution of (1.1) can be got from the functionl.

In what follows, we set

L(t) =

Z _t

0

l(s)ds, t∈_R.

By a simple computation, we see that the inverse function L⁻¹(t) exists and it is an odd function. Moreover, it is very important to note thatL,L⁻¹ ∈C²(_R). The lemma below shows some important properties of the functionslandL⁻¹that will be used in the later part of the paper.

Remark 2.1. From assumption (G₄), if ρ₂ > ρ₁ > 1 and g(t)/t^ρ2 is non-decreasing, then g(t)/t^ρ1 is non-decreasing as well. Thus, if g(t)/t^ρ is non-decreasing for someρ> _{1, we can} assume that ρis sufficiently close to 1, satisfying

4+ ¹ ρ+

s 1 ρ² + ⁸

ρ > √⁸

ρ and 2<2√

ρ<θ. (2.3)

Throughout the paper, we need the following lemma. Its proof can be found in [1] and [2].

Lemma 2.2. The functions l and L⁻¹ satisfy:

(1) limt→0 L⁻¹(t) t =1;

(2) lim_t→_∞ ^L

−1(t)

t =√

ρ;

(3) ^q1

ρt≤ l(t)t≤ L(t)≤t and t≤ L⁻¹(t)≤√

ρt, for all t≥0;

(4) − ^σ2

1−σ² ≤ ^t

l(t)l⁰(t)≤0, for all t≥0;

(₅) ^[L−1(t)]δ

l(L⁻¹(t))t, t >₀is increasing forδ>₁and non-decreasing forδ=1, (6) ^L−1(t)

l(L⁻¹(t))t^ρ, t>0is decreasing forρ>1close to1and ^L−1_t^(t), t>0is non-decreasing.

Now, changing variable by

v=L(u) =

Z _u

0 l(s)ds,

we can observe that the functional I_k can be rewritten in the form J_k(v) = ¹

2 Z

R^N|∇v|²+ ¹ 2 Z

R^NV(x)|L⁻¹(v)|²−

Z

R^NG(L⁻¹(v))_. From Lemma2.2, J_k is well defined inH¹(_R^N)and J_k ∈ C¹(H¹(_R^N),R)with

hJ_k⁰(v),φi=

Z

R^N

∇v∇φ+V(x) ^L

−1(v)

l(L⁻¹(v))^φ− ^g(L⁻¹(v)) l(L⁻¹(v))^φ

, (2.4)

for allv,φ∈ H¹(_R^N).

Lemma 2.3. If v∈ H¹(_R^N)is a critical point of J_k, then u= L⁻¹(v)∈ H¹(_R^N)and additionally it is a weak solution for(2.1)ifsup_x∈RN|u(x)| ≤σ/√

k.

Proof. See [2].

The following embedding result plays an important role in showing that the minimizing function on the partial Nehari manifold are non-trivial functions.

Proposition 2.4. The function L⁻¹is such that:

1. the map v7−→ L⁻¹(v)from E,k · k_E to L^s(_R^N),| · |_s is continuous for2≤ s≤2^∗. 2. under(V₄), the above map is compact for2 ≤s <2^∗, and under(V3)with N ≥ 2, this map is

compact for2<s<2^∗. Proof. See [2].

3 Auxiliary results

Before stating the auxiliary results, let us point out some consequences of our hypotheses.

Remark 3.1. From assumption (G₂), there existsce>0 such that g(t)t≤e|t|²+c_e|t|^q ∀ t∈ _R for eache>0 given.

Remark 3.2. From assumption (G₃), there exists a constantK>0 such that G(_t)≥K|t|^θ for all |t|> δ

for eachδ>0 given.

After these, let us associate to the functional J_k the Nehari manifold N ={v∈ E\{0} | hJ_k⁰(v),vi=0}.

In order to findτ-antisymmetric solutions, we look for critical points of the functional J_k on N^τ ={v∈ N |v(τx) =−v(x)} ⊂ N_.

The involutionτon R^N induces an involutionT_τ :E→Egiven by T_τ(v(x))_:=−v(τ(x))_.

We denote byE^τ := {u∈ E:Tτ(v(x)) =v(x)}the subspace ofτ-invariant functions of E, we have

N^τ = N ∩E^τ.

