bound states to critical quasilinear schrödinger equations

Nonlinear Differ. Equ. Appl. 19 (2012), 19–47c© 2011 Springer Basel AG1021-9722/12/010019-29published online May 27, 2011DOI 10.1007/s00030-011-0116-3

Nonlinear Differential Equationsand Applications NoDEA

Bound states to critical quasilinearSchrodinger equations

Youjun Wang and Wenming Zou

Abstract. In this paper, we consider the critical quasilinear Schrodingerequations of the form

−ε2Δu + V (x)u − ε2[Δ(u2)]u = |u|2(2∗)−2u + g(u), x ∈ RN ,

where N ≥ 3, 2∗ := 2N/(N − 2) and g(u) is of subcritical growth. Weprove the existence of positive bound states which concentrate around alocal minimum point of V as ε → 0+.

Mathematics Subject Classification (2000). 35B33, 35J20, 35J60, 35Q55.

1. Introduction

This paper is motivated by the recent interests on the following type of quasi-linear Schrodinger equations

iε∂ψ

∂t= −ε2Δψ +W (x)ψ − h(|ψ|2)ψ − ε2kΔρ(|ψ|2)ρ′(|ψ|2)ψ, x ∈ R

N , (1.1)

where W is a given potential, k is a real constant, h and ρ are real func-tions of essentially pure power forms. Here we consider the case ρ(s) = sand our special interest is the standing wave solutions, i.e., solutions of typeψ(t, x) = exp(−iEt)u(x), where E ∈ R and u > 0 is a real function. Note thatψ satisfies (1.1) if and only if the function u(x) solves the following equationof elliptic type with the formal variational structure

− ε2Δu+ V (x)u− ε2k[Δ(u2)]u = h(u), x ∈ RN , (1.2)

where V (x) = W (x) − E is the new potential function, h is the new nonlin-earity.

In quantum mechanics, where the number of particles is conserved, abound state is a state in the Hilbert space that corresponds to two or moreparticles whose interaction energy determines whether these particles can be

This article was supported by NSFC (No. 10871109, 11025106) and China Post-Doc ScienceFoundation (No. 20100470175).

20 Y. Wang and W. Zou NoDEA

separated or not. One of the historical puzzles that led to the creation ofquantum mechanics was the stability of a bound state—why electrons in atomsdo not just spiral into the nucleus. It is the bound states of nucleons, atoms,molecules and solids that allow the world and all of life to be what it is.Therefore, the bound state is one of the most important topics in quantummechanics. So much current ongoing research and historical research usingquantum mechanics involve the bound states.

The existence and concentration behavior of positive ground (bound)state solutions for (1.2) with k = 0 has been studied extensively in recentyears. Consider

− ε2Δu+ V (x)u = f(x, u), u > 0, u ∈ H1(RN ). (1.3)

Note that when setting v(x) = u(εx) and Vε(x) = V (εx), then (1.3) is equiv-alent to

−Δv + Vε(x)v = f(x, v), v > 0, v ∈ H1(RN ).

For ε small enough, the solutions to problem (1.3) can induce the stand-ing waves of the Schrodinger equation which are usually referred to as semi-classical states. Some class of solutions of (1.3) concentrate and develop spikelayers and peaks around certain points in R

N but vanishing elsewhere as ε → 0.The existence of single-peak solutions was first studied in [21] where N = 1 andf = u3. A single-peak solution was constructed which concentrates around anygiven non-degenerate critical point of the potential V (x). The higher dimen-sion cases were considered in [35,36]. In particular, in [36], the existence ofmulti-peak solutions which concentrate around any finite subsets of the non-degenerate critical points of V (x) was established. The arguments in [21,35,36]are based on a Lyapunov–Schmidt reduction and heavily rely on the unique-ness and non-degeneracy of the positive ground state solutions (least energysolutions). In [2], they studied (1.3) and considered the concentration phenom-ena at isolated local minima and maxima with polynomial degeneracy. In [27],the author deals with C1-stable critical points of V (x). See also [3,12,25] forrelated results about (1.3). However, as observed by many experts, the unique-ness and non-degeneracy of the positive ground state solutions are usually quitedifficult to verify. They are known so far only for some very restricted caseson nonlinearities f in (1.3). To get the existence of positive solutions with-out these assumptions, the variational approach which was initiated in [40]has proved to be very successful. In [40], by the mountain pass theorem, theauthor proves the existence of positive solutions of (1.3) for small ε > 0 when-ever lim inf |x|→∞ V (x)) > infx∈RN V (x). These solutions concentrate aroundthe global minimum points of V (x) when ε → 0 as was shown in [42]. Laterin [16], by introducing a penalization approach, the authors proved a localizedversion of the result in [40,42], see also [13–15] for related results, they do notassume the uniqueness of a least-energy ground state in a related homogeneousproblem. In [9,24], the monotonicity condition of [16] is not necessary. Also, in[8], the authors develop a new variational approach to construct localized pos-itive solutions to (1.3) which concentrate at an isolated component of positive

Vol. 19 (2012) Critical quasilinear Schrodinger equations 21

local minimum points of V (x) as ε → 0 under certain conditions on f whichare “almost optimal”. Similarly, no uniqueness and non-degeneracy of the pos-itive ground state solutions are required in [8]. In [15,17], they constructed afamily of solutions with several spikes located around any prescribed finite setof local minima of V . For more results of this type, we refer to [6–11,13].

Quasilinear Schrodinger equation of the type with k > 0 arises in var-ious fields of physics, like the theory of superfluids or dissipative quantummechanics, see e.g. [26,33]. Equations with more general dissipative term arisein plasma physics, fluid mechanics, in the theory of Heisenberg ferromagnets,etc. For further physical motivations and a more complete list of referencesdealing with application, we refer the readers to [17–23,29] and their bibliog-raphy.

In [37], the existence of positive ground state solutions for the quasilin-ear Schrodinger equation −u′′ + V (x)u − (u2)′′u = θ|u|p−1u in R were con-structed as minimizers of a constrained minimization problem, with θ beingthe Lagrange multiplier. In [29], by a change of variables the quasilinear prob-lem (1.2), where h(x) is of subcritical growth, was transformed to a semilinearone and an Orlicz space framwork was used as the working space, and theywere able to prove the existence of positive solutions by the mountain-passTheorem. The same method of changing of variables was used in [10], but theusual Sobolev space H1(RN ) framework was used as the working space andthey proved the existence of a spherically symmetric solution from the classicalresults given by Berestycki and Lions [5].

The existence of solutions for problem (1.2) involving subcritical growthwith ε = 1 was considered in [4,10,19,26,28,29] and it was left an open prob-lem for the critical case in [29]. It seems that the existence of solutions forcritical case was first studied by Moameni in [30] when the potential func-tion V satisfies some geometry conditions, and it was established the existenceof multiple solutions in [31] by the fibering method. Recently, do O et al.[20] considered problem (1.2) with ε = 1, h(u) = |u|q−1u + |u|p−1u, where3 < q < p ≤ 2(2∗)− 1. They showed the existence of a positive classic solutionif V (x) is bounded or periodic. In [41], the authors study problem (1.2) withε = 1, h(u) = K(x)|u|2(2∗)−1u + g(x, u), where g(x, u) is an asymptoticallynonlinearity. They also proved one solution if V (x) is periodic. There are fewresults dealing with semi-classical states for quasilinear Schrodinger equationsof the form (1.2) with critical growth, which are families of solutions uε whichdevelop a spike shape around one or more distinguished points of the space,while vanishing asymptotically elsewhere as ε → 0. We only refer to the recentpaper [18] in the case of N = 2.

Based on the above reviews and observations, we know that the existenceof positive bound states along with the concentration behavior of solutions tothe critical case of (1.2) with N ≥ 3 is largely open.

