A positive solution of

A positive solution of

asymptotically periodic Schrödinger equations with local superlinear nonlinearities

Gui-Dong Li, Yong-Yong Li and Chun-Lei Tang

School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China Received 31 October 2019, appeared 11 May 2020

Communicated by Dimitri Mugnai

Abstract. In this paper, we investigate the following Schrödinger equation

−_∆u+V(x)u=λf(u) inR^N,

whereN≥3,λ>0,Vis an asymptotically periodic potential and the nonlinearity term f(u)is only locally defined for|u|small and satisfies some mild conditions. By using Nehari manifold and Moser iteration, we obtain the existence of positive solutions for the equation with sufficiently largeλ.

Keywords: Schrödinger equation, positive solution, locally defined nonlinearity, asymptotically periodic potential.

2020 Mathematics Subject Classification: 35J50, 35A01, 35B09, 35D30.

1 Introduction

In recent years, many researchers consider the following Schrödinger equation

−_∆u+V(x)u= f(x,u), inR^N, (1.1) where N ≥ 3, V is a given potential and f ∈ C(_R^N×_R,R). Knowledge of the solutions of Eq. (1.1) has a great importance for studying standing wave solutions for

ih∂Ψ

∂t =−h²∆Ψ+W(x)_Ψ− f(x,Ψ), for all x∈_Ω, (NLS) where h > 0, W is the real-valued potential and Ω is a domain in R^N. Eq. (NLS) is one of the main objects of the quantum physics, because it appears in problems involving nonlinear optics, plasma physics and condensed matter physics.

Eq. (1.1) has been researched intensively, see [1,3,5,7,10,11,13,14,19,21,22,28] and references therein. In the above works, we observe that many interesting conditions on f have been studied. Notice that, it seems necessary that the condition can be assumed on f at infinity, that is, f is assumed to be subcritical (or critical) at infinity, i.e.,

BCorresponding author. Email: tangcl@swu.edu.cn

(f₀) 0≤_{lim|s|→+∞} ^f(x,s)s

|s|^2∗ < +_∞ uniformly forx ∈_R^N_,

where the number 2^∗ is denoted by _N^2N₋₂ and called the critical Sobolev exponent for the embedding ofH¹(_R^N)intoL^p(_R^N). This aim is to ensure that the associated energy functional would be well defined and of class C¹ on H¹(_R^N), and then its critical points are precisely the solutions of Eq. (1.1) by using variational methods. Certainly, many researchers tried to seek some suitable conditions to replace (f₀). If there does not exist an assumption on f at infinity, can it be proved that there exists a nontrivial solution for Eq. (1.1)? Mathematically this problem is interesting. Accordingly, Costa and Wang [9] have considered the following equation

−_∆u= λf(u), inΩ,

whereλ>0 is a parameter,Ωis a bounded smooth domain inR^N(N≥3), and f :R→_Ris a function of classC¹ satisfying the following conditions:

(f₁) f(−u) =−f(u)_{for any}|u| ≤δ (for someδ >_0);

(f₂) there existsγ∈(2, 2^∗)such that lim sup_|s|→0 ^f_|⁽_s^s_|γ^)s =0;

(f₃) there existsβ∈(2, 2^∗)such that lim inf_|s|→0 ^f_|⁽_s^s_|⁾_β^s >0;

(f₄) there exists µ ∈ (2, 2^∗) such that s f(s) ≥ µF(s) > 0 for all |s| small, where F(s) = Rs

0 f(t)dt.

Motivated by Costa and Wang [9], do Ó et al. [12] have studied the following equation

−_∆u+V(x)u=λf(u) inR^N, (P) whereVsatisfies(V₁)_–[(V₂)_or(V₃)]_,

(V₁) V∈C(_R^N,R)and inf_x∈RNV(x)>0,

(V₂) V(x) → _∞ as |x| → ∞, or more generally, for every M > 0, meas{x ∈ _R^N : V(x) ≤ M}<+_∞,

(V₃) the function [V(x)]⁻¹ belongs to L¹(_R^N),

and f :R→_Ris a function of classC¹ satisfying(f₁⁰)−(f₂⁰)and(f₄), (f₁⁰) there exists p∈(2, 2^∗)such that lim sup_|s|→0 ^f_|⁽_s^s_|⁾p^s <+_∞,

(f₂⁰) there existsq∈ (2, 2^∗)such that lim inf_|s|→0^F_|s⁽_|^sq⁾ >0, where F(s) =Rs

0 f(t)dt.

