Existence and Stability of Steady Waves for the Hasegawa-Mima Equation

Boundary Value Problems

Volume 2009, Article ID 509801,18pages doi:10.1155/2009/509801

Research Article

Existence and Stability of Steady Waves for the Hasegawa-Mima Equation

Boling Guo

and Daiwen Huang

1Institute of Applied Physics and Computational Mathematics, Beijing 100088, China

2Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

Correspondence should be addressed to Daiwen Huang,[email protected] Received 28 September 2008; Accepted 18 February 2009

Recommended by Sandro Salsa

By introducing a compactness lemma and considering a constrained variational problem, we obtain a setG_R² of steady waves for Hasegawa-Mima equation, which describes the motion of drift waves in plasma. Moreover, we prove thatG_R² is a stable set for the initial value problem of the equation, in the sense that a solutionψtwhich starts nearG_R2will remain near it for all time.

Copyrightq2009 B. Guo and D. Huang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

It is commonly believed that drift waves and drift-wave turbulence play a major role in the understanding of anomalous transport at the plasma edge of a tokamak fusion reactor. One- field equation describing the electrostatic potential fluctuations in this regime is Hasegawa- Mima equation:

∂Δψ−ψ

∂t ∂ψ

∂x

∂Δψ−ψ

∂y −∂ψ

∂y

∂Δψ−ψ

∂x 0, 1.1

wherex, y∈R²,ψdescribes the electrostatic fluctuation andψx, y → 0 as|x||y| → ∞.

The derivation of1.1can be found in1.

There are many works about analytical mathematical study for1.1; see, for example, 2,3and references therein. In2, Grauer proved that the energy for a perturbed Hasegawa- Mima equation saturates at a finite level, which was observed by numerical simulations. Guo and Han in3studied the global well-posedness of Cauchy problem for1.1. One of their results is that the solutionψtof1.1withψ0∈W^2,2R²∩W^2,∞R²exists globally and is

unique. However, the global well-posedness of1.1withψ0∈W^2,qR²is still not attacked, whereW^2,qR²is the usual Sobolev space with norm · 2,qand 1< q <2. A natural problem is whether the solution of1.1with the initial dataψ0is close to a steady waveψ₀for all time or not, ifψ0is sufficiently close toψ0inW^2,qR². The problem is concerned with the existence and stability of steady waves of1.1.

Here, we are interested in studying the above problem.ψ₀ is a steady wave for1.1 if and only if there exists some function f such that ψ0 fΔψ0 −ψ0. In order to prove the existence and nonlinear stability of steady waves for1.1, we consider the existence and property of critical points of the so-called energy-Casimir functional:

I_R²ψ ΦR²ψ ΨR²ψ, 1.2

where

Φ_R²ψ 1 2

R²

|∇ψ|²|ψ|² , Ψ_R²ψ

R²FΔψ−ψ

1.3

are two conserved quantities of1.1, called the total energy and the generalized enstrophy, respectively. Here

R²·dx dyis denoted by

R²·, andFis an arbitraryC¹function. The critical pointsψ0ofI_R²are steady waves of1.1, given byψ0fΔψ0−ψ0, wheref F.

A usual approach to prove the existence of stable critical points of I_R² is to find extremum points of it, which is the well-known Liapunov method. Ifψ₀is a global or local extremum point ofI_R²in an appropriate defined function spaceX, then it follows thatψ0is a steady nonlinearly stable solutions of1.1; see, for example,4. There are two examples for Fsuch thatI_R²have a global extremum. One isFsatisfying thatFx≥0 for anyx∈Rand there existsψ0such thatψ0FΔψ0−ψ0. In this case,

I_R²ψ−I_R² ψ₀

≥ 1 2

R²

∇

ψ−ψ₀²ψ−ψ₀²

, 1.4

which implies thatψ0is a global minimizer ofI_R². Therefore, the steady waveψ0is nonlinearly stable in the following sense: for anyε > 0, there existsδ > 0 such that ifψ0−ψ0X < δ andψt ∈ C0, T, X is a solution of1.1with initial dataψ0, then for anyt ∈ 0, T, ψt−ψ01 < ε, where · 1 is the norm ofH¹R²and T > 0. The other example is F, which has the properties:Fx ≤ −c for large enoughc > 0 and there exists ψ0 such that FΔψ0−ψ₀ ψ₀. In this example,

