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Least energy solutions for indefinite biharmonic
problems via modified Nehari-Pankov manifold
M IAOMIAO N IU∗, Z HONGWEI TANG† and L USHUN WANG‡
School of Mathematical Sciences,
Beijing Normal University, Beijing, 100875, P.R. of China
Abstract
In this paper, by using a modified Nehari-Pankov manifold, we prove the existence and
the asymptotic behavior of least energy solutions for the following indefinite biharmonic
equation:
∆2 u + (λV (x) − δ(x))u = |u|p−2 u in RN ,
(Pλ )
where N ≥ 5, 2 < p ≤ N2N
−4 , λ > 0 is a parameter, V (x) is a nonnegative potential
function with nonempty zero set intV −1 (0), δ(x) is a positive function such that the operator
∆2 + λV (x) − δ(x) is indefinite and non-degenerate for λ large. We show that both in
subcritical and critical cases, equation (Pλ ) admits a least energy solution which for λ > 0
localized near the zero set intV −1 (0).
Keywords: Least energy solutions; modified Nehari-Pankov manifold; biharmonic equations;
indefinite potential.
AMS Subject Classification: 35Q55, 35J655
1
Introduction and main results
In the present paper, we are considering the following biharmonic equation:
(
∆2 u + (λV (x) − δ(x))u = |u|p−2 u in RN ,
u ∈ H 2 (RN ),
(1.1)
. We are interested in the case that the operator
where N ≥ 5, λ > 0, 2 < p ≤ 2∗∗ , 2∗∗ := N2N
−4
2
∆ + λV (x) − δ(x) is indefinite and non-degenerate for λ large, we study the existence and the
∗
Email:[email protected]
Corresponding author:[email protected]; The author is also supported by NSFC(11571040)
‡
Email:[email protected]
†
1
asymptotic behavior of least energy solutions to problem (1.1) both in subcritical and critical
cases.
As mathematical model, biharmonic equations can be used to describe some phenomenas appeared in physics and engineering, such as, the problems of nonlinear oscillation in a suspension
bridge (see Lazer and McKenna [23], McKenna and Walter [26]) and the problems of the static
deflection of an elastic plate in a fluid (see Abrahams and Davis [1]). More precisely, when we
consider the compatibility equations of elastic mechanics under small deviation of the thin plates,
or the Von Karma system describing the mechanic behaviors under large deviation of thin plates,
we are forced to study a class of higher order equations or systems with biharmonic operator ∆2 .
Mathematically, the biharmonic operator is closely related to Paneitz operator, which has been
found considerable interests because of its geometry roots.
Recently, Ghergu and Taliaferro [13] proved the nonexistence of positive super-solutions to
some nonlinear biharmonic equations. Although the results they obtained can be seen as an
extension of Armstrong and Sirakov [3] from Laplacian equations to Biharmonic equations, the
methods in [13] and [3] are different. The results in [3] are mainly based on a method which
depends only on properties related to the maximum principle, while the results in [13] are due to
a new representation formula and an a priori point-wise bound of nonnegative super-solutions of
bi-harmonic equations.
We also refer the readers to the paper by Alves and Nóbrega( see [2] ), where the authors
considered the following problem
2
∆ u = f (u), in Ω,
(1.2)
u = Bu = 0, on ∂Ω
and Ω is a smooth bounded domain in RN with N ≥ 1, f is a C 1 function with subcritical
growth. They obtained the existence of nodal solutions for problem (1.2) in the cases Bu = ∆u
(Dirichlet boundary condition) with the unit outer
(Navier boundary condition) and Bu = ∂u
∂ν
norm ν.
There are also some other investigations for the biharmonic problems, for example in the
work of Liu and Chen [20], they obtained the existence of ground state solutions for a class
of biharmonic equation involving critical exponent. In [19], Karachik, Sadybekov and Torebok
proved the uniqueness of solutions to boundary value problems for the biharmonic eequation in
a ball. In [22], Luo proved the uniqueness of the weak extremal solution to biharmonic equation
with logarith mically convex nonlinearities.
We also want to introduce the works by Guo and Wei in [16] and [17], where the authors
firstly discussed the Liouville type results and regularity of the extremal solutions of biharmonic
equation with negative exponents. They also obtained some qualitative properties of entire radial
solutions for a biharmonic equation with supercritical nonlinearity.
For more results related to biharmonic problems, please see [8, 9, 12, 15, 18, 27, 29, 31, 34] and
the references therein.
The study for the Schrödinger equations involving Laplacian with indefinite potentials, we
firstly refer to the paper by Y. Ding and J. Wei [10]. In that paper, the authors considered the
2
following problem
−∆u(x) + λV (x)u(x) = λ|u(x)|p−2 u(x) + λg(x, u),
u(x) → 0
x ∈ RN ,
as |x| → ∞,
(1.3)
where V (x) can be negative in some domains in RN and g(x, u) is a perturbation term. By using
variational methods, the authors proved that there exists Λ > 0 such that for λ > Λ, (1.3) admits
at least one nontrivial solution both for subcritical case and critical case,.
For the indefinite potentials involving Laplacian, we also refer to the work by A. Szulkin and T.
Weth [30], where the authors gave a new minimax characterization of the corresponding critical
value and hence reduced the indefinite problem to a definite one. They also presented a precise
description to the Nehari-Pankov manifold which is useful even for other problems. For the
study of indefinite potential Schrödinger equations, we also refer the readers to T. Bartsch and
the second author [5], where the multi-bump solutions was considered.
More recently, the second author and the third author together with the other coauthor(see Y.
Guo, Z. Tang and L. Wang [14]) considered the existence and the asymptotic behavior of least
energy solutions to problem (1.1) in the case when the operator ∆2 + λV (x) − δ(x) is positively
definite.
The aim of the present paper is to study the existence and the asymptotic behavior of least
energy solutions to (1.1) in the indefinite case. More precisely, we assume that V (x) and δ(x)
satisfy the following conditions:
(V1 ) V (x) ∈ C(RN , R) satisfies V (x) ≥ 0 and 0 < V∞ := lim inf V (x) < +∞;
|x|→∞
(V2 ) Ω := int V −1 (0) is a non-empty bounded domain in RN with smooth boundary and Ω =
V −1 (0);
(V3 ) The operator ∆2 − δ(x) defined in H 2 (Ω) ∩ H01 (Ω) is indefinite and non-degenerate, that
is µk < δ(x) < µk+1 for some k ≥ 1. {µi } is the class of all eigenvalues of the operator
∆2 in H 2 (Ω) ∩ H01 (Ω).
Remark 1.1 According to conditions (V1 ), (V2 ) and (V3 ), we see that
(i) Condition (V1 ) can be replaced by the following:
(V̂1 ) V (x) ∈ C(RN , R), V (x) ≥ 0 and the set {x ∈ RN : 0 ≤ V (x) ≤ M0 } is bounded
in RN for some M0 > 0.
Indeed, take M0 = 12 V∞ , according to conditions (V1 ) and (V2 ), there exists R > 0 such
that
Ω ⊂ {x ∈ RN : V (x) ≤ M0 } ⊂ BR (0),
(1.4)
where BR (0) (or BR ) denotes the ball centered at 0 with radius R.
3
(ii) The regularity of the boundary ∂Ω in (V2 ) can be replaced by a weaker one: ∂Ω is Lipschitz
continuous and satisfies uniformly outer ball condition. Moreover, we can define a new
norm in H(Ω) := H 2 (Ω) ∩ H01 (Ω) by
Z
1/2
2
kuk0 =
|∆u| dx
Ω
which is equivalent to the standard one. For more details, please see F. Gazzola, H.-Ch.
Grunau and G. Sweers [11, Theorem 2.31].
(iii) Under condition (V3 ), as proved in Lemma 2.5 in the next section, we will see that the
operator ∆2 + λV (x) − δ(x) is non-degenerate and indefinite in H 2 (RN ) for λ large.
Namely, for λ large enough, 0 is not an eigenvalue of ∆2 + λV (x) − δ(x) and the principle
eigenvalue of ∆2 + λV (x) − δ(x) is negative.
Before stating our main result, we present some notations first. Let H(Ω) := H 2 (Ω) ∩ H01 (Ω)
be the Hilbert space endowed with the norm
21
Z
|∆u|2 dx .
kuk0 :=
RN
We denote λV (x) − δ(x) by Vλ (x) and define
Z
2
N
X =: u ∈ H (R ) :
V (x)u dx < +∞ .
2
RN
Let us denote
Z
kukλ :=
2
(|∆u| +
RN
Vλ+ (x)u2 )dx
21
,
where Vλ+ = max{Vλ , 0}. It is easy to see that (X, k · kλ ) is a Banach space for each λ > 0 and
we denote it by Xλ for simplicity.
We define the functional Jλ (u) on Xλ by:
Z
Z
1
1
2
2
(|∆u| + Vλ (x)u )dx −
|u|p dx.
(1.5)
Jλ (u) =
2 RN
p RN
It is not difficult to verify that the functional Jλ (u) is C 1 in Xλ and for every w ∈ Xλ ,
Z
Z
0
Jλ (u)w =
(∆u∆w + Vλ uw)dx −
|u|p−2 uwdx.
RN
(1.6)
RN
Let us also denote
Lλ := ∆2 + Vλ (x),
L0 := ∆2 − δ(x)
and {ek }k≥1 be the eigenfunctions of the operator L0 defined in H(Ω), which is an orthogonal
base of H(Ω) and L2 (Ω). By the assumption (V3 ), H(Ω) can be split to orthogonal sum H − (Ω)⊕
H + (Ω) according to the positive and negative eigenfunction spaces of L0 , i.e.
H(Ω) = H − (Ω) ⊕ H + (Ω),
4
where
H − (Ω) = span{e1 , e2 , · · · , ek }, H + (Ω) = span{ek+1 , ek+2 , · · · , }.
As proved in Lemma 2.3, we will see that the essential spectrum σess (Lλ ) of Lλ satisfies
inf σess (Lλ ) ≥ λM0 − kδ(x)kL∞ . Hence Lλ has finite Morse index in Xλ for λ large and
Lλ has finite eigenvalues below inf σess (Lλ ). Since 0 is not an eigenvalue of Lλ for λ large
enough(see Lemma 2.5). Thus Xλ can be split to an orthogonal sum Xλ = Xλ− ⊕ Xλ+ according
to the negative and positive eigenfunction spaces of Lλ for λ large enough. Instead of using the
classical Nehari-Pankov manifold which is defined as
N̂λ = {u ∈ Xλ \ {0} : Pλ− ∇Jλ (u) = 0, Jλ0 (u) · u = 0},
(1.7)
where Pλ− is the orthogonal projection from Xλ to Xλ− . We will define a modified Nehari-Pankov
manifold, to do that, let us denote first
G0 (u) = Jλ0 (u)u, Gi (u) = Jλ0 (u)ei , i = 1, 2, · · · , k.
We define a modified Nehari-Pankov manifold Nλ by:
Nλ = {u ∈ Xλ \ {0} : Gi (u) = 0, i = 0, 1, 2, · · · , k}.
and the corresponding level value
cλ = inf Jλ (u).
u∈Nλ
Let us denote Aλ be the set of all weak solutions to (1.1), then we say u ∈ Aλ \ {0} is a least
energy solution of (1.1) if Jλ (u) ≤ Jλ (v) for any v ∈ Aλ \ {0}.
Remark 1.2 It is easy to see that all weak solutions to (1.1) belong to Nλ , i.e. Aλ ⊂ Nλ . We
will prove later that the minimizer for cλ in Nλ is indeed a weak solution to (1.1), thus u is a
least energy solution if and only if Jλ (u) = cλ with u ∈ Nλ .
Now we consider the following problem defined on Ω = intV −1 (0),