Further results for related problems can be found in [8,15,23,24] and references therein.

Inspired by the above works, we are concerned with the existence of positive solutions for asymptotically periodic Eq. (P) with a locally defined nonlinearity term, namely V satisfies (V₄),

(V₄) there exists a 1-periodic function V_∞(x) ∈ L^∞(_R^N) such that 0 ≤ V(x) ≤ V_∞(x), inf_x∈RNV_∞(x)>0 andV(x)−V_∞(x)∈ F₁, where

F_1:={h(x)_{: for any} ε>_{0, meas}{x ∈B₁(y)_:|h(x)| ≥ε} →_{0 as}|y| →_∞},

and f satisfies(f₅)–(f₆),

(f5) f ∈ C(_R,R)and there exist p > _2,δ ∈ (_{0, 1})such that the function s 7→ ^f(s)

s^p−1 is nonde- creasing and f(s)>0 on (0,δ],

(f₆) there existsq∈(2, 2^∗)such that lim inf_s→0^{+ F}_s⁽q^s) >0, where F(s) =Rs

0 f(t)dt.

As is well known, if f were assumed to be superlinear and subcritical (or critical) at infinity, then the associated energy functional

I(u) = ¹ 2

Z

R^N(|∇u|²+V(x)u²)dx−λ Z

R^NF(u)dx

would be of class C¹ on H¹(_R^N) and has the mountain pass geometry. Classically, it is a minimax principle that shows the mountain pass level is a critical level of the functional (see [4,5,26]). Here, the assumptions(f₅)–(f₆)we make on the nonlinearity f(u)refer solely to its behavior in a neighborhood ofu=0, and we will show that they suffice for the existence of a positive solution of Eq. (P) whenλis large enough. Exactly we give our main result.

Theorem 1.1. Assume that N ≥ 3, (V₄) and (f₅)–(f₆) hold. Then there exists λ₁ > ₀ such that Eq.(P)has a positive solution forλ≥λ₁.

Remark 1.2. In this paper, we study the existence of positive solutions for Schrödinger equa- tions with the assumptions of Theorem 1.1 that has never been investigated. For the case where the nonlinear term is only locally defined for |u| small, we should point out that we refer [8,9,12,15] for references in this direction. Costa and Wang [9] considered Eq. (P) in bound domain. do Ó et al. [12] considered Eq. (P) when V was coercive potential or sat- isfied that [V(x)]⁻¹ _{belongs to} L¹(_R^N). Li and Zhong [15] studied the Kirchhoff equation when the nonlinearity term was sub-linear growth. Chu and Liu [8] investigated quasi-linear Schrödinger equations in the radial space. In these papers, they have the compactness and get certain solutions easily. However, in our cases we do not have compact embedding, which is the main difficulty in this paper. Due to this difficult, the methods in [8,9,12,15] fail in our case, so we will use a different way to overcome the lack of compactness.

We now make some comments on the key ingredients of the analysis in this paper. Fol- lowing the idea of [8,9,12,15], we first extend the nonlinear term f and introduce a modified nonlinear Schrödinger equation. Next, we show by variational methods that the modified nonlinear Schrödinger equation possesses a positive ground state solution. Finally, our ap- proach is inspired by the results of [2,6,9,12,26] and is based on the fact that we can show a priori bound of the form

|u|_∞< Cλ^−β, β>0,

for a class of solutions for the modified nonlinear Schrödinger equation.

The organization of this paper is as follows. In the next section we reserve for setting the framework and establishing some preliminary results. Theorem1.1 is proved in Section 3.

2 Preliminaries

From now on, we will use the following notations.

• H¹(_R^N)is the usual Sobolev space endowed with the usual norm kuk²_H =

Z

R^N(|∇u|²+u²)dx.

