ON A CLASS OF KIRCHHOFF PROBLEMS VIA LOCAL MOUNTAIN PASS

VINCENZO AMBROSIO AND DUAN REPOV

Abstract. In the present work we study the multiplicity and concentration of positive solutions for the following class of Kirchho problems:

−

ε^{2}a+ε b
Z

R^{3}

|∇u|^{2}dx

∆u+V(x)u=f(u) +γu^{5} inR^{3},
u∈H^{1}(R^{3}), u >0inR^{3},

whereε >0is a small parameter,a, b >0are constants,γ ∈ {0,1},V is a continuous positive potential with a local minimum, andfis a superlinear continuous function with subcritical growth. The main results are obtained through suitable variational and topological arguments. We also provide a multiplicity result for a supercritical version of the above problem by combining a truncation argument with a Moser-type iteration. Our theorems extend and improve in several directions the studies made in [18,19,34].

1. Introduction

In this paper we focus our attention on the multiplicity and concentration of positive solutions for the following class of Kirchho problems:

−

ε^{2}a+ε b
Z

R^{3}

|∇u|^{2}dx

∆u+V(x)u=f(u) +γu^{5} inR^{3},
u∈H^{1}(R^{3}), u >0inR^{3},

(1.1)
where ε > 0 is a small parameter, a, b > 0 are constants, and γ ∈ {0,1}. Throughout the paper we
will assume that the potential V : R^{3} → R is a continuous function satisfying the following hypotheses
introduced by del Pino and Felmer [11]:

(V1) there existsV0 >0such thatV0 := inf_{x∈}_{R}^{3}V(x),
(V_{2}) there exists a bounded open setΛ⊂R^{3} such that

V0 <min

∂Λ V and M :={x∈Λ :V(x) =V0} ̸=∅.

We suppose thatf :R→Ris a continuous function such thatf(t) = 0for t≤0and fullls the following conditions:

(f1) f(t) =o(t^{3}) ast→0,

(f_{2}) if γ = 0 then there exists q ∈ (4,6)such that f(t) = o(t^{q−1}) as t → ∞, whereas if γ = 1 then we
suppose that there exist q, σ∈(4,6),C0 >0 such that

f(t)≥C_{0}t^{q−1} for allt >0, lim

t→∞

f(t)
t^{σ−1} = 0,
(f3) there exists ϑ∈(4,6)such that

0< ϑF(t)≤tf(t) for all t >0, whereF(t) :=

Z t 0

f(τ)dτ,

2010 Mathematics Subject Classication. 35J60, 35A15, 58E05.

Key words and phrases. Kirchho problems; Penalization method; Ljusternik-Schnirelmann theory; Critical growth; Su- percritical exponent.

1

Asymptotic analysis, ISSN 0921-7134, Volume 126, Issue 1-2, 2022, pp. 1-43 DOI: https://doi.org/10.3233/ASY-201660

(f_{4}) the map t7→ ^{f(t)}_{t}_{3} is increasing on(0,∞).

When b = 0 and R^{3} is replaced by the more general spaceR^{N}, equation (1.1) reduces to a nonlinear
Schrödinger equation of the type

−ε^{2}∆u+V(x)u=g(u)inR^{N}, (1.2)

which has been widely investigated in the last thirty years. The main motivation for studying (1.2) arises
from seeking standing wave solutions, namely functions of the form ψ(x, t) = u(x)e^{−}^{ıEt}^{ε} , with E ∈ R
constant, for the time-dependent Schrödinger equation

ı ε∂ψ

∂t =−ε^{2}∆ψ+ (V(x) +E)ψ−g(ψ) inR^{N} ×R.

An interesting class of solutions of (1.2), sometimes called semi-classical states, are families of solutions
uε(x) which concentrate and develop a spike shape around one (or more) special points in R^{N}, while
vanishing elsewhere as ε→ 0. We refer the interested reader to [3,10,11,14,16,29] and their references
for several existence and multiplicity results obtained by applying dierent variational and topological
methods.

On the other hand, problem (1.1) is related to the stationary analogue of the Kirchho equation
ρu_{tt}−

p_{0}
h + E

2L Z L

0

|u_{x}|^{2}dx

u_{xx} = 0, (1.3)

which was proposed in 1883by Kirchho [21] as an extension of the classical D'Alembert wave equations for free vibration of elastic strings. The Kirchho model takes into account changes in the length of the string produced by transverse vibrations. In (1.3),u=u(x, t) denotes the transverse string displacement at the spatial coordinate x and timet,L is the length of the string, h is the area of the cross section, E is Young's modulus of the material, ρ is the mass density, andp0 is the initial tension. We refer to [7,27]

for the early classical studies dedicated to (1.3). We also note that nonlocal boundary value problems like (1.3) model several physical and biological systems whereu describes a process which depends on the average of itself, as for example, the population density (see [2,9]). However, only after the pioneering work of Lions [22], where a functional analysis approach was proposed to attack (1.3), problem (1.1) began to attract the attention of several mathematicians (see [2,4,13,15,1820,3134] and the references therein).

In particular, He and Zou [19] obtained the existence and multiplicity of concentrating solutions for small ε >0 of the following perturbed Kirchho equation

−

aε^{2}+bε
Z

R^{3}

|∇u|^{2}dx

∆u+V(x)u=g(u) inR^{3}, (1.4)

assuming thatV :R^{3} →R is a continuous potential satisfying the assumption introduced by Rabinowitz
[29]:

V∞:= lim inf

|x|→∞ V(x)> inf

x∈R^{N}

V(x) =V0, whereV∞≤ ∞, (V)

and g is a C^{1} subcritical nonlinearity. Subsequently, Wang et al. [34] investigated the multiplicity and
concentration phenomenon for (1.4) when g(u) = λf(u) +u^{5}, f is a continuous subcritical nonlinearity
and λis suciently large. Figueiredo and Santos Júnior [15] proved a multiplicity result for a subcritical
Schrödinger-Kirchho equation via the generalized Nehari manifold method, when the potential V has
a local minimum. He et al. [18] considered the existence and multiplicity of solutions for (1.4) when
g(u) =f(u) +u^{5},f ∈C^{1} is a subcritical nonlinearity which does not satises the Ambrosetti-Rabinowitz
condition [5].

