Concentration and multiple normalized solutions for a class of biharmonic Schrödinger equations *

Abstract

In the present paper, we study the existence and concentration of multiple normalized solutions to the following nonlinear biharmonic Schrödinger equation: ${\begin{cases} ε^{4} Δ^{2} u + V (x) u = λ u + h (u), & x \in R^{N}, \\ \int_{R^{N}} | u |^{2} d x = c^{2} ε^{N}, & x \in R^{N}, \end{cases}$ where $ε > 0$ is a positive parameter, $λ \in R$ is unknown and appears as a Lagrange multiplier, and V is a positive potential such that $inf_{Λ} V < inf_{\partial Λ} V$ for some open bounded subset $Λ \subset R^{N} (N ⩾ 5)$ . Applying the penalization techniques and Ljusternik–Schnirelmann theory, we obtain multiple mormalized solutions $u_{ε}$ . When $ε \to 0$ , these solutions concentrates around a local minimum of V.

This paper extends the results of Alves and Thin (2023), which considered the nonlinear Schrödinger equations with general nonlinearities, to the biharmonic Schrödinger equations. We develop a truncated skill to obtain the minimum via careful analysis. Moreover, we also obtain orbital stability of the solutions.

Keywords

Biharmonic Schrödinger equation Normalized solutions Multiplicity Ljusternik–Schnirelmann theory

1. .Introduction and main results

In this paper, we investigate the existence of normalized solutions for the following nonlinear biharmonic Schrödinger equation:

\begin{aligned} {\begin{cases} ε^{4} Δ^{2} u + V (x) u = λ u + h (u), & x \in R^{N}, \\ \int_{R^{N}} | u |^{2} d x = c^{2} ε^{N}, & x \in R^{N}, \end{cases} \end{aligned}

(1.1)

where

ε > 0

is a positive parameter,

λ \in R

is unknown, the function

h \in C (R, R)

and V is a positive potential such that

inf_{Λ} V < inf_{\partial Λ} V

for some open bounded subset

Λ \subset R^{N}

For the classical operator Δ, when $ε = 1, λ = 0$ , problem (1.1) boils down to the following Schrödinger equation

\begin{aligned} - Δ u + V (x) u = g (x, u), x \in R^{N} . \end{aligned}

(1.2)

The model was proposed as a result of expanding the Feynman path integral, from the Brownian like to the Lévy like quantum mechanical paths. In the past few years, it has attracted wide attention from scholars and obtained many interesting results.

For the biharmonic operator, we consider following time-dependent equation:

\begin{aligned} i \partial_{t} ψ - Δ^{2} ψ + f (| ψ |) ψ = 0. \end{aligned}

(1.3)

As everyone knows, if we set

ψ (t, x) = e^{- i λ t} u (x)

in (1.3), then we can get that u is a solution of (1.1). In magnetic materials, [18,32] considered (1.3) to study the stability of solitons. For the relevant results of equation (1.3), please refer to [12,24,25,27,34] and its references. In [19,20], for fourth-order Schrödinger equations of the mixed dispersion, the biharmonic operator

Δ^{2}

was used to indicate the effects of higher order dispersion term.

As the case of $f (| ψ |) ψ = | x |^{- b} | ψ |^{2 σ} ψ$ in (1.3), Cardoso et al. [6] studied the local well-posedness and established a Gagliardo–Nirenberg type inequality of equation

\begin{aligned} i \partial_{t} ψ + Δ^{2} ψ - | x |^{- b} | ψ |^{2 σ} ψ = 0, \end{aligned}

(1.4)

where

σ > 0

and

b > 0

. They used compact embedding of

L^{p} \cap H^{2}

into a weighted

L^{2 σ + 2}

space to study the phenomenon of

L^{σ_{c}}

- norm concentration for finite time blow up solutions. When

f (| u |) u = | u |^{p - 2} u

, (1.3) is

L^{2}

-supercritical if

p > 2 + \frac{8}{N}

L^{2}

-critical if

p = 2 + \frac{8}{N}

and

L^{2}

-subcritical if

p \in (2, 2 + \frac{8}{N})

. Not so long ago, Phan [28] studied the following biharmonic nonlinear Schrödinger functional

I_{a} (u) = \int_{R^{N}} {| Δ u (x) |}^{2} d x + \int_{R^{N}} V (x) {| u (x) |}^{2} d x - a \int_{R^{N}} {| u (x) |}^{q} d x

under constraint

\int_{R^{N}} {| u (x) |}^{2} d x = 1,

where

q = 2 + \frac{8}{N}

. When the potential function is coercive and the nonlinear term is

L^{2}

-critical nonlinearity, the author got that there exists normalized solutions for biharmonic equation. Furthermore, Ma and Chang in [24] considered the mixed nonlinear term, which involving a

L^{2}

-subcritical term and a Sobolev critical term. For the prescribed mass, according to discuss the behavior of the minimum energy, they obtained normalized ground state solutions.

Liang and Zhang [23] dealt with the existence and multiplicity of solutions for the following equation

- ε^{4} Δ^{2} u + ε^{4} (a + b \int_{R^{N}} | \nabla u |^{2} d x) Δ u + V (x) u = | u |^{2^{* *} - 2} u + h (x, u), (t, x) \in R \times R^{N} .

Under some proper assumptions on V and h, combining concentration-compactness principle and variational method, the authors proved that for any

m \in N

, above equation has at least m pairs of solutions. He et al. [17] considered the concentration and existence properties of standing waves for the following singularly perturbed equation with mixed dispersion:

\begin{aligned} ε^{4} Δ^{2} u - β ε^{2} Δ u + V (x) u = | u |^{p - 2} u in R^{N}, u \in H^{2} (R^{N}) . \end{aligned}

(1.5)

They first established a local

W^{4, p}

-estimate which is very important to get the global and uniform

L^{\infty}

-estimate, and then, under some appropriate assumptions on parameter and the potential, they proved that these solutions concentrate around a local minimum of V. When

β = 0

in (1.5), by using a penalization-type method which first introduced by del Pino and Felmer [9], Pimenta and Soares obtained the existence of ground state solution and concentration of nontrivial solutions when

| u |^{p - 2} u

was replaced by a general nonlinearity h.

In the view of physical, it’s more worthy to study the semi-classical normalized solutions of Schrödinger equation with λ arising as a Lagrange multiplier. Very recently, Alves and Thin in [2] studied multiple normalized solutions to equation

\begin{aligned} {\begin{cases} - ε^{2} Δ u + V (x) u = λ u + h (u), & x \in R^{3}, \\ \int_{R^{3}} | u |^{2} d x = c^{2} ε^{3}, & x \in R^{3} \end{cases} \end{aligned}

(1.6)

including non-autonomous cases by variational methods. It is worth mentioning that, h in (1.6) satisfies the

L^{2}

-subcritical growth, and V satisfies global potential conditions. After that, they concerned with (1.6) under the local potential conditions in [3]. Since the conditions of V is not same, the method in [3] needs to be improved. To be precise, in this case, they used penalization-type method and Ljusternik–Schnirelmann theory to obtain the multiplicity of normalized solutions to the related problem.

The existence of multiple solutions for the following equation

\begin{aligned} {\begin{cases} Δ^{2} u = u^{p} + ε f (x, u), & u > 0, in Ω, \\ Δ u = u = 0, & on \partial Ω \end{cases} \end{aligned}

(1.7)

has been proved in [10] when

ε

is small enough. Here

Ω \subset R^{N}

is a smooth bounded domain, f satisfied a critical exponential growth. When

f (x, u) = u

, Mehdi–Selmi [10] got that if

u_{ε}

are solutions of (1.7) concentrating around a point

x_{0}

when

ε \to 0

, then

x_{0}

is a critical point of the Robin’s function ϕ. On the other hand, they also got that any nondegnerate critical point

x_{0}

of ϕ can generates multiple solutions of (1.7) concentrating around

x_{0}

ε \to 0

. Using Ljusternik–Schnirelman category throey, the authors proved that (1.7) has multiple solutions. When

f (x, u) = f (x) ≢ 0

, Selmi [30] constructed multiple solutions concentrating around the well defined points depending on f. We refer to the recent papers [8,11,15,22] for related concentration properties.

Motivated by the above fact, it is quite natural to ask, does (1.1) have multiple normalized solutions? In this paper, we will give an affirmative answer. We are very interested in the numbers of solutions for (1.1) which is respect with the topology of the set ${x \in Λ : V (x) = V_{0}}$ . As far as we know, there are very few results about (1.1) for each $c > 0$ . Our idea is to extend the conclusion in [3] to biharmonic operators. In addition, to make the research more interesting, the orbital stability of the solutions is also investigated. Therefore, we need to introduce new techniques and more precise analysis.

In the present paper, we give some hypothesis about V

$(V 1)$ $0 ⩽ V \in C (R^{N}, R) \cap L^{\infty} (R^{N});$

$(V 2)$ there exists a bounded open set $Λ \subset R^{N}$ such that

0 < V_{0} := min_{\bar{Λ}} V < V_{\infty} := min_{\partial Λ} V .

Moreover, we assume that $0 \in Λ$ without lost of generality.

$h : R \to R$ is continuous and satisfying the following assumptions:

$(h_{1})$ h is odd and $lim_{s \to 0} \frac{| h (s) |}{| s |^{q_{0} - 1}} =: α_{0} > 0$ for some $q_{0} \in (2, 2 + \frac{8}{N})$ ;

$(h_{2})$ there exist constants $c_{1}, c_{2} > 0$ and $p_{1} \in (2, 2 + \frac{4}{N})$ such that, for all $s \in R,$

| h (s) | ⩽ c_{1} + c_{2} | s |^{p_{1} - 1};

$(h_{3})$ $h (s) / s^{q - 1}$ is increasing for $q \in [q_{0}, 2 + \frac{8}{N})$ .

Now we give a useful Lions’ type lemma in $H^{2}$ which can be obtained as that in [33]:

Lemma 1.1.

If ${u_{n}}$ is bounded in $H^{2} (R^{N})$ and satisfies

lim_{n \to \infty} sup_{y \in R^{N}} \int_{B_{R} (y)} {| u_{n} |}^{2} d x = 0, \forall R > 0,

then for all

2 < r < 2^{* *} = \frac{2 N}{N - 4}

u_{n} \to 0

L^{r} (R^{N})

Next, we recall a well-known Gagliardo–Nirenberg inequalities for functions $u \in H^{2} (R^{N})$ , (see, for instance, [13,26]), namely,

\begin{aligned} ‖ u ‖_{l} ⩽ B_{N, l} ‖ Δ u ‖_{2}^{\frac{(l - 2) N}{4 l}} ‖ u ‖_{2}^{1 - \frac{(l - 2) N}{4 l}}, \end{aligned}

(1.8)

where $B_{N, l} > 0$ is an optimal constant and

{\begin{cases} 2 ⩽ l, & N ⩽ 4, \\ 2 ⩽ l ⩽ \frac{2 N}{N - 4}, & N > 4. \end{cases}

The main results of this paper read as follows:

Theorem 1.1.