• L^p(_R^N)is the usual Lebesgue space endowed with the norm

|u|^p_p =

Z

R^N|u|^pdx and |u|_∞ =ess sup

x∈_R^N

|u(x)| for all p ∈[1, +_∞).

• E:= u∈L²(_R^N):|∇u| ∈L²(_R^N)and R

R^NV(x)u²dx< +_∞ has the norm kuk²=

Z

R^N(|∇u|²+V(x)u²)dx.

• measΩdenotes the Lebesgue measure of the setΩ.

• u^± :=_max{±u, 0}_andK:={u∈ E:u⁺ 6=₀}_.

• h·,·idenotes action of dual.

• Br(y):= {x∈_R^N :|x−y| ≤r}andBr :={x∈_R^N :|x| ≤r}.

• Cdenotes a positive constant and is possibly various in different places.

We work in the spaceE and recall some facts that the normsk · k and k · k_H are equivalent andE,→ L^s(_R^N)for anys ∈ [2, 2^∗]is continuous. The proof can be done similarly to that in [19] and details are omitted here. We start by observing that(f5)−(f6)imply that p≤qand

|f(s)s| ≤C|s|^p, for any|s| ≤δ.

In order to prove our main result via variational methods, we need to modify and extend f(u) for outside a neighborhood ofu=0 to get fe(u). We set

ef(s):=











0, s ≤0,

f(s), 0<s≤ δ, C₁s^p−1, δ <s,

and fixC₁ > 0 such that ef ∈ C(_R,R⁺). Combining with the definition of fe, one can easily obtain the following lemma.

Lemma 2.1. Suppose that(f₅)hold. Then (a) lims→+∞ Fe(s)

s² = +_{∞, where} Fe(s) =Rs

0 ef(t)dt,

(b) there exists C>0such that|^ef(s)s| ≤C|s|^pand|Fe(s)| ≤C|s|^p for all s∈_R, (c) there existsµ∈ (_2,p)such that the function s7→ ^ef(s)

s^µ−1 is strictly increasing on(_0,+_∞),

Now let us consider the modified equation of Eq. (P) given by (−_∆u+V(x)u=λef(u),

u∈E. (Pe)

The corresponding energy functional Ie(u) = ¹

2 Z

R^N(|∇u|²+V(x)u²)dx−λ Z

R^NFe(u)dx is of classC¹ by a standard argument and whose derivative is given by

hIe⁰(u),vi=

Z

R^N(∇u· ∇v+V(x)uv)dx−λ Z

R^N ef(u)vdx, v∈ E.

Formally, critical points of Ie are solutions of Eq. (Pe). We note that critical points of Ie with L^∞-norm less than or equal toδare also solutions of the original Eq. (P). We recall the Nehari manifold

N :=ⁿu∈E\ {0}:hIe⁰(u),ui=0o

=ⁿu∈ K:hIe⁰(u),ui=0o , and set

c:= inf

u∈NIe(u). Lemma 2.2. Suppose that(V₄)and(f₅)hold. Then

(a) for any u∈K , there exists a unique t_u >0such that t_uu∈ N. Moreover, the maximum ofIe(tu) for t>0is achieved at tu,

(b) there existsρ>0such thatkuk ≥ρfor all u∈ N,

(c) the functionalI is bounded from below onN by a positive constant.

Proof. (a)For anyu ∈K, we define Ψ(t):=Ie(tu) = ^t

2

2 Z

R^N(|∇u|²+V(x)u²)dx−λ Z

R^N Fe(tu)dx, t∈ (0,+_∞). It follows from (b) of Lemma2.1and the Sobolev inequality that

Z

R^NFe(tu)dx≤ C Z

R^N|tu|^pdx ≤Ct^pkuk^p. Thus one has

Ψ(t)≥ ^t

2

2kuk²−λCt^pkuk^p.

Then there exists t₀ >0 such thatΨ(t₀)>0. We setΩ={x∈_R^N :u(x)>0}. Combining (a) in Lemma2.1with Fatou’s lemma, we have

lim inf

t→_∞ Z

Ω

Fe(tu) (tu)^2u

2dx= +_∞.