I_R²ψ−I_R² ψ0

≤ 1 2

R²

∇

ψ−ψ0²ψ−ψ0²

−c

R²

Δ ψ−ψ0

−

ψ−ψ0²

≤ −c

R²

Δ ψ−ψ0

−

ψ−ψ0²,

1.5

wherec >0. Soψ0is a global maximizer of the functionalI_R² and nonlinearly stable in the above sense.

However, for someF, all critical points ofI_R² are neither global nor local extremum points ofI_R². Among them, some critical points are saddle points regarded as an unstable equilibria or transient excited state of1.1. In the present paper, we consider the existence and stability defined later of saddle points ofI_R²forFx −1/q|x|^q, whereqp/p−1, 2 < p < ∞, andx ∈ R. SinceFx < 0 and there does not exist a positive constant c > 0 such thatFx <−cfor anyx /0,Fis not within the range of the above two examples. As shown in Proposition A.1Proposition A.1, Definition A.2, Proposition A.3, and Remark A.4 are given in the appendix, the functional1.2is neither bounded from above nor from below inW^2,qR², that is, it is impossible to prove the existence of critical points by finding global extremum points ofI_R². However, through studying the constrained variational problem

M_R² inf

ψ∈W^2,qR²,ψ11

R²|Δψ−ψ|^q, M_R² we obtain the existence of critical points of1.2. In fact, ifψ0is a minimizer forM_R², then, according to Lagrange Multiplier Principle, with the transformψ0M^1/2−q_R2 ψ0,ψ0is a steady wave of1.1inR². With Definition A.2 and Proposition A.3,ψ0is a ground state and saddle point ofI_R². LetZ_R²be the set of all minimizers forM_R², that is,

Z_R²

ψ0;

R²

∇ψ0²ψ0² 1,

R²

Δψ0−ψ0^q

M_R² , 1.6

and letG_R²be the set of steady waves of1.1corresponding to minimizers ofM_R², that is,

G_R²

ψ₀;ψ₀M_R^1/2−q₂ ψ₀,ψ₀∈Z_R²

. 1.7

As is presented in Remark A.4, G_R² is the set of all ground states of the functional I_R². Although the elements ofG_R² are saddle points ofI_R²regarded as an unstable state of1.1, we prove thatG_R²is a stable set in the sense ofDefinition 1.1, that is, a solutionψtof1.1 which starts nearG_R²will remain near it for all time.

Definition 1.1. LetEbe a function space with norm · E, andT ∈ 0,∞. A set G ⊂ Eis calledE-stable with respect to1.1inR², if for anyε > 0, there exists aδ > 0 such that if ψ ∈C0, T, Eis a solution to1.1with initial dataψ0satisfying infψ0∈Gψ0−ψ0E≤δ, then for anyt∈0, T, infψ0∈Gψt,·−ψ₀E≤ε.

One gives some explanations for the above definition as follows. IfG has only one elementψ0, then the steady waveψ0is nonlinearly stable in the usual sense. But, in general, the elements of G might not be unique. For example, as shown in Theorem 1.3, G_R² {M^1/2−q_R2 ψ₀·y,−M^1/2−q_R2 ψ₀·y;y ∈ R²}. In this case, ifψ0 is sufficiently close to G_R², then, for anyt >0, the form ofψtis almost similar to that ofM^1/2−q_R2 ψ0or−M^1/2−q_R2 ψ0.

Now one turns to describe his two main results.

Theorem 1.2. If{ψ_m} is a minimizing sequence forM_R², then there exist{ymk} ⊂ R² and a subsequence{ψmk}such that{ψmk·ymk}is a convergent sequence inW^2,qR². In particular, the minimization problemM_R²has a minimizerψ₀.