Suppose that h satisfies the conditions $(h_{1}) - (h_{3})$ and that V satisfies $(V 1) - (V 2)$ . Then for each $δ > 0$ such that $M_{δ} = {x \in R^{N} : dist (x, M) ⩽ δ} \subset Λ$ , there exist $ε_{δ} > 0$ and $μ_{*} > 0$ such that for $0 < ε < ε_{δ}$ and $‖ V ‖_{\infty} < μ_{*}$ , (1.1) admits at least ${cat}_{M_{δ}} (M)$ couples $(u_{j}, λ_{j}) \in H^{2} (R^{N}) \times R$ of weak solutions with $\int_{R^{N}} | u_{j} |^{2} d x = c^{2}, λ_{j} < 0$ , and $I_{ε} (u_{j}) < 0$ . Moreover, if $η_{ε}$ is the global maximum of one of these solutions, then

lim_{ε \to 0} V (η_{ε}) = V_{0} .

Theorem 1.2.

Assume $(V 1) - (V 2)$ and $(h_{1}) - (h_{3})$ hold. Then, under the flow associated with

\begin{aligned} {\begin{cases} i \frac{\partial ψ}{\partial t} - Δ^{2} ψ + V (ε x) ψ + h (| ψ |) ψ = 0, & t > 0, x \in R^{N}, \\ ψ (0, x) = u_{0} (x), \end{cases} \end{aligned}

(1.9)

Ω defined in (7.5) is orbitally stable.

The structure of this document is as follows. In Section 2, we get a variational setting for (1.1) and the associated functional. After that, we consider the autonomous case of problem (1.1) in Section 3. In Sections 4 to 6, we consider the non-autonomous cases of problem (1.1), then get multiplicity and concentration phenomena of solutions for the modified problem. Finally, we obtain orbital stability of the solutions in the last section.

The following notations will be used in the sequel.

$H^{2} (R^{N}) := {u \in L^{2} (R^{N}) : Δ u \in L^{2} (R^{N})}$ ;

$‖ u ‖_{H^{2}} =: ‖ u ‖_{H^{2} (R^{N})}^{2} = {(\int_{R^{3}} (| Δ u |^{2} + | u |^{2}) d x)}^{\frac{1}{2}}$ is a norm on $H^{2}$ which is equivalent to the standard one $‖ u ‖_{H^{2}} := {(\int_{R^{N}} (| D^{2} u |^{2} + | \nabla u |^{2} + | u |^{2}) d x)}^{\frac{1}{2}}$ ;

$L^{p} (R^{N}) (1 ⩽ p ⩽ \infty)$ denotes Lebesgue space with $‖ u ‖_{p} := {(\int_{R^{N}} | u |^{p} d x)}^{\frac{1}{p}}$ ;

$B_{r} (x) := {z \in R^{N} : | z - x | < r}$ , where $r > 0, x \in R^{N}$ ;

$C_{1}, C_{2}, C_{3}, \dots$ denote positive constants whose value can change from line to line.

2. .Variational frameworks

It is easy to see that, after a change of variable, (1.1) is equivalent to

\begin{aligned} {\begin{cases} Δ^{2} u + V (ε x) = λ u + h (u), & x \in R^{N}, \\ \int_{R^{N}} | u |^{2} d x = c^{2}, & x \in R^{N} . \end{cases} \end{aligned}

(2.1)

From now on, we shall refer to it. Once we get a solution

u (x)

of (2.1), then

v_{ε} (x) = u_{ε} (x / ε)

is a solution of (1.1). Moreover, the maximum point

ζ_{ε}

v_{ε}

is related to the maximum point

z_{ε}

u_{ε}

simply by

ζ_{ε} = ε z_{ε}

. Consequently, we need to prove that

lim_{ε \to 0^{+}} V (ζ_{ε}) = V_{0} .

First, we assume that there exist $τ, b_{0} > 0$ satisfying $\frac{h (b_{0})}{b_{0}^{q - 1}} = τ$ , define

\tilde{h} (t) := {\begin{cases} h (t), & | t | ⩽ b_{0}, \\ τ | t |^{q - 2} t, & | t | ⩾ b_{0}, \end{cases}

where q was fixed in $(h_{3})$ . Then we introduce a penalized nonlinear term

g (x, t) := χ_{\bar{Λ}} (x) h (t) + (1 - χ_{\bar{Λ}} (x)) \tilde{h} (t),

where

χ_{\bar{Λ}}

is the characteristic function on

\bar{Λ}

In view of $(h_{1}) - (h_{3})$ , we can get following properties: similar to the proof of [3].

$(g_{1})$ $lim_{t \to 0} \frac{g (x, t)}{t} = 0$ uniformly in $x \in R^{N}$ ;

$(g_{2})$ for each $x \in {\bar{Λ}}^{c}$ and $t \in R, | g (x, t) | ⩽ τ | t |^{q - 1}$ ;

$(g_{3})$ there is $C > 0$ such that for all $x \in R^{N}$ and $t \in R$ , $| g (x, t) | ⩽ C (| t |^{q_{0} - 1} + | t |^{q - 1});$

$(g_{4})$ for each $x \in R^{3}$ and $t ⩾ 0$ , $0 ⩽ g (x, t) ⩽ h (t);$ and for all $t \in R$ and $x \in R^{N}$ , $g (x, t) t ⩽ h (t) t;$

$(g_{5})$ $\forall x \in R^{N} ~and~ t \in R, G (x, t) := \int_{0}^{t} g (x, s) d s ⩽ H (t)$ .

We now remarkable that, if $u_{ε}$ is a solution of the equation

\begin{aligned} {\begin{cases} Δ^{2} u + V (ε x) u = λ u + g (ε x, u), & x \in R^{N}, \\ \int_{R^{N}} | u |^{2} d x = c^{2}, & x \in R^{N} \end{cases} \end{aligned}

(2.2)

with

| u (x) | ⩽ b_{0}

for all

x \in R^{N} ∖ \bar{Λ_{ε}}

, then we get

g (ε x, u_{ε}) = h (u_{ε})

and so,

v_{ε} (x) = u_{ε} (x / ε)

is also a solution of (1.1), where

\bar{Λ_{ε}} := {x \in R^{N} : ε x \in \bar{Λ}} .

The weak solutions of (2.2) are the critical points of $I_{ε} : H^{2} (R^{N}) \to R$ given by

I_{ε} (u) = \frac{1}{2} \int_{R^{N}} (| Δ u |^{2} + V (ε x) u^{2}) d x - \int_{R^{N}} G (ε x, u) d x, u \in H^{2} (R^{N})

restricted to the sphere

S (c) = {u \in H^{2} (R^{N}) : ‖ u ‖_{2} = c}

, here

G_{ε} (x, t) = \int_{0}^{t} g_{ε} (x, s) d s

. It is easy to get that

I_{ε} \in C^{1} (H^{2} (R^{N}))

and

⟨ I_{ε}^{^{'}} (u), φ ⟩ = \int_{R^{N}} (Δ u Δ φ + V (ε x) u φ) d x - \int_{R^{N}} g (ε x, u) φ d x, φ \in H^{2} (R^{N}) .

3. .The limit problem

In this section, we consider the limit problem associated to (2.2) as follows

\begin{aligned} {\begin{cases} Δ^{2} u + μ u = λ u + h (u), & x \in R^{N}, \\ \int_{R^{N}} | u |^{2} d x = c^{2}, & x \in R^{N} . \end{cases} \end{aligned}

(3.1)

A solution u of (3.1) is according to a critical point for functional

I_{μ} (u) = \frac{1}{2} \int_{R^{N}} (| Δ u |^{2} + μ | u |^{2}) d x - \int_{R^{3}} H (u) d x,

restricted to the restrain sphere

S (c)

defined by

S (c) = {u \in H^{2} (R^{N}) : ‖ u ‖_{2} = c} .

Now we give our main result in the present section read as follows:

Theorem 3.1.

Assume that $(h_{1}) - (h_{3})$ hold. Then, there is a $μ_{*} > 0$ such that for $0 ⩽ μ < μ_{*}$ , (3.1) has solution $(u, λ)$ , where $λ < 0$ .

Remark 3.1.

Compared with the Theorem 2.2 of [3], we can not get that the normalized solution is radial Schwarz’s Symmetrization directly because of the biharmonic operator. But combing Fourier transform with Schwarz’s Symmetrization, we also can get the solution of (3.1) is radially symmetric if $q_{0}, p_{1} \in 2 N$ , where $q_{0}, p_{1}$ are given in $(h_{2}), (h_{3})$ respectively. Precisely speaking, for $u \in L^{2} (R^{N})$ , let

u^{♯} := F^{- 1} ((F u)^{*})

be the Fourier rearrangement of

u,

where

F

and

F^{- 1}

denote the Fourier transform and the inverse Fourier transform respectively, and

(F u)^{*}

stands for the Schwarz rearrangement of

F u .

Note that

u^{♯}

is radial and

‖ u^{♯} ‖_{2} = ‖ u ‖_{2} .

Moreover, when

q_{0}, p_{1} \in 2 N,

Lemma A.1 in [5] implies that

\begin{aligned} ‖ Δ u^{♯} ‖_{2} ⩽ ‖ Δ u ‖_{2}, ‖ \nabla u^{♯} ‖_{2} ⩽ ‖ \nabla u ‖_{2}, and \\ ‖ u^{♯} ‖_{q_{0}} ⩾ ‖ u ‖_{q_{0}}, ‖ u^{♯} ‖_{p_{1}} ⩾ ‖ u ‖_{p_{1}} . \end{aligned}

(3.2)

Thus, if

q_{0}, p_{1} \in 2 N,

we denote

(\tilde{u})^{♯}

as the Fourier rearrangement of

\tilde{u},

where

\tilde{u}

is a minimizer of

m_{μ, c} := inf_{u \in S (c)} I_{μ} (u),

it follows from (3.2) that

I_{μ} ((\tilde{u})^{♯}) ⩽ I_{μ} (\tilde{u}),

which indicates that

(\tilde{u})^{♯}

is also a minimizer of

m_{μ, c} .

Hence, if

q_{0}, p_{1} \in 2 N,

(3.1) admits a ground state solution which is radially symmetric.

Theorem 3.1 considers the autonomous case, which is simpler than Theorem 1.1. To be precise, we first show that the energy functional of (3.1) is bounded from below (see Lemma 3.1), which can ensure that the energy functional has a global minimum. Next, in order to obtain the compactness theorem on $S (c)$ , we need to study the properties of global minimum, that is Lemmas 3.2, 3.3 and (3.5). Finally, we use Lagrange multiplier method to obtain the solution.

Lemma 3.1.

Any minimizing sequence ${u_{n}}$ of functional $I_{μ} |_{S (c)}$ is bounded.

Proof.

According to $(h_{1}) - (h_{2})$ , there is $C_{1}, C_{2} > 0$ such that

| H (t) | ⩽ C_{1} | t |^{q_{0}} + C_{2} | t |^{p_{1}}, \forall t \in R .