Motivated by the above works, in this paper we study the multiplicity and concentration of solutions for (1.1) under conditions(V1)-(V2)on the potentialV, and assuming(f1)-(f4)for the continuous nonlinearity f. In order to state our main result more precisely, we recall that ifY is a given closed set of a topological

space X, we denote by cat_{X}(Y) the Ljusternik-Schnirelmann category of Y in X, this is the smallest
number of closed contractible sets inX which coverY (see [25,35] for more details). We are able to prove
the following main result:

Theorem 1.1. Assume that conditions (V_{1})-(V_{2}) and(f_{1})-(f_{4}) hold. Then for any δ >0 such that
M_{δ}:={x∈R^{3} : dist(x, M)≤δ} ⊂Λ,

there exists εδ >0such that for anyε∈(0, εδ), problem (1.1) admits at leastcatMδ(M) positive solutions.

Moreover, if u_{ε} denotes one of these solutions and x_{ε}∈R^{3} is a global maximum point ofu_{ε}, then

ε→0limV(xε) =V0,
and there exist C_{1}, C_{2}>0 such that

0< uε(x)≤C1e^{−C}^{2}

|x−xε|

ε for all x∈R^{3}.

Our proof of Theorem1.1is obtained by applying appropriate variational arguments. First, motivated
by [11], we overcome the lack of information about the behavior of potential V at innity by making
a suitable modication on the nonlinearity, solve the modied problem and then check that, for ε > 0
small enough, the solutions of the modied problem are indeed solutions of the original one. Due to
the fact that f is only continuous, the Nehari manifold associated with the modied problem is not
dierentiable, so we cannot apply standard variational arguments for C^{1}-Nehari manifolds developed, for
example, in [3,10,18,19]. For this reason we use certain versions of critical point theorems due to Szulkin
and Weth [30]. We also note that the presence of the Kirchho term creates some diculties in getting
the compactness of the modied functional J_{ε}. Indeed, it is not clear that weak limits of bounded (P S)
sequences are critical points of J_{ε}. Moreover, when γ = 1, problem (1.1) presents an extra diculty
due to the presence of the critical exponent, and in order to recover some compactness properties for
J_{ε}, we invoke the Concentration-Compactness Lemma [24]. Since we are interested in obtaining multiple
critical points, we use a technique introduced by Benci and Cerami [6], which consists in making precise
comparisons between the category of some sublevel sets of J_{ε} and the category of the set M. Then we
apply Ljusternik-Schnirelmann theory to deduce a multiplicity result for the modied problem. Finally,
we show that the solutions of the modied problem are also solutions for (1.1), whenε >0is small enough,
by using the Moser iteration technique [26].

In the last part of this paper we consider a supercritical version of problem (1.1). In this case, we deal with the sum of two homogeneous nonlinearities and add a new positive parameterµ. More precisely, we consider the following problem:

−

ε^{2}a+ε b
Z

R^{3}

|∇u|^{2}dx

∆u+V(x)u=u^{p−1}+µu^{r−1} inR^{3},
u >0 inR^{3}, u(x)→0 as|x| → ∞,

(1.5) whereε, µ >0 and the exponents satisfy4< p <6< r. Our multiplicity result for the supercritical case can be stated as follows.

Theorem 1.2. Assume that conditions (V_{1})-(V_{2}) hold. Then there existsµ_{0} >0 such that for anyδ > 0
satisfying

M_{δ}={x∈R^{3} : dist(x, M)≤δ} ⊂Λ,

and for any µ ∈ (0, µ0), there exists ε_{δ,µ} > 0 such that for any ε ∈ (0, ε_{δ,µ}), problem (1.5) admits at
least cat_{M}_{δ}(M) positive solutions. Moreover, if u_{ε} denotes one of these solutions and x_{ε} ∈R^{3} is a global
maximum point of u_{ε}, then

ε→0limV(x_{ε}) =V_{0}.

The main diculty in the study of (1.5) is due to the fact that r > 6 is supercritical, and we cannot
directly use variational techniques because the corresponding functional is not well-dened on the Sobolev
space H^{1}(R^{3}). In order to overcome this obstacle, we use some arguments inspired by [8,14,28] which
can be summarized as follows. We rst truncate in a suitable way the nonlinearity on the right hand side
of (1.5), so we deal with a new problem but with subcritical growth. In the light of Theorem 1.1, we
know that a multiplicity result for this truncated problem is available. Then we deduce a priori bound
(independent ofµ) for these solutions and by using an appropriate Moser iteration technique [26], we show
that, for µ >0 suciently small, the solutions of the truncated problem also solve the original one.

We stress that our theorems complement and improve the main results in [18,19,34], in the sense that we are considering multiplicity results for subcritical, critical and supercritical Kirchho problems involving continuous nonlinearities and imposing local conditions on the potentialV.

The paper is organized as follows. In Section 2 we collect some notations and basic results. We also modify the nonlinearity and prove some useful lemmas to overcome the non dierentiability of the Nehari manifold. In Section 3 we provide our rst existence result. In Section 4 we deal with the autonomous problems. In Section5we introduce some tools which are needed to establish a multiplicity result. Section 6 is devoted to the proof of Theorem 1.1. In Section 7 we study the multiplicity of positive solutions for the supercritical problem.