The main purpose of the present paper is to show the existence andconcentration behavior of positive bound state solutions for (1.2) when the


nonlinearity has the critical nonlinearity. More precisely, we consider the prob-lem

− ε2Δu+ V (x)u− ε2[Δ(u2)]u = |u|2(2∗)−2u+ g(u), x ∈ RN , (1.4)

where N ≥ 3, 2∗ = 2N/(N − 2). As in [16], we also assume the followingconditions on V :(V1) The function V : R

N → R is locally Holder continuous and V (x) ≥ α > 0for all x ∈ R

N .(V2) There is a bounded domain Ω ⊂ R

N such that V0 := infΩV < inf

∂ΩV.

The nonlinear term g : R+ → R is of class C1 and satisfies

(g1) g(s) = o(s) as s → 0.(g2) There are q1, q2 ∈ (2(N +2)/(N −2), 2(2∗)) and a λ ∈ (0,+∞) such that

g(s) ≥ λsq1−1 for all s > 0 and lims→∞

g(s)sq2−1

= 0.

(g3) There exists θ ∈ (4, q2) such that

0 < G(s) =∫ s

0

g(t) dt ≤ 1θg(s)s for all s > 0.

(g4) The function g(s)s−3 is increasing for s > 0.Our main result in the current paper is the following

Theorem 1.1. Suppose that (V1)–(V2) and (g1)–(g4) hold. Then there is anε0 > 0 such that problem (1.4) possesses a positive bound state solution uε ∈C2,α

loc (RN ) for all 0 < ε < ε0 and some α ∈ (0, 1). Moreover, uε has at mostone local (global) maximum zε in R

N , which is inside Ω such that

limε→0+

V (zε) = V0 := infΩV.

Besides, there exist C, ξ > 0 such that

uε(z) ≤ C exp(

−ξ∣∣∣∣z − zε

ε

∣∣∣∣).

2. Reformulation of the problem

It should be pointed out that we may not apply directly the variational methodto study (1.4) since the natural associated functional I given by

I(u) =12ε2

∫RN

(1 + 2u2)|∇u|2 dx+12

∫RN

V (x)u2 dx− 12(2∗)

∫RN

u2(2∗) dx

−∫

RN

G(u) dx

is not well defined in general, for instance, in H1(RN ). To overcome this diffi-culty, we employ an argument developed by Colin and Jeanjean [10] (see also[19,29]). We make the changing of variables v = f−1(u), where f is definedby: f ′(t) = 1√

1+2f2(t)on [0,+∞) and f(t) = −f(−t) on (−∞, 0].


Lemma 2.1. We have that

(i) the functional f(t)f ′(t)t−1 is decreasing for t > 0;(ii) the functional fq(t)f ′(t)t−1 is increasing for q ≥ 3 and t > 0.

Proof. (i) was proved in [18]. To prove (ii), by the definition of f , we havef(t)/2 ≤ tf ′(t) ≤ f(t) for all t ≥ 0 and |f(t)f ′(t)| ≤ 1/

√2 for all t ∈ R (see

[19]). Then,

d

dt

[fq(t)f ′(t)

t

]=qfq−1(t)(f ′(t))2t− 2fq+1(t)(f ′(t))4t− fq(t)f ′(t)

t2

≥ fq−1(t)f ′(t)[qf ′(t)t− 2f2(t)(f ′(t))3t− f(t)]t2

≥ fq−1(t)f ′(t)[(q − 1)f ′(t)t− f(t)]t2

> 0.

The lemma is proved. �

After the change of variables (u = f(v)), we can rewrite I(u) as

J(v) =12

∫RN

(ε2|∇v|2 + V (x)f2(v)) dx− 12(2∗)

∫RN

|f(v)|2(2∗)−2f(v) dx

−∫

RN

G(f(v)) dx.

Then, the critical points of J are weak solutions of the equation

−ε2Δv = f ′(v)(g(f(v)) + |f(v)|2(2∗)−2f(v) − V (x)f(v)

), x ∈ R

N .

Note that if v is a critical point of J(v), then, u = f(v) is a weak solution ofthe equation (1.4). As in [19], we define the Orlicz space

E :={v ∈ H1(RN ) :

∫RN

V (x)f2(v) dx < ∞}

which is endowed with the norm ||v|| = ||∇v||2 + infξ>0

1ξ [1+

∫RN V (x)f2(ξv) dx].

Let X := {u ∈ H1(RN ) :∫

RN V (x)u2 dx < ∞}, which is endowed with theinner product 〈u, v〉 =

∫RN (∇u∇v + V (x)uv) dx and the corresponding norm

||u||X =( ∫

RN

(|∇u|2 + V (x)u2) dx)1/2

.


Proposition 2.1. The functional J has the following properties:(1) J is well defined in E.(2) J is continuous in E.(3) J is Gateaux-differentiable in E.

Proof. The proof is similar to the proof of Proposition 2.5 in [30]. �

The following result can be found in [19].

Proposition 2.2.

(1) E is a Banach space.(2) There exists a C > 0 such that∫

RN V (x)f2(v) dx1 + (

∫RN V (x)f2(v) dx)1/2

≤ C‖v‖, ∀v ∈ E. (2.1)

(3) The embedding X ↪→ E is continuous.(4) The map v → f(v) from E to Lq(RN ) is continuous for 2 ≤ q ≤ 2(2∗).(5) The embedding E ↪→ Lr(RN ) is continuous for 2 ≤ r ≤ 2(2∗).(6) The embedding E ↪→ H1(RN ) is continuous. Moreover, C∞

0 (RN ) is densein E.

Before closing this section, we establish the following lemma.

Lemma 2.2. Let {vn} be a bounded sequence in E such that

lim infn→∞ sup

y∈RN

∫BR(y)

f2(vn) dx → 0,

for some R > 0. Then f(vn) → 0 in Ls(RN ) for 2 < s < 2(2∗).

Proof. Since E ↪→ H1(R) is continuous, {vn} is also bounded in H1(RN ). Itfollows from the Holder and Sobolev inequalities, we have

∫BR(y)

|f(v)|sdx ≤(∫

BR(y)

f2(v) dx

) (1−α)s2

(∫BR(y)

(f2(v))2∗dx

) αs2(2∗)

≤ C

(∫BR(y)

f2(v) dx

) (1−α)s2

(∫BR(y)

(|∇v|2 + v2) dx

)αs4

,

where α = 2∗s−2(2∗)(2∗−1)s . If s ≥ 4

N , then αs ≥ 4 and

∫BR(y)

|f(v)|s dx ≤ C

(∫BR(y)

f2(v) dx

) (1−α)s2

‖v‖αs−4

2H1

∫BR(y)

(|∇v|2 + v2) dx.

Covering RN by a family of balls {BR(yi)} such that each point is contained

in at most k such balls and summing up these inequalities over this family ofballs we obtain

∫RN

|f(v)|s dx ≤ kC supy∈RN

(∫BR(y)

f2(v) dx

) (1−α)s2

‖v‖αs2

H1 .


Substituting v = vn the result follows for s ≥ 4N . If 2 < s < 4

N , we writes = 2τ + (1 − τ) 4

N for some τ ∈ (0, 1). By the Holder inequality, we get

‖f(v)‖ss ≤ ‖f(v)‖2τ

2 ‖f(v)‖(1−τ) 4N

4N

and the result in this case follows the conclusion already established above. �

2.1. Penalized nonlinearity

We consider the following Caratheodory function:

h(x, s) ={χΩ(g(s) + s2(2

∗)−1) + (1 − χΩ)g(s) if s ≥ 0,0 if s < 0,

where

g(s) ={g(s) + s2(2

∗)−1 if s ≤ a,k−1αs if s > a

with k > 2(θ−2)θ−4 > 2, a > 0 such that g(a) + a2(2∗)−1 = k−1aα, χΩ is the

characteristics function on Ω.Using (g1)−(g4), it is easy to check that h(x, s) satisfies the following

conditions:

(h1) h(x, s) = o(s) as s → 0;(h2) h(x, s) ≤ g(s) + s2(2

∗)−1 for all s > 0, x ∈ RN ;

(h3) 0 < θH(x, s) ≤ h(x, s)s for all x ∈ Ω, s > 0 or x �∈ Ω, s ≤ a; 0 ≤2H(x, s) ≤ h(x, s)s ≤ k−1V (x)s2 for all x �∈ Ω, s > 0, where H(x, s) =∫ s

0h(x, t)dt;

(h4) The function h(x, s)s−3 is increasing for s > 0.