Combining (1.8), we get

\begin{aligned} I_{μ} (u_{n}) & = \frac{1}{2} \int_{R^{N}} (| Δ u_{n} |^{2} + μ u_{n}^{2}) d x - \int_{R^{N}} H (u_{n}) d x \\ ⩾ \frac{1}{2} \int_{R^{N}} | Δ u_{n} |^{2} d x - C C_{1} \int_{R^{N}} | u_{n} |^{q_{0}} d x - C_{2} \int_{R^{N}} | u_{n} |^{p_{1}} d x \\ ⩾ \frac{1}{2} \int_{R^{N}} | Δ u_{n} |^{2} d x - C C_{1} c^{(1 - β_{q_{0}}) q_{0}} {(\int_{R^{N}} | Δ u_{n} |^{2} d x)}^{\frac{β_{q_{0}} q_{0}}{2}} \\ - C C_{2} c^{(1 - β_{p_{1}}) p_{1}} {(\int_{R^{N}} | Δ u_{n} |^{2} d x)}^{\frac{β_{p_{1}} p_{1}}{2}} . \end{aligned}

q_{0}, p_{1} \in (2, 2 + \frac{8}{N})

, clearly

β_{q_{0}} q_{0}, β_{p_{1}} p_{1} < 2

, which ensures the boundedness of

{u_{n}}

From above lemma, we get that

m_{μ, c^{2}} := inf_{u \in S (c)} I_{μ} (u)

is well defined.

Lemma 3.2.

There is a $μ_{*} > 0$ such that for $0 ⩽ μ < μ_{*}$ , $m_{μ, c^{2}} < 0$ .

Proof.

From $(h_{1})$ , $lim_{t \to 0} \frac{q_{0} H (t)}{t^{q_{0}}} = α_{0} > 0$ , we get that there is a $0 < t_{0}$ such that

\begin{aligned} \frac{q_{0} H (t)}{t^{q_{0}}} ⩾ \frac{α_{0}}{2} for all t \in [0, t_{0}] . \end{aligned}

(3.3)

Let

ω_{0} \in S (c) \cap L^{\infty} (R^{N})

be a nonnegative function, define

γ (ω_{0}, θ) (x) := e^{\frac{N θ}{2}} ω_{0} (e^{θ} x) for all x \in R^{N}, θ \in R .

Then, through direct computation,

\int_{R^{N}} {| γ (ω_{0}, θ) (x) |}^{2} d x = c^{2}

and

\int_{R^{N}} H (γ (ω_{0}, θ) (x)) d x = e^{- N θ} \int_{R^{N}} H (e^{\frac{N θ}{2}} ω_{0} (x)) d x .

Moreover, if we let

θ < 0

and

| θ |

large, we can get

0 ⩽ e^{\frac{N θ}{2}} ω_{0} (x) ⩽ δ, \forall x \in R^{N},

then by (3.3)

\int_{R^{N}} H (γ (ω_{0}, θ) (x)) d x ⩾ \frac{α_{0}}{2 q_{0}} e^{\frac{(q_{0} - 2) N θ}{2}} \int_{R^{N}} {| ω_{0} (x) |}^{q_{0}} d x,

and so,

I_{μ} (γ (ω_{0}, θ)) ⩽ \frac{e^{4 θ}}{2} \int_{R^{N}} | Δ ω_{0} |^{2} d x + \frac{μ c^{2}}{2} - \frac{α_{0} e^{\frac{(q_{0} - 2) N θ}{2}}}{2 q_{0}} \int_{R^{N}} | ω_{0} (x) |^{q_{0}} d x .

Since

q_{0} \in (2, 2 + \frac{8}{N})

, we can increase

| θ |

if necessary, then we get that

\frac{e^{4 θ}}{2} \int_{R^{N}} | Δ ω_{0} |^{2} d x - \frac{α_{0} e^{\frac{(q_{0} - 2) N}{2} θ}}{2 q_{0}} \int_{R^{N}} | ω_{0} (x) |^{q_{0}} d x =: A_{θ} < 0,

thus,

I_{μ} (γ (ω_{0}, θ)) ⩽ A_{θ} + \frac{μ c^{2}}{2} .

Now, we fix

μ_{*} > 0

such that

A_{θ} + \frac{μ_{*} c^{2}}{2} < 0,

then,

I_{μ} (γ (ω_{0}, θ)) < 0, \forall μ \in [0, μ_{*}) .

Hence

m_{μ, c^{2}} < 0

Lemma 3.3.

The function $c \mapsto m_{μ, c^{2}}$ is continuous on $(0, + \infty)$ .

Proof.

It is equivalent to prove that if $ρ_{n} \to ρ$ as $n \to \infty$ , then $lim_{n \to \infty} m_{μ, ρ_{n}^{2}} = m_{μ, ρ^{2}}$ . For every $n \in N$ , let $w_{n} \in S (ρ_{n})$ such that $I_{μ} (w_{n}) < m_{μ, ρ_{n}^{2}} + \frac{1}{n} < \frac{1}{n}$ . Therefore, like Lemma 3.1, we get

\begin{aligned} \frac{1}{2} \int_{R^{N}} (| Δ w_{n} |^{2} + μ w_{n}^{2}) d x - C C_{1} ρ_{n}^{(1 - β_{q_{0}}) q_{0}} {(\int_{R^{N}} | \nabla w_{n} |^{2} d x)}^{\frac{β_{q_{0}} q_{0}}{2}} \\ - C C_{2} ρ_{n}^{(1 - β_{p_{1}}) p_{1}} {(\int_{R^{N}} | \nabla w_{n} |^{2} d x)}^{\frac{β_{p_{1}} p_{1}}{2}} \\ ⩽ \frac{1}{2} \int_{R^{N}} (| Δ w_{n} |^{2} + μ w_{n}^{2}) d x - \int_{R^{N}} H (w_{n}) d x = I_{μ} (w_{n}) < \frac{1}{n} . \end{aligned}

Since

β_{p_{1}} p_{1}, β_{q_{0}} q_{0} < 2

and

{ρ_{n}}

is bounded, we deduce that

{w_{n}} \subset H^{2} (R^{N})

is bounded. In particular

‖ Δ w_{n} ‖_{2}

and

H (w_{n})

are bounded sequences. Since

\frac{ρ}{ρ_{n}} w_{n} \in S (ρ)

, we easily find

\begin{aligned} m_{μ, ρ^{2}} & ⩽ I_{μ} (\frac{ρ}{ρ_{n}} w_{n}) \\ = \frac{1}{2} {(\frac{ρ}{ρ_{n}})}^{2} \int_{R^{N}} (| Δ w_{n} |^{2} + μ w_{n}^{2}) d x - \int_{R^{N}} H (\frac{ρ}{ρ_{n}} w_{n}) d x \\ = I_{μ} (w_{n}) + o_{n} (1) ⩽ m_{μ, ρ_{n}^{2}} + \frac{1}{n} + o_{n} (1) . \end{aligned}

(3.4)

On the other hand, given a minimum sequence

{v_{n}} \subset S (ρ)

for

m_{μ, ρ^{2}}

, we have

\begin{aligned} m_{μ, ρ_{n}^{2}} ⩽ I_{μ} (\frac{ρ_{n}}{ρ} v_{n}) = I_{μ} (v_{n}) + o_{n} (1) = m_{μ, ρ^{2}} + o_{n} (1) . \end{aligned}

(3.5)

Combining (3.4) and (3.5), we get

lim_{n \to \infty} m_{μ, ρ_{n}^{2}} = m_{μ, ρ^{2}}

Lemma 3.4.

Fix $μ \in [0, μ_{*})$ , then $m_{μ, ρ^{2}} < m_{μ, ρ_{0}^{2}} + m_{μ, ρ^{2} - ρ_{0}^{2}}$ for any $0 < ρ_{0} < ρ$ .

Proof.

Let $(u_{n}) \subset S (ρ_{0})$ be a minimizing sequence to the $m_{μ, ρ_{0}^{2}}$ , then from Lemma 3.1, there are $0 < k_{1} < k_{2}$ independent of n such that

k_{1} ⩽ ‖ Δ u_{n} ‖_{2}^{2} ⩽ k_{2} .

Set

u_{n}^{ς} (x) = u_{n} (ς^{- \frac{2}{N}} x)

with

ς > 1

. Then,

u_{n}^{ς} \in S (ς ρ_{0})

. So we have that

\begin{aligned} I_{μ} (u_{n}^{ς}) & = \frac{ς^{2 - \frac{8}{N}}}{2} \int_{R^{N}} | Δ u_{n} |^{2} d x + \frac{ς^{2}}{2} \int_{R^{N}} μ | u_{n} |^{2} d x - ς^{2} \int_{R^{N}} H (u_{n}) d x \\ = ς^{2} I_{μ} (u_{n}) + \frac{ς^{2 - \frac{8}{N}} - ς^{2}}{2} \int_{R^{N}} | Δ u_{n} |^{2} d x \end{aligned}

Since

ς > 1, \frac{ς^{2 - \frac{8}{N}} - ς^{2}}{2}

is strictly negative and then as

n \to \infty

m_{μ, ς^{2} ρ_{0}^{2}} < ς^{2} m_{μ, ρ_{0}^{2}} .

That is, for

ς = \frac{ρ}{ρ_{0}} > 1

with

ρ_{0} \in (0, ρ)

m_{μ, ρ^{2}} < \frac{ρ^{2}}{ρ_{0}^{2}} m_{μ, ρ_{0}^{2}} \Rightarrow \frac{ρ_{0}^{2}}{ρ^{2}} m_{μ, ρ^{2}} < m_{μ, ρ_{0}^{2}}

Similarly, we can get $\frac{ρ^{2} - ρ_{0}^{2}}{ρ^{2}} m_{μ, ρ^{2}} < m_{μ, ρ^{2} - ρ_{0}^{2}}$ . Hence, $m_{μ, ρ^{2}} < m_{μ, ρ_{0}^{2}} + m_{μ, ρ^{2} - ρ_{0}^{2}}$ .

Proof Proof of Theorem 3.1.

Let ${u_{n}} \subset S (c)$ be a minimizing sequence, it follows from Lemma 3.1 that ${u_{n}}$ is bounded in $H^{2} (R^{N})$ . Set

σ := \underset{n \to + \infty}{\lim \sup} sup_{y \in R^{N}} \int_{B_{1} (y)} | u_{n} |^{2} d x .