2. The functional setting

2.1. Notations and basic results. We start by giving some notations and collecting useful preliminary
results. If A⊂R^{3} and 1≤p≤ ∞, we denote by∥u∥_{L}^{p}_{(A)} theL^{p}(A)-norm of a function u:R^{3}→R. Let
us dene D^{1,2}(R^{3}) as the completion of C_{0}^{∞}(R^{3}) with respect to the norm

∥∇u∥^{2}_{L}2(R^{3})=
Z

R^{3}

|∇u|^{2}dx.

Then we can consider the Sobolev space
H^{1}(R^{3}) =n

u∈L^{2}(R^{3}) :∥∇u∥^{2}_{L}2(R^{3}) <∞o
endowed with the norm

∥u∥^{2}_{H}1(R^{3})=∥∇u∥^{2}_{L}2(R^{3})+∥u∥^{2}_{L}2(R^{3}).
We have the following well-known main Sobolev embeddings.

Theorem 2.1. (see [1]) H^{1}(R^{3}) is continuously embedded in L^{p}(R^{3}) for any p ∈ [2,6] and compactly
embedded in L^{p}_{loc}(R^{3}) for any p∈[1,6).

We denote byS∗ the best constant of the Sobolev embeddingH^{1}(R^{3})⊂L^{6}(R^{3}), that is
S∗:= inf

(∥∇u∥^{2}_{L}2(R^{3})

∥u∥^{2}_{L}6(R^{3})

:u∈D^{1,2}(R^{3})\ {0}

) . We also recall the following classical lemma of Lions:

Lemma 2.1. (see [23]) If{u_{n}}n∈N is a bounded sequence in H^{1}(R^{3}) and

n→∞lim sup

y∈R^{3}

Z

B_{R}(y)

|u_{n}|^{2}dx= 0
for someR >0, then u_{n}→0 in L^{p}(R^{3}) for all p∈(2,6).

2.2. The modied problem. In order to study (1.1), we use the change of variable x7→ε xand we look for positive solutions to

−

a+b Z

R^{3}

|∇u|^{2}

∆u+V(ε x)u=f(u) +γu^{5} inR^{3},
u∈H^{1}(R^{3}), u >0 inR^{3}.

(2.1) In what follows, we introduce a penalized function [11] which will be useful to obtain our results. Let K >2 andα >0be such that

f(α) +γα^{5}= V_{0}

Kα (2.2)

and dene

f˜(t) :=

f(t) +γ(t^{+})^{5} if t≤α,

V0

Kt if t > α, and

g(x, t) :=χ_{Λ}(x)(f(t) +γ(t^{+})^{5}) + (1−χ_{Λ}(x)) ˜f(t).

It is easy to check that g satises the following properties:

(g1) limt→0 g(x,t)

t^{3} = 0 uniformly with respect to x∈R^{3},
(g2) g(x, t)≤f(t) +γt^{5} for allx∈R^{3},t >0,

(g_{3}) (i) 0< ϑG(x, t)≤g(x, t)t for allx∈Λ,t >0,

(ii) 0≤2G(x, t)≤g(x, t)t≤ ^{V}_{K}^{0}t^{2} for allx∈R^{3}\Λ,t >0,

(g_{4}) for eachx∈Λthe function ^{g(x,t)}_{t}3 is increasing on(0,∞), and for each x∈R^{3}\Λ, the function ^{g(x,t)}_{t}3

is increasing on(0, α).

Let us consider the following modied problem

−

a+b Z

R^{3}

|∇u|^{2}

∆u+V(ε x)u=g(ε x, u) inR^{3},
u∈H^{1}(R^{3}), u >0 inR^{3}.

(2.3)
It is clear that ifuis a positive solution of (2.3) withu(x)≤αfor all x∈R^{3}\Λε, thenuis also a positive
solution for (2.1), whereΛ_{ε}:={x∈R^{3}:ε x∈Λ}.

The energy functional associated with (2.3) is given by
J_{ε}(u) = 1

2∥u∥^{2}_{ε}+b

4∥∇u∥^{4}_{L}2(R^{3})−
Z

R^{3}

G(ε x, u)dx, which is well-dened on the space

H_{ε}:=

u∈H^{1}(R^{3}) :
Z

R^{3}

V(ε x)u^{2}dx <∞

endowed with the norm

∥u∥^{2}_{ε}:=a∥∇u∥^{2}_{L}2(R^{3})+
Z

R^{3}

V(ε x)u^{2}dx.

Clearly, H_{ε} is a Hilbert space with inner product
(u, v)ε:=

Z

R^{3}

a∇u∇v+V(ε x)uv dx.

It is easy to check that J_{ε}∈C^{1}(H_{ε},R) and its dierential is given by

⟨J_{ε}^{′}(u), v⟩= (u, v)_{ε}+b∥∇u∥^{2}_{L}2(R^{3})

Z

R^{3}

∇u∇v dx− Z

R^{3}

g(ε x, u)v dx

for any u, v∈ H_{ε}. Let us introduce the Nehari manifold associated with (2.3), that is,
N_{ε}:={u∈ H_{ε}\ {0}:⟨J_{ε}^{′}(u), u⟩= 0},

and we denote

H^{+}_{ε} :=

u∈ H_{ε}:|supp(u^{+})∩Λε|>0 and S^{+}ε :=Sε∩ H_{ε}^{+},

where Sε is the unit sphere in H_{ε}. Note that S^{+}_{ε} is a non-complete C^{1,1}-manifold of codimension one,
modelled on H_{ε} and contained in the open H^{+}_{ε} (see [30]). Then we have that H_{ε} = TuS^{+}ε ⊕Ru for all
u∈ H^{+}_{ε}, whereT_{u}S^{+}ε :={v∈ H_{ε}: (u, v)_{ε} = 0}.

Now we prove thatJ_{ε} possesses a mountain-pass geometry [5]:

Lemma 2.2. The functionalJ_{ε} satises the following properties:

(a) there exist η, ρ >0 such that J_{ε}(u)≥η with ∥u∥_{ε}=ρ;

(b) there existse∈ H_{ε} with ∥e∥_{ε}> ρsuch that J_{ε}(e)<0.