We introduce the penalized functional J : E → R as following

J(v) =12

∫RN

(ε2|∇v|2 + V (x)f2(v)) dx−∫

RN

H(x, f(v)) dx.

Using Proposition 2.2, it is easy to check that J is well defined and thatJ ∈ C1(E,R). Moreover, the critical points of J are solutions of the modifiedproblem

− ε2Δv = f ′(v) (h(x, f(v)) − V (x)f(v)) , x ∈ RN . (2.2)

Next, we assume that ε = 1.

2.2. Mountain pass geometry

Consider the set S(ρ) = {v ∈ E : Φ(v) = ρ2}, where Φ : E → R is given by

Φ(v) =∫

RN

(|∇v|2 + V (x)f2(v)) dx.

Since Φ(v) is continuous then S(ρ) is a closed subset and disconnects the spaceE for ρ > 0.


Lemma 2.3. There exist ρ0, a0 > 0 such that J(v) ≥ a0 for all v ∈ S(ρ0).

Proof. For v ∈ S(ρ), by the Holder and the Sobolev inequalities, we have

∫RN

|f(v)|q2 dx ≤(∫

RN

f2(v) dx) rq2

2(∫

RN

f2(2∗)(v) dx)1− rq2

2

≤ C(ρ2)rq22

(∫RN

|∇f2(v)|2 dx) (1−rq2/2)2∗

2

≤ C(ρ2)rq22

(∫RN

|∇v|2 dx) (1−rq2/2)2∗

2

≤ Cρ2N+2q2

N+2 ,

where r = 2(2∗)−q2q2(2∗−1) . Therefore, by (h2), (g1) and (g2), we have

J(v) =12

∫RN

(|∇v|2 + V (x)f2(v)) dx−∫

RN

H(x, f(v)) dx

≥ 12

∫RN

(|∇v|2 + V (x)f2(v)) dx−∫

RN

(G(f(v)) +

12(2∗)

f2(2∗)(v))dx

≥ 12

∫RN

(|∇v|2 + V (x)f2(v)) dx

−∫

RN

(εf2(v) + Cε|f(v)|q2 +

12(2∗)

f2(2∗)(v))dx

≥ 14ρ2 − Cρ

2N+2q2N+2 − Cρ2∗

, for ε small enough.

Since (2N + 2q2)/(N + 2) > 2, we choose ρ0 > 0 small enough such that

β0 :=14

− Cρ2(q2−2)

N+2 − Cρ4

N−2 > 0,

we have J(v) ≥ a0 := ρ20β0 > 0 for all v ∈ S(ρ0). �


Lemma 2.4. There exists an v ∈ E such that Φ(v) > ρ20 and J(v) < 0.

Proof. Given a ϕ ∈ C∞0 (RN , [0, 1]) with suppϕ = B1, we prove that J(tϕ) →

−∞ as t → ∞, which proves our thesis if we take v = tϕ with t large enough.Since tf ′(t) ≤ f(t), it follows that the function f(t)/t is decreasing for t > 0.Since 0 < tϕ(x) ≤ t for x ∈ B1, we have f(tϕ(x)) ≥ f(t)ϕ(x), which impliesthat

J(tϕ) ≤ 12

∫B1

(|∇(tϕ)|2 + V (x)f2(tϕ)) dx− 12(2∗)

∫B1

f2(2∗)(tϕ) dx

≤ t2

2

∫B1

(|∇ϕ|2 + V (x)ϕ2) dx− 12(2∗)

∫B1

f2(2∗)(tϕ) dx

≤ t2[12

∫B1

(|∇ϕ|2 + V (x)ϕ2) dx− f2(2∗)(t)2(2∗)t2

∫B1

ϕ2(2∗) dx

]

→ −∞, as t → ∞,

since limt→+∞

f2(2∗)(t)t2 = +∞. �

Now, in view of Lemmas 2.3, 2.4, we can apply a version of mountainpass Theorem without (PS)c condition due to Ambrosetti–Rabinowitz [39],it follows that there exists a (PS)c sequence {vn} ⊂ E, i.e., a sequence suchthat J(vn) → c and J ′(vn) → 0, where c is the mountain pass level of Jcharacterized by

c = infγ∈Γ

supt∈[0,1]

J(γ(t)), (2.3)

where Γ = {γ ∈ C([0, 1], E) : γ(0) = 0, J(γ(1)) < 0, γ(1) �= 0}. Let us considerthe Nehari manifold N = {v ∈ E \{0} : 〈J ′(v), v〉 = 0}. We have the followinglemma.

Lemma 2.5. For every v ∈ E \ {0}, there exists a unique t0 > 0 such thatt0v ∈ N . Moreover, J(t0v) = max

t≥0J(tv).

Proof. Let v ∈ E \ {0} be fixed and define the function I(t) = J(t|v|) fort ≥ 0. We notice that I ′(t) = 〈J ′(t|v|), |v|〉 = 0 if and only if tv ∈ N . More-over, I ′(t) = 0 is equivalent to

∫RN

|∇v|2 dx =∫

RN

[h(x, f(t|v|))f ′(t|v|)

t|v| − V (x)f(t|v|)f ′(t|v|)t|v|

]v2 dx.(2.4)

The right hand side of Eq. (2.4) is an increasing function of t. Indeed, for xfixed, we consider the function z : (0,∞) → R given by

z(s) =h(x, f(s))f ′(s)

s− V (x)f(s)f ′(s)

s.


If x ∈ Ω, we have

z(s) =

[g(f(s)) + f2(2∗)−1(s)

]f3(s)

f3(s)f ′(s)s

− V (x)f(s)f ′(s)s

.

If x �∈ Ω, we have

z(s) =

[g(f(s)) + f2(2∗)(s)

]f3(s)

f3(s)f ′(s)s

− V (x)f(s)f ′(s)s

, s ≤ f−1(a),

and

z(s) =kV (x) − α

k

(−f(s)f ′(s)

s

), s > f−1(a).

Thus, Lemma 2.1 and (g4) imply the claim. Since v is fixed, we may chooseδ > 0 such that

∫RN

|∇v|2 dx− δ

∫RN

|v|2 dx > 0. (2.5)

By the definition of h(x, s), (h2), (g1), (g2) and (f2), we have

limt→0+

∫RN

h(x, f(tv))f(tv)t2

dx ≤ δ limt→0+

∫RN

f2(tv)t2

dx

+C limt→0+

∫RN

f2(2∗)(tv)t2

dx

≤ δ

∫RN

v2 dx+ C limt→0+

t2(2∗)−2

∫RN

v2(2∗) dx

= δ

∫RN

v2 dx. (2.6)

By (h3), we have

I(t) = t2(

12

∫RN

|∇v|2 dx+12

∫RN

V (x)f2(tv)t2

dx−∫

RN

H(x, f(tv))t2

dx

)

≥ 12t2

(∫RN

|∇v|2 dx−∫

RN

h(x, f(tv))f(tv)t2

dx

).

It follows by (2.5) and (2.6) that I(t) > 0 for small t > 0. Moreover, I(0) = 0and I(t) < 0 for t large. Therefore, there exists a unique t0 > 0 such thatI ′(t0) = 0, that is, t0v ∈ N . Furthermore, I(t0) = max

t≥0I(t). �

Define

c∗ = infv∈N

J(v), c∗∗ = infv∈E\{0}

maxt≥0

J(tv).


Lemma 2.6. c = c∗ = c∗∗.

Proof. This kind of result has already been studied for Eq. (1.2) in [16,40,43]with k = 0 and [18] with k �= 0. We sketch the proof as following. It followsfrom Lemma 2.4 that c∗ = c∗∗. Note J(t0v) < 0 for v ∈ E \ {0} and t0 largeenough. Defining γ : [0, 1] → E by γ(t) = tt0v, it follows that γ ∈ Γ andconsequently, c ≤ c∗∗. Next, we show that c∗ ≤ c. The manifold N separatesE into two components. As in the proof of Lemma 2.3, it follows that

〈J ′(v), v〉 =∫

RN

(|∇v|2 + V (x)f(v)f ′(v)v2) dx−∫

RN

h(x, f(v))f ′(v)vdx

≥ 14

∫RN

(|∇v|2 + V (x)f(v)f ′(v)v2) dx

−C(∫

RN

(|∇v|2 + V (x)f(v)f ′(v)v2) dx)N+q2

N+2

−C(∫

RN

(|∇v|2 + V (x)f(v)f ′(v)v2) dx) 2∗

2

.