σ = 0

, then from Lemma 1.1,

u_{n} \to 0

L^{ν} (R^{N})

(

2 < ν < 2^{* *})

. So,

0 ⩽ lim_{n \to \infty} \frac{1}{2} \int_{R^{N}} (| Δ u |^{2} + μ | u |^{2}) d x = lim_{n \to \infty} I_{μ} (u_{n}) = m_{μ, c^{2}} < 0,

which is a contradiction. Therefore,

σ > 0

, so there is a sequence

{y_{n}} \subset R^{N}

satisfying

\begin{aligned} \int_{B_{1} (y_{n})} | u_{n} |^{2} d x ⩾ \frac{σ}{2} > 0. \end{aligned}

(3.6)

Denote

{\tilde{u}}_{n} (x) = u_{n} (x + y_{n})

, we can see that

{{\tilde{u}}_{n}} \subset S (c)

is a bounded minimizing sequence to

m_{μ, c^{2}}

. So, for some

\tilde{u} \in H^{2} (R^{N})

\begin{aligned} {\begin{cases} {\tilde{u}}_{n} ⇀ \tilde{u}, & in H^{2} (R^{N}), \\ {\tilde{u}}_{n} \to \tilde{u}, & in L_{loc}^{ν} (R^{N}), ν \in [1, 2^{* *}), \\ {\tilde{u}}_{n} (x) \to \tilde{u} (x), & a . e . in R^{N} . \end{cases} \end{aligned}

(3.7)

It follows from (3.6) that

\tilde{u} \neq 0

. Hence,

‖ \tilde{u} ‖_{2} \in (0, c]

. Then we will prove that

\tilde{u} \in S (c)

. If

α^{2} := ‖ \tilde{u} ‖_{2}^{2} < c^{2}

. From (3.7), we get

\begin{aligned} ‖ {\tilde{u}}_{n} ‖_{2}^{2} = ‖ {\tilde{u}}_{n} - \tilde{u} ‖_{2}^{2} + ‖ \tilde{u} ‖_{2}^{2} + o_{n} (1) . \end{aligned}

(3.8)

It follows from the Brézis-Lieb Lemma, (3.8) and Lemma 3.3 that

\begin{aligned} m_{μ, c^{2}} & = lim_{n \to + \infty} I_{μ} ({\tilde{u}}_{n}) ⩾ I_{μ} (\tilde{u}) + lim_{n \to + \infty} I_{μ} ({\tilde{u}}_{n} - \tilde{u}) \\ ⩾ m_{μ, α^{2}} + m_{μ, c^{2} - α^{2}}, \end{aligned}

which contradicts Lemma 3.4. Hence

‖ \tilde{u} ‖_{2} = c

, and then from (1.8)

{\tilde{u}}_{n} \to \tilde{u}

L^{p} (R^{N})

. Then, by (3.7),

m_{μ, c^{2}} ⩽ I_{μ} (\tilde{u}) ⩽ lim_{n \to + \infty} I_{μ} ({\tilde{u}}_{n}) = m_{μ, c^{2}} .

Thus,

\tilde{u} \in S (c)

is a minimizer of

m_{μ, c^{2}}

. Hence, there exists

λ_{c} \in R

from lagrange multiplier such that

\begin{aligned} I_{μ}^{^{'}} (\tilde{u}) = λ_{c} \int_{R^{N}} | \tilde{u} |^{2} d x, in {(H^{2} (R^{N}))}^{^{'}} \end{aligned}

(3.9)

by the Lagrange multiplier. Thereby, according to (3.9),

\begin{aligned} Δ^{2} \tilde{u} + μ \tilde{u} = λ_{c} \tilde{u} + h (\tilde{u}) in R^{3} . \end{aligned}

(3.10)

Furthermore, from

(h_{3})

we have

q H (t) ⩽ h (t) t (q > 2)

, thus it follows from (3.10) that

\begin{aligned} \frac{1}{2} λ_{c} \int_{R^{N}} | \tilde{u} |^{2} d x & = \frac{1}{2} \int_{R^{N}} (| Δ \tilde{u} |^{2} + μ | \tilde{u} |^{2}) d x - \frac{1}{2} \int_{R^{N}} | h (\tilde{u}) \tilde{u} d x \\ = I_{μ} (\tilde{u}) + \int_{R^{N}} H (\tilde{u}) d x - \frac{1}{2} \int_{R^{N}} | h (\tilde{u}) \tilde{u} d x \\ < I_{μ} (\tilde{u}) = m_{μ, c^{2}} < 0, \end{aligned}

we must have

λ_{c} < 0

. Hence

(\tilde{u}, λ_{c})

is a couple of solution to problem (3.1).

From above theorem, we get following conclusion. Corollary 3.1.

For the fix $c > 0$ and $0 ⩽ μ_{1} < μ_{2} ⩽ μ_{*}$ , we get that $m_{μ_{1}, c^{2}} < m_{μ_{2}, c^{2}}$ .

Proof.

If $u \in S (c)$ satisfying $I_{μ_{2}} (u) = m_{μ_{2}, c^{2}}$ , we have that

m_{μ_{1}, c^{2}} ⩽ I_{μ_{1}} (u) < I_{μ_{2}} (u) = m_{μ_{2}, c^{2}} .

4. .The nonautonomous problem

In the present section, we investigated the non-autonomous case. First, we prove some properties of the functional $I_{ε} |_{S (c)} : H^{2} (R^{N}) \to R$ defined by

I_{ε} (u) = \frac{1}{2} \int_{R^{N}} (| Δ u |^{2} + V (ε x) | u |^{2}) d x - \int_{R^{N}} G (ε x, u) d x .

Hereafter, we are assuming that

‖ V ‖_{\infty} < μ_{*}

, where

μ_{*}

was given in Section 3. Denote

J_{0}, J_{\infty} : H^{2} (R^{N}) \to R, Θ_{0, c}, Θ_{\infty, c}

as follows:

\begin{aligned} J_{0} (u) & := \frac{1}{2} \int_{R^{N}} | Δ u |^{2} d x + \frac{1}{2} \int_{R^{N}} V_{0} | u |^{2} d x - \int_{R^{N}} H (u) d x, \\ Θ_{0, c} & := inf_{u \in S (c)} J_{0} (u); \end{aligned}

and

\begin{aligned} J_{\infty} (u) & := \frac{1}{2} | Δ u |^{2} d x + \frac{1}{2} \int_{R^{N}} V_{\infty} | u |^{2} d x - \int_{R^{N}} H (u) d x, \\ Θ_{\infty, c} & := inf_{u \in S (c)} J_{\infty} (u) . \end{aligned}

Similar to Lemma 3.1, the definition

Θ_{ε, c} := inf_{u \in S (c)} I_{ε} (u)

is well defined.

Since $0 < V_{\infty} = min_{\partial Λ} V (x) < + \infty$ , Corollary 3.1 leads that

\begin{aligned} Θ_{0, c} < Θ_{\infty, c} < 0. \end{aligned}

(4.1)

In what follows, we fix $0 < ρ_{1} = \frac{1}{2} (Θ_{\infty, c} - Θ_{0, c})$ .

The following lemma involving relations between $Θ_{ε, c}$ , $Θ_{\infty, c}$ and $Θ_{0, c}$ is one of the keys of our argument.

Lemma 4.1.

\underset{ε \to 0^{+}}{\lim \sup} Θ_{ε, c} < Θ_{\infty, c} < 0.

Proof.

Choosing $u \in S (c)$ as the minimum of $Θ_{ε, c}$ , and define ${\tilde{u}}_{ε} (x) := u (x - \frac{x_{0}}{ε})$ with $x_{0} \in Λ$ and $V (x_{0}) = V_{0}$ . A simple calculus gives

Θ_{ε, c} ⩽ I_{ε} ({\tilde{u}}_{ε}) = \frac{1}{2} \int_{R^{N}} (| Δ u |^{2} + V (ε x + x_{0}) | u |^{2}) d x - \int_{R^{N}} G (ε x + x_{0}, u) d x .

Letting

ε \to 0^{+}

, then

\underset{ε \to 0^{+}}{\lim \sup} Θ_{ε, c} ⩽ \underset{ε \to 0^{+}}{\lim \sup} I_{ε} (u) = J_{0} (u) = Θ_{0, c},

which combined with (4.1) and then we obtain the result.

From above lemma, we know $ε \to 0^{+}$ , so we assume $ε \in (0, ε_{1})$ in the following.

Lemma 4.2.

There exists $τ^{*} > 0$ independent of $ε \in (0, ε_{1})$ , for all $τ \in (0, τ^{*})$ , if ${u_{n}} \subset S (c)$ such that $I_{ε} (u_{n}) \to d$ with $d < Θ_{0, c} + ρ_{1} < 0$ and $u_{n} ⇀ u_{ε}$ in $H^{2} (R^{N})$ , then $u_{ε} ≢ 0$ . Moreover, if $u_{n} ↛ u_{ε}$ in $H^{2} (R^{N})$ , there is $β > 0$ such that

\underset{n \to + \infty}{\lim \sup} ‖ u_{n} - u_{ε} ‖_{2}^{2} ⩾ β .

Proof.

From Lemma 3.1 and $(g_{1}) - (g_{3})$ , for all $v \in H^{2} (R^{N})$ , we derive that

\begin{aligned} I_{ε} (v) & = \frac{1}{2} \int_{R^{N}} | Δ v |^{2} d x + \frac{1}{2} \int_{R^{N}} V (ε x) | v |^{2} d x - \int_{R^{N}} G (ε x, v) d x \\ ⩾ \frac{1}{2} \int_{R^{N}} | Δ v |^{2} d x - τ \int_{Λ_{ε}^{c}} | v |^{q} d x - C \int_{Λ_{ε}} (| v |^{q_{0}} + | v |^{q}) d x \\ ⩾ \frac{1}{2} \int_{R^{N}} | Δ v |^{2} d x - τ C_{q} c^{(1 - β_{q}) q} {(\int_{R^{N}} | Δ v |^{2} d x)}^{\frac{β_{q} q}{2}} - C \int_{Λ_{ε}} (| v |^{q_{0}} + | v |^{q}) d x . \end{aligned}

Now, setting the function

\tilde{Υ} (t) := \frac{1}{2} t - τ C_{q} c^{(1 - β_{q}) q} t^{\frac{β_{q} q}{2}}, \forall t ⩾ 0

, according to the fact

\frac{β_{q} q}{2} < 1

, we get that there exists

{\tilde{t}}_{τ} > 0

such that

min_{t ⩾ 0} \tilde{Υ} (t) = \tilde{Υ} ({\tilde{t}}_{τ}) < 0

, and as

τ \to 0

\tilde{Υ} ({\tilde{t}}_{τ}) \to 0

. So, there exists

τ^{*} > 0

satisfying

\begin{aligned} 0 > \tilde{Υ} ({\tilde{t}}_{τ}) > \frac{1}{2} (Θ_{\infty, c} + Θ_{0, c}), \forall τ \in (0, τ^{*}) . \end{aligned}

(4.2)

By contradiction, if

u_{ε} \equiv 0,

then,

\begin{aligned} \frac{1}{2} (Θ_{\infty, c} + Θ_{0, c}) & = Θ_{0, c} + ρ_{1} > d = I_{ε} (u_{n}) + o_{n} (1) \\ ⩾ \tilde{Υ} ({\tilde{t}}_{τ}) + o_{n} (1) - C \int_{Λ_{ε}} (| u_{n} |^{q_{0}} + | u_{n} |^{q}) d x \\ = \tilde{Υ} ({\tilde{t}}_{τ}) + o_{n} (1), \end{aligned}

that is

\frac{1}{2} (Θ_{\infty, c} + Θ_{0, c}) ⩾ \tilde{Υ} ({\tilde{t}}_{τ}), as n \to + \infty,

which contradicts (4.2). Thus,

u_{ε} ≢ 0

Similar to Lemma 3.1, for n large enough, we get $‖ Δ u_{n} ‖_{2} < R_{1}$ . It follows from Willem ([33], Proposition 5.12) that there is ${λ_{n}} \subset R$ satisfying, as $n \to + \infty$ ,

\begin{aligned} {‖ I_{ε}^{^{'}} (u_{n}) - λ_{n} \int_{R^{N}} | u_{n} |^{2} d x ‖}_{(H^{2} (R^{N}))^{^{'}}} \to 0. \end{aligned}

(4.3)

Since

{u_{n}}

is bounded in

H^{2} (R^{N})