Proof. (a) By assumptions(g_{1}) and (g_{2}), we deduce that for anyξ >0 there existsC_{ξ}>0such that
J_{ε}(u)≥ 1

2∥u∥^{2}_{ε}−
Z

R^{3}

G(ε x, u)dx≥ 1

2∥u∥^{2}_{ε}−ξC∥u∥^{2}_{ε}−C_{ξ}C∥u∥^{6}_{ε}.
Then we can nd η, ρ >0 such thatJ_{ε}(u)≥η with ∥u∥_{ε}=ρ.

(b) Using (g_{3})-(i), we deduce that for anyu∈ H^{+}_{ε} andt >0
J_{ε}(tu) = t^{2}

2∥u∥^{2}_{ε}+bt^{4}

4∥∇u∥^{4}_{L}2(R^{3})−
Z

Λε

G(ε x, tu)dx

≤ t^{2}

2∥u∥^{2}_{ε}+bt^{4}

4∥∇u∥^{4}_{L}2(R^{3})−C_{1}t^{ϑ}
Z

Λε

(u^{+})^{ϑ}dx+C_{2}|supp(u^{+})∩Λ_{ε}|, (2.4)
for some constants C_{1}, C_{2} > 0. Recalling that ϑ ∈ (4,6), we can conclude that J_{ε}(tu) → −∞ ast →

+∞. □

Sincef is only continuous, the next results will be very useful to overcome the non-dierentiability of N_{ε}
and the incompleteness of S^{+}ε.

Lemma 2.3. Assume that conditions(V_{1})-(V_{2})and(f_{1})-(f_{4})hold. Then the following assertions are true.

(i) For each u∈ H^{+}_{ε}, let h:R^{+} →Rbe dened by hu(t) =J_{ε}(tu). Then, there is a unique tu >0 such
that

h^{′}_{u}(t)>0 for all t∈(0, tu) and h^{′}_{u}(t)<0 for all t∈(tu,∞).

(ii) There exists τ >0 independent ofu, such that t_{u} ≥τ for any u∈S^{+}ε. Moreover, for each compact
setK⊂S^{+}_{ε} there is a positive constant C_{K} such that t_{u} ≤C_{K} for any u∈K.

(iii) The mapmˆε:H^{+}_{ε} → N_{ε} given bymˆε(u) =tuu, is continuous andmε := ˆmε|

S^{+}ε is a homeomorphism
betweenS^{+}ε andN_{ε}. Moreover, m^{−1}_{ε} (u) = _{∥u∥}^{u}

ε.

(iv) If there is a sequence {u_{n}}_{n∈}_{N} ⊂ S^{+}ε such that dist(un, ∂S^{+}ε) → 0, then ∥m_{ε}(un)∥_{ε} → ∞ and
J_{ε}(m_{ε}(u_{n}))→ ∞.

Proof. (i) Let us observe that hu ∈ C^{1}(R^{+},R). By Lemma 2.2, we can infer that hu(0) = 0, hu(t) > 0
for t > 0 small enough and h_{u}(t) < 0 for t > 0 suciently large. Then there exists t_{u} > 0 such that
h^{′}_{u}(t_{u}) = 0 and t_{u} is a global maximum for h_{u}. Hence we can deduce that t_{u}u ∈ N_{ε}. Now we can prove
the uniqueness of tu. Assume by contradiction that there are two positive numbers t1 and t2 such that
t_{1} > t_{2} and h^{′}_{u}(t_{1}) =h^{′}_{u}(t_{2}) = 0. Hence

t_{1}∥u∥^{2}_{ε}+bt^{3}_{1}∥∇u∥^{4}_{L}2(R^{3})=
Z

R^{3}

g(ε x, t_{1}u)u dx (2.5)

and

t2∥u∥^{2}_{ε}+bt^{3}_{2}∥∇u∥^{4}_{L}2(R^{3})=
Z

R^{3}

g(ε x, t2u)u dx. (2.6)

Exploiting (2.5), (2.6),t1 > t2 and(g4), we can see that 1

t^{2}_{1} − 1
t^{2}_{2}

∥u∥^{2}_{ε} =
Z

R^{3}

g(ε x, t1u)

(t_{1}u)^{3} −g(ε x, t2u)
(t_{2}u)^{3}

u^{4}dx

= Z

R^{3}\Λ_{ε}

g(ε x, t_{1}u)

(t_{1}u)^{3} −g(ε x, t_{2}u)
(t_{2}u)^{3}

u^{4}dx+
Z

Λε

g(ε x, t_{1}u)

(t_{1}u)^{3} −g(ε x, t_{2}u)
(t_{2}u)^{3}

u^{4}dx

≥ Z

R^{3}\Λ_{ε}

g(ε x, t1u)

(t_{1}u)^{3} −g(ε x, t2u)
(t_{2}u)^{3}

u^{4}dx

=I1+I2+I3, where

I1:=

Z

(R^{3}\Λε)∩{t2u>α}

g(ε x, t1u)

(t1u)^{3} −g(ε x, t2u)
(t2u)^{3}

u^{4}dx,
I_{2}:=

Z

(R^{3}\Λε)∩{t2u≤α<t1u}

g(ε x, t_{1}u)

(t1u)^{3} − g(ε x, t_{2}u)
(t2u)^{3}

u^{4}dx
and

I3 :=

Z

(R^{3}\Λε)∩{t1u<α}

g(ε x, t1u)

(t1u)^{3} −g(ε x, t2u)
(t2u)^{3}

u^{4}dx.

Now we estimate eachIi,i∈ {1,2,3}. ConsideringI1, from the denition ofgand using(g3)-(ii), we have

I1 ≥ Z

(R^{3}\Λε)∩{t_{2}u>α}

V0

K 1

(t_{1}u)^{2} −V0

K
1
(t_{2}u)^{2}

u^{4}dx= 1
K

1
t^{2}_{1} − 1

t^{2}_{2}
Z

(R^{3}\Λε)∩{t_{2}u>α}

V0u^{2}dx.