Theorem 1.3. According toDefinition 1.1, the ground state setG_R²isW^2,qR²-stable with respect to1.1inR². Moreover, there is a unique, up to translation, positive radially symmetricC²minimizer

ψ₀ofM_R², and

G_R²

M^1/2−q_R2 ψ0·y,−M^1/2−q_R2 ψ0·y; y∈R²

. 1.8

The important step to obtain Theorems1.2 and 1.3is to prove that the infimum is achieved. If{ψ_m}is a minimizing sequence ofM_R², then

ψm

11,

R²

Δψm−ψm^q−→M_R², m−→∞. 1.9

Going if necessary to a subsequence, we may assumeψm → ψ0weakly inW^2,qR², so that

R²

Δψ₀−ψ₀^q ≤lim inf

R²

Δψ_m−ψ_m^qM_R². 1.10

Thus ψ₀ is a minimizer of M_R² provided ψ₀1 1. Since W^2,qR² → H¹R² is not compact, we cannot deriveψ01 1 fromψm1 1 andψm → ψ0 weakly in W^2,qR². Therefore, we cannot directly derive the existence of minimizer from any minimizing sequence. However, if we obtain the result that for any minimizing sequence{ψ_m}there exist {ymk} ⊂ R² and a subsequence{ψ_mk}such that{ψ_mk·y_mk}is a convergent sequence in W^2,qR², which is the first part ofTheorem 1.2, then we prove that the infimum is achieved.

In order to prove Theorem 1.2, we construct Lemma 2.1, which is used to study the behavior at infinity of the minimizing sequence {ψ_m} and to overcome the loss of compactness of MR².Theorem 1.2 is proved by two steps. Firstly, using Lemma 2.4, we prove that for any minimizing sequence{ψ_m}, there exist a subsequence{ψ_mk}and{ymk} ⊂ R² such thatψ_mk·y_mk → ψ₀/0 weakly inW^2,qR², which denotes 0 ≤ α_∞ < 1. Here α_∞ is a quantity related to {ψmk· ymk} defined in Lemma 2.1. Secondly, according to Lemmas 2.1 and 2.3 based on Ekeland Principle, we know that if α_∞ > 0, then α_∞ ≥ 1.

Therefore, putting together the results of the above steps, we obtainα_∞ 0, which implies that there exists a sequence ymk ⊂ R² such that the sequence{ψmk·ymk} is convergent inW^2,qR². Applying Theorem 1.2, we prove that G_R² is a stable set with respect to1.1, which is the first part ofTheorem 1.3. The second part about the structure ofG_R²is obtained by studying the properties of the elliptic equation satisfied byM^1/2−q_R2 ψ₀. Our method in proving the existence and stability of steady waves for 1.1 is different from that in 5.

In5, Albert considered constrained variational problems with concentration-compactness Lemma introduced by Lions6,7and proved the existence and stability of solitary waves to Kdv equation and some nonlocal equations.

The paper is organized as follows. InSection 2, we establish three lemmas for proving Theorem 1.2. In Section 3, we give the proofs of Theorems 1.2 and 1.3. In Section 4, we consider the existence and stability of steady waves for Hasegawa-Mima equation in general periodic domains and give the application of Lemmas 2.1 and 2.3 to study the existence and stability of steady waves for two-dimensional incompressible fluid in an infinite strip channel. Two propositions about the properties of the functionalI_R² forFx −1/q|x|^q and the definition of the ground state ofI_R2are presented in the appendix.

2. Three Lemmas

At first, we give some notations used later. LetL^qΩbe the usual Lebesgue space with norm

| · |_q, whereΩis an unbounded domain inR²andqp/p−1,2< p <∞. The spaceW₀^2,qΩ is the completion ofDΩwith respect to · 2,q, whereDΩis the set of allC^∞-functions with compact support inΩ. The spaceH₀¹Ωis the completion ofDΩwith respect to · 1. Now we giveLemma 2.1, which is used to study the behavior at infinity of minimizing sequence{ψm}forM_R².