, we have

{λ_{n}}

is also bounded, so, there exists

λ_{ε}

such that

λ_{n} \to λ_{ε}

n \to + \infty

for some subsequence. Combining this with (4.3), we get

I_{ε}^{^{'}} (u_{ε}) - λ_{ε} \int_{R^{N}} | u_{ε} |^{2} d x = 0 in (H^{2} (R^{N}))^{^{'}},

and then

\begin{aligned} {‖ I_{ε}^{^{'}} (v_{n}) - λ_{ε} \int_{R^{N}} | v_{n} |^{2} d x ‖}_{(H^{2} (R^{N}))^{^{'}}} \to 0 as n \to + \infty, \end{aligned}

(4.4)

where

v_{n} := u_{n} - u_{ε}

. By direct calculations and

(h_{3}), q H (t) ⩽ h (t) t

for all

t ⩾ 0,

and then we get that

\begin{aligned} Θ_{\infty, c} & > lim_{n \to + \infty} I_{ε} (u_{n}) \\ = lim_{n \to + \infty} (I_{ε} (u_{n}) - \frac{1}{2} ⟨ I_{ε}^{^{'}} (u_{n}), u_{n} ⟩ + \frac{1}{2} λ_{n} c^{2} + o_{n} (1)) \\ = lim_{n \to + \infty} (\frac{1}{2} λ_{n} c^{2} + o_{n} (1) - \int_{R^{N}} G (ε x, u_{n}) d x + \frac{1}{2} \int_{R^{N}} g (ε x, u_{n}) d x) \\ ⩾ lim_{n \to + \infty} (\frac{1}{2} λ_{n} c^{2} + o_{n} (1)) \\ = \frac{1}{2} λ_{ε} c^{2}, \end{aligned}

which implies that

\begin{aligned} λ_{ε} ⩽ \frac{2 Θ_{\infty, c}}{c^{2}} < 0 for all ε \in (0, ε_{1}) . \end{aligned}

(4.5)

From (4.4), we get

‖ Δ v_{n} ‖_{2}^{2} + V (ε x) \int_{R^{N}} | v_{n} |^{2} d x - λ_{ε} \int_{R^{N}} | v_{n} |^{2} d x - \int_{R^{N}} g_{ε} (x, v_{n}) v_{n} d x = o_{n} (1),

this together with (4.5) obtains that

\int_{R^{N}} g_{ε} (x, v_{n}) v_{n} d x + o_{n} (1) ⩾ ‖ Δ v_{n} ‖_{2}^{2} - \frac{2 Θ_{\infty, c}}{c^{2}} \int_{R^{N}} | v_{n} |^{2} d x .

u_{n} ↛ u_{ε}

H^{2} (R^{N})

, then from

(g_{2})

,we deduce that

‖ Δ v_{n} ‖_{2}^{2} ⩽ ‖ Δ v_{n} ‖_{2}^{2} - \frac{2 Θ_{\infty, c}}{c^{2}} \int_{R^{N}} | v_{n} |^{2} d x ⩽ C \int_{R^{N}} | v_{n} |^{q} d x + o_{n} (1) ⩽ C_{1} ‖ v_{n} ‖_{H^{2} (R^{N})}^{q} + o_{n} (1) .

Since

q > q_{0} > 2

, we can get that there exists

C_{2} > 0

such that

‖ v_{n} ‖_{H^{2} (R^{N})} ⩾ C_{2} .

Thus,

\begin{aligned} \underset{n \to + \infty}{\lim \sup} (\int_{R^{N}} | v_{n} |^{q} d x) ⩾ C_{3} . \end{aligned}

(4.6)

By using (4.6) and (1.8), there is a

β > 0

independent of

ε \in (0, ε_{1})

satisfing

β ⩽ \underset{n \to + \infty}{\lim \sup} ‖ v_{n} ‖_{2} .

From now on, we fix $0 < ρ < ρ_{1}$ . Lemma 4.3.

Decreasing if necessary $τ^{*} > 0$ . Let $ε \in (0, ε_{1}), β$ be as in Lemma 4.2, then $I_{ε} |_{S (c)}$ satisfies the $(P S)_{d}$ condition if $d < Θ_{0, c} + ρ$ .

Proof.

Let ${u_{n}}$ be a $(P S)_{d}$ sequence of $I_{ε} |_{S (c)}$ . By Lemma 3.1, ${u_{n}} \subset H^{2} (R^{N})$ is a bounded $(P S)_{d}$ sequence, so there is $u_{ε} \in H^{2} (R^{N})$ satisfying $u_{n} ⇀ u_{ε}$ in $H^{2} (R^{N})$ . Noting that $d < Θ_{0, c} + ρ < Θ_{0, c} + ρ_{1} < Θ_{\infty, c} < 0$ . According to Lemma 4.2, $u_{ε} ≢ 0$ . Define $v_{n} := u_{n} - u_{ε}$ . If $u_{n} \to u_{ε}$ in $H^{2} (R^{N})$ , we finish our proof. On the other hand, if $u_{n} ↛ u_{ε}$ in $H^{2} (R^{N})$ , from Lemma 4.2,

\underset{n \to + \infty}{\lim \sup} ‖ v_{n} ‖_{2} ⩾ β .

Since

{u_{n}}

is a

(P S)_{d}

sequence of

I_{ε} |_{S (c)}

, for any

φ \in H^{2} (R^{N})

, there exists

λ_{n} \in R

from the Lagrange multipliers rule such that

\begin{aligned} \int_{R^{N}} Δ u_{n} Δ φ d x + \int_{R^{N}} V (ε x) u_{n} φ d x - \int_{R^{N}} h (u_{n}) φ d x \\ = λ_{n} \int_{R^{N}} u_{n} φ d x + o_{n} (1) . \end{aligned}

(4.7)

Because of

u_{n} ⇀ u ≢ 0

H^{2} (R^{N})

and (4.7), for all

φ \in H^{2} (R^{N})

\begin{aligned} \int_{R^{N}} Δ u Δ φ d x + \int_{R^{N}} V (ε x) u φ d x - \int_{R^{N}} h (u) φ d x = λ_{ε} \int_{R^{N}} u φ d x . \end{aligned}

(4.8)

Test (4.7)–(4.8) with

φ = u_{n} - u_{ε}

, we obtain

‖ Δ (u_{n} - u_{ε}) ‖_{2}^{2} - λ_{ε} ‖ u_{n} - u_{ε} ‖_{2}^{2} \to 0

. That is

‖ u_{n} - u_{ε} ‖_{2} \to 0

, which is absurd. Hence,

u_{n} \to u_{ε}

H^{2} (R^{N})

5. .Multiple normalized solutions for the modified problem (2.2)

Let w be a solution of the following problem

{\begin{cases} Δ^{2} u + V_{0} u = h (u) + λ u, & in R^{N}, \\ \int_{R^{N}} | u |^{2} d x = c^{2}, & in R^{N} \end{cases}

with

J_{0} (w) = Θ_{0, c}

. Let

δ > 0

be fixed and

η (s) = {\begin{cases} 1, & if 0 ⩽ s ⩽ \frac{δ}{2}, \\ 0, & if s ⩾ δ, \end{cases}

which is a smooth function defined in

[0, \infty)

. For any

y \in M

, we define

Ψ_{ε, y} (x) = w ((ε x - y) / ε) η (| ε x - y |), {\tilde{Ψ}}_{ε, y} (x) = c Ψ_{ε, y} (x) / ‖ Ψ_{ε, y} ‖_{2} .

Letting

Φ_{ε} : M \to S (c)

Φ_{ε} (y) = {\tilde{Ψ}}_{ε, y}

, then we get that, for any

y \in M

Φ_{ε} (y)

has compact support.

Lemma 5.1.

$lim_{ε \to 0} I_{ε} (Φ_{ε} (y)) = Θ_{0, c} uniformly in y \in M .$

Proof.
If the conclusion is not true, there is $δ_{0} > 0, {y_{n}} \subset M$ with $y_{n} \to y \in M$ and $ε_{n} \to 0$ satisfying
$| I_{ε_{n}} (Φ_{ε_{n}} (y_{n})) - Θ_{0, c} | ⩾ δ_{0}, \forall n \in N .$

Considering the change of variable $z = (ε_{n} x - y_{n}) / ε_{n}$ , if $z \in B_{δ / ε_{n}} (0)$ , then $ε_{n} z \in B_{δ} (0)$ and $ε_{n} z + y_{n} \in B_{δ} (y_{n}) \subset M_{δ} \subset Λ$ . According to the Lebesgue’s theorem,
$\begin{aligned} lim_{n \to + \infty} \int_{R^{N}} {| Φ_{ε_{n}} (y_{n}) |}^{2} d x = lim_{n \to + \infty} \int_{R^{N}} {| η (ε_{n} z) w (z) |}^{2} d z = \int_{R^{N}} | w |^{2} d x, \\ lim_{n \to \infty} \int_{R^{N}} G (ε_{n} x, Φ_{ε_{n}} (y_{n})) d x = lim_{n \to + \infty} \int_{R^{N}} G (ε_{n} z + y_{n}, c \frac{η (ε_{n} z) w (z)}{‖ Ψ_{ε_{n}, y_{n}} ‖_{2}}) d z = \int_{R^{N}} H (w) d x, \\ lim_{n \to + \infty} \int_{R^{N}} {| Δ Φ_{ε_{n}} (y_{n}) |}^{2} d x = lim_{n \to + \infty} \int_{R^{N}} \frac{c^{2}}{‖ Ψ_{ε_{n}, y_{n}} ‖_{2}^{2}} {| Δ (η (ε_{n} z) w (z)) |}^{2} d z = \int_{R^{N}} | Δ w |^{2} d x, \\ lim_{n \to + \infty} \int_{R^{N}} V (ε_{n} x) {| Φ_{ε_{n}} (y_{n}) |}^{2} d x = V_{0} \int_{R^{N}} | w |^{2} d x . \end{aligned}$
Consequently,
$lim_{n \to + \infty} I_{ε_{n}} (Φ_{ε_{n}} (y_{n})) = J_{0} (w) = Θ_{0, c},$
which is absurd.