From the denition of gand using (g_{2}), we can infer

I_{2} ≥
Z

(R^{3}\Λ_{ε})∩{t_{2}u≤α<t_{1}u}

V_{0}
K

1

(t_{1}u)^{2} −f(t_{2}u) +γ(t_{2}u^{+})^{5}
(t_{2}u)^{3}

u^{4}dx.

Finally, let us observe that by (g_{4}) and fromt_{1}> t_{2}, it follow thatI_{3} ≥0. Thus we have
1

t^{2}_{1} − 1
t^{2}_{2}

∥u∥^{2}_{ε} ≥ 1
K

1
t^{2}_{1} − 1

t^{2}_{2}
Z

(R^{3}\Λε)∩{t2u>α}

V_{0}u^{2}dx
+

Z

(R^{3}\Λ_{ε})∩{t_{2}u≤α<t_{1}u}

V_{0}
K

1

(t_{1}u)^{2} −f(t_{2}u) +γ(t_{2}u^{+})^{5}
(t_{2}u)^{3}

u^{4}dx,

from which, multiplying both sides by _{t}^{t}2^{2}^{1}^{t}^{2}^{2}

2−t^{2}_{1} <0 and using assumption(f4)and (2.2), we obtain

∥u∥^{2}_{ε}≤ 1
K

Z

(R^{3}\Λε)∩{t2u>α}

V_{0}u^{2}dx+ t^{2}_{1}t^{2}_{2}
t^{2}_{2}−t^{2}_{1}

Z

(R^{3}\Λε)∩{t2u≤α<t1u}

V_{0}
K

1

(t1u)^{2} −f(t_{2}u) +γ(t_{2}u^{+})^{5}
(t2u)^{3}

u^{4}dx

= 1 K

Z

(R^{3}\Λε)∩{t_{2}u>α}

V0u^{2}dx

− t^{2}_{2}
t^{2}_{1}−t^{2}_{2}

Z

(R^{3}\Λε)∩{t2u≤α<t1u}

V0

Ku^{2}dx+ t^{2}_{1}
t^{2}_{1}−t^{2}_{2}

Z

(R^{3}\Λε)∩{t2u≤α<t1u}

f(t2u) +γ(t2u^{+})^{5}
t_{2}u u^{2}dx

≤ 1 K

Z

R^{3}\Λ_{ε}

V_{0}u^{2}dx≤ 1
K∥u∥^{2}_{ε}.

Since u̸= 0 and K >2, we get a contradiction.

(ii)Let u∈S^{+}ε. By(i), there exists t_{u}>0such thath^{′}_{u}(t_{u}) = 0, that is
tu+bt^{3}_{u}∥∇u∥^{4}_{L}2(R^{3})=

Z

R^{3}

g(ε x, tuu)u dx. (2.7)

Using assumptions (g1) and(g2), (2.7) and Theorem 2.1, givenξ > 0, there exists a positive constantC_{ξ}
such that

tu ≤ Z

R^{3}

g(ε x, tuu)tuu dx≤ξt^{3}_{u}C1+Cξt^{5}_{u}C2.

This implies that there existsτ >0, independent ofu, such thattu ≥τ. Now, letK⊂S^{+}ε be a compact set.

We prove thatt_{u} ≤C_{K} for anyu∈K. Assume to the contrary, that there exists a sequence{u_{n}}_{n∈}_{N}⊂K
such thattn:=tun → ∞. SinceKis compact, there existsu∈Ksuch thatun→uinH_{ε}. It follows from
(2.4) thatJ_{ε}(t_{n}u_{n})→ −∞. Now, x v∈ N_{ε} and usingϑ∈(4,6)and (g_{3}), we can deduce that

J_{ε}(v) =J_{ε}(v)− 1

ϑ⟨J_{ε}^{′}(v), v⟩

=

ϑ−2 2ϑ

∥v∥^{2}_{ε}+b

ϑ−4 4ϑ

∥∇v∥^{4}_{L}2(R^{3})+ 1
ϑ

Z

R^{3}\Λε

[g(ε x, v)v−ϑG(ε x, v)]dx + 1

ϑ Z

Λε

[g(ε x, v)v−ϑG(ε x, v)]dx

≥

ϑ−2 2ϑ

∥v∥^{2}_{ε}+ 1
ϑ

Z

R^{3}\Λ_{ε}

[g(ε x, v)v−ϑG(ε x, v)]dx

≥

ϑ−2 2ϑ

∥v∥^{2}_{ε}−

ϑ−2 2ϑ

1 K

Z

R^{3}\Λ_{ε}

V(ε x)v^{2}dx

≥

ϑ−2

2ϑ 1− 1 K

∥v∥^{2}_{ε}. (2.8)

Takingv=t_{u}_{n}u_{n}∈ N_{ε} in (2.8) and using the facts∥v_{n}∥_{ε}=t_{n} and K >2, we get
0<

ϑ−2

2ϑ 1− 1 K

≤ J_{ε}(tnun)
t^{2}_{n} ≤0
for nlarge, and this gives a contradiction.