2
p−2

in Ω,
∆ u − δ(x)u = |u| u,
u 6= 0,
in Ω,


u = 0, ∆u = 0,
on ∂Ω,
(1.8)
which is a kind of limit problem of the original problem (1.1). The corresponding energy functional to (1.8) is defined on H(Ω) by
Z
Z
1
1
2
2
JΩ (u) =
(|∆u| − δ(x)u )dx −
|u|p dx.
2 Ω
p Ω
Moreover for any v ∈ H(Ω),
JΩ0 (u)v
Z
Z
(∆u∆v − δ(x)uv)dx −
=
Ω
Ω
5
|u|p−2 uvdx.
We want to point out that in the case of p = 2∗∗ , problem (1.8) is close to the famous BrezisNirenberg problem and our method to prove the existence of least energy solutions to problem
(1.8) also follows the methods developed by Brezis and Nirenberg (see [7]).
Let P0 denote the orthogonal projection from H(Ω) to H − (Ω), we define the following NehariPankov manifold NΩ by
NΩ := u ∈ HΩ \ H − (Ω) : P0 ∇JΩ (u) = 0, JΩ0 (u)u = 0
=
u ∈ HΩ \ H − (Ω) : JΩ0 (u)ei = 0, i = 1, 2, · · · k, JΩ0 (u)u = 0 ,
(1.9)
where as above mentioned, ei (i = 1, 2 · · · , k) denote the negative eigenfunctions of the operator
L0 . The corresponding level c(Ω) is defined by
c(Ω) := inf JΩ (u).
NΩ
According to A. Szulkin and T. Weth [30], we knew that u is a least energy solution to (1.8) if
JΩ (u) = c(Ω) with u ∈ NΩ .
Remark 1.3 The reason why we introduce a modified Nehari-Pankov manifold Nλ for the functional Jλ instead of using the Nehari-Pankov manifold N̂λ of Jλ directly is that for any u ∈ NΩ
which is the Nehari-Pankov manifold related to the limit functional JΩ , one can not say that
u ∈ N̂λ which is the Nehari-Pankov manifold related to the functional Jλ . Thus to consider the
asymptotic behavior of the least energy solution of (1.1), we introduce a modified Nehari-Pankov
manifold Nλ and it is easy to see that NΩ ⊂ Nλ .
Our main result is:
for N ≥ 5 or p = 2∗∗ for
Theorem 1.4 Suppose (V1 ), (V2 ) and (V3 ) hold, 2 < p < 2∗∗ := N2N
−4
N ≥ 8. Then for λ large, (1.1) has a least energy solution uλ (x) which achieves cλ . Moreover,
for any sequence λn → ∞, there exists a subsequence of {uλn (x)} ( still denoted by {uλn (x)} )
such that uλn (x) converges in H 2 (RN ) to a least energy solution u(x) of (1.8).
The paper is organized as follows: In Section 2, we give some preliminary results. In Section
3, we study the limit equation (1.8) and prove the existence of least energy solutions. In Section
4, we prove the existence of least energy solutions to (1.1) for λ large enough. In Section 5, we
study the limit of cλ as λ → +∞ and finalize the paper by proving Theorem 1.4.
2
Preliminary results
In this section, we present some preliminary results which we need in proving our main result
and we divided them into two subsections. More precisely, in Subsection 2.1, we introduce the
spectrum of the operators ∆2 + λV − δ and ∆2 − δ. In Subsection 2.2, we give some properties
of the Nehari-Pankov manifold NΩ and also the modified Nehari-Pankov manifold Nλ .
6
2.1
Eigenvalues and eigenfunction spaces
In this subsection, we mainly discuss the eigenvalues and eigenfunction spaces of the operator
Lλ defined in Xλ . To do that we firstly give the following embedding result which is
Lemma 2.1 Assume (V1 ), (V2 ) and (V3 ) hold, then there exists Λ0 > 0 such that for each λ > Λ0
and u ∈ Xλ , we have
kukH 2 (RN ) ≤ Ckukλ
(2.1)
for some C > 0 which does not depend on λ.
Proof: Let M0 = 12 V∞ , by (1.4), we know that
V (x) ≥ M0 , ∀x ∈ RN \ BR (0) and suppVλ− ⊂ BR (0), ∀λ >
µk+1
,
M0
where suppVλ− denotes the support set of Vλ− . Thus for each u ∈ Xλ and λ >
we have
Z
Z
1
2
u dx ≤
(λV (x) − δ)u2 dx
M0 RN \BR (0)
RN \BR (0)
Z
1
≤
V + u2 dx
M0 RN \BR (0) λ
Z
1
≤
(|∆u|2 + Vλ+ u2 )dx.
M0 RN
M0 +µk+1
,
M0
(2.2)
by (2.2),
(2.3)
By Hölder’s inequality and Sobolev inequality, we obtain that
Z
2
Z
|u|
u dx ≤
BR (0)
2N
N −4
NN−4
dx
4
|BR | N
BR
≤ C1 |BR |
4
N
4
Z
N
ZR
≤ C1 |BR | N
RN
|∆u|2 dx
(|∆u|2 + Vλ+ u2 )dx.
(2.4)
Combining (2.3) and (2.4), we have
Z
Z
4
M0 + 1
2
2
(|∆u| + u )dx ≤
+ C1 |BR | N
(|∆u|2 + Vλ+ u2 )dx.
M0
RN
RN
Thus (2.1) holds for Λ0 =
lemma.
M0 +µk+1
M0
and C =
M0 +1
M0
4
+ C1 |BR | N . This completes the proof of this
2
Remark 2.2 As a result of Lemma 2.1, one can see that Xλ can be continuously imbedded
into Lp (RN ) for 2 < p ≤ 2∗∗ := N2N
and the embedding Xλ ,→ Lploc (RN ) is compact for
−4
7
2 < p < N2N
when λ > Λ0 . Moreover, for any u ∈ Xλ , when λ > Λ0 and 2 < p ≤ 2∗∗ , there
−4
exists a C > 0 independent of λ such that
Z
p
|u| dx
p2
Z
≤C
RN
RN
(|∆u|2 + Vλ+ u2 )dx.
Now we come to study the eigenvalue problems for the operator Lλ as λ large, we have the
following lemma.
Lemma 2.3 Under the conditions (V1 ), (V2 ) and (V3 ), for each λ > Λ0 , we have
σess (Lλ ) ⊂ [λM0 − kδ(x)kL∞ , +∞).
Furthermore, inf σess (Lλ ) → +∞ as λ → +∞.
Proof: The proof of this lemma is similar to the proof of Proposition 2.3 in [4]. For readers’
convenience, we give the details.
We set Wλ = Vλ − λM0 + δ = λ(V (x) − M0 ) and write Wλ1 = max{Wλ , 0}, Wλ2 =
min{Wλ , 0}. Obviously, for λ > Λ0 ,
σ(∆2 + Wλ1 + λM0 − δ) ⊂ [λM0 − kδ(x)kL∞ , +∞)
(2.5)
for Wλ1 ≥ 0. Let Hλ = ∆2 + Wλ1 + λM0 − δ, then Lλ = Hλ + Wλ2 .
We claim that
Wλ2 is a relative form compact perturbation of Lλ for λ > Λ0 .
Indeed, since Wλ2 is bounded, then the form domain of Hλ is the same as the form domain Xλ
of Lλ . Thus we have to show that
Xλ 7→ Xλ∗ , u 7→ Wλ2 · u is compact.
Here Xλ∗ is the dual space of Xλ . Take a bounded sequence {un }n≥1 in Xλ , then according to
Lemma 2.1, {un }n≥1 is also a bounded sequence in H 2 (RN ). Thus for some u ∈ H 2 (RN ), up to
a subsequence,

 un * u weakly in H 2 (RN ),
un → u strongly in L2loc (RN ),
(2.6)