Let $r_{0} = r_{0} (δ) > 0$ for any $δ > 0$ satisfying that $M_{δ} \subset B_{r_{0}} (0)$ , $χ : R^{N} \to R^{N}$ is defined as $χ (x) = x$ if $| x | ⩽ r_{0}; χ (x) = r_{0} \frac{x}{| x |}$ if $| x | > r_{0}$ . Set $β_{ε} : H^{2} (R^{N}) ∖ {0} \to R^{N}$ as following,
$β_{ε} (u) = \frac{\int_{R^{N}} χ (ε x) | u |^{2} d x}{\int_{R^{N}} | u |^{2} d x} .$

Since $M_{δ} \subset B_{r_{0}} (0)$ , we conclude that
$\begin{aligned} lim_{ε \to 0} β_{ε} (Φ_{ε} (y)) = y uniformlyin y \in M . \end{aligned}$
(5.1)
Proposition 5.1.
Let $ε_{n} \to 0$ , for all $n \in N$ , $I_{ε_{n}} |_{S (c)} (u_{n}) \to Θ_{0, c}$ and $I_{ε_{n}}^{^{'}} |_{S (c)} (u_{n}) = o_{n} (1)$ . There exists a sequence ${{\tilde{y}}_{n}} \subset R^{N}$ satisfying that $v_{n} (x) := u_{n} (x + {\tilde{y}}_{n}) \subset H^{2} (R^{N})$ has a strongly convergent subsequence. Moreover, $y_{n} := ε_{n} {\tilde{y}}_{n} \to y_{0} \in M$ up to a subsequence.
Proof.
Similar to Lemma 3.1, we obtain that $‖ u_{n} ‖_{H^{2}} ⩽ C$ . Notice that $Θ_{0, c} > 0$ , and $‖ u_{n} ‖_{H^{2}} \to 0$ would implies $I_{ε_{n}}^{^{'}} (u_{n}) \to 0$ , we can argue as (3.6) to obtain that there are constants $R, ϱ > 0$ and a sequence ${{\tilde{y}}_{n}} \subset R^{N}$ satisfying
$lim_{n \to \infty} \int_{B_{R} ({\tilde{y}}_{n})} | u_{n} |^{2} d x ⩾ ϱ > 0.$
Setting $v_{n} (x) := u_{n} (x + {\tilde{y}}_{n})$ , since the boundedness of $u_{n}$ , ${v_{n}}$ is also bounded. Therefore, up to a subsequence,
$v_{n} ⇀ v ≢ 0 in H^{2} (R^{N}) .$
Hence
$\begin{aligned} Θ_{0, c} & ⩽ J_{0} (v_{n}) = \frac{1}{2} \int_{R^{N}} | Δ v_{n} |^{2} d x + \frac{1}{2} \int_{R^{N}} V_{0} | v_{n} |^{2} d x - \int_{R^{N}} H (v_{n}) d x \\ ⩽ \frac{1}{2} \int_{R^{N}} | Δ v_{n} |^{2} d x + \frac{1}{2} \int_{R^{N}} V (ε_{n} (x + {\tilde{y}}_{n})) | v_{n} |^{2} d x - \int_{R^{N}} H (v_{n}) d x \\ = \frac{1}{2} \int_{R^{N}} (| \nabla u_{n} |^{2} + V (ε_{n} x) | u_{n} |^{2}) d x - \int_{R^{N}} G (ε_{n} x, u_{n}) d x \\ = I_{ε_{n}} (u_{n}) \\ = Θ_{0, c} + o_{n} (1), \end{aligned}$
which implies $lim_{n \to \infty} J_{0} (v_{n}) = Θ_{0, c}$ , then according to Lemma 4.3 we have $v_{n} \to v$ in $H^{2} (R^{N})$ . Moreover,
$J_{0} (v) = Θ_{0, c}, ⟨ J_{0}^{^{'}} (v), v ⟩ = 0 and ‖ J_{0} |_{S (c)}^{^{'}} (v) ‖_{(H^{2})^{^{'}}} = 0.$

Next, we will prove that ${y_{n}} = {ε_{n} {\tilde{y}}_{n}}$ has a subsequence, still denoted by ${y_{n}}$ , satisfying $y_{n} \to y_{0} \in M$ . First we prove that $y_{n} \subset R^{N}$ is a bounded sequence. In fact, if not, there is a subsequence, still denoted by $y_{n}$ , satisfying $| y_{n} | \to \infty$ . From $‖ I_{ε_{n}}^{^{'}} |_{S (c)} (u_{n}) ‖_{(H^{2})^{^{'}}} \to 0$ , we have
$\begin{aligned} \int_{R^{N}} (| Δ u_{n} |^{2} + V (ε_{n} x) | u_{n} |^{2}) d x & = \int_{R^{N}} g (ε_{n} x + λ_{n} \int_{R^{N}} | u_{n} |^{2} d x, u_{n}) u_{n} d x \\ \Rightarrow \int_{R^{N}} (| Δ v_{n} |^{2} + V (ε_{n} (x + {\tilde{y}}_{n})) | v_{n} |^{2}) d x = \int_{R^{N}} g (ε_{n} (x + {\tilde{y}}_{n}) + λ_{n} \int_{R^{N}} | v_{n} |^{2} d x, v_{n}) v_{n} d x, \end{aligned}$
where ${λ_{n}} \subset R$ satisfies
$\underset{n \to + \infty}{\lim \sup} λ_{n} < 0.$
So, we can get that there is $n_{0} \in N$ and $λ^{} < 0$ such that $λ_{n} < λ^{}, \forall n ⩾ n_{0}$ . Taking into account $ε_{n} \to 0^{+}$ , we have that
$\int_{R^{N} ∖ B_{\frac{r}{ε_{n}}} (0)} | h (v_{n}) | v_{n} d x \to 0 as n \to \infty .$
It follows from $(g_{2})$ that
$\begin{aligned} \int_{R^{N}} g (ε_{n} (x + {\tilde{y}}_{n}), v_{n}) v_{n} d x \\ = \int_{B_{\frac{r}{ε_{n}}} (0) \cup R^{N} ∖ B_{\frac{r}{ε_{n}}} (0)} g (ε_{n} (x + {\tilde{y}}_{n}), v_{n}) v_{n} d x \\ ⩽ τ \int_{B_{\frac{r}{ε_{n}}} (0)} | v_{n} |^{q} d x + \int_{R^{N} ∖ B_{\frac{r}{ε_{n}}} (0)} | h (v_{n}) v_{n} | d x, \end{aligned}$
Then by the definition of $V_{0}$ , we obtain that
$\begin{aligned} \int_{R^{N}} | Δ v_{n} |^{2} d x - λ_{} \int_{R^{N}} | v_{n} |^{2} d x & ⩽ \int_{R^{N}} | Δ v_{n} |^{2} d x + \int_{R^{N}} V_{0} | v_{n} |^{2} d x - λ_{} \int_{R^{N}} | v_{n} |^{2} d x \\ ⩽ \int_{R^{N}} | Δ v_{n} |^{2} d x + \int_{R^{N}} V (ε_{n} x + ε_{n} {\tilde{y}}_{n}) | v_{n} |^{2} d x - λ_{n} \int_{R^{N}} | v_{n} |^{2} d x \\ ⩽ τ \int_{R^{N}} | v_{n} |^{q} d x \\ ⩽ τ C_{q} c^{(1 - β_{q}) q} {(\int_{R^{N}} | Δ v_{n} |^{2} d x)}^{\frac{β_{q} q}{2}} . \end{aligned}$

Let $Υ (t) = \frac{1}{2} t - τ C_{q} c^{(1 - β_{q}) q} t^{\frac{β_{q} q}{2}}$ , then it follows from $\frac{β_{q} q}{2} < 1$ that there is $t_{τ} > 0$ satisfying $Υ (t_{τ}) = min_{t ⩾ 0} Υ (t) < 0$ . Therefore,
$Υ (t_{τ}) - λ_{} c^{2} ⩽ \int_{R^{N}} | Δ v_{n} |^{2} d x - τ C_{q} c^{(1 - β_{q}) q} {(\int_{R^{N}} | Δ v_{n} |^{2} d x)}^{\frac{β_{q} q}{2}} - λ_{} c^{2} ⩽ 0,$

A direct computation obtains that as $τ \to 0$ , $Υ (t_{τ}) \to 0$ . Since $λ_{} < 0$ independent on τ, decreasing τ if necessary, we have $Υ (t_{τ}) - λ_{} c^{2} > 0$ , which is absurd. Therefore, $y_{n} \to y_{0} \in R^{N}$ up to a subsequence if necessary.

Next we want to prove that $y_{0} \in M$ . If $y_{0} \notin M$ , since $v_{n} \to v$ in $H^{2} (R^{N})$ , it follows from Fatou’s lemma and $(g_{5})$ that
$\begin{aligned} Θ_{0, c} + o_{n} (1) & = \underset{n \to \infty}{\lim \inf} I_{ε_{n}} (u_{n}) \\ = \underset{n \to \infty}{\lim \inf} \frac{1}{2} \int_{R^{N}} (| Δ v_{n} |^{2} + V (ε_{n} (x + {\tilde{y}}_{n})) | v_{n} |^{2}) d x - \int_{R^{N}} G (ε_{n} (x + {\tilde{y}}_{n}), v_{n}) d x \\ ⩾ \underset{n \to \infty}{\lim \inf} \frac{1}{2} \int_{R^{N}} (| Δ v_{n} |^{2} + V (ε_{n} (x + {\tilde{y}}_{n})) | v_{n} |^{2}) d x - \int_{R^{N}} H (v_{n}) d x \\ ⩾ \frac{1}{2} \int_{R^{N}} (| Δ v |^{2} + V (y_{0}) | v |^{2}) d x - \int_{R^{N}} H (v) d x \\ > J_{0} (v) = Θ_{0, c}, \end{aligned}$
which is a contradiction.

Let $ξ \in C ([0, + \infty))$ such that as $ε \to 0, ξ (ε) \to 0$ . Define
$\begin{aligned} \tilde{S} (c) := {u \in S (c) : I_{ε} (u) ⩽ Θ_{0, c} + ξ (ε)} . \end{aligned}$
(5.2)
Thanks to Lemma 5.1, as $ε \to 0$ , the function $ξ (ε) = sup_{y \in M} | I_{ε} (Φ_{ε} (y)) - Θ_{0, c} | \to 0$ . Hence, for all $y \in M$ , $Φ_{ε} (y) \in \tilde{S} (c)$ .
Lemma 5.2.

$lim_{ε \to 0} sup_{u \in \tilde{S} (c)} inf_{y \in M_{δ}} | β_{ε} (u) - y | = 0.$

Proof.
Let $ε_{n} \to 0$ and $u_{n} \in \tilde{S} (c)$ such that
$inf_{y \in M_{δ}} | β_{ε_{n}} (u_{n}) - y | = sup_{u \in \tilde{S} (c)} inf_{y \in M_{δ}} | β_{ε_{n}} (u) - y | + o_{n} (1) .$
From the equality above, it is necessary to find a sequence ${y_{n}} \subset M_{δ}$ satisfing
$lim_{n \to + \infty} | β_{ε_{n}} (u_{n}) - y_{n} | = 0.$
Since $u_{n} \in \tilde{S} (c)$ ,
$Θ_{0, c} ⩽ Θ_{ε_{n}, c} ⩽ I_{ε_{n}} (u_{n}) ⩽ Θ_{0, c} + ξ (ε_{n}), \forall n \in N,$
where $ξ (ε_{n}) = sup_{y \in M} | I_{ε_{n}} (Φ_{ε_{n}} (y)) - Θ_{0, c} |$ satisfies $ξ (ε_{n}) \to 0$ as $n \to \infty$ by Lemma 5.1. And so
$u_{n} \in S (c) and I_{ε_{n}} (u_{n}) \to Θ_{0, c} .$
From Proposition 5.1, there is a sequence $({\tilde{y}}_{n}) \subset R^{N}$ satisfying that $v_{n} (x) = u_{n} (x + {\tilde{y}}_{n}) \in H^{2} (R^{N})$ has a strongly convergent subsequence. Moreover, $y_{n} = ε_{n} {\tilde{y}}_{n} \to y_{0} \in M$ up to a subsequence if necessary. Therefore,
$\begin{aligned} β_{ε_{n}} (u_{n}) & = \frac{\int_{R^{N}} (χ (ε_{n} z) | u_{n} (z) |^{2} d z}{c^{2}} \\ = \frac{\int_{R^{N}} (χ (ε_{n} z) | v_{n} (z - {\tilde{y}}_{n}) |^{2} d z}{c^{2}} \\ = \frac{\int_{R^{N}} (χ (ε_{n} z + ε_{n} {\tilde{y}}_{n}) | v_{n} (z) |^{2} d z}{c^{2}} \\ = y_{n} + \frac{\int_{R^{N}} (χ (ε_{n} z + y_{n}) - y_{n}) | v_{n} |^{2} d z}{c^{2}} . \end{aligned}$
Consequently, there exists $y_{n} = ε_{n} {\tilde{y}}_{n} \in M_{δ}$ satisfying that $| β_{ε_{n}} (u_{n}) - y_{n} | = o_{n} (1)$ since $ε_{n} z + y_{n} \to y_{0} \in M$ , the we finish the proof of the lemma.