(iii) First, we note that mˆ_{ε},m_{ε} and m^{−1}_{ε} are well dened. Indeed, by (i), for each u∈ H^{+}_{ε} there exists a
unique mε(u)∈ N_{ε}. On the other hand, if u∈ N_{ε} thenu∈ H^{+}_{ε}. Otherwise, if u /∈ H^{+}_{ε}, we have

|supp(u^{+})∩Λ_{ε}|= 0,

which together with (g_{3})-(ii)implies that

∥u∥^{2}_{ε}+b∥∇u∥^{4}_{L}2(R^{3})=
Z

R^{3}

g(ε x, u)u dx

= Z

R^{3}\Λε

g(ε x, u)u dx+ Z

Λε

g(ε x, u)u dx

= Z

R^{3}\Λε

g(ε x, u^{+})u^{+}dx

≤ 1 K

Z

R^{3}\Λ_{ε}

V(ε x)u^{2}dx≤ 1

K∥u∥^{2}_{ε} (2.9)

and this yields a contradiction becauseu̸= 0 andK >2. As a consequence,m^{−1}_{ε} (u) = _{∥u∥}^{u}

ε ∈S^{+}ε,m^{−1}_{ε} is
well dened and continuous. Moreover, for all u∈S^{+}_{ε} we have

m^{−1}_{ε} (m_{ε}(u)) =m^{−1}_{ε} (t_{u}u) = t_{u}u

∥t_{u}u∥_{ε} = u

∥u∥_{ε} =u

from which we deduce that m_{ε} is a bijection. Now we prove that mˆ_{ε} is a continuous function. Let
{u_{n}}_{n∈}_{N} ⊂ H_{ε}^{+} and u∈ H^{+}_{ε} be such that un → u in H^{+}_{ε}. Since mˆε(tu) = ˆmε(u) for any t > 0, we may
assume that {u_{n}}_{n∈}_{N} ⊂S^{+}ε. Then by (ii), there existst_{0} >0 such thatt_{n} =t_{u}_{n} → t_{0}. Sincet_{n}u_{n}∈ N_{ε},
we obtain

t^{2}_{n}∥u_{n}∥^{2}_{ε}+bt^{4}_{n}∥∇u_{n}∥^{4}_{L}2(R^{3})=
Z

R^{3}

g(ε x, t_{n}u_{n})t_{n}u_{n}dx,
and passing to the limit as n→ ∞, we get

t^{2}_{0}∥u∥^{2}_{ε}+bt^{4}_{0}∥∇u∥^{4}_{L}2(R^{3}) =
Z

R^{3}

g(ε x, t_{0}u)t_{0}u dx
which yieldst0u∈ N_{ε}. This shows that

ˆ

m_{ε}(u_{n})→mˆ_{ε}(u) inH_{ε}.
Therefore,mˆ_{ε} andm_{ε} are continuous functions.

(iv)Let {u_{n}}_{n∈}_{N}⊂S^{+}ε be such thatdist(un, ∂S^{+}ε)→0. Since for eachv∈∂S^{+}ε andn∈Nwe have
u^{+}_{n} ≤ |u_{n}−v|a.e. inΛ_{ε},

it follows that

∥u^{+}_{n}∥^{p}_{L}_{p}_{(Λ}

ε)≤ inf

v∈∂S^{+}ε

∥u_{n}−v∥^{p}_{L}_{p}_{(Λ}

ε) for allp∈[2,6], for all n∈N. Hence, by (V1),(V2) and Theorem2.1, there is a constantCp>0such that

∥u^{+}_{n}∥_{L}p(Λε)≤ inf

v∈∂S^{+}_{ε}

∥u_{n}−v∥_{L}p(Λε)≤Cp inf

v∈∂S^{+}_{ε}

∥u_{n}−v∥_{ε} ≤Cpdist(un, ∂S^{+}_{ε})^{p} for all n∈N.

Using (g_{1}),(g_{2})and (g_{3})-(ii), we can infer that, for each t >0
Z

R^{3}

G(ε x, tu_{n})dx=
Z

R^{3}\Λ_{ε}

G(ε x, tu_{n})dx+
Z

Λε

G(ε x, tu_{n})dx

≤ t^{2}
K

Z

R^{3}\Λε

V(ε x)u^{2}_{n}dx+
Z

Λε

F(tun) +γt^{6}(u^{+}_{n})^{6}dx

≤ t^{2}

K∥u_{n}∥^{2}_{ε}+C1t^{4}
Z

Λε

(u^{+}_{n})^{4}dx+C2t^{6}
Z

Λε

(u^{+}_{n})^{6}dx

≤ t^{2}

K +C_{1}^{′}t^{4}dist(u_{n}, ∂S^{+}ε)^{4}+C_{2}^{′} dist(u_{n}, ∂S^{+}ε)^{6}
from which,

lim sup

n→∞

Z

R^{3}

G(ε x, tun)dx≤ t^{2}

K for allt >0. (2.10)

Recalling the denition of m_{ε}(u_{n}) and using (2.10) we get
lim inf

n→∞ J_{ε}(mε(un))≥lim inf

n→∞ J_{ε}(tun)

= lim inf

n→∞

t^{2}

2∥u_{n}∥^{2}_{ε}+bt^{4}

4∥∇u_{n}∥^{4}_{L}2(R^{3})−
Z

R^{3}

G(ε x, tu_{n})dx

≥ 1

2 − 1 K

t^{2}
which implies that

lim inf

n→∞

1

2∥m_{ε}(u_{n})∥^{2}_{ε}+ b

4∥∇m_{ε}(u_{n})∥^{4}_{L}2(R^{3}) ≥lim inf

n→∞ J_{ε}(m_{ε}(u_{n}))≥
1

2− 1 K

t^{2}.

Since K > 2 and t >0 is arbitrary, we obtain that J_{ε}(mε(un)) → ∞and ∥m_{ε}(un)∥_{ε} → ∞ as n → ∞.

This completes the proof of the lemma. □

Now, we dene the maps

ψˆε:H^{+}_{ε} →R and ψε:S^{+}_{ε} →R,
by ψˆ_{ε}(u) := J_{ε}( ˆm_{ε}(u)) and ψ_{ε} := ˆψ_{ε}|

S^{+}ε. The next result is a direct consequence of Lemma 2.3 and
Corollary 2.3 in [30].

Proposition 2.1. Assume that conditions(V1)-(V2) and(f1)-(f4)hold. Then the following assertions are true.