Hence lim sup

t→_∞

Ψ(t) t² = ¹

2kuk²−λlim inf

t→_∞ Z

Ω

Fe(tu)

t² dx= ¹

2kuk²−λlim inf

t→_∞ Z

Ω

Fe(tu) (tu)^2u

2dx=−_∞.

One could deduce Ψ(t) → −_∞ as t → +∞. So there exists t_u > 0 such that Ψ(t_u) = maxt>0Ψ(t) and Ψ⁰(tu) = 0, i.e., Ie(tuu) = maxt>0Ie(tu) and tuu ∈ N. Suppose that there existst₁ >t₂>0 such thatt_iu∈ N,i=1, 2, one has

Z

Ω

ef(t₁u)u² t₁u dx=

Z

Ω

ef(t₂u)u² t₂u dx,

which contradicts (c) of Lemma2.1. Thus we can conclude thatt_uis unique.

(b)_{For any}u∈ N, combining the Sobolev embedding and (b) of Lemma2.1, one obtains kuk²= λ

Z

R^N ef(u)udx≤ Cλ Z

R^N|u|^pdx≤Cλkuk^p. (2.1) It follows from (2.1) that there existsρ>0 independent ofusuch that

ρ≤ kuk_.

(c)Also from (b) of Lemma2.1 and the Sobolev inequality, we have Ie(u)≥ ¹

2kuk²−Cλkuk^p.

Sincep>2, there existsσ >0 such thatIe(u)≥ ^σ₄² >0 forkuk=σ >0. For anyv∈ N, there existst⁰ >0 such thatt⁰kvk=σ. Combining with (a)-(b) of Lemma2.2, one obtains

Ie(v)≥Ie(t⁰v)≥ ^σ

2

4 . This completes the proof.

From Lemmas2.1–2.2, one can easily know (see also [19,26]) c= inf

u∈NI(e u) = inf

u∈Ksup

t>0

Ie(tu) =min

γ∈_Γ max

t∈[0,1]

Ie(γ(t)),

whereΓ={γ∈C([0, 1],E):γ(0) =0,Ie(γ(t))<0}. Notice that,c>0 from (c) of Lemma2.2.

In order to prove our results, we introduce the following equation

−_∆u+V_∞(x)u=λfe(u), (P_∞) and it follows from [16,19,26,27] that Eq. (P_∞) has a positive ground state solutionω. From Lemma2.2 and(V₄), there exists a uniquet_ω >0 such thatt_ωω∈ N and

c≤Ie(t_ωω)≤Ie_∞(t_ωω)≤Ie_∞(ω):=c_∞, (2.2) whereIe_∞ is the energy functional associated with Eq. (P_∞).

Lemma 2.3. Suppose that(V₄)and(f5)hold. If u∈ N andIe(u) =c, then u is a nontrivial solution of Eq.(Pe).

Proof. Inspired by the method in [18], one supposes by contradiction thatuis not a nontrivial solution of Eq. (Pe). Then there existsφ∈ Esuch that

hIe⁰(u),φi=

Z

R^N(∇u· ∇φ+V(x)uφ)dx−λ Z

R^N ef(u)φdx<−1.

Letε ∈(0, 1)be small enough. Then hIe⁰(tu+sφ),φi ≤ −¹

2, for any|t−1| ≤ε, |s| ≤ε. (2.3) We set a curve

γ(t) =tu+sτ(t)_φ, t >_0,

where τ∈C(_R,[0, 1])is a smooth cut-off function such thatτ(t) =1 for|t−1| ≤ ₂^ε,τ(t) =0 for|t−1| ≥ε. Obviously,γis a continuous. We can claim thatIe(γ(t))<cfor anyt∈ (0,+_∞). Indeed, it follows from Lemma2.2that Ie(γ(t)) =Ie(tu)<Ie(u) =cfor|t−1| ≥ε. When

|t−1|< ε, owing toΦ(s):= Ie(tu+sτ(t)φ)is ofC¹on [0,ε], there exists ¯s∈ (0,ε)such that Ie(tu+sτ(t)φ) =Ie(tu) +hIe⁰(tu+sτ¯ (t)φ),ετ(t)φi ≤Ie(tu)− ¹

2ετ(t)<c, where the inequality holds from (2.3). HenceIe(γ(t))<cfor anyt∈ (0,+_∞).