Now, we are going to introduce the differentiable continuous functionh^v_k : [0,∞)→ _Rby settingh^v_k(t) =J_k(tv), that is,

h^v_k(t):= ¹ 2

Z

R^N|t∇v|²+ ¹ 2

Z

R^NV(x)|L⁻¹(tv)|²−

Z

R^NG(L⁻¹(tv)), for each v∈Ewith v6=0.

Lemma 3.3. Assume that(G₁)–(G₃)hold. If v∈E^τ with v6=0, then there existα>0such that hJ_k⁰(αv),vi=0,

that is,αv∈ N^τ, andα∈ (0,∞)is a critical point of h^v_k. Proof. It follows from the definition ofh^v_k, that

∂h^v_k(t)

∂t =t Z

R^N|∇v|²+

Z

R^NV(x) ^L

−1(tv) l(L⁻¹(tv))^v−

Z

R^N

g(L⁻¹(tv)) l(L⁻¹(tv))^v

=hJ_k⁰(tv),vi.

(3.1)

So, it follows from Remark3.1and(3)of Lemma2.2, that hJ_k⁰(tv),tvi ≥t²

Z

R^N|∇v|²−

Z

R^N

g(L⁻¹(tv)) l(L⁻¹(tv))^tv

≥t² Z

R^N|∇v|²−

Z

R^N

e|L⁻¹(tv)|²+c_e|L⁻¹(tv)|^q p1/ρ|L⁻¹(tv)| |tv|

≥t²|∇v|²₂−ρet²|v|²₂−√

ρ^qcet^q|v|^q_q, which means there existst_m >0 sufficiently small such that

hJ_k⁰(t_mv)_,t_mvi>_0, since q>2.

On the other hand, it follows from Hypothesis(G₃)that hJ_k⁰(tv)_,tvi ≤t²

Z

R^N|∇v|²+

Z

R^NV(x) ^L

−1(tv)

l(L⁻¹(tv))(tv)−θ Z

R^N

G(L⁻¹(tv))

l(L⁻¹(tv))L⁻¹(tv)(tv)_. Setδ >0 such that the set

A={x∈_R^N; |v(x)| ≥δ} ⊂_R^N is not empty. By Remark3.2;l(t)>1/√

ρ,t>0; and(3)of Lemma2.2, we get hJ_k⁰(tv),tvi ≤t²

Z

R^N|∇v|²+√ ρt²

Z

R^NV(x)v²−θKt^θ Z

A|v|^θ

fort >0.

As a consequence, we obtaintM >0 sufficiently large such that hJ_k⁰(t_Mv),t_Mvi<0,

sinceθ >2. Hence, the lemma follows from intermediate value theorem.

Lemma 3.4. If v∈ N and(G₄)hold, then

∂h^v_k

∂t (t)>0 for0<t<1, ∂h^v_k

∂t (t)<0 for t>1, In particular, h^v_k(t)< h^v_k(₁) = J_k(v)for all t≥0such that t6=1.

Proof. By the facts of lbeing even andL odd, it is sufficiently to prove the case of thatv≥ 0.

First, it follows from (3.1) that

∂h^v_k(t)

∂t =t^ρ Z

R^N

|∇v|² t^ρ−1 −

Z

R^N

g(L⁻¹(tv))

l(L⁻¹(tv))(tv)^ρ − ^V(x)L⁻¹(tv) l(L⁻¹(tv))(tv)^ρ

v^ρ+1

. Now, by using(G₄), (5), (6) of Lemma2.2, and the monotonicity ofl,L⁻¹, we obtain

g(L⁻¹(tv))

l(L⁻¹(tv))(tv)^ρ − ^V(x)L⁻¹(tv) l(L⁻¹(tv))(tv)^ρ

= ^g(L⁻¹(tv)) (L⁻¹(tv))^ρ

(L⁻¹(tv)) tv

ρ

1

l(L⁻¹(tv))−V(x) ^L

−1(tv) l(L⁻¹(tv))(tv)^ρ

< ^g(L⁻¹(v)) (L⁻¹(v))^ρ

(L⁻¹(v)) v

ρ

1

l(L⁻¹(v))−V(x) ^L

−1(v) l(L⁻¹(v))(v)^ρ

= ^g(L⁻¹(v))

l(L⁻¹(v))(v)^ρ −V(x) ^L

−1(v) l(L⁻¹(v))(v)^ρ

for 0<t <1, and in a similar way, we obtain g(L⁻¹(tv))

l(L⁻¹(tv))(tv)^ρ − ^V(x)L⁻¹(tv)

l(L⁻¹(tv))(tv)^ρ > ^g(L⁻¹(v))

l(L⁻¹(v))(v)^ρ − ^V(x)L⁻¹(v) l(L⁻¹(v))(v)^ρ fort >1.