Using (2.1), notice that if ‖v‖ → 0 then∫

R2(|∇v|2 +V (x)f2(v)) dx → 0, whichimplies that there exists a δ > 0 such that 〈J ′(v), v〉 > 0 when 0 < ‖v‖ < δ.This proves that the component including the origin also contains a small ballaround the origin. Moreover, J(v) ≥ 0 for all v in this component, because〈J ′(tv), v〉 > 0 for all 0 ≤ t ≤ t0. Thus, γ(0) = 0 and γ(1) are in differentcomponents, which shows that every path γ ∈ Γ has to cross N . Therefore,we must have c∗∗ ≤ c and consequently c∗ ≤ c. The lemma is proved. �

2.3. Estimates for the minimax Level

Proposition 2.3. There exits a ϑ ∈ E \ {0} such that maxt>0

J(tϑ) < 12N S

N/2.

Therefore, the mountain pass value c defined in (2.3) satisfies c < 12N S

N/2,

where S is the best constant for the embedding D1,2(RN ) ↪→ L2∗(RN ).

Proof. Consider the functional I : E → R given by

I(u) =12

∫RN

(1 + 2u2)|∇u|2 dx+12

∫RN

V (x)u2 dx−∫

RN

H(x, u) dx

where E := {u ∈ X :∫

RN |∇u2|2 dx < ∞}. Then, by Proposition 2.2 (3), itsuffices to show that there is 0 �= ϑ ∈ E such that

supt≥0

I(tϑ) <1

2NSN/2.

Indeed, since J(f−1(tϑ)) = −∞ as t → ∞, there exists a t0 �= 0 such thatJ(f−1(t0ϑ)) < 0. Set γ1(t) := f−1(tt0ϑ). It follows from the definition of themountain pass value that


c = infγ∈Γ

supt∈[0,1]

J(γ(t)) ≤ supt∈[0,1]

J(γ1(t)) = supt∈[0,1]

J(f−1(tt0ϑ))

≤ supt≥0

I(tϑ) <1

2NSN/2.

Let R > 0 be fixed and ϕ ∈ C∞0 (RN ) be a cut-off function with suppϕ(x) ⊂

B2R ⊂ Ω such that ϕ = 1 if x ∈ BR and ϕ(x) = 0 for x ∈ Bc2R; 0 ≤ ϕ(x) ≤ 1

for all x. It is known that the function

Uε =[N(N − 2)ε](N−2)/4

(ε+ |x|2)(N−2)/2, ∀ε > 0

satisfies the equation −Δu = |u|2∗−2u in RN . We now consider the following

function

wε := ϕ(x)U1/2ε =

ϕ(x)[N(N − 2)ε](N−2)/8

(ε+ |x|2)(N−2)/4, ∀ε > 0.

From the definition of wε, we have∫

BR

|∇w2ε |2 dx ≤

∫BR

|w2ε |2∗

dx,

∫Bc

R

|w2ε |2∗

dx = O(ε(N−2)/2), as ε → 0.

Using the similar arguments as that in Brezis and Nirenberg [6], if we letϑε = wε

||w2ε ||

122∗, then we have

||∇ϑ2ε||22 ≤ S +O(ε(N−2)/2).

On the other hand, limt→∞ I(tϑε) = −∞, which implies that there exists atε ≥ 0 such that sup

t≥0I(tϑε) = I(tεϑε). By (g2), we have

I(tϑε) ≤ t2

2

∫RN

(|∇ϑε|2 + ‖V ‖L∞(Ω)ϑ2ε) dx+

t4

4

∫RN

|∇ϑ2ε|2 dx

−λtq1

q1

∫RN

|ϑε|q1 dx− t2(2∗)

2(2∗),

then tε satisfies

0 = tε

∫RN

(|∇ϑε|2 + ‖V ‖L∞(Ω)ϑ2ε) dx+ t3ε

∫RN

|∇ϑ2ε|2 dx

−λtε∫

RN

|ϑε|q1 dx− t2(2∗)−1

ε . (2.7)

We claim that there is a positive constant d > 0 such that Cε := tq1ε /q1 ≥

d > 0,∀ε > 0. Otherwise, we could find a sequence εn → 0 such that tεn→ 0.

Up to a subsequence (still denote by εn), we have tεnϑεn

→ 0. Therefore,0 < c ≤ supt≥0 I(tεn

ϑεn) = I(0) = 0, which is a contradiction. The claim is

true. Next there are two cases to be considered: either tε ≤ 1 or tε > 1. For


the case tε ≤ 1, we have

t2ε2

∫RN

(|∇ϑε|2 + ‖V ‖L∞(Ω)ϑ2ε) dx− λtq1

ε

q1

∫RN

|ϑε|q1 dx

≤ 12

∫RN

(|∇ϑε|2 + ‖V ‖L∞(Ω)ϑ2ε) dx− λCε

∫RN

|ϑε|q1 dx.

Note the following inequality:

(a+ b)α ≤ aα + α(a+ b)α+1b, for all a, b > 0 and α ≥ 1.

Since the function t �→ t4∫

RN |∇ϑ2ε|2 dx/4 − t2(2

∗)/2(2∗) attains its maximumat t0 := (

∫RN |∇ϑ2

ε|2 dx)2(2∗)/(2(2∗−4)), we conclude that

I(tεϑε) ≤ t4ε4

∫RN

|∇ϑ2ε|2 dx− t

2(2∗)ε

2(2∗)+

12

∫RN

(|∇ϑε|2 + ‖V ‖L∞(Ω)ϑ2ε) dx

−λCε

∫RN

|ϑε|q1 dx

≤ 12N

SN/2 +O(ε(N−2)/2) +12

∫RN

(|∇ϑε|2 + ‖V ‖L∞(Ω)ϑ2ε) dx

−λCε

∫RN

|ϑε|q1 dx. (2.8)

For the case tε > 1, by (2.7), we have

tε ≤(∫

RN

(|∇ϑε|2 + ‖V ‖L∞(Ω)ϑ2ε) dx+

∫RN

|∇ϑ2ε|2 dx

)1/(2(2∗)−4)

≤(

2∫

RN

(|∇ϑε|2 + ‖V ‖L∞(Ω)ϑ2ε) dx+

∫RN

|∇ϑ2ε|2 dx

)1/(2(2∗)−4)

:= t1.

Since the function t → (t4t(2(2∗)−4)

1 )/4− t2(2∗)/2(2∗) is increasing on [0, t1), we

have,

I(tεϑε) ≤ t4ε4

∫RN

2(|∇ϑε|2 + ‖V ‖L∞(Ω)ϑ2ε) dx+

t4ε4

∫RN

|∇ϑ2ε|2 dx

−λtq1ε

q1

∫RN

|ϑε|q1 dx− t2(2∗)ε

2(2∗)

≤[t4εt

(2(2∗)−4)1

4− t

2(2∗)ε

2(2∗)

]− λCε

∫RN

|ϑε|q1 dx

≤ 12N

SN/2 +O(ε(N−2)/2) − λCε

∫RN

|ϑε|q1 dx. (2.9)

Therefore, combining (2.8) and (2.9), if can we show that for small ε > 0

O(ε(N−2)/2) + C

∫RN

(|∇ϑε|2 + ‖V ‖L∞(Ω)ϑ2ε) dx− λCε

∫RN

|ϑε|q1 dx < 0,(2.10)

then we have I(tεϑε) < 12N S

N/2, which concludes the proposition.