Lemma 2.1compactness lemma. Let{ψ_m} ⊂W₀^2,qΩbe a sequence such thatψ_m → ψweakly inW₀^2,qΩand define

α_∞: lim

R→ ∞lim sup

m→ ∞

Ω∩{|x|>R}

∇ψm²ψm² ,

β_∞: lim

R→ ∞lim sup

m→ ∞

Ω∩{|x|>R}

Δψm−ψm^q.

2.1

Then one has

1lim sup_{m→ ∞}

Ω|∇ψm|²|ψm|²

Ω|∇ψ|²|ψ|² α_∞, 2lim sup_{m→ ∞}

Ω|Δψ_m−ψ_m|^q≥

Ω|Δψ−ψ|^qβ_∞, 3 α∞^q/2M_Ω≤β_∞,where

M_Ω inf

ψ∈W₀^2,qΩ,ψ11

Ω

Δψ−ψ^q. MΩ

Proof. iFor anyR >0, letBR{x;|x|< R}, B_R^c {x;|x| ≥R}:

lim sup

m→ ∞

Ω

∇ψ_m²ψ_m²

lim sup

m→ ∞

Ω∩BR

∇ψ_m²ψ_m²

Ω∩B^c_R

∇ψ_m²ψ_m² .

2.2

Since the imbeddingW^2,qΩ∩B_R→H¹Ω∩BRis compact andψ_m → ψweakly inW₀^2,qΩ,

mlim→ ∞

Ω∩BR

∇ψ_m²ψ_m²

Ω∩BR

∇ψ²ψ²

. 2.3

Combining2.2with2.3, we get

lim sup

m→ ∞

Ω

∇ψm²ψm²

Ω∩BR

∇ψ²ψ²

lim sup

m→ ∞

Ω∩B^c_R

∇ψ_m²ψ_m² .

2.4

LettingR → ∞in the above formula, we obtain1.

iiUsing the weakly lower semicontinuity of a norm, we have

lim inf

m→ ∞

Ω∩BR

Δψm−ψm^q≥

Ω∩BR

Δψ−ψ^q, for any R >0. 2.5

Applying2.5, we have

lim sup

m→ ∞

Ω

Δψm−ψm^q

≥lim inf

m→ ∞

Ω∩BR

Δψm−ψm^qlim sup

m→ ∞

Ω∩B^c_R

Δψm−ψm^q

≥

Ω∩BR

Δψ−ψ^qlim sup

m→ ∞

Ω∩B^c_R

Δψm−ψm^q.

2.6

LettingR → ∞in2.6, we deduce2.

iii Applying elementary inequalities, we can prove that the norm |Δ−1· |q is equivalent to · 2,qinW₀^2,qΩ. LetϕR∈C^∞R²such that

ϕ_Rx

0 |x|< R,

1 |x|> R1 2.7

and 0 ≤ ϕRx ≤ 1 on R². It follows from the definition of M_Ω and the convexity of the functiongx |x|^qx∈Rthat