We denote J(u) = hIe⁰(u),ui. According to Lemma2.2 and the definition of γ, we have J(γ(₁−ε)) = J((₁−ε)u) > _{0 and} J(γ(₁+ε)) = J((₁+ε)u) < 0. By the continuity of t 7→ J(γ(t)) there exists t⁰ ∈ (1−ε, 1+ε) such that J(γ(t⁰)) = 0. Thus γ(t⁰) ∈ N and I(e γ(t⁰))<c, which is a contradiction. This completes the proof.

Lemma 2.4. Suppose that(V₄)and(f₅)hold. Then the Cerami sequence forIeat level m>0(shortly:

(Ce)_msequence) is bounded in E.

Proof. We recall the(Ce)_m sequence{un}, that is,

Ie(un)→m, kIe⁰(un)k(1+kunk)→0.

Then

o(1) =hIe⁰(un),u⁻_ni=−ku⁻_nk².

Consequently we could deduce that {u⁺_n} is also a (Ce)_m sequence. For the sake of con- venience, we denote u⁺_n by un. By a contradiction, we assume that kunk → +_∞ and set v_n = _k^u_uⁿ

nk. Obviously up to a subsequence, there exists a nonnegative function v ∈ E such that v_n → v ∈ E,v_n → v ∈ L²_loc(_R^N) and v_n(x) → v(x) a.e. in R^N. We denote Ω1 ={x ∈_R^N :v(x)>0}. If measΩ1>0, Fatou’s lemma and (a) of Lemma2.1imply

lim inf

n→_∞ Z

R^N

Fe(un)

u²_n v²_ndx≥lim inf

n→_∞ Z

Ω1

Fe(un)

u²_n v²_ndx= +_∞.

Then

0=lim sup

n→_∞

Ie(un) ku_nk² = ¹

2−λlim inf

n→_∞ Z

R^N

Fe(un)

u²_n v²_ndx=−_∞, which is a contradiction. Thusv=0. We denote

α:= _lim

n→_∞ sup

z∈_R^N Z

B1(_z)v²_ndx. (2.4)

If α = 0, we have v_n → 0 in L^p(_R^N) from the Lions lemma [17,26]. Combining with (b) of Lemma2.1, we obtainR

R^NFe(₂√

mv_n)dx= o(₁). By the continuity ofIe, there existst_n ∈[_{0, 1}]

such thatIe(tnun) =max_t∈[0,1]Ie(tun). Sincekunk →+_{∞, one has} ²

√m

kunk ≤1 asnlarge enough.

We observe that

Ie(t_nu_n) +o(1)≥Ie 2√

m ku_nk^un

+o(1) =2mkv_nk²−λ Z

R^NFe(2√

mv_n)dx+o(1)

=2m+o(₁)_.

In view ofIe(u_n)→mand (a) of Lemma2.2, we can see thatt_n∈(0, 1)andhIe⁰(t_nu_n),t_nu_ni= 0 asnlarge enough. Hence by Lemma 2.3 in [20], one has

m=Ie(u_n) +o(1)

=Ie(u_n)− ¹

µhIe⁰(u_n),u_ni+o(1)

= 1

2 − ¹ µ

_Z

R^N |∇u_n|²+V(x)u²_n

dx+λ Z

R^N

1 µ

ef(u_n)u_n−Fe(u_n)

dx+o(1)

≥ ^µ−₂ 2µ t²_n

Z

R^N |∇un|²+V(x)u²_n

dx+λ Z

R^N

1 µ

fe(tnun)tnun−Fe(tnun)dx+o(1)

=Ie(tnun)− ¹

µhIe⁰(tnun),tnuni+o(1)

=Ie(t_nu_n) +o(1)

≥2m+o(1), which is a contradiction.

Ifα>0, there exists{z_n} ⊂_R^N such that α 2 ≤

Z

B₁(zn)v²_ndx.