To show (2.10), we change the variables and obtain∫BR

(|∇ϑε|2 + ‖V ‖L∞(Ω)ϑ2ε) dx− λCε

∫BR

|ϑε|q1 dx

≤ CεN−2

4

∫BR

[ |x|2(ε+ |x|2)(N+2)/2

+1

(ε+ |x|2)(N−2)/2

]dx

−λCε (N−2)q18

∫BR

1(ε+ |x|2)(N−2)q1/4

dx

≤ CNωNεN−2

4 [1 + ln(Rε−1/2)]

−λCNωNεN2 − (N−2)q1

8

∫ R/ε1/2

0

rN−1

(1 + r2)(N−2)q1/4dr.

Furthermore, we also have∫B2R\BR

(|∇ϑε|2 + ‖V ‖L∞(Ω)ϑ2ε) dx− λCε

∫B2R\BR

|ϑε|q1 dx ≤ CεN−2

4 .

Therefore, we choose some proper σ > 0, whose value is to be determinedlater, such that

limε→0

[O(ε(N−2)/2) + C

∫RN

(|∇ϑε|2 + ‖V ‖L∞(Ω)ϑ2ε) dx− λCε

∫RN

|ϑε|q1 dx

]

≤ limε→0

ε(N−2)/4−σ

{Cε(N−2)/4+σ + Cεσ[1 + ln(Rε−1/2)]

−λCεNωNεN/2−(N−2)q1/8−(N−2)/4+σ

∫ R/ε1/2

0

rN−1

(1 + r2)(N−2)q1/4dr

}.

For any σ > 0, we have limε→0

εσ ln(Rε−1/2) = 0. Since σ1 := N/2−(N−2)q1/8−(N − 2)/4 < 0, choose σ = −σ1/2, then we get (2.10). �

Lemma 2.7. Any (PS)c sequence {vn} of J is bounded in E.

Proof. Note that

J(vn) =12

∫RN

|∇vn|2 dx+12

∫RN

V (x)f2(vn) dx−∫

RN

H(x, f(vn)) dx → c,

and for any ω ∈ E,

〈J ′(vn), ω〉 =∫

RN

∇vn∇ω dx+∫

RN

V (x)f ′(vn)f(vn)ω dx

−∫

RN

h(x, f(vn))f ′(vn)ω dx → 0.

We choose ω = ωn = f(vn)/f ′(vn), we can deduce that

|∇ω| =(

1 +2f2(vn)

1 + 2f2(vn)

)|∇vn|.


By (h3), we obtain,

J(vn) − 1θ〈J ′(vn), ωn〉 =

∫RN

[12

− 1θ

(1 +

2f2(vn)1 + 2f2(vn)

)]|∇vn|2 dx

+(

12

− 1θ

) ∫RN

V (x)f2(vn) dx

+1θ

∫RN

[h(x, f(vn))f(vn) − θH(x, f(vn))] dx

≥ θ−42θ

∫RN

[|∇vn|2+

(1 − 2(θ − 2)

k(θ − 4)

)V (x)f2(vn)

]dx,

which implies that∫

RN (|∇vn|2 +V (x)f2(vn)) dx ≤ C. We deduce that {vn} isbounded. �

Lemma 2.8. There is a sequence {zn} ⊂ RN and R > 0, β > 0 such that∫

BR(zn)

f2(vn) dx ≥ β.

Proof. Let {vn} be a (PS)c sequence. Suppose by a contradiction that thelemma does not hold. Then by Lemma 2.2 it follows that∫

RN

|f(vn)|q dx = o(1), 2 < q < 2(2∗),

and then by (g1) and (g2),∫RN

G(f(vn)) dx =∫

RN

g(f(vn))f(vn) dx = o(1).

By the definition of h, we have∫RN

H(x, f(vn)) dx ≤ 12(2∗)

∫Ω∪{vn≤f−1(a)}

f2(2∗)(vn) dx

+α

2k

∫Ωc∩{vn>f−1(a)}

f2(vn) dx+ o(1)

and ∫RN

h(x, f(vn))f(vn) dx =∫

Ω∪{vn≤f−1(a)}f2(2∗)(vn) dx

+α

k


f2(vn) dx+ o(1).

Then since 〈J ′(vn), f(vn)/f ′(vn)〉 = o(1), we have∫

RN

[1 +

2f2(vn)1 + 2f2(vn)

]|∇vn|2 dx+

∫RN

V (x)f2(vn) dx

−α

k


f2(vn) dx =∫

Ω∪{vn≤f−1(a)}f2(2∗)(vn) dx+ o(1).


Let l ≥ 0 be such that∫

RN

(1 +

2f2(vn)1 + 2f2(vn)

)|∇vn|2 dx+

∫RN

V (x)f2(vn) dx

−α

k


f2(vn) dx

→ l.

Then l > 0. Otherwise, we have vn → 0 which contradicts c > 0. So,∫

Ω∪{vn≤f−1(a)}f2(2∗)(vn) dx → l.

Since∫

RN

V (x)f2(vn) dx− α

k


f2(vn) dx ≥ 0,

by the definition of the best constant S,

S ≤∫

RN |∇f2(vn)|2 dx(∫RN |f2(vn)|2∗ dx

)2/2∗

≤∫

RN

(1 + 2f2(vn)

1+2f2(vn)

)|∇vn|2 dx+

∫RN V (x)f2(vn) dx

(∫Ω∪{vn≤f−1(a)} f

2(2∗)(vn) dx)2/2∗

−αk

∫Ωc∩{vn>f−1(a)} f

2(vn) dx

(∫Ω∪{vn≤f−1(a)} f

2(2∗)(vn) dx)2/2∗

→ l1−2/2∗, as n → ∞.

Therefore, l ≥ SN2 . It follows that

c = limn→∞

[12

∫RN

|∇vn|2 dx+12

∫RN

V (x)f2(vn) dx−∫

RN

H(x, f(vn)) dx]

≥ limn→∞

[14

∫RN

(1 +

2f2(vn)1 + 2f2(vn)

)|∇vn|2 dx+

14

∫RN

V (x)f2(vn) dx

− α

4k


f2(vn) dx− 12(2∗)

∫Ω∪{vn≤f−1(a)}

(f(vn))2(2∗) dx

]

→(

14

− 12(2∗)

)l

≥ 12N

SN2 ,

which is a contradiction since c < 12N S

N2 . This completes the proof of the

lemma. �


Lemma 2.9. The sequence {zn} is bounded in RN .

Proof. For each R > 0 we set ψR ∈ C∞0 (R, [0, 1]) such that ψR(x) = 0 if |x| ≤

R,ψR(x) = 1 if |x| ≥ 2R and |∇ψ′(x)| ≤ CR−1. Since 〈J ′(vn), ψnvn〉 = o(1),we have,∫

RN

|∇vn|2ψR dx+∫

RN

∇vn∇ψRvndx+∫

RN

V (x)f(vn)f ′(vn)ψRvn dx

−∫

RN

h(x, f(vn))f ′(vn)vnψR dx = o(1).

Taking R large enough such that Ω ⊂ BR(0), by (g3), we have,∫RN

|∇vn|2ψR dx+∫

RN

∇vn∇ψRvn dx+∫

RN

V (x)f(vn)f ′(vn)ψRvn dx

≤ 1k

∫Ωc∪{vn≤f−1(a)}

V (x)f ′(vn)vnψR dx+ o(1).

Then, by (f6), we have(12

− 12k

)α

∫RN

f2(vn)ψR dx

≤∫

RN

|∇vn|2ψR dx+(

1 − 1k

)∫RN

V (x)f(vn)f ′(vn)vnψR dx

= −∫

RN

∇vn∇ψRvn dx+ o(1)

≤ C

R‖vn‖2‖∇vn‖2 + o(1),

which implies the result. �

Lemma 2.10. If vn ⇀ v in E, then v is a nontrivial critical point of J.

Proof. Arguing as in [41], let ϕ ∈ C∞0 (RN ,R) with Q := supp(ϕ). Since E ↪→

H1(RN ) is continuous, we see that∫RN

∇vn∇ϕdx →∫

RN

∇v∇ϕdx.