We would like to point out that if Y is a closed subset of a topological space X, the Lusternik–Schnirelmann category ${cat}_{X} (Y)$ is the least number of closed and contractible sets in X which cover Y. If $X = Y$ , we use the notation $cat (X)$ . We aim to prove that there are at least ${cat}_{M_{δ}} (M)$ couples critical points $(u_{j}, λ_{j}) \in H^{2} (R^{N}) \times R$ for the energy functional $I_{ε} (u)$ when study the concentration phenomina when $ε \to 0$ .

Before proving the multiplicity result, we recall the following useful abstract result from Benci and Cerami [4]. Lemma 5.3.
Let $I, I_{1}$ and $I_{2}$ be closed with $I_{1} \subset I_{2}$ , $φ : I_{1} \to I$ and $π : I \to I_{2}$ be non continuous maps satisfying that $π \circ φ$ is homotopically equivalent to $j : I_{1} \to I_{2}$ . Then ${cat}_{I} (I) ⩾ {cat}_{I_{2}} (I_{1})$ .
Theorem 5.1.
Asume that $(V 1) - (V 2)$ , $(h_{1}) - (h_{3})$ hold. Then, for the given $δ > 0$ , there exists ${\bar{ε}}_{δ} > 0$ such that there exist at least ${cat}_{M_{δ}} (M)$ positive solutions to (2.2) for any $ε \in (0, {\bar{ε}}_{δ})$ .
Proof.
From Proposition 5.1, there exists $\bar{ε} = {\bar{ε}}_{δ} > 0$ satisfying the diagram of continuous mappings $M \to Φ_{ε} (M) \to M_{δ}$ is well defined if $ε \in (0, \bar{ε})$ . From (5.1), choosing a function $ψ (ε, z)$ with $| ψ (ε, z) | < \frac{δ}{2}$ uniformly in $z \in M$ such that $β_{ε} (Φ_{ε} (z)) = z + ψ (ε, z)$ for all $z \in M, ε \in (0, \bar{ε})$ . Define $\tilde{Γ} (t, z) = z + (1 - t) ψ (ε, z)$ , then we have $\tilde{Γ} : [0, 1] \times M \to M_{δ}$ is continuous. It is easy to check that $\tilde{Γ} (0, z) = β_{ε} (ε (z)), \tilde{Γ} (1, z) = z$ for all $z \in M$ . And then, $\tilde{Γ} (t, z)$ is a homotopy between $β_{ε} \circ Φ_{ε}$ and the inclusion map $I d : M \to M_{δ}$ , which implies that
$\begin{aligned} cat (\tilde{S} (c)) ⩾ {cat}_{M_{δ}} (M) . \end{aligned}$
(5.3)
It follows from Lemma 4.3 and from the abstract category Lemma 5.3, Lemma 3.1, we have that $I_{ε} |_{S (c)}$ is bounded from below and for $d \in (Θ_{0, c}, Θ_{0, c} + ξ (ε))$ , $I_{ε} |_{S (c)}$ satisfies the $(P S)_{d}$ condition. From the Lusternik–Schnirelmann category of critical points ([14]), there are at last ${cat}_{M_{δ}} (M)$ critical points $u_{ε}$ for $I_{ε} |_{S (c)}$ .

6. .Multiplicity and concentration phenomena of solutions for problem (2.1)

In the present section we shall prove the main result by showing that the solutions $u_{ε}$ obtained in Theorem 5.1 are indeed solutions of the original problem (2.1) if $| u_{ε} | ⩽ b_{0}, \forall x \in {\bar{Λ_{ε}}}^{c}$ for $ε$ small enough.

By the uniform $L^{\infty}$ estimates to solutions given in [29], we deduce a fundamental $L^{\infty}$ -estimate.

Lemma 6.1.
Let $ε_{n} \to 0$ as $n \to \infty$ and ${x_{n}} \subset {\bar{Λ}}_{ε_{n}}$ , ${u_{ε_{n}}}$ be the sequence defined as in Theorem5.1. Then, $v_{n} := u_{ε_{n}} (\cdot + x_{n}) \in L^{\infty} (R^{N})$ and there is $C > 0$ such that for all $n \in N$ , $‖ v_{n} ‖_{\infty} ⩽ C$ .
Lemma 6.2.
$v_{n} (x) \to 0$ as $| x | \to \infty$ uniformly in n.
Proof.
Through Theorem 5.1, we have $v_{n} \in L^{s} (B_{1} (x))$ for all $s ⩾ 1$ , then by [[1], Theorem 7.1],
$\begin{aligned} ‖ v_{n} ‖_{W^{4, s} (B_{1} (x))} & ⩽ C ({‖ h (v_{n}) ‖}_{L^{s} (B_{2} (x))} + ‖ v_{n} ‖_{L^{s} (B_{2} (x))}) \\ ⩽ C ‖ v_{n} ‖_{L^{s} (B_{2} (x))} ⩽ C ‖ v_{n} ‖_{L^{\infty} (R^{N})}^{\frac{s - 2^{* }}{s}} ‖ v_{n} ‖_{L^{2^{ }} (B_{2} (x))}^{2^{ }} ⩽ C ‖ v_{n} ‖_{L^{2^{ }} (B_{2} (x))}^{2^{ }} . \end{aligned}$
If $s > N$ , the imbedding $W^{4, s} (B_{1} (x)) ↪ C^{3, α} (\bar{B_{1} (x)})$ is continuous for $α \in (0, 1 - \frac{N}{s})$ . Hence, by lemma 6.1,
$‖ v_{n} ‖_{C^{3, α} (\bar{B_{1} (x)})} ⩽ ‖ v_{n} ‖_{W^{4, s} (B_{1} (x))} ⩽ C ‖ v_{n} ‖_{L^{2^{ }} (B_{2} (x))}^{2^{ *}} .$
It follows from the convergence $v_{n} \to v$ in $H^{2} (R^{N})$ that as $| x | \to \infty$ , $| v_{n} (x) | \to 0$ uniformly in n.
Proof Proof of Theorem 1.1.

Assume that $u_{ε}$ is a solution of (1.1) with $I_{ε} (u_{ε}) ⩽ Θ_{0, c} + ξ (ε)$ , where ξ was given in (5.2). Arguing as the same as that in Proposition 5.1, there exists ${\tilde{y}}_{n} \in R^{N}$ satisfying $y_{n} = ε_{n} {\tilde{y}}_{n} \to y$ with $y \in M$ as $ε_{n} \to 0$ . Set $v_{n} (x) := u_{n} (x + {\tilde{y}}_{n})$ , we get that there is $v \neq 0$ in $H^{2} (R^{N})$ such that $v_{n} (x) \to v$ in $H^{2} (R^{N})$ . Since $v_{n}$ is a solution of

Δ^{2} v_{n} + V (ε_{n} x + y_{n}) v_{n} = λ_{n} v_{n} + h (v_{n}), in R^{N},

with

\underset{ε \to 0}{\lim \sup} λ_{n} ⩽ \frac{2 (ρ_{1} + Θ_{0, c})}{c^{2}} < 0,

lemma 6.2 shows that, there are

R_{1} > 0

and

n_{0} \in N

satisfying

| v_{n} (x) | ⩽ \frac{b}{2} for~ | x | ⩾ R_{1} ~and~ n ⩾ n_{0} .

Now, according to the fact

y_{n} = ε_{n} {\tilde{y}}_{n} \to y \in Λ

, if

x \notin Λ_{ε_{n}}

, there exists

n_{0} \in N

satisfying that

| x - {\tilde{y}}_{n} | = \frac{1}{ε_{n}} | ε_{n} x - y_{n} | ⩾ \frac{r_{1}}{2 ε_{n}} > R_{1}, \forall n ⩾ n_{0},

where

r_{1} = d i s t (y, \partial Λ)

. Hence

| u_{n} (x) | = | v_{n} (x - {\tilde{y}}_{n}) | ⩽ \frac{b_{0}}{2}, \forall x \notin Λ_{ε_{n}} ~and~ n ⩾ n_{0},

which shows that for

n ⩾ n_{0}

u_{n}

is a solution of (2.1).

We claim that $‖ v_{n} ‖_{\infty} ↛ 0$ . Otherwise we will have that $v_{n} \to 0$ in $H^{2} (R^{N})$ , which is impossible since $‖ v_{n} ‖_{2} = c$ for all $n \in N$ . Now, let us fix $τ > 0$ such that $‖ v_{n} ‖_{\infty} ⩾ 2 τ$ and $z_{n} \in N$ satisfying $| v_{n} (z_{n}) | = ‖ v_{n} ‖_{\infty}$ for all $n \in N$ . The above analysis ensures that $| z_{n} | ⩽ R_{1}$ for all $n \in N$ . Now, let us consider $η_{n} \in N$ such that $| u_{n} (η_{n}) | = ‖ u_{n} ‖_{\infty}$ . Then, $η_{n} = z_{n} + {\tilde{y}}_{n}$ and

lim_{n \to + \infty} V (ε_{n} η_{n}) = lim_{n \to + \infty} V (ε_{n} z_{n} + ε_{n} {\tilde{y}}_{n}) = V_{0},

this proves the desired result.

7. .Orbital stability of the solutions

We will consider problem

\begin{aligned} {\begin{cases} i \frac{\partial ψ}{\partial t} - Δ^{2} ψ - V (ε x) ψ + h (| ψ |) ψ = 0, & t > 0, x \in R^{N}, \\ ψ (0, x) = u_{0} (x), \end{cases} \end{aligned}

(7.1)

and get the orbital stability.

Recalling that

β_{ε} (u) := \frac{\int_{R^{N}} χ (ε x) | u |^{2} d x}{\int_{R^{N}} | u |^{2} d x},

in what follows, we define:

$θ_{ε} = {u \in S (c); | β_{ε} (u) - c | ⩽ ϱ_{0}}$ ,

$\partial θ_{ε} = {u \in S (c); | β_{ε} (u) - c | = ϱ_{0}}$ ,

$m_{ε} = inf_{u \in θ_{ε}} J_{ε} (u)$ ,

${\tilde{m}}_{ε} = inf_{u \in \partial θ_{ε}} J_{ε} (u)$ .

Lemma 7.1.

Assume $0 < ρ < ρ_{1}$ . Decreasing if necessary $ε_{0}$ , we have

\begin{aligned} m_{ε} < Θ_{0, c} + ρ \end{aligned}

(7.2)

and

\begin{aligned} m_{ε} < {\tilde{m}}_{ε} \end{aligned}

(7.3)

for all

ε \in (0, ε_{0})

Proof.

Let $u \in H^{2} (R^{N})$ such that

J_{0} (u) = Θ_{0, c}, J_{0}^{^{'}} (u) = 0.