un → u a.e. in RN
8
as n → +∞. According to (2.2), we know that suppWλ2 ⊂ BR for any λ > Λ0 . Thus by Hölder’s
inequality, Sobolev inequality and Lemma 2.1, for any λ > Λ0 , v ∈ Xλ , we have
Z
Z
2
2
= (u
−
u)vdx
(u
−
u)vdx
W
W
n
n
λ
λ
N
BR
R
Z
≤ kδkL∞
|(un − u)v|dx
BR
Z
2
≤ µk+1
21 Z
|un − u| dx
21
v dx
BR
BR
Z
2
≤ µk+1
2
21
kvkH 2 (RN )
|un − u| dx
BR
Z
2
≤ C
21
|un − u| dx
kvkλ .
(2.7)
BR
Hence by (2.6) and (2.7), we have as n → +∞,
kWλ2 un
−
Wλ2 ukXλ∗
Z
2
|un − u| dx
≤C
21
→ 0.
BR
Thus Wλ2 is a relative form compact perturbation of Lλ .
According to the classical Weyl theorem (see Example 3 in [28], page 117), σess (Lλ ) =
σess (Hλ ). Thus by (2.5), for λ > Λ0 , we have
σess (Lλ ) ⊂ [λM0 − kδkL∞ , +∞).
Moreover,
inf σess (Lλ ) → +∞ as λ → +∞.
2
Thus the proof of this lemma is completed.
Remark 2.4 Let
R
RN
µn (L0 ) := max min
S∈Σ0,n−1 u∈S ⊥
and
R
µn (Lλ ) := max min
S∈Σλ,n−1 u∈S ⊥
RN
(|∆u|2 + Vλ u2 )dx
R
u2 dx
RN
(|∆u|2 + Vλ u2 )dx
R
,
u2 dx
RN
where Σ0,n−1 and Σλ,n−1 denote the collection of (n − 1)−dimensional subspaces in H(Ω) and
Xλ respectively. It is easy to see that µn (Lλ ) ≤ µn (L0 ), according to the above lemma and the
min-max principle in spectral analysis (see Theorem XIII.1 and Theorem XIII.2 in [28]), we
obtain that µn (Lλ ) is indeed an eigenvalue of Lλ for λ large enough.
9
Finally, let {µi (Lλ )} be the class of all distinct eigenvalues of Lλ := ∆2 + Vλ in Xλ and
{µi (L0 )} be the class of all distinct eigenvalues of L0 := ∆2 − δ(x) in H(Ω). Without loss of
generality, we may assume that
µ1 (Lλ ) < µ2 (Lλ ) < µ3 (Lλ ) < · · · < µkλ (Lλ ) < inf σess (Lλ ),
and
µ1 (L0 ) < µ2 (L0 ) < µ3 (L0 ) < · · · < µk (L0 ) < 0 < µk+1 (L0 ) < · · · .
Moreover, µkλ (Lλ ) → +∞ as λ → +∞ and µi (L0 ) → +∞ as i → +∞.
Let Vi (Lλ ) be the eigenfunction space of µi (Lλ ) and Vi (L0 ) be the eigenfunction space of
µi (L0 ). We say that Vi (Lλ ) converges to Vi (L0 ), i.e. Vi (Lλ ) → Vi (L0 ) as λ → +∞, if for
any sequence λn → ∞ and normalized eigenfunctions ψn ∈ Vi (Lλn ), there exists a normalized
eigenfunction ψ ∈ Vi (L0 ) such that ψn → ψ strongly in H 2 (RN ) along a subsequence.
The following Lemma concerns the asymptotic behavior of µi (Lλ ) and Vi (Lλ ) as λ → +∞.
Lemma 2.5 For i = 1, 2, · · · , we have
µi (Lλ ) → µi (L0 ) and Vi (Lλ ) → Vi (L0 ), as λ → +∞.
Moreover by assumption (V3 ), there exists Λ1 > Λ0 such that for any λ > Λ1 , we have
µ1 (Lλ ) < µ2 (Lλ ) < · · · < µk (Lλ ) < 0 < µk+1 (Lλ ) < · · · < µkλ (Lλ ) < inf σess (Lλ ).
Proof: We prove this lemma by induction.
Step 1: We prove the case for i = 1, i.e.
µ1 (Lλ ) → µ1 (L0 ) and V1 (Lλ ) → V1 (L0 ) as λ → +∞.
Let ψn ∈ Xλn be an eigenfunction corresponding to µ1 (Lλn ) which satisfies
Z
Z
2
ψn dx = 1 and
(|∆ψn |2 + Vλn ψn2 )dx = µ1 (Lλn ).
RN
(2.8)
RN
As µ1 (Lλn ) is increasing in λn and µ1 (Lλn ) ≤ µ1 (L0 ), by (2.8) we have
Z
Z
2
2
2
||ψn ||λn =
(|∆ψn | + Vλn ψn )dx +
Vλ−n ψn2 dx
RN
RN
Z
= µ1 (Lλn ) +
Vλ−n ψn2 dx ≤ µ1 (L0 ) + kδ(x)kL∞
RN
≤ µ1 (L0 ) + µk+1 .
According to Lemma 2.1, {ψn } is bounded in H 2 (RN ). Up to a subsequence, there is ψ ∈
H 2 (RN ) such that as λn → +∞, we have

 ψn * ψ weakly in H 2 (RN ),
ψn → ψ strongly in L2loc (RN ),
(2.9)

N
ψn → ψ a.e. in R .
10
Firstly, we prove that
ψ ∈ H 2 (Ω) ∩ H01 (Ω).
In fact, we just need to verify that
ψ(x) = 0 a.e. in RN \ Ω.
For each integer m ≥ 1, we denote
1
N
Cm := x ∈ R : V (x) >
.
m
Fix m, by (2.8), as λn → +∞, we have
Z
Z
m
2
ψn dx ≤
λn V (x)ψn2 dx
λn RN
Cm
Z
m
≤
(|∆ψn |2 + λn V (x)ψn2 )dx
λn RN
m
m
(µ1 (Lλn ) + kδ(x)kL∞ ) ≤
(µ1 (L0 ) + µk+1 ) → 0.
≤
λn
λn
N
N
Thus ψ(x) = 0 a.e. in Cm . Note that ∪∞
m=1 Cm = R \ Ω, we have ψ(x) = 0 a.e. in R \ Ω.
Secondly, we prove that
Z
ψ 2 dx = 1.
Ω
In fact, according to (2.2) and (2.8), we have
Z
Z
1
2
ψn dx ≤
λn V (x)ψn2 dx
M
λ
N
N
0 n R \BR (0)
R \BR (0)
Z
1
≤
(|∆ψn |2 + λn V (x)ψn2 )dx
M0 λn RN
1
≤
(µ1 (L0 ) + kδ(x)kL∞ ) → 0
M0 λn
as λn → +∞. Thus
Z
lim
n→+∞
RN \BR (0)
ψn2 dx = 0.
Combining (2.8), (2.9) and (2.10), we have
Z
Z
2
ψ dx = lim
ψn2 dx
n→+∞
Ω
Z
ZBR (0)
2
= lim
ψn dx − lim
n→+∞
n→+∞
RN
RN \BR (0)
Finally, we prove that
µ1 (Lλn ) → µ1 (L0 ) as n → +∞.
11
(2.10)
ψn2 dx = 1.
In fact, ψn → ψ strongly in L2 (RN ) as n → +∞. Thus by (2.8), we have
Z
2
2
2
1
µ1 (L0 ) =: inf
(|∆u| − δ(x)u )dx : u ∈ H (Ω) ∩ H0 (Ω), kukL2 (Ω) = 1
Ω
Z
≤
(|∆ψ|2 − δ(x)ψ 2 )dx
Ω Z
|∆ψn |2 + (λn V (x) − δ(x))ψn2 dx
≤ lim
n→∞
=
RN
lim µ1 (Lλn ) ≤ µ1 (L0 ),
n→∞
which implies that µ1 (Lλn ) → µ1 (L0 ) as n → ∞. Since µ1 (Lλ ) is increasing in λ, then
µ1 (Lλ ) → µ1 (L0 ) as λ → +∞.
Step 2: Suppose that the results hold up to k − 1 for k ≥ 2, we want to prove that the same
result is true for the k-th eigenvalue.
Since Lλ ψ = L0 ψ for any ψ ∈ H(Ω), then by the k-th Rayleigh quotient descriptions of
µk (Lλ ) and µk (L0 ), we have
lim sup µk (Lλ ) ≤ µk (L0 ).
λ→+∞
Just like the case when k = 1, we can take λn → +∞ and the normalized eigenfunctions
ψn ∈ Vk (Lλn ) which is the eigenfunction space corresponding to µk (Lλn ), such that
Z
(|∆ψn |2 + Vλn ψn2 )dx = µk (Lλn ),
RN
Z
ψn2 dx = 1, ψn ⊥Vj (Lλn ), j = 1, 2, 3 · · · , k − 1.
R
Similar to the proof in Step 1, we have for some ψ ∈ H(Ω) with Ω |ψ|2 dx = 1,