M_Ω

Ω

∇ψ_mϕ_R²ψ_mϕ_R^2q/2

≤

Ω

Δ ψ_mϕ_R

−ψ_mϕ_R^q

Ω

ϕR

Δψm−ψm

ψmΔϕR2∇ψm· ∇ϕR^q

Ω

1−ε ϕR

1−ε

Δψm−ψm

ε

ψmΔϕR

ε 2∇ψm· ∇ϕR

ε ^q

≤1−ε

Ω

ϕR

1−ε _q

Δψm−ψm^qε

Ω

ψmΔϕR

ε 2∇ψm· ∇ϕR

ε ^q,

2.8

where 0 < ε < 1. Sinceψm → ψ weakly inW₀^2,qΩand the embeddingW^2,qΩ∩ {x :R <

|x|< R1}→H¹Ω∩ {x:R <|x|< R1}is compact,

ψ_mΔϕ_R

ε −→ψΔϕ_R

ε , ∇ψ_m∇ϕ_R

ε −→ ∇ψ∇ ϕ_R

ε inL^q Ω∩

x:R <|x|< R1

, 2.9

which implies

lim

R→ ∞lim sup

m→ ∞

Ω

ψmΔϕR

ε 2∇ψm· ∇ϕR

ε

^q0. 2.10

With the definitions ofα_∞, β_∞,

lim

R→ ∞lim sup

m→ ∞

Ω

∇ψmϕR²ψmϕR^2q/2

α_∞q/2 ,

lim

R→ ∞lim sup

m→ ∞

Ωϕ^q_RΔψm−ψm^qβ_∞.

2.11

We derive from2.8–2.11

M_Ω

α_∞q/2≤1−ε^1−qβ_∞. 2.12

Sinceεis arbitrary, we have

M_Ω α_∞q/2

≤β_∞. 2.13

Remark 2.2. In8, Huang and Li have used a concentration-compactness principle at infinity, similar to Lemma 2.1, to study the existence of positive solutions for some quasilinear equations on unbounded domains inR^N.

In the following, we giveLemma 2.3, which is used to find a Palais-Smale sequence {ψ_m¹}ofI_R²through the minimizing sequence{ψ_m}forM_R². Firstly, we give some notations.

Let

I_Ωψ 1 2

Ω

|∇ψ|²|ψ|²

−1 q

Ω|Δψ−ψ|^q, J_Ω

ψ

ΩΔψ−ψ^q

Ω∇ψ²ψ^2q/2.

2.14

By the definition of Fr´echet derivative, it is easy to verify that I_Ω ∈ C¹W₀^2,qΩ,R,J_Ω ∈ C¹W₀^2,qΩ\ {0},Rand

I_Ωψ, h

Ω∇ψ∇hψh−

Ω|Δψ−ψ|^q−2Δψ−ψΔh−h, J_Ωψ, h

qψ^q−2₁ ψ²

1

ΩAψ^q−2AψAh −ψ^q

2,q

Ω

∇ψ∇hψh

Ω∇ψ²ψ^2q ,

2.15

whereψ,ψ, h ∈W₀^2,qΩandAψ Δψ−ψ.

Lemma 2.3. If{ψm}is a minimizing sequence ofMΩ, then there is a minimizing sequence{ψ_m¹} such thatψ_m¹ −ψm2,q<1/m,J_Ωψ_m¹ → M_Ω,J_Ωψ_m¹ → 0 inW^−2,qΩasm → ∞, and

I_Ω ψm

−→

1 2−1

q

M^2/2−q_Ω , I_Ω ψm

−→0 in W^{−2, q}Ω, asm−→ ∞, 2.16

whereψ_m M_Ω^1/2−qψ_m¹, andW^−2,qΩis the dual space ofW₀^2,qΩ, 1/q1/q1. Moreover, if α_∞, β_∞are quantities related toψ¹_minLemma 2.1, then

M_Ω·α_∞β_∞. 2.17

Proof. Using the definitions ofM_ΩandJ_Ω, we have

M_Ω inf

0/ψ∈W ₀^{2, q}ΩJ_Ω ψ

lim

m→ ∞J_Ω ψ_m

. 2.18

Applying the Ekeland Variational Principlecf.9, page 51to2.18, we get a Palais-Smale sequence{ψ_m¹}, which satisfies

ψ_m¹ −ψ_m

2, q< 1

m, J_Ω ψ_m¹

−→M_Ω, J_Ω

ψ_m¹

−→0 inW^−2,qΩ, asm−→ ∞.