Besides, we also have vn → v in Lploc(R

N ) for p ∈ [2, 2∗). Then, vn → v a.e.on Q as n → ∞ and |vn(x)| ≤ |ωp(x)| for every n ∈ N and a.e. on Q withwp(x) ∈ Lp(RN ) ( see e.g. Lemma A.1, [44]). Therefore,

V (x)f(vn)f ′(vn)ϕ → V (x)f(v)f ′(v)ϕ, a.e. on Q, as n → ∞,

f2(2∗)−1(vn)f ′(vn)ϕ → f2(2∗)−1(v)f ′(v)ϕ, a.e. on Q, as n → ∞,

g(f(vn))f ′(vn)ϕ → g(f(v))f ′(v)ϕ, a.e. on Q, as n → ∞.

Since

|V (x)f(vn)f ′(vn)ϕ| ≤ |V (x)f(vn)ϕ| ≤ supQV (x)|ω2||ϕ|,


the Lebesgue Dominated Convergence Theorem implies that∫RN

V (x)f(vn)f ′(vn)ϕdx →∫

RN

V (x)f(v)f ′(v)ϕdx.

Similarly, since |f2(2∗)−1(vn)f ′(vn)ϕ| ≤ C|ω2∗−1|2∗−1|ϕ|, we have∫RN

f2(2∗)−1(vn)f ′(vn)ϕdx →∫

RN

f2(2∗)−1(v)f ′(v)ϕdx.

For |vn| ≤ 1, by (g1) and (g2), we have

|g(f(vn))f ′(vn)ϕ| ≤ ε|f(vn)||ϕ| + Cε|f(vn)|q2−1|ϕ|≤ c(ε+ Cε)|ϕ|, (2.11)

and for |vn| > 1,

|g(f(vn))f ′(vn)ϕ| ≤ ε|f(vn)||ϕ| + Cε|f(vn)|q2−1|f ′(vn)||ϕ|≤ ε|f(vn)||ϕ| + Cε|f(vn)|q2−1 |f(vn)|

|vn| |ϕ|

≤ ε|ω2||ϕ| + Cε|ω2∗−1|2∗−1|ϕ|. (2.12)

Combining (2.11) and (2.12), the Lebesgue Dominated Convergence Theoremimplies ∫

RN

g(f(vn))f ′(vn)ϕdx →∫

RN

g(f(v))f ′(v)ϕdx.

Hence, v is a critical point of J because C∞0 (RN ) is dense in E. From Lem-

mas 2.8 and 2.9, v is nontrivial. �Lemma 2.11. Any critical point of J is of class C2,α

loc (RN ).

Proof. Let v be a critical point of J . Then

−Δv = w, x ∈ RN .

where w := f ′(v)[h(x, f(v))f(v) − V (x)f(v)]. By (h1), (h2) and |f(t)| ≤21/4|t|1/2, |f(t)f ′(t)| ≤ 1/

√2 for all t ∈ R, we have

|w| ≤ f ′(v)(C1 + C2|f(v)|2(2∗)−1) ≤ C3 + C4|v|(2(2∗)−2)/2

in any ball BR. Setting p0 := 2(2∗)/(2(2∗) − 2) > 1. As v ∈ L2∗(RN ), we

conclude that w ∈ Lp0(BR). By the elliptic regularity theory, w ∈ W 2,p0(BR).Using a standard bootstrap argument, we have v ∈ W 2,p(BR) for all p ≥ 2(see [23]). Hence, v ∈ C1,1

loc (RN ) and this implies that w is locally Holder con-tinuous. Consequently, v ∈ C2,α

loc (RN ) for some α ∈ (0, 1). �Up to now, we have the following result:

Proposition 2.4. For all ε > 0, there is a positive critical point vε ∈ Eassociated to the functional

Jε(v) =12

∫RN

(ε2|∇v|2 + V (x)f2(v)) dx−∫

RN

H(x, f(v)) dx

at the level cε = infv∈E\{0}

maxt≥0

Jε(tv) > 0.


Proof. Let {vn} be a (PS)c sequence such that vn ⇀ vε in E. By the semi-continuity of norm, we have∫

RN

|∇vε|2 dx ≤ limn→∞

∫RN

|∇vn|2 dx. (2.13)

We show only the equality holds in (2.13). Otherwise, we have

cε ≤ Jε(vε) − 1θJ ′

ε(vε)vε =(

12

− 1θ

)∫RN

|∇vε|2 dx

=∫

RN

[12V (x)f2(vε) − 1

θV (x)f ′(vε)f(vε)vε

]dx

+∫

RN

[1θh(x, f(vε))f ′(vε)vε −H(x, f(vε))

]dx

< limn→∞

(12

− 1θ

) ∫RN

|∇vn|2 dx

+ limn→∞

∫RN

[12V (x)f2(vn) − 1

θV (x)f ′(vn)f(vn)vn

]dx

+ limn→∞

∫RN

[1θh(x, f(vn))f ′(vn)vn −H(x, f(vn))

]dx

= limn→∞

[Jε(vn) − 1

2J ′

ε(vn)vn

]= cε,

which is a contradiction. Therefore, we have

limn→∞

∫RN

|∇vn|2 dx =∫

RN

|∇vε|2 dx. (2.14)

By a similar argument as Proposition 4.2 [18], we have

limn→∞

∫RN

V (x)f2(vn) dx =∫

RN

V (x)f2(vε) dx.

This together (2.14) and (4) in Proposition 2.4 [18] implies that vn → v in E.We complete the proof. �

3. Proof of Theorem 1.1

We suppose that ∂Ω is smooth and 0 ∈ Ω. Furthermore, without loss of gen-erality, we assume

V (0) = V0 := infΩV (x).

Denote by J0 : E → R the functional given by

J0(v) =12

∫RN

(|∇vn|2 + V0f2(v)) dx−

∫RN

(G(f(v)) +

12(2∗)

f2(2∗)(v))dx,

associated to the problem

− Δv = f ′(v)(g(f(v)) + f2(2∗)−1(v) − V0f(v)), x ∈ RN . (3.1)


Under the conditions (g1) − (g4), similar to the arguments of [41], (3.1) pos-sesses a nontrivial solution ω at the level

c0 = infv∈H1(RN )\{0}

maxt≥0

J0(tv).

Moreover,

0 < c0 <1

2NSN/2.

Let Jε : E → R denote the functional

Jε(v) =12

∫RN

(|∇v|2 + V (εx)f2(v)) dx−∫

RN

H(εx, f(v)) dx

associated to the equation

− Δv = f ′(v) (h(εx, f(v)) − V (εx)f(v)) , x ∈ RN , (3.2)

and defined in the space

Eε ={v ∈ H1(RN ) :

∫RN

V (εx)f2(v) dx < ∞}

equipped with the norm

||v||ε = ||∇v||2 + infξ>0

1ξ

[1 +

∫RN

V (εx)f2(ξv) dx].

Let vε(x) = vε(z), z = εx, be a critical point of Jε at the level

bε = infv∈Eε\{0}

maxt≥0

Jε(tv).

It is easy to check that bε = ε−Ncε. Furthermore, from Proposition 2.3, foreach ε > 0, we have bε < 1

2N SN/2.


Lemma 3.1. limε→0

bε ≤ c0.