 ψn * ψ weakly in H 2 (RN ),
ψn → ψ strongly in L2loc (RN ),

ψn → ψ a.e. in RN .
RN
Since ψn ⊥Vj (Lλn ), j = 1, 2, · · · , k−1, and Vj (Lλn ) → Vj (L0 ) as n → +∞, then ψ⊥Vj (L0 ), j =
1, 2, · · · , k − 1 and
Z
µk (L0 ) ≤
(|∆ψ|2 − δ(x)ψ 2 )dx
Ω Z
≤ lim
|∆ψn |2 + (λn V (x) − δ(x))ψn2 dx
n→∞
≤
RN
lim µk (Lλn ) ≤ µk (L0 ).
n→∞
This induces that µk (Lλn ) → µk (L0 ) and Vk (Lλn ) → Vk (L0 ) as n → +∞.
2
Remark 2.6 By assumption (V3 ), for λ large enough, the operator ∆2 + Vλ defined in Xλ is
non-degenerate and indefinite whose Morse index is dj = dimXλ− uniformly in λ.
12
2.2
The modified Nehari-Pankov manifold
In this subsection, we consider the modified Nehari-Pankov manifold Nλ and the corresponding
level value cλ .
Firstly, we use the following lemma to collect some properties of the Nehari-Pankov manifold
NΩ and the corresponding level c0 , which are
Lemma 2.7 Let Ω := intV −1 (0), for any w ∈ H(Ω) \ H − (Ω), set
Hw := {v + tw : v ∈ H − (Ω), t > 0}.
Then the following properties hold:
(i) NΩ = {w ∈ H(Ω) \ H − (Ω) : ∇(JΩ (w)|Hw ) = 0}.
(ii) For every w ∈ H + (Ω) \ {0} there exists tw > 0 and ϕ(w) ∈ H − (Ω) such that
Hw ∩ NΩ = {ϕ(w) + tw · w}.
(iii) For every w ∈ NΩ and every u ∈ Hw \ {w} there holds JΩ (u) < JΩ (w).
(iv) c(Ω) = inf u∈NΩ JΩ (u) > 0.
Proof: The similar proof can be found in the paper by A. Szulkin and T. Weth [30] which is
concerned about the Laplacian operator. For the completeness of the paper, we give the detail of
the proof.
(i) Take ω ∈ NΩ , according to the definition of NΩ , we have
ω ∈ H(Ω) \ H − (Ω), P0− ∇JΩ (ω) = 0, JΩ0 (ω)ω = 0.
For any φ = ψ + tω ∈ Hω , we obtain
h∇JΩ (ω), φiH(Ω) = h∇JΩ (ω), P0− ψiH(Ω) + th∇Jλ (ω), ωiH(Ω) = hP0− ∇JΩ (ω), ψiH(Ω) + 0 = 0.
Thus ∇(JΩ (ω)|Hω ) = 0.
Next, we take ω ∈ H(Ω)\H − (Ω) and ∇(JΩ (ω)|Hω ) = 0. For any φ = ψ +tω ∈ Hω , we have
h∇JΩ (ω), φiH(Ω) = 0. Let φ = ω, we obtain that JΩ0 (ω)ω = 0. Note that P0− ψ = ψ ∈ H − (Ω),
then we can easily get that P0− ∇JΩ (ω) = 0. Thus ω ∈ NΩ .
(iii) Take u = v + tω ∈ Hω \ {ω}, a direct computation shows that
JΩ (u) − JΩ (ω)
Z
Z
Z
1
1
2
2
2
2
=
(|∆u| − δu )dx − (|∆ω| − δω )dx +
(|ω|p − |u|p )dx
2 Ω
p
Ω
Ω
Z
Z
Z
2
1
t
−
1
2
2
2
2
=
(|∆v| − δv )dx +
(|∆ω| − δω )dx + t (∆v∆ω − δvω)dx
2 Ω
2
Ω
Ω
Z
1
+
(|ω|p − |v + tω|p )dx
p Ω
13
Since ω ∈ NΩ , then
Z
Z
Z
Z
2
2
p
(|∆ω| − δω )dx =
|ω| dx, (∆ω∆v − δωv)dx =
|ω|p−2 ωvdx.
Ω
Ω
Ω
Ω
Thus
JΩ (u) − JΩ (ω)
Z
Z 2
1
1 p 1
t −1 p
p−2
p
2
2
=
|ω| + t|ω| ωv + |ω| − |v + tω| dx.
(|∆v| − δv )dx +
2 Ω
2
p
p
Ω
Let
φ(t) =
1
1
t2 − 1 p
|ω| + t|ω|p−2 ωv + |ω|p − |v + tω|p .
2
p
p
Then φ(t) ≤ 0 if ω(v + tω) ≤ 0. For ω(v + tω) > 0, i.e. t > − ωv , it is easy to see that
1 1
1
φ(0) = −( − )|ω|p − |v|p ≤ 0, lim φ(t) = −∞.
t→+∞
2 p
p
Assume φ(t0 ) =
sup
v
t≥max{0,− ω
}
φ(t) > 0 for some t0 ≥ max{0, − ωv }, then t0 > max{0, − ωv }.
Thus φ0 (t0 ) = 0 implies that
|ω|p−2 ω(v + t0 ω) − |v + t0 ω|p−2 (v + t0 ω)ω = 0, i.e. |ω| = |v + t0 ω|.
Therefore,
|ω|p−2 v 2
t20 − 1 p
|ω| + t0 |ω|p−2 ωv = −
≤ 0,
2
2
which leads to a contradiction. Hence φ(t) ≤ 0 for t > 0.
If v 6= 0, since v ∈ Xλ− , then we can easily obtain that JΩ (u) < JΩ (ω). If v = 0, then t 6= 1
and we can also obtain that JΩ (u) < JΩ (ω). Thus JΩ (u) < JΩ (ω) for any u ∈ Hω \ {ω} where
ω ∈ NΩ .
(iv) Denote
Z
φ(t0 ) =
Sα =
u ∈ H + (Ω) :
(|∆u|2 − δ(x)u2 )dx = α2 .
Ω
Note that for any u ∈ Sα , we have
Z
Z
Z
2
2
2
|∆u| dx =
(|∆u| − δ(x)u )dx + δ(x)u2 dx
Ω
Ω
Z
Ω
kδ(x)kL∞
(|∆u|2 − δ(x)u2 )dx
≤
1+
µk+1 (L0 )
Ω
kδ(x)kL∞
=
1+
α2 ,
µk+1 (L0 )
14
where µk+1 (Lλ ) > 0. Then by Sobolev imbedding theorem, we have
Z
Z
1
1
2
2
JΩ (u) =
(|∆u| − δ(x)u )dx −
|u|p dx
2 Ω
p Ω
Z
p2
Z
1
1
2
2
2
|∆u| dx
≥
(|∆u| − δ(x)u )dx − · C
2 Ω
p
Ω
p
1 2 C
kδ(x)kL∞ 2 p
≥
α −
1+
α >0
2
p
µk+1 (L0 )
for α > 0 small enough. Let u ∈ NΩ , then u = u− + u+ , where u− ∈ H − (Ω) and 0 6= u+ ∈
H + (Ω). Since tu+ ∈ Hu ∩ Sα for some t > 0, then according to (iii) in this Lemma, we have
Jλ (u) > Jλ (tu+ ) ≥ inf Jλ (u) > 0.
Sα
(ii) Take ω ∈ H + (Ω) \ {0}. If ω ∈ NΩ , then by (iii), we have φ(ω) = 0 and tω = 1. If
ω∈
/ NΩ , we may assume that
Z
(|∆ω|2 − δ(x)ω 2 )dx = 1.
Ω
For any u ∈ Hω which is contained in a finite space, we have
Z
Z
1
1
2
2
(|∆u| − δ(x)u )dx −
|u|p dx → −∞.
JΩ (u) =
2 Ω
p Ω
as kuk0 → +∞ due to the fact that all norms in a finite space are equivalent. Thus for any
u ∈ Hω \ BR (0), we have JΩ (u) < 0 for some large R > 0. According to the proof of
(iv), we obtain that JΩ (tω) > 0 for some small t > 0. Note that Hω is contained in a finite
dimensional space, then there exists some u0 ∈ Hω ∩ BR (0) such that JΩ (u0 ) = sup JΩ (u).
Hω
Thus ∇JΩ (u0 )|Hω = 0 which implies that u0 ∈ NΩ ∩ Hω due to (i). According to (iii), we