2.19

Then, according to the definitions ofI_Ωandψm, for anyh∈W₀^2,qΩ, we get

I_Ω ψ_m

1

2M_Ω^2/2−q

Ω

∇ψ¹_m²ψ_m¹²

−1

qM_Ω^q/2−q

Ω

Δψ¹_m−ψ_m^1q

−→ 1

2M^2/2−q_Ω −1

qM^q/2−q1_Ω

1 2−1

q

M^2/2−q_Ω , I_Ωψm, h

M^1/2−q_Ω

Ω

∇ψ_m¹∇hψ_m¹h

−M^q−1/2−q_Ω

Ω

Aψ¹_m^q−2Aψ_m¹Ah M^q−1/2−q_Ω

M_Ω

Ω

∇ψ¹_m∇hψ¹_mh

−

Ω

Aψ_m^1q−2Aψ_m¹Ah

−→0

because of J_Ω

ψ_m¹

, h

−→0

, asm−→ ∞.

2.20

Since2.20implies I_Ω

M_Ω^1/2−qψ¹_m

−→0 in W^−2,qΩ, asm−→ ∞, 2.21

it follows that I_Ω

M_Ω^1/2−qψ¹_m

, M^1/2−q_Ω ψ_m¹ϕ_R

−→0, asm−→ ∞uniformly for R≥1, 2.22

whereϕRis the function defined in the proof ofLemma 2.1. With the definition ofI_Ω, we have I_Ω

M^1/2−q_Ω ψ_m¹

, M^1/2−q_Ω ψ_m¹ϕ_R M^2/2−q_Ω

Ω

∇ψ_m¹∇ ψ_m¹ϕ_R

ψ_m¹ψ_m¹ϕ_R

−M^q/2−q_Ω

Ω

Δψ_m¹ −ψ_m^1q−2

Δψ_m¹ −ψ_m¹ Δ

ψ¹_mϕ_R

−ψ_m¹ϕ_R .

2.23

Using2.22, and lettingR → ∞afterm → ∞in2.23, we obtain

M_Ω·α_∞β_∞. 2.24

At last, we give another lemma for provingTheorem 1.2.

Lemma 2.4. If{ψm}is bounded inW^2,qR², and

sup

y∈R²

By,Rψ_m^q −→0, for some R >0, 2.25 thenψ_m → 0 inH¹R².

Proof. Applying interpolation inequalities, forψ∈W^2,qR², we have ∇ψ²

L²By,R≤cψ^21−γ1

L^qBy,Rψ^2γ1

W^2,qBy,R, 2.26

ψ²

L²By,R≤cψ^21−γ2

L^qBy,Rψ^2γ2

W^2,qBy,R, 2.27

whereγ11/q, γ21/q−1/2, andcis a positive constant. LettingB1B0, R,B2By2, R wherey₂ ∈ ∂B0, R,B₃ By3, R, B4 By4, Rwhere{y3, y₄} ∂B₁∩∂B₂, . . . ,we cover R²by the above balls of radiusRsuch that each point ofR²is contained in at most 3 balls.

Therefore, combining2.26with2.27, we obtain ∇ψm²

L²R²≤3c

sup

y∈R²

By,Rψm^q_21−γ1

ψm^2γ1

W^2,qR², ψm²

L²R²≤3c

sup

y∈R²

By,Rψm^q_21−γ2

ψm^2γ2

W^2,qR².

2.28

According to the assumptions ofLemma 2.4 and the above two inequalities, ψ_m → 0 in H¹R².

3. Proof of Theorems 1.2 and 1.3

Now we turn to prove our main results.

Proof ofTheorem 1.2. Using Lemma 2.3 and I_R² ∈ C¹W^2,qR²,R, for any minimizing sequence{ψ_m}ofM_R², we have

I_R²

M^1/2−q_R2 ψm

−→

1 2 −1

q

M^2/2−q_R2 , I_R2

M^1/2−q_R2 ψm

−→0 inW^−2,q R²

asm−→ ∞.

3.1

Moreover,{ψ_m}is bounded inW^2,qR².