If{zn}is bounded, there exists R>0 such that α 2 ≤

Z

BR

v²_ndx.

which is a contradiction with vn → 0 in L²_loc(_R^N). Then {zn} is unbounded, up to a subse- quence,|z_n| →_{∞. We set}w_n(x):=v_n(x+z_n), wherew_nsatisfies

α 2 ≤

Z

B1

w²_ndx,

up to a subsequence, there exists w ∈ E such that wn * _{w in E, wn} → w in L²_loc(_R^N) _and wn(x)→ w(x)a.e. inR^N. Evidently, measΩ2 > 0 whereΩ2 = {x ∈ _R^N : w(x) >0}. In fact wn(_x) = ^un(_k^x+zn)

unk . Also from Fatou’s lemma and (a) of Lemma2.1, one obtains lim inf

n→_∞

1 ku_nk²

Z

R^NFe(u_n)dx

=lim inf

n→_∞

1 ku_nk²

Z

R^NFe(u_n(x+z_n))dx

≥lim inf Z

Ω2

Fe(u_n(x+z_n)) [u_n(x+z_n)]^{2 w}

2ndx

= +_∞.

Hence

0=lim sup

n→_∞

Ie(un) ku_nk = ¹

2 −λlim inf

n→_∞

1 ku_nk²

Z

R^NFe(u_n)dx

=−_∞, which is a contradiction. In a word, the(Ce)_{m sequence} {u_n}is bounded inE.

Proposition 2.5. Suppose that(V₄)and(f₅)hold. Then Eq.(Pe)has a positive ground state solution.

Proof. Notice that 0<c≤c_∞. Therefore, one of the two cases occurs:

Case 1. c=c_∞. It follows from (2.2) that

c_∞ ≤Ie(t_ωω)≤ Ie_∞(t_ωω)≤Ie_∞(ω) =c_∞.

Thenω is also a positive ground state solution of Eq. (Pe) from Lemma2.3.

Case 2. 0 < c < c_∞. We see easily Ie satisfies the mountain pass geometry. From the mountain pass theorem [25,26] and Lemma 2.4, there exists a nonnegative and bounded sequence {u_n} ∈Esuch that

Ie(un)→c, kIe⁰(un)k(1+kunk)→0.

Then there exists a nonnegative function u ∈ E such that up to a subsequence, u_n * u in E, un → u in L²_loc(_R^N) and un(x) → u(x) a.e. in R^N. For any ϕ ∈ C₀^∞(_R^N), one has 0 = hIe⁰(u_n)_,ϕi+o(₁) = hIe⁰(u)_,ϕi_{, i.e.,} uis a nonnegative solution of Eq. (Pe_{). If}u 6= _{0 in}E, combining Lemma 2.3 in [20] with Fatou’s lemma one obtains

c=Ie(u_n) +o(1)

=Ie(u_n)− ¹

µhIe⁰(u_n),u_ni+o(1)

= 1

2 − ¹ µ

Z

R^N |∇un|²+_V(_x)_u²_ndx+λ Z

R^N

1 µ

ef(_un)_un−Fe(_un)

dx+_o(₁)

≥ 1

2 − ¹ µ

_Z

R^N |∇u|²+V(x)u²

dx+λ Z

R^N

1 µ

fe(u)u−Fe(u)

dx+o(1)

=Ie(u)− ¹

µhIe⁰(u),ui+o(1)

=Ie(u) +o(1). (2.5)

At the same time, one knows c ≤ Ie(u) from the definition of c and u ∈ N. Applying the strongly maximum principle, we could deduce that u is a positive ground state solution of Eq. (Pe).

We assume that u = 0 (otherwise we complete the proof). Then there exists α ≥ 0 such that

nlim→_∞ sup

z∈_R^N Z

B1(z)

|u_n|²dx =α.

Indeed, ifα=0, applying the Lions lemma [17,26] we obtain

un →0 in L^p(_R^N). (2.6)

HenceIe(u_n)→_{0 as}n→_∞from (b) in Lemma2.1, which contradictsc>0. Then there exists {zn} ⊂_R^N such thatR

B1(zn)|un|²dx≥ ^α₂ >0.

If{zn}is bounded, there existsR >0 such that R

BR(0)|un|²dx ≥ ^α₂ >0, which is a contra- diction with u_n →_{0 in} L²_loc(_R^N)_{. Then}{z_n}is unbounded. After extracting a subsequence if

necessary, we have (i) |z_n| →+_∞,

(ii) un(·+zn)*v6=0 inE.