+
know that u0 is unique. Put φ(ω) = u−
0 and tω ω = u0 , we have Hω ∩ NΩ = {ϕ(ω) + tω · ω}. 2
Remark 2.8 By Lemma 2.7, we conclude that for each w ∈ H(Ω)\H − (Ω), the set NΩ intersects
Hw in exactly one point τ (w) := ϕ(w) + tw · w which is the unique global maximum point of
JΩ |Hw . Moreover, similar to the proof in A. Szulkin and T. Weth [30], the map τ : w → ϕ(w) +
tw · w is continuous and the restriction of τ to the unit sphere SΩ+ in H + (Ω) is a homeomorphism
between SΩ+ to NΩ . Thus the least energy level c(Ω) has a minimax characterization given by
c(Ω) =
inf
max JΩ (u).
w∈H + (Ω)\{0} u∈Hw
Now we are going to give some properties of the modified Nehari-Pankov manifold Nλ and
the corresponding level cλ .
15
Lemma 2.9 For w ∈ Xλ \ H − (Ω), set
Ĥw := {v + tw : v ∈ H − (Ω), t > 0}.
Then there exists a constant Λ2 (Λ2 > Λ1 > 0) such that for any λ > Λ2 we have the following
properties hold:
(i) Nλ = {w ∈ Xλ \ H − (Ω) : ∇(Jλ (w)|Ĥw ) = 0}.
(ii) Let
Eλ+
Z
+
:= w ∈ Eλ :
wei dx = 0, i = 1, 2, · · · , k .
RN
Then for every w ∈ Eλ+ \ {0} there exists tω > 0 and ϕ(w) ∈ H − (Ω) such that
Ĥw ∩ Nλ = {ϕ(w) + tω · w}.
(iii) For every w ∈ Nλ and every u ∈ Ĥw \ {w} there holds Jλ (u) < Jλ (w).
(iv) cλ = inf u∈Nλ Jλ (u) ≥ τ > 0 for some small τ > 0 which is independent on λ.
Proof: The proofs of (i) − (iii) are similar to the corresponding proofs of Lemma 2.7, we omit
them and we only need to prove (iv).
Firstly, we claim that there exists a Λ2 (Λ2 > Λ1 > 0) such that for any λ > Λ2 and u ∈ Eλ+ ,
we have the following inequality
Z
Z
2
2
(|∆u| + Vλ u )dx ≥ C
(|∆u|2 + Vλ+ u2 )dx
RN
RN
holds for some C > 0 which is independent on λ.
In fact, for any u ∈ Eλ+ , we have
−
+
+
−
−
u = u+
λ + uλ , uλ ∈ Xλ , uλ ∈ Xλ .
Note that
Z
RN
2
[|∆u+
λ|
+
2
Vλ+ (u+
λ ) ]dx
Z
=
RN
≤
2
[|∆u+
λ|
+
2
Vλ (u+
λ ) ]dx
kδ(x)kL∞
1+
µk+1 (Lλ )
16
Z
RN
Z
+
RN
2
Vλ− (u+
λ ) dx
+ 2
2
|∆u+
|
+
V
(u
)
dx.
λ
λ
λ
Then a direct computation gives us that
Z
(|∆u|2 + Vλ u2 )dx
N
ZR
Z
+ 2
+ 2
− 2
2
=
(|∆uλ | + Vλ (uλ ) )dx +
(|∆u−
λ | + Vλ (uλ ) )dx
RN
RN
Z
Z
µk+1 (Lλ )
+ 2
+ + 2
− 2
2
(|∆uλ | + Vλ (uλ ) )dx +
(|∆u−
≥
λ | + Vλ (uλ ) )dx
kδ(x)kL∞ + µk+1 (Lλ ) RN
RN
Z
µk+1 (Lλ )
≥
(|∆u|2 + Vλ+ u2 )dx
kδ(x)kL∞ + µk+1 (Lλ ) RN
(
Z
kδ(x)kL∞
− 2
2
(|∆u−
+
λ | + Vλ (uλ ) )dx
∞
kδ(x)kL + µk+1 (Lλ ) RN
)
Z
µk+1 (Lλ )
+ −
2
V − (u−
−
λ ) + 2uλ uλ dx .
kδ(x)kL∞ + µk+1 (Lλ ) RN λ
Since
u−
λ
=
k Z
X
i=1
ueλ,i dx eλ,i =
RN
k Z
X
i=1
u(eλ,i − ei )dx eλ,i ,
RN
where ei and eλ,i are the eigenfunction of L0 and Lλ corresponding to µi (L0 ) and µi (Lλ ) respectively and eλ,i → ei in H 2 (RN ). Then as λ → +∞, we have
Z
− 2
− 2
(|∆u
|
+
V
(u
)
)dx
λ
λ
λ
N
R Z
2 Z
k
X
ueλ,i dx
(|∆eλ,i |2 + Vλ e2λ,i )dx
= N
N
R
R
i=1 Z
2
k
X
= u(eλ,i − ei )dx µi (Lλ )
RN
i=1
≤ o(1)kuk2λ .
Thus as λ → +∞, we have
Z
kδ(x)kL∞
− 2
2
(|∆u−
λ | + Vλ (uλ ) )dx
kδkL∞ + µk+1 (Lλ ) RN
Z
µk+1 (Lλ )
−
− 2
+ −
−
Vλ [(uλ ) + 2uλ uλ ]dx = o(1)kuk2λ .
∞
kδ(x)kL + µk+1 (Lλ ) RN
17
Therefore, there exists a constant Λ2 > Λ1 such that
Z
(|∆u|2 + Vλ u2 )dx
RN
Z
µk+1 (L0 )
≥
(|∆u|2 + Vλ+ u2 )dx
2(kδ(x)kL∞ + µk+1 (L0 )) RN
Z
µk+1 (L0 )
(|∆u|2 + Vλ+ u2 )dx
−
4(kδ(x)kL∞ + µk+1 (L0 )) RN
Z
µk+1 (L0 )
(|∆u|2 + Vλ+ u2 )dx.
=
∞
4(kδ(x)kL + µk+1 (L0 )) RN
Thus taking C =
µk+1 (L0 )
> 0, we have
4(kδ(x)kL∞ + µk+1 (L0 ))
Z
Z
2
2
(|∆u| + Vλ u )dx ≥ C
(|∆u|2 + Vλ+ u2 )dx.
RN
RN
Secondly, let
Z
+
Sα := u ∈ Eλ :
2
2
(|∆u| dx + Vλ u )dx = α
2
.
RN
Then for any u ∈ Sα , by Sobolev inequality, we have
Z
Z
1
1
2
2
(|∆u| + Vλ u )dx −
|u|p dx
Jλ (u) =
2 RN
p RN
p2
Z
Z
C
1
+ 2
2
2
2
(|∆u| + Vλ u )dx −
(|∆u| + Vλ u )dx
≥
2 RN
p
RN
p2
Z
Z
1
2
2
2
2
(|∆u| + Vλ u )dx
≥
(|∆u| + Vλ u )dx − C
2 RN
RN
1
1 2
=
α − Cαp ≥ α2 > 0
2
4
for α > 0 small enough. Thus inf Sα Jλ (u) > 0.
Finally, for any w ∈ Nλ , w+ = −w− + w ∈ Hw , we take t > 0 small enough such that
tw+ ∈ Hw ∩ Sα , thus by taking τ = 41 α2 and (iii) in this Lemma, we have
Jλ (w) > Jλ (tw+ ) ≥ inf Jλ (u) ≥ τ > 0,
Sα
which implies cλ ≥ τ > 0.
2
At the end of this section, we use the following Lemma to prove that the minimizer for cλ in
Nλ is indeed a weak solution (also a least energy solution) of (1.1).
Lemma 2.10 For λ > Λ2 , assume u is an achieved function for cλ in Nλ , i.e. cλ = Jλ (u) and u ∈
Nλ . Then u is a least energy solution to (1.1).
18
Proof: We just need to verify that u is a weak solution to (1.1), i.e. Jλ0 (u) = 0 in Xλ .
In fact, according to Lagrange multiplier theorem, there exist (λ0 , λ1 , · · · , λk ) ∈ Rk such that
Jλ0 (u) + λ0 G00 (u) + λ1 G01 (u) + · · · + λk G0k (u) = 0.
Multiplying u and ei (i = 1, 2, · · · , k) on both sides of the above equation respectively, we have
the following system holds:

a00 λ0 + a01 λ1 + a02 λ2 + · · · + a0k λk = 0,




 a10 λ0 + a11 λ1 + a12 λ2 + · · · + a1k λk = 0,
a20 λ0 + a21 λ1 + a22 λ2 + · · · + a2k λk = 0,


···



ak0 λ0 + ak1 λ1 + ak2 λ2 + · · · + akk λk = 0,
where
Z
|u|p dx,
Z
|u|p−2 uei dx, i = 1, 2, · · · , k,
a0,i = ai,0 = (p − 2)
N
Z R
aij = aji = (p − 1)
|u|p−2 ei ej dx, i, j = 1, 2, · · · , k,
N
R
Z
Z
p−2 2
e2i dx, i = 1, 2, · · · , k.
|u| ei dx − µi (L0 )
aii = (p − 1)
a00 = (p − 2)
RN
RN
RN
Denote the coefficient matrix of the above system by A = (aij )0≤i≤k,0≤j≤k . we define f (y) =
y T Ay for any y ∈ Rk+1 , where y T denotes the transposition of the vector y in Rk+1 . For any
y ∈ Rk+1 , by a simple computation, we have
Z
Z
k
X
p−2
p
2
|u| uei dx y0 yi
f (y) = (p − 2)
|u| dx y0 + 2
(p − 2)
RN
Z
k X
+
(p − 1)
RN
i=1
k X
X
Z
Z
p−2
|u|
p−2
|u|
+
RN
RN
yi2
|u|p−2 ei ej dx yi yj
y0 u +
RN
Z
− µi (L0 )
e2i dx
RN
i=1 j>i
= (p − 2)
Z
|u|p−2 e2i dx
(p − 1)
+2
RN
i=1
k
X
!2
yi ei
dx
i=1
k
X
!2
yi ei
dx −
i=1
k
X
i=1
Z
µi (L0 )
RN
e2i dx
yi2 .
Note that p > 2, µi (L0 ) < 0, i = 1, 2 · · · , k and u, e1 , e2 , · · · , ek are linear independent, then
for any y ∈ Rk+1 , we have f (y) > 0. Thus the matrix A is positively definite. Therefore the
19
solution of the above system is (λ0 , λ1 , · · · , λk ) = 0 which implies that Jλ0 (u) = 0 in Xλ , i.e. u
is a least energy solution to (1.1).
2
3
Limit problem
In this section, we consider the limit problem defined in Ω, where Ω is the interior part of the
zero set V −1 (0):
(
∆2 u − δu = |u|p−2 u,
x ∈ Ω,
(3.1)
u = 0, ∆u = 0,
x ∈ ∂Ω.
Recall that the corresponding functional to (3.1) is
Z
Z
1
1
2
2
(|∆u| − δu )dx −
|u|p dx,
JΩ (u) =
2 Ω
p Ω
(3.2)
the Nehari-Pankov manifold is
NΩ := u ∈ H(Ω) \ {0} : P0− ∇JΩ (u) = 0, JΩ0 (u)u = 0
and the corresponding level value is defined by
c(Ω) = inf JΩ (u).
NΩ
We say that {un } is a (P S)c sequence of JΩ if JΩ (un ) → c and JΩ0 (un ) → 0 in H 0 (Ω), the
dual space of H(Ω), as n → +∞. JΩ satisfies the (P S)c condition if any (P S)c sequence {un }
contains a convergent subsequence.
Lemma 3.1 For 2 < p ≤ 2∗∗ , N ≥ 5, {un } is a (P S)c(Ω) sequence, i.e. as n → +∞,
JΩ (un ) → c(Ω), JΩ0 (un ) → 0
in H 0 (Ω),
where H 0 (Ω) is the dual space of H(Ω). Then {un } is bounded in H(Ω).
Proof: For n large enough, we have
1
c(Ω) + 1 + kun k0 ≥ JΩ (un ) − JΩ0 (un )un =
p
and
1 1
−
2 p
Z
(|∆un |2 − δu2n )dx
Ω
Z
1 0
1 1
c(Ω) + 1 + kun k0 ≥ JΩ (un ) − JΩ (un )un =
−
|un |p dx.
2
2 p
Ω
By Hölder’s inequality, we have
Z
p2
Z
1− p2
2
p
|un | dx ≤ |Ω|
|un | dx .
Ω
Ω
Thus {un } is bounded in H(Ω).
20
Lemma 3.2 For 2 < p < 2∗∗ and N ≥ 5, c(Ω) is achieved by a nontrivial solution u of (3.1) in
NΩ .
Proof: Since the proof is quite standard, for readers’ convenience, we give the sketch of the
proof.
Indeed, from the definition of c(Ω) and thanks to Ekeland’s Variational Principle, we know
that there exists a sequence {un } ⊂ NΩ such that
JΩ (un ) → c(Ω) and JΩ0 (un ) → 0 in H 0 (Ω).
(3.3)
Thus by Lemma 3.1 and the fact that H(Ω) ,→ Lp (Ω) is compact , we immediately obtain that
JΩ (un ) satisfies Palais-Smale condition. Namely (3.3) indicate that there is a subsequence of
{un }( still denote it as itself) and u ∈ NΩ such that un → u in H(Ω) and
JΩ0 (u) = 0.
JΩ (u) = c(Ω) > 0,
2
Thus we complete the proof of this lemma.
Now we focus on the existence of least energy solution of (3.1) in the critical case. We want
to point out that in this case problem (3.1) is close to the famous Brezis-Nirenberg problem
(
∗
−∆u − δu = |u|2 −2 u,
x ∈ Ω,
(3.4)
u = 0,
x ∈ ∂Ω.
where 2∗ is the critical Sobolev exponent which is 2∗ = N2N
for N ≥ 3 and 2∗ = +∞ for
−2
N = 1, 2. And our method to prove the existence of least energy solutions to problem (3.1) in
critical case also follows the methods developed by Brezis and Nirenberg (see [7]).
Firstly we have the following estimate for the least energy c(Ω) when p = 2∗∗ .
Lemma 3.3 For N ≥ 8, p = 2∗∗ , we have
0 < c(Ω) <
where
Z
2
2 N
S4,
N
2
N
Z
|∆u| dx : u ∈ H (R ),
S = inf
RN
2∗∗
|u|
dx = 1 .
RN
Proof: It was shown by P.L.Lions (see Corollary I.2 in [25]) that there is an nonnegative minimizer for S which is radial symmetric and decreasing in |x|. In 1998, C.S. Lin [21] showed that
any positive extremal function of S has the form
Uε = c
ε
2
ε + |x|2
N2−4
for each ε > 0. One also can refer to the paper by J.Wei and X.Xu [32], where the authors
extended C.S.Lin’s results to more general case.
21
We may assume that 0 ∈ Ω. Let η be a smooth cutoff function satisfying
η(x) = 1 for x ∈ Br (0) and supp η ⊂ Ω.
Defining uε (x) = η(x)Uε (x) ∈ H(Ω). By a direct calculation, we have
Z
Z
Z
N
2
2
|∆uε | dx =
|∆Uε | dx +
|∆uε |2 dx = S 4 + O(εN −4 ),
Ω
RN \Br (0)
Br (0)
N2−4
N −4
ε
dx = O(ε 2 ),
uε dx =
ηUε dx =
η 2
2
ε + |x|
Ω
Ω
Ω
Z
Z
Z
N
∗∗
∗∗
∗∗
|uε |2 dx =
|Uε |2 dx +
|uε |2 dx = S 4 + O(εN ),
Z
Z
Ω
Z
RN \Br (0)
Br (0)
and
Z
Z
2
2
|uε | dx =
Z
|Uε | dx +
Ω
Bε (0)
≥ c
Z
2
Bε (0)
ε
ε2 + ε2
Z
Br (0)\Bε (0)
N −4
dx + c
Ω\Br (0)
2
Z
Br (0)\Bε (0)
η2
Ω\Br (0)
≥
|uε |2 dx
|Uε | dx +
+c2 εN −4
(
Z
2
(ε2
ε
|x|2 + |x|2
N −4
1
dx
+ |x|2 )N −4
dε4 | ln ε| + O(ε4 ),
if N = 8,
dε4 + O(εN −4 ),
if N ≥ 9.
Let us define
Z
2
Z
2
(|∆u| − δ(x)u )dx : u ∈ Huε \ {0},
Mε := max
Ω
2∗∗
|u|
dx = 1 .
Ω
We claim that
for ε > 0 small enough and N ≥ 8, we have Mε < S.
In fact, take
zε = uε −
k Z
X
uε ei dx ei ,
Ω
i=1
and we may assume that u = y + tuε = y + tzε with t > 0 and kukL2∗∗ (Ω) = 1 such that
Z
(|∆u|2 − δ(x)u2 )dx = Mε .
Ω
By Hölder’s inequality, we have
kykL2 (Ω) ≤ c1 kukL2∗∗ (Ω) = c1 .
22
Since dimH − (Ω) < +∞, then
kykL2∗∗ (Ω) ≤ CkykL2 (Ω) ≤ c2 .
Note that
ktzε kL2∗∗ (Ω) = ku − ykL2∗∗ (Ω) ≤ kukL2∗∗ (Ω) + kykL2∗∗ (Ω) ≤ 1 + c2
and
kzε kL2∗∗ (Ω)
k Z
X
uε ei dx ei ≥ kuε kL2∗∗ (Ω) − RN
L2∗∗ (Ω)
i=1
1 N −4
≥ S 4 >0
2
for ε > 0 small enough. Then we can easily obtain that 0 < t ≤ c3 . Thus
Z
Z "
|y|2 dx =
y−t
Ω
Ω
Ω
i=1
Z
2
2
|y| dx + 2t
≤ 2
#2
uε ei dx ei dx
k Z
X
Ω
k Z
X
i=1
2
uε ei dx
Ω
≤ c4 .
Again by dimH − (Ω) < +∞, we get that
|y| ≤ kykL∞ (Ω) ≤ CkykL2 (Ω) ≤ c5 .
Thus we have
1=
∗∗
kuk2L2∗∗ (Ω)
2∗∗
∗∗
kuε k2L2∗∗ (Ω)
∗∗ 2∗∗ −1
Z
∗∗
+2 t
u2ε −1 ydx
Ω
Z
∗∗
∗∗
∗∗
≥ t2 kuε k2L2∗∗ (Ω) − c6 |uε |2 −1 dxkykL2 (Ω) .
≥ t
Ω
Note that
kuε k20 − µk kuε k2L2 (Ω)
kuε k2L2∗∗ (Ω)
 4
S − µk dε4 | ln ε| + O(ε4 )


, if N = 8,

1

(S 2 + O(ε4 )) 2
N
=
S 4 − µk dε4 + O(εN −4 )



if N ≥ 9
N
2

(S 4 + O(εN )) 2∗∗
S − µk dS −1 ε4 | ln ε| + O(ε4 ), if N = 8,
4−N
=
S − µk dS 4 ε4 + O(εN −4 ), if N ≥ 9,
23
which conclude that
Mε ≤
µk (L0 )kyk2L2 (Ω)
+
≤ µk (L0 )kyk2L2 (Ω) +
kuε k20 − µk kuε k2L2 (Ω)
kuε k2L2∗∗ (Ω)
ktuε k2L2∗∗ (Ω) + c7 kuε kL1 (Ω) kykL2 (Ω)
kuε k20 − µk kuε k2L2 (Ω)
Z
|uε |
1 + c6
kuε k2L2∗∗ (Ω)
2∗∗ −1
2
2∗∗
dxkykL2 (Ω)
Ω
+c7 kuε kL1 (Ω) kykL2 (Ω)

 (S − µk dε4 S −1 ) 1 + O(ε2 )kykL2 (Ω) + O(ε4 ),
≤
4−N
N −4
 (S − µk dε4 S 4 ) 1 + O(ε 2 )kykL2 (Ω) + O(εN −4 ),
if N = 8,
if N ≥ 9,
< S.
Since for each u ∈ Huε \ {0},
2
JΩ (u) ≤ max JΩ (tu) =
t≥0
N
N
!
R
kuk20 − Ω δ(x)u2 dx
.
kuk2L2∗∗ (Ω)
N
Then maxu∈Huε JΩ (u) ≤ N2 Mε4 < N2 S 4 for ε > 0 small and N ≥ 8. Remark 2.8 immediately
N
implies that c(Ω) < N2 S 4 for N ≥ 8.
Lemma 3.4 For p = 2∗∗ , N ≥ 8, c(Ω) is achieved by a nontrivial solution u of (3.1) in NΩ .
Proof: : By Ekeland’s Variational Principle and the definition of c(Ω), we can easily get a
(P S)c(Ω) sequence {un }. Moreover, {un } is bounded in H(Ω). Then up to a subsequence, we
may assume that