So, it follows from above informations, and the hypothesisv∈ N, that

∂h^v_k

∂t (t)>0 for 0<t<1, and ∂h^v_k

∂t (t)<0 fort >1. (3.2) That is,h^v_k(t)< h^v_k(1) = J_k(v). So, the lemma is proved.

It follows from above informations, that:

Remark 3.5. Ifv ∈ N, then 1 is an unique critical point ofh^v_k.

Remark 3.6. If v ∈ E with v 6= 0, then the critical point α = α_v ∈ (0,+_∞) of h^v_k, given by Lemma3.3, is unique.

In fact, by Lemma 3.3 there is α > 0 such thatα is a critical point of h^v_k. Finally, assume that α₁ andα₂ are two critical points ofh^v_k, then

α₂ α₁

(α₁v) =α₂v.

Sinceα₁v∈ N, then by the Remark3.5, we haveα₂/α1=_{1, and so} α₁=α₂.

The following two lemmas are important to prove our theorem, the proofs can be found in [2]

Lemma 3.7. Assume that V is continuous such that V(x)≥ V₀ >0for all x ∈_R^N and(G₁)–(G₃) hold. Then:

(i) for all v∈ N, we have

J_k(v)≥ ^θ−2√ ρ 2θ

_Z

R^N|∇v|²+

Z

R^NV(x)|L⁻¹(v)|²

,

(ii) there isγ>0such that Z

R^N|∇v|²+

Z

R^NV(x)|L⁻¹(v)|² ≥γ, for all v∈ N.

Lemma 3.8. Assume the same hypotheses of Lemma3.7, and(v_n)being a sequence inN. Then lim inf

n→_∞ Z

R^N|L⁻¹(v_n)|^qdx>0 for some q∈(2, 2^∗).

Remark 3.9. By Lemma3.8and(₃)_{of Lemma}2.2, there exists a constantγ₁ >0 such that Z

R^N|vn|^q ≥γ₁>0.

Lemma 3.10. Assume that(G₄)hold. IfV ⊂ S^τ is a compact subset of E^τ, then there exists R > 0 such that J_k ≤0on(_R⁺V)\B_R(0), where S^τ :={u∈ E^τ; kuk_E =1}.

Proof. Arguing by contradiction, suppose there exitsun∈ Vandwn =tnunsuch thatJ_k(wn)≥ 0 andt_n→_∞asn→_∞.

By the definition of J_k and(3)of Lemma2.2 have J_k(w_n)≤ ^ρ

2kw_nk_E−

Z

R^NG(L⁻¹(w_n)) = ^ρ 2t²_n−

Z

R^NG(L⁻¹(w_n)). Using(G₄), we havet7−→ ^G(t)

t^ρ+1,t >0 is non-decreasing for someρ>1 and G(L⁻¹(w))

L⁻¹(w)² →_∞ uniformly inx as|w| →_∞. (3.3)

Passing to a subsequence, we may assume thatu_n →u ∈ S^τ. Since|w_n(x)| →_∞ ifu(x)6= 0, it follows from(3)of Lemma2.2, (3.3) and Fatou’s lemma that

Z

R^N

G(L⁻¹(w_n)) t²_n =

Z

R^N

G(L⁻¹(w_n))u²_n w²_n

=

Z

R^N

G(L⁻¹(w_n)) L⁻¹(w_n)²

L⁻¹(w_n)²

w²_n u²_n →_∞ Hence

0≤ J_k(_wn)≤t²_n ρ

2 −

Z

R^N

G(L⁻¹(w_n)) t²_n

→ −_∞, a contradiction.

Recall thatSis the unit sphere inEand define the mappingm:S→ N by setting m(w):=t_ww,

wheret_wis asαin Lemma3.3. Moreover,km(w)k_E = t_w.

Recall thatS^τ is the unit sphere in E^τ, and consider the mappingm^τ :S^τ → N^τ by setting m^τ :=m|_Sτ.

We shall consider the functional

ψ_k^τ(w):= J_k(m^τ(w)).