(a) ˆψε∈C^{1}(H^{+}_{ε},R) and

⟨ψˆ_{ε}^{′}(u), v⟩= ∥mˆε(u)∥_{ε}

∥u∥_{ε} ⟨J_{ε}^{′}( ˆmε(u)), v⟩

for every u∈ H^{+}_{ε}, v∈ H_{ε}.
(b) ψε∈C^{1}(S^{+}ε,R) and

⟨ψ^{′}_{ε}(u), v⟩=∥m_{ε}(u)∥_{ε}⟨J_{ε}^{′}(m_{ε}(u)), v⟩,
for every v∈TuS^{+}_{ε}.

(c) If{u_{n}}_{n∈}_{N}is a(P S)_{d}sequence forψε, then{m_{ε}(un)}_{n∈}_{N}is a(P S)_{d}sequence forJ_{ε}. If{u_{n}}_{n∈}_{N}⊂ N_{ε}
is a bounded(P S)_{d} sequence forJ_{ε}, then {m^{−1}_{ε} (u_{n})}_{n∈}_{N} is a(P S)_{d} sequence for the functional ψ_{ε}.
(d) u is a critical point of ψε if, and only if, mε(u) is a nontrivial critical point for J_{ε}. Moreover, the

corresponding critical values coincide and inf

u∈S^{+}ε

ψε(u) = inf

u∈Nε

J_{ε}(u).

Remark 2.1. As in [30], we have the following variational characterization of the inmum of J_{ε} over
N_{ε}:

cε:= inf

u∈Nε

J_{ε}(u) = inf

u∈H^{+}_{ε}

maxt>0 J_{ε}(tu) = inf

u∈S^{+}ε

maxt>0 J_{ε}(tu).

Remark 2.2. Let us note that if u∈ N_{ε}, it follows from (g_{1})-(g_{2}) that
0 =∥u∥^{2}_{ε}+b∥∇u∥^{4}_{L}2(R^{3})−

Z

R^{3}

g(ε x, u)udx≥ 1

2∥u∥^{2}_{ε}−C∥u∥^{6}_{ε}
which implies that∥u∥_{ε}≥r >0 for some r independent of u.

3. An existence result for the modified problem

In this section we focus our attention on the existence of positive solutions to (2.3) for suciently small
ε >0. We begin showing that the functional J_{ε} satises the Palais-Smale condition at any level d∈Rif
γ = 0, and d < c∗ for some suitable c∗ >0 depending onS∗, when γ = 1. This last fact is motivated by
the following result:

Lemma 3.1. Let γ = 1. Then
c_{ε}< 1

4abS_{∗}^{3}+ 1

24b^{3}S_{∗}^{6}+ 1

24(b^{2}S_{∗}^{4}+ 4aS∗)^{3}^{2} =:c∗

for all ε >0.

Proof. One can argue as in the proof of Lemma 2.1 in [18]. □

In view of Lemma 2.2, we can apply a version of the mountain-pass theorem without (PS) condition
(see [35]) to obtain a sequence {u_{n}}_{n∈}_{N}⊂ H_{ε} such that

J_{ε}(un)→cε and J_{ε}^{′}(un)→0. (3.1)
We start with the following result:

Lemma 3.2. Every sequence satisfying (3.1) is bounded.

Proof. Arguing as in the proof of Lemma2.3-(ii) (see formula (2.8) there), we can deduce that
C(1 +∥u_{n}∥_{ε})≥ J_{ε}(u_{n})− 1

ϑ⟨J_{ε}^{′}(u_{n}), u_{n}⟩

≥

ϑ−2

2ϑ 1− 1 K

∥u_{n}∥^{2}_{ε}.

Since ϑ >4 andK >2, we can conclude that {u_{n}}_{n∈}_{N}is bounded in H_{ε}. □
Lemma 3.3. There is a sequence {z_{n}}_{n∈}_{N}⊂R^{3} andR, β >0 such that

Z

BR(zn)

u^{2}_{n}dx≥β.

Proof. Assume to the contrary, that the conclusion of lemma is not true. By Lemma2.1, we then have
u_{n}→0inL^{r}(R^{3}) for any r∈(2,6),

so, in view of (f_{1}) and(f_{2}), we get
Z

R^{3}

F(un)dx= Z

R^{3}

f(un)undx=on(1)asn→ ∞. (3.2)
Since {u_{n}}_{n∈}_{N} is bounded inH_{ε}, we may assume that u_{n}⇀ u inH_{ε}.

If γ = 0, then we can use ⟨J_{ε}^{′}(u_{n}), u_{n}⟩ = o_{n}(1) and (3.2) to deduce that ∥u_{n}∥_{ε} → 0, which in turn
implies that J_{ε}(un)→0, and this is impossible because cε>0.

Now assume that γ= 1. Using the denition ofg and (3.2), we can deduce that Z

R^{3}

G(ε x, u_{n})dx≤ 1
6

Z

Λε∪{un≤α}

(u^{+}_{n})^{6}dx+ V_{0}
2K

Z

(R^{3}\Λε)∩{un>α}

u^{2}_{n}dx+o_{n}(1) (3.3)
and

Z

R^{3}

g(ε x, un)undx= Z

Λε∪{un≤α}

(u^{+}_{n})^{6}dx+V_{0}
K

Z

(R^{3}\Λε)∩{un>α}

u^{2}_{n}dx+on(1). (3.4)
From ⟨J_{ε}^{′}(un), un⟩=on(1)we have

∥u_{n}∥^{2}_{ε}−V_{0}
K

Z

(R^{3}\Λ_{ε})∩{u_{n}>α}

u^{2}_{n}dx+b∥∇u_{n}∥^{4}_{L}2(R^{3})=
Z

Λε∪{u_{n}≤α}

(u^{+}_{n})^{6}dx+o_{n}(1). (3.5)
Letℓ_{1}, ℓ_{2} ≥0 be such that

∥u_{n}∥^{2}_{ε}−V0

K Z

(R^{3}\Λε)∩{un>α}

u^{2}_{n}dx→ℓ1 (3.6)

and

b∥∇u_{n}∥^{4}_{L}2(R^{3})→ℓ2. (3.7)