Then, consider the function

{\hat{u}}_{ε} : R^{N} \to R

as follows:

{\hat{u}}_{ε} (x) = u (x - \frac{x_{0}}{ε}) \in S (c)

with

x_{0} \in Λ

and

V (x_{0}) = V_{0}

. Next, we will prove that

\begin{aligned} \underset{ε \to 0}{\lim \sup} I_{ε} ({\hat{u}}_{ε}) ⩽ Θ_{0, c} . \end{aligned}

(7.4)

Indeed, according to a simple change of variable, we get

\begin{aligned} I_{ε} ({\hat{u}}_{ε}) & = \frac{1}{2} \int_{R^{N}} | Δ {\hat{u}}_{ε} |^{2} + V (ε x) | {\hat{u}}_{ε} |^{2} d x - \int_{R^{N}} G (ε x, {\hat{u}}_{ε}) d x \\ = \frac{1}{2} \int_{R^{N}} | Δ u |^{2} + V (ε x + x_{0}) | u |^{2} d x - \int_{R^{N}} G (ε x + x_{0}, u) d x, \end{aligned}

lim_{ε \to 0} I_{ε} ({\hat{u}}_{ε}) = J_{0} (u) = Θ_{0, c} .

Due to $ε \to 0$ , $β_{ε} ({\hat{u}}_{ε}) \to x_{0}$ , we know that for all $ε$ small enough, ${\hat{u}}_{ε} \in θ_{ε}$ . Moreover, by using (7.4), we get that, for $ε$ small enough,

I_{ε} ({\hat{u}}_{ε}) < Θ_{0, c} + \frac{δ_{0}}{4} .

Decreasing

ε_{0}

if necessary, we have

m_{ε} < Θ_{0, c} + \frac{δ_{0}}{4}, \forall ε \in (0, ε_{0}) .

Decreasing

δ_{0}

if necessary, we obtain that

m_{ε} < Θ_{0, c} + ρ, \forall ε \in (0, ε_{0}) .

This implies that (7.2) holds.

In order to get (7.3), recalling that if $u \in \partial θ_{ε}, u \in S (c)$ and

| β_{ε} (u) - x_{0} | = ϱ_{0} > \frac{ρ}{2},

that is

β_{ε} (u) \notin B_{\frac{ϱ_{0}}{2}}

. Thereby, by Proposition 5.1

I_{ε} (u) > Θ_{0, c} + \frac{δ_{0}}{2}, for all u \in \partial θ_{ε} ~and~ ε \in (0, ε_{0}),

{\tilde{m}}_{ε} = inf_{u \in \partial θ_{ε}} I_{ε} (u) ⩾ Θ_{0, c} + \frac{δ_{0}}{2}, \forall ε \in (0, ε_{0}) .

By using (7.2), we have that

\forall ε \in (0, ε_{0})

m_{ε} < {\tilde{m}}_{ε} .

Proof Proof of Theorem 1.2.

From Gagliardo–Nirenberg inequalities,

\begin{aligned} I_{ε} (u) & ⩾ \frac{1}{2} \int_{R^{N}} | Δ u |^{2} d x - \frac{C c^{q_{0} (1 - β_{q_{0}})}}{q_{0}} {(\int_{R^{N}} | Δ u |^{2} d x)}^{\frac{β_{q_{0}} q_{0}}{2}} \\ - \frac{C c^{p_{1} (1 - β_{p_{1}})}}{p_{1}} {(\int_{R^{N}} | Δ u |^{2} d x)}^{\frac{β_{p_{1}} p_{1}}{2}} \\ = Q_{c} (‖ Δ u ‖_{2}), \end{aligned}

where

Q_{c} (r) = \frac{1}{2} r^{2} - \frac{C c^{(1 - β_{q_{0}}) q_{0}}}{q_{0}} r^{β_{q_{0}} q_{0}} - \frac{C c^{(1 - β_{p_{1}}) p_{1}}}{p_{1}} r^{β_{p_{1}} p_{1}}

. Define Ω as follows:

\begin{aligned} Ω : & = {v \in θ_{ε} : I_{ε} |_{S (c)}^{^{'}} (v) = 0, I_{ε} (v) = m_{ε}} \\ = {v \in θ_{ε} : I_{ε} |_{S (c)}^{^{'}} (v) = 0, I_{ε} (v) = m_{ε}, ‖ Δ v ‖_{2} ⩽ R_{0}, Q_{c} (R_{0}) = 0} . \end{aligned}

(7.5)

Next we prove the stability of the sets Ω. With the initial data $u_{0}$ , we denote $ψ (t, \cdot)$ be the solution to (7.1) and its maximal existence interval is $[0, T_{max})$ , then either $‖ Δ ψ (t, \cdot) ‖_{2} = + \infty$ as $t \to T_{max}^{-}$ , or for positive times, $ψ (t, \cdot)$ is globally defined, see ([31], Section 3).

First of all, we show that $ψ (t, \cdot)$ is globally defined. Assuming that $r_{1}$ satisfies $Q_{c} (r_{1}) = m_{ε}$ , by using $m_{ε} < 0$ and the definition of Ω, we have

‖ Δ v ‖_{2} ⩽ r_{1} < R_{0}, \forall v \in Ω .

Note that, since

Q_{c}

is monotonically decreasing with respect to c, we can take

c_{1} > c

to satisfy

Q_{c_{1}} (R_{0}) = \frac{m_{ε}}{2}

. There is

δ > 0

such that if

u_{0} \in H^{2} (R^{N})

and

{dist}_{H^{2} (R^{N})} (u_{0}, Ω) < δ

, then

‖ u_{0} ‖_{2} ⩽ c_{1}, ‖ Δ u_{0} ‖_{2} ⩽ r_{1} + \frac{R_{0} - r_{1}}{2}, I_{ε} (u_{0}) ⩽ \frac{2}{3} β_{ε} .

Let

\tilde{c} = ‖ u_{0} ‖_{2}

. Note that by the conservation of the mass,

‖ ψ (t, \cdot) ‖_{2} = ‖ u_{0} ‖_{2}

for all

t \in (0, T_{max})

. If there is

\tilde{t} \in (0, T_{max})

satisfying

‖ Δ ψ (\tilde{t}, \cdot) ‖_{2} =

R_{0}

, then

I_{ε} (ψ (\tilde{t}, \cdot)) ⩾ Q_{\tilde{c}} (R_{0}) ⩾ Q_{c_{1}} (R_{0}) = \frac{m_{ε}}{2},

which is a contradiction to $I_{ε} (ψ (t, \cdot)) = I_{ε} (u_{0}) ⩽ \frac{2}{3} m_{ε}, t \in (0, T_{max})$ . Thus,

\begin{aligned} ‖ Δ ψ (t, \cdot) ‖_{2} < R_{0}, t \in [0, T_{max}), \end{aligned}

(7.6)

so we get that for positive times,

ψ (t, \cdot)

is globally defined.

Next we prove that Ω is orbitally stable. Using Theorem 3.1 in [16], we can get that for complex valued function, Lemma 4.2 is also true. Therefore, by modifying the classical Cazenave–Lions argument [7,21], we will obtain the orbital stability of $Ω_{i}$ . Assume that the conclusion is invalid, there exist sequences ${u_{0, n}} \subset H^{2} (R^{N})$ and ${t_{n}} \subset R^{+}$ and a constant $θ_{0} > 0$ satifying

\begin{aligned} inf_{v \in Ω} ‖ u_{0, n} - v ‖_{H^{2}} < \frac{1}{n}, \forall n ⩾ 1 \end{aligned}

(7.7)

and

\begin{aligned} inf_{v \in Ω} ‖ ψ_{n} (t_{n}, \cdot) - v ‖_{H^{2}} ⩾ θ_{0}, \end{aligned}

(7.8)

here with initial data

u_{0, n}

ψ_{n} (t, \cdot)

is the solution of (7.1). Due to (7.6), there is

n_{0}

such that

‖ Δ ψ_{n} (t, \cdot) ‖_{2} < R_{0}

for all

t ⩾ 0, n > n_{0}

According to (7.7), there is ${v_{n}} \subset Ω$ such that

\begin{aligned} ‖ u_{0, n} - v_{n} ‖_{H^{2}} < \frac{2}{n} . \end{aligned}

(7.9)

According to the fact

{v_{n}} \subset Ω

{v_{n}} \subset θ_{ε}

is a

(P S)_{m_{ε}}

sequence of

I_{ε} |_{S (c)}

. And then, there is

v \in Ω

satisfying

lim_{n \to + \infty} ‖ v_{n} - v ‖_{H^{2}} = 0.

And because (7.9), we have

\begin{aligned} lim_{n \to + \infty} ‖ u_{0, n} - v ‖_{H^{2}} = 0. \end{aligned}

(7.10)

Thereby, as

n \to + \infty

‖ u_{0, n} ‖_{2} \to ‖ v ‖_{2} = c, I_{ε} (u_{0, n}) \to I_{ε} (v) = m_{ε} < {\tilde{m}}_{ε} .

Hence,

\begin{aligned} ‖ ψ_{n} (t, \cdot) ‖_{2} \to c, I_{ε} (ψ_{n} (t, \cdot)) \to m_{ε}, \forall t ⩾ 0. \end{aligned}

(7.11)

Define

φ_{n} (t, \cdot) = \frac{c ψ_{n} (t, \cdot)}{‖ ψ_{n} (t, \cdot) ‖_{2}}, t ⩾ 0,

then

φ_{n} (t, \cdot) \in S (c)

\begin{aligned} lim_{n \to + \infty} ‖ φ_{n} (t, \cdot) - ψ_{n} (t, \cdot) ‖ = 0 uniformly in t ⩾ 0. \end{aligned}

(7.12)

Combining (7.10) with (7.12), we lead that

lim_{n \to + \infty} ‖ φ_{n} (0, \cdot) - v ‖_{H^{2}} = lim_{n \to + \infty} {‖ \frac{c u_{0, n}}{‖ u_{0, n} ‖_{2}} - v ‖}_{H^{2}} = 0.

Therefore, for n large enough,

φ_{n} (0, \cdot) \in θ_{ε} ∖ \partial θ_{ε}

because of

v \in θ_{ε} ∖ \partial θ_{ε}

. And then

lim_{n \to + \infty} I_{ε} (φ_{n} (t, \cdot)) = m_{ε}

for all

t ⩾ 0

, and

m_{ε} < {\tilde{m}}_{ε}

, then

\begin{aligned} φ_{n} (t, \cdot) \in θ_{ε} ∖ \partial θ_{ε} for all t ⩾ 0 ~for~ n large enough . \end{aligned}

(7.13)

It follows from (7.11)–(7.13) that

{φ_{n} (t_{n}, \cdot) t} \subset θ_{ε}

is a minimizing sequence of

I_{ε}

at level

m_{ε}

. As the same proof of Theorem 1.1, there is

\tilde{v} \in θ_{ε}

satisfying

lim_{n \to + \infty} {‖ φ_{n} (t_{n}, \cdot) - \tilde{v} ‖}_{H^{2}} = 0,

which together with (7.12) leads that

lim_{n \to + \infty} {‖ ψ_{n} (t_{n}, \cdot) - \tilde{v} ‖}_{H^{2}} = 0.

That contradicts (7.8). Hence Ω is orbitally stable.

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