Lemma 2.4implies that there exists{ymk} ⊂R²such that

ψ_m¹_k :ψmk

·ymk

−→ψ0/0 weakly inW^{2, q} R²

. 3.2

In fact, sinceψm11, byLemma 2.4, there existsν >0 such that

lim inf

m→ ∞ sup

y∈R²

By,Rψ_m^q> ν. 3.3

Then, the above inequality and the boundedness of{ψm}inW^2,qR²imply that there exist a subsequence{ψ_mk}andy_mk ∈R²such that

By_mk,Rψ_mk^q > ν

2 3.4

and ψ_mk·y_mk → ψ₀ weakly inW^{2, q}R². Letting ψ_m¹_k : ψ_mk·y_mk, we know that {ψ_m¹_k} is a minimizing sequence of M_R² and

B0,R|ψ_m¹_k|^q > ν/2. Since the embedding W^2,qB0, R→L^qB0, Ris compact,ψ_m¹_k → ψ0inL^qB0, R, which impliesψ0^q_LqB0,R

≥ν/2. Soψ¹_mk → ψ0/0 weakly inW^2,qR².

LetΩ R²andα_∞, β_∞be quantities related toψ_m¹_k inLemma 2.1. WithLemma 2.1, we have

R²

∇ψ0²ψ0²

α_∞1. 3.5

In order to proveTheorem 1.2, we have to show α_∞ 0. Since ψ₀/0in W^2,qR² implies 0≤α_∞<1, arguing by contradiction, we assume 0< α_∞ <1. Using Lemmas2.1and2.3, we have

M_R² α_∞q/2

≤β_∞M_R²α_∞. 3.6

Since Sobolev Theorem impliesM_R² >0, we derive from3.6α_∞≥1,which contradicts the assumption that 0 < α_∞ < 1.Thusψ0 is a minimizer for the minimization problemMR². SinceW^2,qR²is a uniformly convex space,ψ_m¹_k2,q → ψ02,q andψ_m¹_k → ψ0 weakly in W^2,qR², we know thatψ_m¹_k → ψ₀inW^2,qR².

Proof ofTheorem 1.3. We divide the proof into two steps.

Step 1. We prove thatG_R²is a stable set. Assume that the ground state setG_R²is notW^2,qR²- stable. Then there existε₀>0,ψ_m⁰ ⊂W^2,qR²andt_m∈0, Tsuch that

ψ0inf∈G_R2

ψ_m⁰ −ψ₀

E≤ 1

m, 3.7

ψ0inf∈G_R2

ψ_m t_m

−ψ₀

E≥ε₀, 3.8

whereψ_m∈C0, T, Eis a solution to1.1withψ_m0 ψ_m⁰. Letψ_mM^−1/2−q_R2 ψ_m⁰. Equation 3.7implies that

R²

∇ψm²ψm²

−→1,

R²

Δψm−ψm^q−→M_R². 3.9

Then there exists{rm} ⊂R,rm → 1 asm → ∞such that rmψm

is a minimizing sequence forM_R². 3.10

Using1.3, we have ψ_m⁰

1ψ_m t_m

1, ψ_m⁰

2, qψ_m t_m

2, q. 3.11

With 3.10 and 3.11, we know that {rmM^−1/2−q_R2 ψ_mtm} is a minimizing sequence for M_R². ByTheorem 1.2, there existy_mk ∈R²andψ_m¹_k ∈Z_R²such that

rmkM^−1/2−q_R2 ψmk

tmk

·ymk

−ψ_m¹_k

2, q

rmkM_R^−1/2−q2 ψmk

tmk

−ψ_m¹_k

· −ymk

2, q

≤ ε₀ 2M^1/2−q_R2

,

3.12

for sufficiently largem_k. Sincer_m → 1 andψmtm2,qis bounded, we derive from3.8and 3.12

ε0≤ψmk

tmk

−M^1/2−q_R2 ψ_m¹_k

· −ymk

2, q

≤ψmk

tmk

−rmkψmk

tmk

2,qrmkψmk

tmk

−M_R^1/2−q2 ψ_m¹_k

· −ymk

2, q

≤ψmk

tmk

−rmkψmk

tmk

2, qM^1/2−q_R2 rmkM^−1/2−q_R2 ψmk

tmk

−ψ_m¹_k

· −ymk

2, q

≤ 3 4ε0,

3.13 for sufficiently large mk. Equation 3.13is a contradiction. Therefore, the ground state set G_R²isW^2,qR²-stable with respect to1.1.