Note thatℓ1 >0, otherwise (3.5) would yield∥u_{n}∥_{ε}→0asn→ ∞and thenJ_{ε}(un)→0, which contradicts
c_{ε}>0. Hence, putting together (3.5), (3.6) and (3.7), we have

Z

Λε∪{un≤α}

(u^{+}_{n})^{6}dx→ℓ_{1}+ℓ_{2}. (3.8)

By (3.3), (3.6), (3.7), (3.8) and J_{ε}(un) =cε+on(1), it follows that
c_{ε}≥ 1

3ℓ_{1}+ 1

12ℓ_{2}. (3.9)

On the other hand, from the denition ofS∗ we can see that

∥u_{n}∥^{2}_{ε}−V_{0}
K

Z

(R^{3}\Λ_{ε})∩{u_{n}>α}

u^{2}_{n}dx≥aS∗

Z

Λε∪{u_{n}≤α}

(u^{+}_{n})^{6}dx

!^{1}_{3}

and

b∥∇u_{n}∥^{4}_{L}2(R^{3}) ≥bS_{∗}^{2}
Z

Λε∪{un≤α}

(u^{+}_{n})^{6}dx

!^{2}_{3}
.
This, together with (3.6), (3.7) and (3.8), implies that

ℓ1 ≥aS∗(ℓ1+ℓ2)^{1}^{3} and ℓ2≥bS_{∗}^{2}(ℓ1+ℓ2)^{2}^{3}, (3.10)
which yields

ℓ_{1}+ℓ_{2}≥aS∗(ℓ_{1}+ℓ_{2})^{1}^{3} +bS_{∗}^{2}(ℓ_{1}+ℓ_{2})^{2}^{3}.
Consequently,

(ℓ1+ℓ2)^{1}^{3} ≥ bS_{∗}^{2}+ (b^{2}S^{4}_{∗}+ 4aS∗)^{1}^{2}

2 . (3.11)

Combining (3.9), (3.10), (3.11), it follows that
c_{ε}≥ 1

3ℓ_{1}+ 1

12ℓ_{2}≥ 1

3aS∗(ℓ_{1}+ℓ_{2})^{1}^{3} + 1

12bS_{∗}^{2}(ℓ_{1}+ℓ_{2})^{2}^{3}

≥ 1

4abS_{∗}^{3}+ 1

24b^{3}S_{∗}^{6}+ 1

24(b^{2}S^{4}_{∗}+ 4aS∗)^{3}^{2}

and by Lemma3.1, this is a contradiction. □

Lemma 3.4. The sequence {z_{n}}_{n∈}_{N} given in Lemma 3.3is bounded in R^{3}.

Proof. For any ρ > 0, let ψρ ∈ C^{∞}(R^{3}) be such that ψρ = 0 in Bρ(0) and ψρ = 1 in R^{3}\B2ρ(0), with
0≤ψρ≤1 and |∇ψ_{ρ}| ≤ ^{C}_{ρ}, whereC is a constant independent of ρ. Since {ψ_{ρ}un}_{n∈}_{N} is bounded inH_{ε},
it follows that⟨J_{ε}^{′}(un), ψρun⟩=on(1), namely

a Z

R^{3}

|∇u_{n}|^{2}ψρdx+a
Z

R^{3}

∇u_{n}∇ψ_{ρ}undx+b∥∇u_{n}∥^{2}_{L}2(R^{3})

Z

R^{3}

|∇u_{n}|^{2}ψρdx+
Z

R^{3}

∇u_{n}∇ψ_{ρ}undx

+ Z

R^{3}

V(ε x)u^{2}_{n}ψρdx=on(1) +
Z

R^{3}

g(ε x, un)unψρdx.

Takeρ >0such thatΛ_{ε}⊂B_{ρ}(0). Then, using(g_{3})-(ii)and Lemma 3.2, we get

1− 1 K

V_{0}

Z

{|x|≥2ρ}

u^{2}_{n}dx

≤ −a Z

R^{3}

∇u_{n}∇ψ_{ρ}u_{n}dx−b∥∇u_{n}∥^{2}_{L}2(R^{3})

Z

R^{3}

∇u_{n}∇ψ_{ρ}u_{n}dx

+o_{n}(1)

≤ C ρ

Z

R^{3}

|∇u_{n}||u_{n}|dx+C

ρ∥∇u_{n}∥^{2}_{L}2(R^{3})

Z

R^{3}

|∇u_{n}||u_{n}|dx

+on(1)

≤ C

ρ +on(1), which implies that

Z

{|x|≥2ρ}

u^{2}_{n}dx≤ C

ρ +on(1). (3.12)

Now, if{z_{n}}n∈N is unbounded, it follows by Lemma3.3and (3.12), that0< β≤ ^{C}_{ρ} →0 asρ→ ∞, which

gives a contradiction. □

The next results will be essential for obtaining the compactness of bounded Palais-Smale sequences.

Lemma 3.5. Let {u_{n}}_{n∈}_{N} be a (P S)cε sequence for J_{ε}. Then for each ζ >0, there exists R=R(ζ)>0
such that

lim sup

n→∞

"

Z

R^{3}\BR(0)

a|∇u_{n}|^{2}+V(ε x)u^{2}_{n}dx

#

< ζ. (3.13)

Proof. LetR >0 be such thatΛε ⊂BR

2(0), and η_{R}∈C^{∞}(R^{3}) such thatη_{R}= 0inBR

2(0) andη_{R}= 1in
R^{3}\B_{R}(0), with 0≤η_{R}≤1 and |∇η_{R}| ≤ ^{C}_{R}, whereC is a constant independent of R. Since {η_{R}u_{n}}_{n∈}_{N}