By Lemma3.3, Lemma3.4, Remark3.5, Lemma3.7and Lemma3.10, we have the following two lemmas, similar to the results in [23].

Lemma 3.11. The mapping m^τ is a homeomorphism between S^τ andN^τ, and the inverse of m^τ is given by(m^τ)⁻¹(u) = _k^u

uk_E. Lemma 3.12.

(1) ψ^τ_k ∈ C¹(S^τ,R)and

h(ψ^τ_k)⁰(w),zi=km^τ(w)k_EhJ_k⁰(m^τ(w)),zi for all z∈ Tw(S^τ)⊂ E^τ.

(2) If (w_n)is a Palais–Smale sequence for ψ^τ_k, then (m^τ(w_n)) is a Palais–Smale sequence for J_k. If (u_n) ⊂ N^τ is a bounded Palais–Smale sequence for J_k, then ((m^τ)⁻¹(u_n)) is a Palais–Smale sequence forψ^τ_k.

(3) w is a critical point ofψ^τ_k if and only if m^τ(w)is a nontrivial critical point of J_k|_Eτ. Moreover, the corresponding values ofψ^τ_k and J_kcoincide andinf_S^τψ_k^τ =infN^τ J_k.

(4) If J_k is even, then so isψ^τ_k.

4 Proof of Theorem 1.1

Now, we are ready to prove Theorem1.1by applying the auxiliary results in Section 3.

Proof of Theorem1.1. It follows from Lemma3.7that there existsc₀>0 such that c₀ = inf

w∈N^τJ_k(w).

Moreover, if u0 ∈ N^τ satisfies J_k(u0) = c0, then (m^τ)⁻¹(u0) ∈ S^τ is a minimizer of ψ^τ_k and therefore a critical point of ψ^τ_k, so that u₀ is a critical point of J_k in E^τ by Lemma 3.12. We will show that there exists a minimizer v ∈ N^τ of J_k|_Nτ. By Ekeland’s variational principle [27], there exists a sequence(wn)⊂ S^τ with ψ^τ_k(wn)→c0 and(ψ^τ_k)⁰(wn) →0 as n→ _{∞. Put} u_n=m^τ(w_n)∈ N^τ forn∈_{N. Then}J_k(u_n)→c₀andJ_k⁰(u_n)→0 asn →_∞by Lemma3.12(2).

Claim:(u_n)⊂ E^τ is bounded.

In fact, assume by contradiction thatkunk →+_∞up to subsequence, that is, Z

R^N|∇u_n|²+

Z

R^NV(x)u²_n =ku_nk²_E →_∞.

So, at least one of the two terms goes to infinity. If _Z

R^N|∇u_n|² 1/2

→_∞, it would follow from Lemma3.7that

J_k(u_n)≥ ^θ−2√ ρ 2θ

Z

R^N|∇u_n|² →_∞, which is a contradiction, because(J_k(u_n))⊂_Ris bounded. Now, if

Z

R^NV(x)u²_n→_∞,

then it would follow from Lemma3.7again and(3)of Lemma2.2, that J_k(un)≥ ^θ−2√

ρ 2θ

Z

R^NV(x)|L⁻¹(un)|²

≥ ^θ−2√ ρ 2θ

Z

R^NV(x)u²_n→_∞,

which is a contradiction again. Henceun *vafter passing to a subsequence.

Claim: v 6=0 and J_k⁰(v) =0 inE^τ.

If(V₂)is fulfilled, then lety_n∈_R^N satisfy Z

B1(yn)

u²_ndx= _max

y∈_R^N Z

B1(y)

u²_ndx.

Using once more that J_k andN^τ are invariant under translations of the form u 7−→ u(· −k) with k∈_Z^N, we may assume that(y_n)is bounded inR^N. If

Z

B1(yn)u²_ndx→0 asn→_∞, (4.1)

then u_n → 0 in L^p(_R^N), 2 < p < 2^∗, by Lemma 1.21 in [27]. From Proposition 2.4 and (G₂), we infer that

Z

R^N

g(L⁻¹(un))un

l(L⁻¹(u_n)) ^dx=o(ku_nk_E) asn→_{∞, hence}

o(ku_nk_E) = J_k⁰(u_n)u_n =

Z

R^N|∇u_n|²+

Z

R^NV(x)^L

−1(u_n)u_n l(L⁻¹(u_n))−

Z

R^N

g(L⁻¹(u_n))u_n l(L⁻¹(u_n)) ^dx

=

Z

R^N|∇u_n|²+

Z

R^NV(x)^L

−1(un)un

l(L⁻¹(u_n))−o(ku_nk_E)

and thereforeku_nk_E → 0, contrary to Lemma3.7. It follows that (4.1) cannot hold, so u_n * v6=0 and J_k⁰(v) =0.