 un * u in H(Ω),
∗∗
un * u in L2 (Ω),

un → u in L2 (Ω).
Let vn = un − u, by Brézis-Lieb’s Lemma, we have
Z
Z
Z
2
2
|∆un | dx =
|∆u| dx + |∆vn |2 dx + o(1),
Ω
Ω
Z
2∗∗
|un |
Z
2∗∗
|u|
dx =
Ω
Ω
Z
dx +
Ω
∗∗
|vn |2 dx + o(1).
Ω
A direct computation shows that
1
JΩ (un ) = JΩ (u) +
2
and
JΩ0 (un )un
=
JΩ0 (u)u
Z
1
|∆vn | dx − ∗∗
2
Ω
Z
2
1
+ |∆vn | dx − ∗∗
2
Ω
2
24
Z
∗∗
|vn |2 dx + o(1)
Ω
Z
Ω
∗∗
|vn |2 dx + o(1).
It is easy to see that JΩ0 (u) = 0 and JΩ (u) ≥ 0. We may assume that
Z
Z
∗∗
2
b = lim
|∆vn | dx = lim
|vn |2 dx > 0.
n→+∞
n→+∞
Ω
Ω
On one hand,
Z
|vn |
b = lim
n→+∞
2∗∗
Ω
Z
2∗∗
|∆vn | dx ≥ S lim
dx = lim
n→+∞
Z
2
n→+∞
Ω
|vn |
2
2∗∗
2
dx
= Sb 2∗∗ .
Ω
N
Thus b ≥ S 4 . But on the other hand,
2 N
1
S 4 > c(Ω) ≥
lim
N
2 n→+∞
Thus b < S
for c(Ω).
4
N
4
Z
1
|∆vn | dx − ∗∗ lim
2 n→+∞
Ω
2
Z
∗∗
|vn |2 dx =
Ω
2
b.
N
which leads to a contradiction. Therefore, un → u in H(Ω) and u is a minimizer
2
Existence of least energy solutions
In this section, we consider the existence of least energy solutions for (1.1). We use the same
notations as in Section 1. Recall that {un } ⊂ Xλ is called a Palais-Smalec sequence ((P S)c
sequence in short) for functional Jλ (u) if
Jλ (un ) → c and Jλ0 (un ) → 0 in Xλ0 , as n → +∞
where Xλ0 is the dual space of Xλ . We say that the functional Jλ (u) satisfies (P S)c condition if
any of the (P S)c sequence (up to a subsequence, if necessary){un } converges strongly in Xλ .
In the following subsections, we firstly present some properties of the (P S)c sequence of Jλ (u)
and then we prove the existence of least energy solutions of (1.1) both in subcritical and critical
cases.
4.1
Properties of (P S)c sequence
Lemma 4.1 For 2 < p ≤ 2∗∗ , λ > Λ2 , if {un } is a (P S)c sequence for Jλ (u), then {un } is
bounded in Xλ . Furthermore, if un * 0 in Xλ , then up to a subsequence,
lim sup kun k2λ ≤
n→+∞
2p
c.
p−2
(4.1)
Proof: Since {un } is a (P S)c sequence of Jλ (u), then for λ > Λ2 , we have
1
c + o(1) + o(kun kλ ) = Jλ (un ) − Jλ0 (un )un
p
Z
1 1
=
−
(|∆un |2 dx + Vλ u2n )dx
2 p
N
R
25
(4.2)
and
Z
1 0
1 1
c + o(1) + o(kun kλ ) = Jλ (un ) − Jλ (un )un =
−
|un |p dx.
2
2 p
N
R
By Hölder’s inequality, we have
Z
p2
Z
Z
1− p2
−
p
2
2
Vλ |un | dx ≤ kδ(x)kL∞
|un |
.
|un | dx ≤ kδ(x)kL∞ |Ω|
RN
(4.3)
(4.4)
BR
BR
Thus by (4.2), (4.3) and (4.4), we can easily obtain that un is bounded in Xλ for λ > Λ2 .
Furthermore, if unR* 0 in Xλ as n → +∞, then up to a subsequence, by Lebesgue Dominated
theorem, we have RN Vλ− |un |2 dx → 0 as n → +∞. Thus (4.1) holds directly from (4.2).
2
Lemma 4.2 For 2 < p ≤ 2∗∗ , λ > Λ2 , {un } is a (P S)c sequence of Jλ , if un * 0 in Xλ as
n → +∞. Then there exists a subsequence such that one of the following statements holds:
R
(i) lim inf n→+∞ RN |un |p dx = 0;
(ii) There exists σ > 0 which is independent of λ such that
Z
lim inf
|un |p dx ≥ σ.
n→+∞
RN
Proof: Since {un } is a (P S)c sequence of Jλ and un * 0 in Xλ as n → +∞, then up to a
subsequence, by Lebesgue Dominated theorem, we have
Z
Z
Z
p
2
2
|un | dx + o(1) =
(|∆un | + Vλ |un | )dx =
(|∆un |2 + Vλ+ |un |2 )dx + o(1).
RN
RN
RN
By Sobolev embedding theorem, for λ > Λ2 we have
Z
Z
+
2
2
(|∆un | + Vλ |un | )dx ≥ Λ
RN
p
|un | dx
p2
,
RN
R
where Λ does not depend on λ. Thus if lim inf n→+∞ RN |un |p dx 6= 0, then
Z
p
lim inf
|un |p dx ≥ Λ p−2 .
n→+∞
RN
p
We complete the proof of this lemma by taking σ = Λ p−2 .
2
Lemma 4.3 Let 2 < p < 2∗∗ , N ≥ 5 and M > 0 be a constant which does not depend on λ,
then for any ε > 0, there exist Λε > Λ2 such that for any λ > Λε , c < M , {un } is a (P S)c
sequence of Jλ and un * 0 in Xλ as n → +∞, up to a subsequence, we have
Z
lim sup
|un |p dx ≤ ε,
n→∞
c
BR
where BRc = x ∈ RN : |x| ≥ R . Especially, there exists Λ3 > Λ2 such that
Z
σ
lim sup
|un |p dx ≤ .
c
2
n→∞
BR
26
Proof: For λ > Λ2 , by (2.2), we have
Z
Z
1
2
un dx ≤
(λV (x) − δ(x))u2n dx
c
λM0 − kδ(x)kL∞ BRc
BR
Z
1
≤
(|∆un |2 + Vλ (x)u2n )dx
λM0 − kδ(x)kL∞ BRc
Z
1
(|∆un |2 + Vλ+ |un |2 )dx
≤
λM0 − kδ(x)kL∞ RN
2pM
≤
→ 0 as λ → ∞.
(p − 2)(λM0 − kδ(x)kL∞ )
By using Hölder’s inequality and Sobolev imbedding theorem, as λ → +∞ we have
Z
|un |p dx ≤ C
=
θ
2∗∗
+
1−θ
.
2
!
≤ Ckun kpθ
λ
Z
pθ
2pM
p−2
Z
dx
c
BR
≤C
1
p
2∗∗
|un |
c
BR
where
! N2N−4 pθ
Z
! p(1−θ)
2
|un |2 dx
c
BR
p(1−θ)
2
|un |2 dx
c
BR
Z
! p(1−θ)
2
|un |2 dx
→ 0,
c
BR
Thus the proof of the lemma is completed.
2
We complete this subsection by showing the following lemma which compare cλ and c(Ω).
Lemma 4.4 For λ > Λ2 , 2 < p ≤ 2∗∗ , the following estimate holds:
0 < τ < cλ ≤ c(Ω).
Proof:Since NΩ ⊂ Nλ , then cλ ≤ c(Ω). According to (iv) in Lemma 2.9, we know that cλ >
τ > 0. Thus we complete the proof of this lemma.
2
4.2
Existence of least energy solutions in subcritical case
In this subsection, we are concerned with the existence of least energy solutions for subcritical
case. To begin with, we give the following proposition.
Proposition 4.5 For any λ > Λ3 , 2 < p < 2∗∗ , cλ := inf Nλ Jλ (u) is achieved by some u 6= 0.
Proof: For any λ > Λ3 , 2 < p < 2∗∗ , by the definition of cλ and Ekeland Variational Principle,
there exits a (P S)cλ sequence {un } of Jλ (u). By Lemma 4.1, we know that {un } is bounded in
27
Xλ . Then up to a subsequence, we have

un



un
un



un
* u in Xλ ,
* u in Lp (RN ),
→ u in Lploc (RN ),
→ u a.e. in RN
as n → ∞. Thus Jλ0 (u) = 0 and
1
Jλ (u) = Jλ (u) − Jλ0 (u)u =
2
1 1
−
2 p
Z
|u|p dx ≥ 0.
RN
Let vn = un − u, by Brezis-Lieb’s Lemma( see [6]), we obtain that
kun k2λ = kuk2λ + kvn k2λ ,
kun kpLp (RN ) = kukpLp (RN ) + kvn kpLp (RN ) .
It is easy to obtain that
Jλ (un ) = Jλ (u) + Jλ (vn ) + o(1),
Jλ0 (un )un = Jλ0 (u)u + Jλ0 (vn )vn + o(1).
According to Lemma 8.1 and Lemma 8.2 in [33], we know that {vn } is a (P S)d sequence of Jλ
where d = cλ − Jλ (u).
We may assume lim kvn kpLp (RN ) = b. If b = 0, we easily obtain that vn → 0 in Xλ , which
n→+∞
implies un → u in Xλ . If b > 0, then by Lemma 4.2, we have b ≥ σ. On the other hand, if we
take M = c(Ω), then by Lemma 4.3 we immediately have
Z
σ
p
|vn |p dx ≤ ,
b = lim kvn kLp (RN ) = lim
n→+∞ B c
n→+∞
2
R
which leads to a contradiction. Thus un → u in Xλ and Jλ (u) = cλ > 0. This implies u ∈ Nλ .
Therefore, cλ is achieved by some u ∈ Nλ and u is a nontrivial least energy solution to (1.1) for
any λ > Λ3 .
2
4.3
Existence of least energy solutions in critical case
In this subsection, we consider the existence of least energy solutions for (1.1) in the critical case
p = 2∗∗ . We have the following proposition.
Proposition 4.6 For p = 2∗∗ , λ > Λ3 , then cλ := inf Nλ Jλ (u) is achieved by some u 6= 0.
Proof: For any λ > Λ3 , by the definition of cλ and Ekeland Variational Principle, there exists
a (P S)cλ sequence {un } of Jλ (u). According to Lemma 4.1, we know that {un } is bounded in
Xλ . Then up to a subsequence, we have