Proof. We consider the function ωε = φ(εx)ω(x), where φ ∈ C∞0 (RN , [0, 1]) is

defined by

φ(x) ={

1 if x ∈ Bρ,0 f x ∈ R

N \B2ρ,

for some ρ > 0. We will assume that B2ρ ⊂ Ω. It is easy to see that ωε → ω inH1(RN ) as ε → 0. Furthermore, supp(ωε) ⊂ Ωε := {x ∈ R

N : εx ∈ Ω} and∫RN

V (εx)f2(ωε) dx =∫

supp(ωε)

V (εx)f2(ωε) dx

≤∫

Ωε

V (εx)f2(ωε) dx ≤ supΩV

∫Ωε

ω2ε dx

≤ C supΩV,

which implies that ωε ∈ Eε. For each ε > 0, let tε satisfies maxt≥0

Jε(tωε) =

Jε(tεωε). Then

bε ≤ maxt≥0

Jε(tωε) = Jε(tεωε)

≤ t2ε2

∫RN

(|∇ωε|2 + V (εx)ω2ε) dx−

∫RN

H(εx, f(tεωε)) dx

=t2ε2

∫RN

(|∇ωε|2 + V (εx)ω2ε) dx

−∫

RN

(G(f(tεωε)) +

12(2∗)

f2(2∗)(tεωε))dx. (3.3)

We claim that tε → 1 as ε → 0. First, we show that tε is bounded. Otherwise,since 〈J ′

ε(tεωε), tεωε〉 = 0, we have∫RN

t2ε|∇ωε|2 dx+∫

RN

V (εx)f(tεωε)f ′(tεωε)tεωε dx

=∫

RN

(g(f(tεvε))f ′(tεωε)tεωε + f2(2∗)−1(tεωε)f ′(tεωε)tεωε

)dx. (3.4)

Then by (3.4),∫

RN

(|∇ωε|2 + V (εx)ω2ε) dx ≥ 1

2

∫RN

f2(2∗)(tεωε)t2ε

dx

≥ 12

[f(tεωε(x0))tεωε(x0)

]2

f2(2∗)−2(tεωε(x0))ω2ε(x0)|Ω|

→ ∞,

where ωε(x0) = minΩωε(x) > 0, which is a contradiction. Thus, up to a

subsequence, we assume that tε → t1 ≥ 0. Moreover, by (3.3),we see thatt2ε ≥ 2bε/C. Furthermore, we may assume, up to a sequence of ε → 0, that


bε ≥ c0. (otherwise, we are done). Therefore, t2ε ≥ 2bε/C ≥ 2c0/C > 0 andthis implies that t1 > 0. Passing to the limit in (3.4), we obtain

t21

∫RN

|∇ω|2 dx+ V0

∫RN

f(t1ω)f ′(t1ω)t1ω dx

=∫

RN

f2(2∗)−1(t1ω)f ′(t1ω)t1ω dx+∫

RN

g(f(t1ω))f ′(t1ω)t1ω dx.

On the other hand, since∫RN

|∇ω|2 dx+ V0

∫RN

f(ω)f ′(ω)ω dx

=∫

RN

f2(2∗)−1(ω)f ′(ω)ω dx+∫

RN

g(f(ω))f ′(ω)ω dx,

then

V0

∫RN

[f(t1ω)f ′(t1ω)

t1ω− f(ω)f ′(ω)

ω

]ω2 dx

=∫

RN

[f2(2∗)−1(t1ω)f ′(t1ω)

t1ω− f2(2∗)−1(ω)f ′(ω)

ω

]ω2 dx

+∫

RN

[g(f(t1ω))f ′(t1ω)

t1ω− g(f(ω))f ′(ω)

ω

]ω2 dx.

By (g4) and Lemma 2.1, it is easy to see that

L1(s) =f2(2∗)−1(s)f ′(s) − g(f(s))f ′(s)

sand L2(s) =

f(s)f ′(s)s

are increasing and decreasing for s > 0 respectively. Hence, we must havet1 = 1. Moreover,

Jε(tεωε) = J0(tεωε) +12

∫RN

(V (εx) − V0)f2(tεωε) dx.

Taking the limit as ε → 0 and using the definition of ωε and the LebesgueDominated Convergence Theorem, we conclude that∫

RN

(V (εx) − V0)f2(tεωε) dx → 0 as ε → 0

and therefore lim supε→0

bε ≤ lim supε→0

J(tεωε) = J(ω) = c0. �

Using similar arguments as that of Lemma 2.8, we have the followingresult.

Lemma 3.2. There are ε0 > 0, a family {yε} ⊂ RN and R > 0, β > 0 such that

∫BR(yε)

f2(vε) dx ≥ β for 0 < ε ≤ ε0.


Lemma 3.3. {εyε} is bounded in RN . Moreover, dist(εyε,Ω) ≤ εR.

Proof. We modify the arguments in [18]. For δ > 0, we define Kδ :={x ∈ R

N : dist(x,Ω) ≤ δ}. We consider the function φε = φ(εx), whereφ ∈ C∞(RN , [0, 1]) is defined by

φ(x) ={

1 if x �∈ Kδ,0 if x ∈ Ω,

and |∇φ| ≤ Cεδ−1. Since 〈J ′ε(vε), φεvε〉 = o(1), using (g3) and the fact that

the support of φε does not intercept Ωε, we obtain

α

(12

− 12k

)∫RN

f2(vε)φε dx ≤∫

RN

[|∇vε|2 +

(V (εx) − α

2k

)f2(vε)

]φε dx

= −∫

RN

vε∇vε∇φε dx

≤ Cδ−1ε.

If for some sequence εn → 0 we have

BR(yεn) ∩ {x ∈ R

N : εnx ∈ Kδ} = ∅, (3.5)

then∫

BR(yεn )f2(vεn

) dx ≤ Cδ−1εn, which contradicts Lemma 3.2. Thus (3.5)does not hold, that is, for all ε there is an x such that εx ∈ Kδ and |x−yε| ≤ R.It is easy to verify that this implies dist(εyε,Ω) ≤ εR + δ. From this fact weconclude the proof. �

Remark 3.1. From Lemma 3.3 we can suppose that the family {yε}, definedin Lemma 3.2, can be taken in such a way that εyε ∈ Ω for all ε ∈ (0, ε0].Indeed, if not, we replace yε by ε−1xε, where xε comes from Lemma 3.3, sothat |εyε − xε| ≤ εR. Observing that |yε − xε/ε| ≤ R, we can replace R by 2Rin Lemma 3.2.

Lemma 3.4. limε→0

V (εyε) = V0. Moreover, ωε(x) = vε(x + yε) converges uni-

formly to a nontrivial solution of problem (3.1) over compact subsets of RN .

Proof. If yn := yεnare such that εnyn → x0 as εn → 0, we must prove that

V (x0) = V0. By Lemma 3.3, we know that x0 ∈ Ω, that is, V (x0) ≥ V0. Let usset vn(x) := vεn

(x), ωn(x) := vεn(x+ yn). Then

− Δωn =f ′(ωn) (h(εnx+εnyn, f(ωn))−V (εnx+εnyn)f(ωn)) , x ∈ RN . (3.6)

On the other hand, since ‖ωn‖H1 = ‖vn‖H1 is bounded, let ω ∈ H1(RN )such that ωn ⇀ ω in H1(RN ). By Lemma 3.2, ω ≥ 0, ω �= 0. Denote χ(x) =lim

n→∞χΩ(εn +x+εnyn) a.e. in RN , h(x, f(ω)) = χ(x)(f2(2∗)−1(ω)+g(f(ω)))+

(1 − χ(x))g(f(ω)). By (3.6), we have∫RN

(∇ωn∇φ+ V (εnx+ εnyn)f(ωn)f ′(ωn)φ) dx

=∫

RN

h(εnx+ εnyn, f(ωn))f ′(ωn)φdx, (3.7)


for all φ ∈ C∞0 (RN ). Since ‖vn‖∞ ≤ C for all n (see e. g. [18]), by the Lebesgue

Dominated Convergence Theorem we obtain

limn→∞

∫RN

h(εnx+ εnyn, f(ωn))f ′(ωn)φdx =∫

RN

h(x, f(ω))f ′(ω)φdx,

for all φ ∈ C∞0 (RN ). Taking the limit in (3.7) we see that ω satisfies

∫RN

(∇ω∇φ+ V (x0)f(ω)f ′(ω)φ) dx =∫

RN

h(x, f(ω))f ′(ω)φdx,

for all φ ∈ C∞0 (RN ). Therefore, ω is a critical point of the functional

F(v) =12

∫RN

(|∇v|2 + V (x0)f2(v)) dx−∫

RN

H(x, f(v)) dx,

where H(s) =∫ s

0h(x, t)dt. If x0 ∈ Ω, we have εnx + εnyn ∈ Ω as n → ∞.

Hence, χ(x) = 1 for all x ∈ RN and so ω is a critical point of the following

functional

Jx0(v)=12

∫RN

(|∇v|2 + V (x0)f2(v)) dx− 12(2∗)

∫RN

f2(2∗)(v) dx−∫

RN

G(f(v)) dx.