Suppose that(V₃)_or (V₄)is satisfied. Then it follows from Proposition2.4, that L⁻¹(u_n)→ L⁻¹(v) _inL^γ(_R^N)_{for all}γ∈ (_{2, 2}^∗)_.

Then by Lemma3.8, we conclude thatv6=0 and J_k⁰(v) =0 inE^τ.

Hence, we conclude that v ∈ N^τ is a critical point of J_k in E^τ. Now we will show that J_k(v) =c₀. By Lemma2.2, Fatou’s lemma and since(u_n)⊂ E^τ is bounded,

c₀+o(1) = J_k(un)− ¹

θhJ_k⁰(un),uni

= ¹ 2 Z

R^N|∇un|²dx+ ¹ 2

Z

R^NV(x)|L⁻¹(un)|²dx−

Z

R^NG(L⁻¹(un))dx

− ¹ θ Z

R^N|∇un|²dx− ¹ θ

Z

R^NV(x)^L

−1(un)un

l(L⁻¹(u_n))^dx+¹ θ

Z

R^N

g(L⁻¹(un))un

l(L⁻¹(u_n)) ^dx

= ^θ−2 2θ

Z

R^N|∇u_n|²dx+¹ 2

Z

R^NV(x)|L⁻¹(u_n)|²dx−

√ρ

θ V(x)|L⁻¹(u_n)|²dx +

√ρ

θ V(x)|L⁻¹(un)|²dx− ¹ θ

Z

R^NV(x)^L

−1(un)un

l(L⁻¹(u_n))^dx +

Z

R^N

1 θ

g(L⁻¹(u_n))u_n

l(L⁻¹(un)) −G(L⁻¹(un))

dx

= ^θ−2 2θ

Z

R^N|∇u_n|²dx+^θ−2√ ρ 2θ

Z

R^NV(x)|L⁻¹(u_n)|²dx + ¹

θ Z

R^NV(x) √

ρ|L⁻¹(u_n)|²− ^L

−1(u_n)u_n l(L⁻¹(un))

dx +

Z

R^N

1 θ

g(L⁻¹(u_n))u_n

l(L⁻¹(un)) −G(L⁻¹(u_n))

dx

≥ ^θ−2 2θ

Z

R^N|∇v|²dx+ ^θ−₂√ ρ 2θ

Z

R^NV(x)|L⁻¹(v)|²dx + ¹

θ Z

R^NV(x) √

ρ|L⁻¹(v)|²− ^L

−1(v)v l(L⁻¹(v))

dx +

Z

R^N

1 θ

g(L⁻¹(v))v

l(L⁻¹(v)) −G(L⁻¹(v))

dx+o(1)

= J_k(v)−¹

θhJ_k⁰(v),vi+o(1) = J_k(v) +o(1).

On the other hand, since J_k(v)≥c₀, hence J_k(v) =c₀.

Now, by using a quantitative deformation lemma and adapting the arguments in [4,11], we are going to show J_k⁰(v) =0 inE.

Suppose, by contradiction, thatJ_k⁰(v)6=0. Then there existδ>0 andν>0 such that kJ_k⁰(w)k ≥ν for every w∈ E withkw−vk ≤2δ.

Sincev6=0, we can takeL=kvk_E >0 and, without loss of generality, we may assume 6δ <L.

Let I =¹₂,³₂

. Since,hJ_k⁰(v),vi=0 and by Lemma3.4, J_k(tv)< J_k(v) =c₀, holds fort∈ I witht6=1, we obtain that

˜

c=max

∂I J_k(tv)< c₀.