From Lemma 2.4 in [19], we have 0=hIe⁰(un),ϕ(· −zn)i+o(1)

=

Z

R^N[∇un· ∇ϕ(· −zn) +V(x)unϕ(x−zn)]dx−λ Z

R^N fe(un)ϕ(x−zn)dx+o(1)

=

Z

R^N[∇u_n· ∇ϕ(· −z_n) +V_∞(x)u_nϕ(x−z_n)]dx−λ Z

R^N ef(u_n)ϕ(x−z_n)dx+o(1)

=hIe_∞⁰ (v),ϕi+o(1).

Thenv is a nontrivial solution of Eq. (P_∞). Notice that, also from [19], we obtain c=Ie(_un)− ¹

µhIe⁰(_un)_,uni+_o(₁)

= 1

2− ¹ µ

_Z

R^N |∇u_n|²+V(x)u²_n

dx+λ Z

R^N

1 µ

ef(u_n)u_n−Fe(u_n)

dx+o(1)

= 1

2− ¹ µ

_Z

R^N |∇u_n|²+V_∞(x)u²_n dx +λ

Z

R^N

1 µ

fe(un(·+zn))un(·+zn)−Fe(un(·+zn))

dx+o(1)

=Ie_∞(u_n(·+z_n))− ¹

µhIe_∞⁰ (u_n(·+z_n)),u_n(·+z_n)i+o(1)

=Ie_∞(v) +o(1)

≥c_∞+o(₁)_, which is a contradiction.

In conclusion, whether Case 1 occurs or Case 2 occurs, we can prove Proposition2.5.

3 Proof of Theorem 1.1

Lemma 3.1. Suppose that (V₄) and (f₅) hold. If u is a critical point of Ie, then u ∈ L^∞(_R^N). Furthermore, there exists a positive constant C independent ofλsuch that

|u|_∞ ≤Cλ^{2∗ −1p} _Z

R^N|∇u|²dx ₂₍²₂^{∗ −}_{∗ −}²_p)

.

Proof. We prove the result by using the Moser iteration. For eachk>0, we define u_k(_x) =

(u(x), if|u(x)| ≤k,

±k, if ±u(x)>k.

Forβ>1, we use ϕ_k = |u_k|^2(β−1)uas a test function inhIe⁰(u),ϕ_kito obtain Z

R^N|u_k|^2(β−1)|∇u|²dx+2(β−1)

Z

R^N|u_k|^2(β−2)uu_k∇u· ∇u_kdx +

Z

R^NV(x)|u_k|^2(β−1)u²dx=λ Z

R^N ef(u)|u_k|^2(β−1)udx. (3.1)

Then we use the Sobolev inequality to yield β²

Z

R^N

|u_k|^2(β−1)|∇u|²dx+2(β−1)|u_k|^2(β−2)uu_k∇u· ∇u_k dx

≥

Z

R^N|u_k|^2(β−1)|∇u|²+ (β−1)²|u_k|^2(β−2)u²|∇u_k|²+2(β−1)|u_k|^2(β−2)uu_k∇u· ∇u_kdx

≥

Z

R^N

∇|u_k|^β−1u

2dx

≥C _Z

R^N

|u_k|^β−1u

2^∗

dx ₂²_∗

, (3.2)

where we also have used the facts thatu²|∇u_k|² ≤u²_k|∇u|²_andβ>1. From (b) in Lemma2.1, we deduce

Z

R^N ef(u)|u_k|^2(β−1)udx≤C Z

R^N|u|^p|u_k|^2(β−1)dx. (3.3) Combining (3.1), (3.2) and (3.3), we obtain

Z

R^N

|u_k|^β−1u

2^∗

dx ₂²∗

≤Cβ²λ Z

R^N|u|^p−2|u_k|^2(β−1)u²dx

≤Cβ²λ _Z

R^N|u|^2∗dx

^p₂⁻_∗² _Z

R^N

|u_k|^2(β−1)u²

2∗ 2∗ −_p+2

dx

!^{2∗ −}_2∗^p+2 . Lettingk→_{∞, we have}

|u|_β·2∗ ≤ Cβ²λ_2β¹ _Z

R^N|∇u|²dx ^p_4β⁻²

|u| 2·2∗_β 2∗ −p+2

. (3.4)