 un * u in Xλ ,
∗∗
un * u in L2 (RN ),

un → u a.e. in RN .
28
Thus Jλ0 (u) = 0 and
1
1
1
Jλ (u) = Jλ (u) − Jλ0 (u)u = ( − ∗∗ )
2
2 2
Z
∗∗
|u|2 dx ≥ 0.
RN
Let vn = un − u, by Brézis Lieb’s lemma, we have
kun k2λ = kuk2λ + kvn k2λ + o(1),
∗∗
∗∗
∗∗
kun k2L2∗∗ (RN ) = kuk2L2∗∗ (RN ) + kvn k2L2∗∗ (RN ) + o(1).
It is easy to obtain that
Jλ (un ) = Jλ (u) + Jλ (vn ) + o(1),
and
Jλ0 (un )un = Jλ0 (u)u + Jλ0 (vn )vn + o(1).
According to Lemma 8.1 and Lemma 8.2 in [33], we know that {vn } is a (P S)d sequence of Jλ
where d = cλ − Jλ (u). We may assume that
∗∗
lim kvn k2λ = lim kvn k2L2∗∗ (RN ) = b > 0.
n→+∞
n→+∞
On the one hand, we have
Z
b =
=
≥
∗∗
|vn |2 dx
lim
n→+∞
N
ZR
lim
n→+∞
N
ZR
lim
n→+∞
(|∆vn |2 + Vλ+ vn2 )dx
|∆vn |2 dx
RN
Z
≥ S lim
n→+∞
2∗∗
|vn |
2
2∗∗
dx
2
= Sb 2∗∗ ,
RN
N
Thus b ≥ S 4 .
By Lemma 2.7, Lemma 3.3 and Lemma 3.4, we know that
0 < cλ ≤ c(Ω) <
2 N
S4.
N
Then we have
2 N
S 4 > cλ ≥ lim
n→+∞
N
Z
Z
1
1
1
1
+ 2
2
2∗∗
(|∆vn | + Vλ vn )dx − ∗∗
vn dx =
−
b,
2 RN
2
2 2∗∗
RN
N
Thus b < S 4 which leads to a contradiction. This implies that un → u strongly in Xλ and cλ is
achieved by u in Nλ . Thus u ∈ Nλ is a least energy solution of (1.1).
2
29
5
Asymptotic behavior of least energy solutions
In this section, we study the asymptotic behavior of cλ as λ → +∞.
We firstly give the asymptotic behavior of cλ in the subcritical case, we have the following
lemma.
Lemma 5.1 Let 2 < p < 2∗∗ , N ≥ 5, then for any λn → +∞, up to a subsequence (still denoted
by λn ), we have
lim cλn = c(Ω).
λn →+∞
Proof: Since 0 < τ ≤ cλ ≤ c(Ω) < +∞ for λ > Λ3 , then up to a subsequence, we may assume
0<τ ≤
lim cλn = k ≤ c(Ω).
λn →+∞
For n = 1, 2, · · · , let un ∈ Xλn satisfies Jλn (un ) = cλn and Jλ0 n (un ) = 0. According to
Lemma 4.1, {kun kλn } is bounded. By Lemma 2.1, {un } is also bounded in H 2 (RN ). Up to a
subsequence, we have

un * u in H 2 (RN ),



un → u in Lploc (RN ),
un * u in Lp (RN ),



un → u a.e. in RN .
Firstly, we claim that u|Ωc = 0, where Ωc =: x : x ∈ RN \ Ω .
If not, we have u|Ωc 6= 0. Then there exists a compact subset F ⊂ Ωc with dist {F, ∂Ω} > 0
such that u|F 6= 0 and
Z
Z
u2n dx →
F
u2 dx > 0, as n → ∞.
F
Moreover, by assumption (V2 ), there exists ε0 > 0 such that V (x) ≥ ε0 for any x ∈ F. Since
{un } is bounded in H 2 (RN ) and
Z
Z
2
2
(|∆un | + Vλn un )dx =
|un |p dx,
RN
RN
then
Z
Z
1
1
2
2
(|∆un | + Vλn un )dx −
|un |p dx
Jλn (un ) =
2 RN Z
p RN
1 1
=( − )
(|∆un |2 + Vλn u2n )dx
2 p RN
Z
Z
1 1
2
2
≥( − )
λn V (x)un dx − kδ(x)kL∞
un dx
2 p
N
N
R
R
Z
Z
1 1
2
2
2
≥( − )
λn ε0 un dx − kδ(x)kL∞
(|∆un | + un )dx
2 p
F
RN
→ +∞ as n → +∞.
This contradiction shows that u|Ωc = 0, by the smooth assumption on ∂Ω we have u ∈ H(Ω).
30
Now we are going to show that
un → u in Lp (RN ).
(5.1)
Suppose (5.1) is not true, then by the Concentration Compactness Principle of P. L. Lions (see
[24]), there exist δ > 0, ρ > 0 and xn ∈ RN with |xn | → +∞ such that
Z
lim sup
|un − u|2 dx ≥ δ > 0.
(5.2)
n→∞
Bρ (xn )
By the choice of {un } and the facts that u|Ωc = 0, we have
Z
1 1
Jλn (un ) = ( − )
(|∆un |2 + Vλn u2n )dx
2 p RN
!
Z
Z
1 1
≥( − )
λn V (x)u2n dx − kδ(x)kL∞
u2n dx
c (0)
2 p
Bρ (xn )∩BR
RN
!
Z
Z
1 1
≥ ( − ) λn M0
|un − u|2 dx − kδkL∞
u2n dx
2 p
Bρ (xn )
RN
→ +∞.
This contradiction induce that un → u in Lp (RN ).
Since Jλ0 n (un ) = 0, then for any ψ ∈ H(Ω), we have
Z
Z
(∆un ∆ψ + Vλn un ψ)dx =
|un |p−2 un ψdx.
RN
RN
Let n → +∞, we have
Z
Z
|u|p−2 uψdx.
(∆u∆ψ − δuψ)dx =
Ω
Ω
Thus JΩ0 (u) = 0. Since
Jλn (un ) =
Then k = ( 21 − p1 )
R
RN
1 1
−
2 p
Z
p
|un | dx =
RN
1 1
−
2 p
Z
|u|p dx + o(1).
Ω
|u|p dx > 0 which implies u 6= 0. Thus u ∈ NΩ and
JΩ (u) =
1 1
−
2 p
Z
|u|p dx = k ≥ c(Ω).
RN
This implies that k = c(Ω). Furthermore, by Brézis-Lieb’s Lemma, we obtain that kun −uk2λn →
0 as n → +∞. Thus according to Lemma 2.1, we have un → u in H 2 (RN ).
2
Now we give the asymptotic behavior of cλ in the critical case and which is
31
Lemma 5.2 Let N ≥ 8, p = 2∗∗ , then for any λn → +∞, up to a subsequence (still denoted by
λn ), we have
lim cλn = c(Ω).
λn →+∞
Proof: Since 0 < τ ≤ cλ ≤ c(Ω) < +∞ for λ > Λ3 , then up to a subsequence, we may assume
0<τ ≤
lim cλn = k ≤ c(Ω).
λn →+∞
For n = 1, 2, · · · , let un ∈ Xλn satisfies Jλn (un ) = cλn and Jλ0 n (un ) = 0. As proved in Lemma
4.1, we can easily get that {un } is bounded in Xλn , namely kun kλn ≤ C for some C > 0.
According to Lemma 2.1, {un } is also bounded in H 2 (RN ). Then up to a subsequence, we have

un * u in H 2 (RN ),


∗∗

un * u in L2 (RN ),
un → u in L2loc (RN ),



un → u a.e. in RN .
Similar to the proof of Lemma 5.1, we have u = 0 on RN \ Ω. Thus for each φ ∈ H(Ω), as
n → +∞, we have
0 = Jλ0 n (un )φ
Z
Z
∗∗
u2n φdx
(∆un ∆φ + Vλn un φ)dx −
=
n
RN
ZR
Z
∗∗
→
(∆u∆φ − δu2 φ)dx − u2 φdx
Ω
Ω
= JΩ0 (u)φ.
Thus JΩ0 (u) = 0. Furthermore, we have
1
JΩ (u) = JΩ (u) − JΩ0 (u)u =
2
1
1
− ∗∗
2 2
Let vn = un − u, by Brézis-Lieb’s Lemma, we have
Z
Z
Z
2
2
|∆un | dx =
|∆u| dx +
RN
2∗∗
|un |
Z
RN
2∗∗
|u|
dx =
∗∗
|u|2 dx ≥ 0.
Ω
|∆vn |2 dx + o(1),
RN
Ω
Z
Z
Z
dx +
∗∗
|vn |2 dx + o(1)
RN
Ω
and
Z
RN
Vλn u2n dx
Z
Z
2
=
Vλn vn2 dx
Z
Vλn u dx +
+
2Vλn uvn dx
N
N
Z R
ZR
= − δu2 dx +
Vλn vn2 dx − 2 δuvn dx
N
Ω
ZΩ
ZR
= − δu2 dx +
Vλn vn2 dx + o(1).
N
RZ
RN
Ω
32
Thus we can easily get that
Jλn (un ) = JΩ (u) + Jλn (vn ) + o(1), Jλ0 n (un )un = JΩ0 (u)u + Jλ0 n (vn )vn + o(1).
We may assume that
Z
(|∆vn | +
b = lim
n→+∞
2
RN
Vλn vn2 )dx
Z
∗∗
|vn |2 dx > 0.
= lim
n→+∞
RN
On the one hand, by Sobolev inequality, we have
Z
∗∗
b = lim
|vn |2 dx
n→+∞ RN
Z
= lim
(|∆vn |2 + Vλn vn2 )dx
n→+∞ RN
Z
= lim
(|∆vn |2 + Vλ+n vn2 )dx
n→+∞ RN
Z
≥ lim
|∆vn |2
n→+∞
RN
Z
≥
|vn |
lim S
n→+∞
2∗∗
2
2∗∗
dx
.
RN
N
Thus b ≥ S 4 . Recall that
1
Jλn (vn ) = Jλn (vn ) − Jλn (vn )vn + o(1) =
2
1
1
− ∗∗
2 2
Z
∗∗
|vn |2 dx + o(1),
RN
then
2 N
S4
N
> c(Ω) ≥ k ≥ lim Jλn (vn )
n→+∞
Z
1
1
1
1
2∗∗
=
−
lim
|vn | dx =
−
b.
2 2∗∗ n→+∞ RN
2 2∗∗
N
∗∗
Thus b < S 4 which leads to a contradiction. This implies that un → u in L2 (RN ). According
to Lemma 2.1, we known that un → u in H 2 (RN ). Furthermore,
Z
Z
1
1
1
1
∗∗
2∗∗
− ∗∗
|u| dx = ( − ∗∗ ) lim
|un |2 dx
JΩ (u) =
2 2
2 2 n→+∞ RN
Ω
= lim Jλn (un ) = k > 0,
n→+∞
which implies u 6= 0. Hence, u ∈ NΩ and
c(Ω) ≤ JΩ (u) = k ≤ c(Ω)
2
which implies JΩ (u) = c(Ω).
33
Finally, we complete our paper by proving our main result Theorem 1.4.
Proof of Theorem 1.4: The existence of least energy solutions to (1.1) is proved by Proposition
4.5 and Proposition 4.6 for λ > Λ2 . The asymptotic behavior of least energy solutions follows
from Lemma 5.1 and Lemma 5.2 for λ → +∞. Thus we complete the proof of our main result
Theorem 1.4.
2
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