Applying Theorem A.4 in [28] with e = min{(c₀−c˜)/2,νδ/8}and S = B(v,δ), there exists η∈ C([0, 1]×E,E)such that

(i) η(θ,u) =uifθ =0 or ifu∈/ J_k⁻¹[c0−2e,c0+2e]∩B(v, 2δ); (ii) η(1,J_k^c0+e)∩B(v,δ)⊂ J_k^c0−e;

(iii) J_k(η(1,w))≤ J_k(w)for every w∈E, where J_k^a ={w∈E; J_k(w)≤a}, (iv) η(t,u)is odd inu.

Consequently, we have

maxt∈I J_k(η(_1,tv))<_{c0. (4.2)} On the other hand, we claim that there exists t₀ ∈ I such that

η(1,t₀v)∈ N^τ.

In fact, by (iv) for η, we know η(1,tv) ∈ E^τ for each t. Now we will prove that there exists t0 ∈ I such thatt0v∈ N. Define ϕ(t) =η(1,tv)and

Ψ(t) =hJ_k⁰(ϕ(t))_,ϕ(t)i fort >0. Since,

kv−tvk_E = |1−t|kvk_E = |1−t|L ≥6δ|1−t|>2δ (4.3) if only if t < ²₃ or t > ⁴₃. It follows from property (i) for η and inequality (4.3) that ϕ(t) = η(1,tv) =tv∈ E^τ ift ∈[¹₂,²₃)∪(⁴₃,³₂].

Thus,

Ψ(¹₂) =J_k⁰ ϕ ¹₂

,ϕ(¹₂)=J_k^{0 1}₂v ,¹₂v

, and it follows from (3.2) that

J_k^{0 1}₂v ,¹₂v

= ¹₂^∂h_∂t^{vk 1}₂ >0. (4.4) On the other hand,

Ψ(³₂) =hJ_k⁰(ϕ(³₂))_,ϕ(³₂)i=hJ_k⁰(³₂v)_,³₂vi_,

and it follows from (3.2) that

J_k^{0 3}₂v ,³₂v

= ³₂^∂h_∂t^{vk 3}₂<0. (4.5) Noting that the function Ψ is continuous on I and taking (4.4) and (4.5) into account, we can apply the intermediate value theorem again to conclude that there existst₀ ∈ I such that Ψ(t₀) = 0. This and (4.2) lead to a contradiction. Hence, we conclude that v is a critical point of J_k. So, by Lemma2.3, we just need to show that|u|_∞ = |L⁻¹(v)|_∞ ≤ σ/√

k holds to conclude thatuis a solution of problem (1.1).

Now, set ϕ=L⁻¹(v)l(L⁻¹(v)). It follows from Lemma2.2that

|ϕ|= |L⁻¹(v)l(L⁻¹(v))| ≤ |v|, and |∇ϕ|=

1+ ^L

−1(v)l⁰(L⁻¹(v)) l(L⁻¹(v))

|∇v| ≤ |∇v|, that is,ϕ∈ H¹(_R^N). So, by taking ϕas a test function in (2.4), we obtain

Z

R^N

1+ ^L

−1(v)l⁰(L⁻¹(v)) l(L⁻¹(v))

|∇v|²+V(x)|L⁻¹(v)|²−g(L⁻¹(v))L⁻¹(v) =_0.

As a consequence of (4) of Lemma2.2, we have Z

R^N|∇v|²+V(x)|L⁻¹(v)|²−g(L⁻¹(v))L⁻¹(v)≥0.

Sincevis a critical point of J_k, it follows that

θc₀= θJ_k(v)− hJ_k⁰(v),L⁻¹(v)l(L⁻¹(v))i

≥ ^θ−2 2

Z

R^N|∇v|²+V(x)|L⁻¹(v)|². Then, by (3) of Lemma2.2,

kvk²_E ≤ ^2θc0

θ−2. (4.6)

For eachm∈_Nandβ>1 given, define

A_m ={x ∈_R^N_; |v|^β−1≤m}_andB_m =_R^N\A_m, and

v_m =

(v|v|^2(β−1) in A_m, m²v in B_m. We knowv_m ∈ H¹(_R^N),v_m ≤v_m+1,v_m ≤ |v|^2β−1, and

∇vm =

((2β−1)|v|^2(β−1)∇v in A_m,

m²∇v inBm,

that is,vm can be used as a test function. Besides this, we have Z

R^N∇v∇vm = (2β−1)

Z

Am

|v|^2(β−1)|∇v|²+m² Z

Bm

|∇v|². (4.7) Letting

w_m =

(v|v|^β−1 _in A_m, mv inB_m,