To carry out an iteration process, we set β_m =

2^∗−p+2 2

m+1

, m=0, 1, . . . Then we have

2·₂^∗β_m

2^∗−p+2 =₂^∗β_m−1. By (3.4), one obtains

|u|_βm·2^∗ ≤ Cβ²_mλ_2βm¹ _Z

R^N|∇u|²dx _4βm^p−2

|u|2·2∗_βm 2∗ −p+2

= (Cλ)^2βm1 β

1

mβm

Z

R^N|∇u|²dx _4βm^p−2

|u|_βm−1·2^∗. By the Moser iteration, we have

|u|_βm·2^∗ ≤ (Cλ)^∑mi=02β1i

∏

β

1 βi

i

_Z

R^N|∇u|²dx

^p−₄²_∑^m_i=0 ¹

βi

|u|₂^∗. (3.5)

Sinceβ₀ =^2∗−₂^p+2>1 andβ_i =βⁱ₀⁺¹, we observe that

∑

1 β_i

=

∑

1 βⁱ₀⁺¹

,

∏

β

1 βi

i =

∏

βⁱ₀⁺¹

¹

βi+₁

0 = (β₀)^∑

m i=0 i+1

βi+₁

0 .

One can easily see

∑

i+1 βⁱ₀⁺¹

= β^∗ <+_∞,

∑

1 βⁱ₀⁺¹

= ²

2^∗−p. Lettingm→_∞in (3.5), we conclude thatu ∈L^∞(_R^N)and

|u|_∞ ≤Cλ^{2∗ −1p}β^β

∗ 0

_Z

R^N|∇u|²dx ₂₍₂^p_{∗ −}⁻²_p)

|u|₂^∗ ≤Cλ^{2∗ −1p} _Z

R^N|∇u|²dx ₂₍²₂^{∗ −}_{∗ −}²_p)

. (3.6) This completes the proof.

Proof of Theorem1.1. By proposition2.5, Eq. (Pe) has a positive ground solutionu. Combining the Sobolev embedding and (b) of Lemma2.1, one obtains

c=Ie(u)− ¹

µhIe⁰(u),ui ≥ 1

2 − ¹ µ

kuk². (3.7)

We can see that there existsv ∈K∩L^∞(_R^N)such that|v|_∞<1. Since (f6), there existsC>0 independent ofλsuch that

Fe(tv)≥C|tv|^q, t∈[0, 1].

At the same time there existsλ₀ >0 such that Ie(_v)<_{0 for} λ≥λ₀. Then from the definition ofc, we have

c≤ _max

t∈[0,1]

Ie(tv)

= max

t∈[0,1]

t² 2

Z

R^N(|∇v|²+V(x)v²)dx−λ Z

R^NFe(tv)dx

≤ max

t∈[0,1]

t² 2

Z

R^N(|∇v|²+V(x)v²)dx−Ct^qλ Z

R^N|v|^qdx

≤Cλ^−q−22. (3.8)

Combining (3.6), (3.7) and (3.8), we have

|u|_∞ ≤Cλ^{2∗ −1p}kuk^{22∗ −∗ −2p} ≤Cλ^{2∗ −1p}λ

1 2−q·₂^{2∗ −}_{∗ −}²_p

. Sincep,q∈ (2, 2^∗), there existsλ₁≥ λ₀such that

|u|_∞ ≤Cλ

2∗ −q (2∗ −p)(2−q)

1 ≤ δ.

Therefore, from the definition of ef, we can conclude thatuis also a positive solution of Eq. (P) forλ≥λ₁. This completes the proof of Theorem1.1.

Acknowledgment

This work was partially supported by Postgraduate research and innovation project of Chongqing (No. CYB19082), Fundamental Research Funds for the Central Universities (No.

XDJK2020D032) and National Natural Science Foundation of China (No.11971393). The au- thors express their gratitude to the handling editor and anonymous reviewers for careful reading and helpful suggestions for the improvement of the original manuscript.

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