Step 2. Show the structure of G_R². Ifψ₀is a minimizer forM_R², then|ψ₀|is also a minimizer forM_R². We can assume thatψ₀is a nonnegative minimizer forM_R². Letψ₀M^1/2−q_R2 ψ₀, thenI_R2ψ0 0 inW^−2,qR², that is,

−Δψ0ψ0ψ₀^p−1, ψ0x−→0 as|x| −→∞, 3.14

wherep > 2 is given inSection 2. In fact, using the implicit function theorem, we know that M{ψ : ψ ∈W^2,qR²,

R²|∇ψ|²|ψ|²1}is aC¹-submanifold ofW^2,qR². Then, according to the Lagrange Multiplier Principle, there existsλ∈Rsuch that

−qΨ_R² ψ₀

2λΦ_R² ψ₀

inW^−2,q R²

, 3.15

where Ψ_R²ψ₀ −1/q

R²|Δψ₀ −ψ₀|^q,Φ_R²ψ₀ 1/2

R²|∇ψ₀|² |ψ₀|². Since ψ₀ is a minimizer forMR², with the definition ofM_R², we obtainλ q/2MR². So, we conclude I_R2ψ0 0.

Applying elliptic regularity theorycf., e.g., 10, Lemma 1.30, we prove that ψ0 ∈ C²R². According to the strong maximum principle,ψ₀is positive. Using the moving plane methodcf.11, we show thatψ₀is radially symmetric. Moreover, by the uniqueness result in12,ψ0is unique up to translations. Therefore,ψ0 is a unique, up to translation, positive radially symmetricC²minimizer ofM_R², and

G_R²

M^1/2−q_R2 ψ0·y,−M^1/2−q_R2 ψ0·y;y∈R²

. 3.16

Remark 3.1. The existence of nontrivial critical points of 1.2 for Fx −1/q|x|^q can also be proved by the following method. Since W_r^2,qR² → H_r¹R² is compact, where W_r^2,qR² and H_r¹R² are the set of the radially symmetric functions of W^2,qR² and H¹R², respectively, with Principle of Symmetric Criticality proposed by Palais cf. 13 and Fountain Theorem cf. 10, Chapter 3, we can prove that there are infinitely many solutions to1.2. However, we do not know how to consider the stability of these solutions.

So, we cannot study the existence of nontrivial critical points of 1.2 with this method.

Through considering the constrained variational problemM_R², we obtain the existence of a set G_R² of steady waves for 1.1 and know thatG_R² isW^2,qR²-stable with respect to 1.1.

4. Hasegawa-Mima Equation in Periodic Domains

In this section, we consider the existence and stability of steady waves for Hasegawa-Mima equation in periodic domains:

∂Δψ−ψ

∂t ∂ψ

∂x

∂Δψ−ψ

∂y −∂ψ

∂y

∂Δψ−ψ

∂x 0, 4.1

where x, y ∈ Ω,Ωis a periodic domain defined later, ψx, y → 0 as |x||y| → ∞, andψ|x,y∈∂Ω 0. Moreover, as a byproduct, we prove the existence and stability of steady two-dimensional incompressible flows in infinite strip channel.

At first, we give the definition and two examples of periodic domain.

Definition 4.1periodic domain. IfΩis a domain inR², and there are a partition{Ωn}ofΩ and points{yn}inR²satisfying the following conditions:1{yn}forms a subgroup ofR², 2 Ω0is a bounded domain inR²,3 Ωny_n Ω0,thenΩis called a periodic domain.

It is clear thatR² is a periodic domain. The other example of periodic domain isQ {x, y∈R×R: |y|<1}.

Similarly to the method used in studying1.1, in order to prove the existence and stability of steady waves for4.1, we consider the following minimization problem:

M_Ω inf

ψ∈W₀^2,qΩ,ψ11

Ω

Δψ−ψ^q. M_Ω