Denoting by cx0 the mountain pass level associated to the functional Jx0 andby c the mountain pass level associated to the functional F . Since H(x, s) ≤G(s) + s2(2

∗)−1 for all x ∈ RN and s ∈ R, we obtain Jx0(v) ≤ F(v) for all v ∈

H1(RN ) and this implies that cx0 ≤ c. Set An := {x ∈ RN : εnx+ εnyn ∈ Ω}and

Ψn(x) =θ

2V (εnx+ εnyn)f2(ωn(x)) − V (εnx+ εnyn)f(ωn(x))f ′(ωn(x))ωn(x)

+h(εnx+ εnyn, f(ωn(x)))f ′(ωn(x))ωn(x)−θH(εnx+ εnyn, f(ωn(x))).

If x ∈ An, using (h3) and (g3) we have

Ψn(x) ≥(θ

2− 1

)V (εnx+ εnyn)f2(ωn(x))

+12

(g(f(ωn(x)) − 2θG(f(ωn(x)))) ≥ 0.

If x �∈ An,

Ψn(x) ≥(θ

2− 1 − θ

2k

)V (εnx+ εnyn)f2(ωn(x)) ≥ 0.

Since c0 ≤ cx0 ≤ c ≤ F(ω) in RN and Ψn → Ψ a.e., where

Ψ(x) :=θ

2V (x0)f2(ω) − V (x0)f(ω)f ′(ω)ω + h(x, f(ω))f ′(ω)ω − θH(x, f(ω)).


By the Fatou’s Lemma and semicontinuity of norm, we obtain

θc0 ≤ θF(ω) = θF(ω) − 〈F ′(ω), ω〉=

(θ

2− 1

) ∫RN

|∇ω|2 dx+∫

RN

Ψ(x) dx

≤ lim infn→∞

(θ

2− 1

) ∫RN

|∇ωn|2 dx+ lim infn→∞

∫RN

Ψn(x) dx

= lim infn→∞

(θ

2− 1

) ∫RN

|∇ωn|2 dx+ lim infn→∞

∫RN

(θ

2V (εnx)f2(vn)

−V (εnx)f(vn)f ′(vn)vn + h(εnx, f(vn)f ′(vn)vn

−θH(εnx, f(vn)))) dx= lim inf

n→∞ (θJεn(vn) − 〈J ′

εn(vn), vn〉) = θ lim inf

n→∞ bε ≤ θc0.

Thus, F(ω) = c0 and limε→0

bε = c0. Moreover, if V (x0) > V0, by the fact thatthe dependence of the mountain pass value c on the constant potential V0 iscontinuous and increasing [34], we have c0 < cx0 = c ≤ F(ω) = c0, whichis a contradiction. Therefore, V (x0) = V0 and this implies that x0 ∈ Ω andF = Jx0 = J . Hence, ω is a nontrivial solution of (3.1). Furthermore,

−Δ(ωn − ω) = Hn, x ∈ RN ,

where

Hn(x) = V0f(ω(x))f ′(ω(x)) − V (εnx+ εnyn)f(ω(x))f ′(ω(x))+h(εnx+ εnyn, f(ω(x)))f ′(ω(x)) − g(f(ω(x)))f ′(ω(x))

−f2(2∗)−1)(ω(x))f ′(ω(x)).

As ωn → ω a.e. in RN , this implies that Hn → 0 a.e. in R

N . Notice thatfor each compact subset D in R

N , we have |Hn|, |ω| ≤ C since ‖ωn‖∞ ≤ Cand |εnx + εnyn| ≤ C for all n and x ∈ D. Thus the Lebesgue DominatedConvergence Theorem implies that Hn → 0 in Ls

loc for all s ≥ 1. By ellipticregularity theory [23], ωn → ω in C2

loc(RN ) and the lemma is proved. �

Using an similar argument as in Lemma 5.8 of [18], we have,

lim|x|→∞

vε(x) = 0. (3.8)

Thus, there exists an R > 0 such that vε(x) < a for all |x| ≥ R. Choosingε0 > 0 sufficiently small such that BR ⊂ Ωε0 , since Ωε := {x ∈ R

N : εx ∈ Ω}and Ω is bounded, if ε small enough, we deduce that Ωε large enough. So,BR ⊂ Ωε can hold. We conclude that for all ε ∈ (0, ε0),

−Δvε + V (εx)f(vε)f ′(vε) = h(vε)f ′(vε), x ∈ RN .

Thus,

−ε2vε + V (x)f(vε)f ′(vε) = h(vε)f ′(vε), x ∈ RN ,

which implies that uε = f(vε) is a positive solution of (1.4) for all ε ∈ (0, ε0).Now, we show the concentration behavior of the solutions. By (3.8),

ωε possesses a global maximum point xε ∈ Bρ for all ε ∈ (0, ε0) and some


ρ > 0. Considering the translation ωε(x) = ωε(x + xε), we may assume thatωε achieves its global maximum at the origin of R

N . Using the fact that ωis spherically symmetric, ∂ω/∂r < 0 for all r > 0 and ωε → ω in C2

loc(RN ).

By Lemma 4.2 in [34] we can conclude that ωε possesses no critical pointother than the origin for all ε ∈ (0, ε0). Notice that the maximum value ofvε(z) = vε(εx) = vε(x) = ωε(x− yε) is achieved at the point zε = εyε ∈ Ω. Asthe function f is strictly increasing, the maximum value of uε(z) = f(vε(z)) isalso achieved in this point. As ∇uε = f ′(vε)∇vε, uε possesses no critical pointother zε. Finally, we show the exponential decay of solution. Since ωε(x) → 0as |x| → +∞ and f(t)/t → 1 as t → 0, we have

lim|x|→∞

f(ωε(x))f ′(ωε(x))ωε(x)

= 1,

and

lim|x|→∞

(g(f(ωε(x)))f(ωε(x)) + f2(2∗)−1(ωε(x)))f ′(ωε(x))ωε(x)

= 0.

Then, we can choose R0 > 0 such that for all ε ∈ (0, ε0] and for all |x| ≥ R0,

f(ωε(x))f ′(ωε(x)) ≥ 34ωε(x)

and

(g(f(ωε(x)))f(ωε(x)) + f2(2∗)−1(ωε(x)))f ′(ωε(x)) ≤ V0

2ωε(x).

Next, we shall apply the maximum principle (see [38]) to derive the propertiesof exponential decay of the bound state solution. Similar arguments have beenused in [1,18]. We define ψ(x) = M exp(−ξ|x|), where ξ and M are such that4ξ2 < V0 and

M exp(−ξ|x|) ≤ ξ2ψ for all |x| = R0. (3.9)

It is easy to verify that

Δψ ≤ ξ2ψ for all x �= 0. (3.10)

Define ψε = ψ − ωε, then using (3.9), (3.10) and

−Δωε+V (εx+ εyε)f(ωε)f ′(ωε)=g(f(ωε))f(ωε)f ′(ωε)+(f(ωε))2(2∗)−1f ′(ωε)

we obtain,

−Δψε +V0

4ψε ≥ 0 in |x| ≥ R0,

ψε ≥ 0 in |x| = R0,

lim|x|→∞

ψε(x) = 0.

Then, the maximum principle implies that ψε ≥ 0 for all |x| ≥ R0. Hence,ψε(x) ≤ M exp(−ξ|x|) for |x| ≥ R0 and ε ∈ (0, ε0]. We then conclude that


uε(z) = f(vε(z)) ≤ vε(z) = v(zε

)= ωε

(z − zε

ε

)≤ C exp

(−ξ

∣∣∣∣z − zε

ε

∣∣∣∣).

We have complete the proof of Theorem 1.1. �

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Youjun Wang and Wenming ZouDepartment of Mathematical SciencesTsinghua UniversityBeijing 100084Chinae-mail: [email protected]

Wenming Zoue-mail: [email protected]

Received: 2 October 2010.

Accepted: 5 May 2011.

bound states to critical quasilinear schrödinger equations

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