# Nonexistence Results of Semilinear Elliptic Equations Coupled with the Chern-Simons Gauge Field

Authors: Hyungjin Huh
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We discuss the nonexistence of nontrivial solutions for the Chern-Simons-Higgs and Chern-Simons-Schrödinger equations. The Derrick-Pohozaev type identities are derived to prove it.


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## Body

## 1. Introduction and Main Results

In this paper, we are concerned with the nonexistence of nontrivial solutions to some elliptic equations coupled with Chern-Simons gauge field. More precisely, let us first consider the following system:(1)-(ω+A0)2ϕ-D1D1ϕ-D2D2ϕ+V′(|ϕ|2)ϕ=0,(2)∂1A0=-Im(ϕ-D2ϕ),(3)∂2A0=Im(ϕ-D1ϕ),(4)∂1A2-∂2A1=-(ω+A0)|ϕ|2,
which is derived from the system (5) with stationary solution ansatz ψ(t,x)=eiωtϕ(x), ϕ∈ℂ, and Aμ(t,x)=Aμ(x) for μ=0,1,2. Consider
(5)D0D0ψ-(D1D1+D2D2)ψ+V′(|ψ|2)ψ=0,∂0A1-∂1A0=Im(ψ-D2ψ),∂0A2-∂2A0=-Im(ψ-D1ψ),∂1A2-∂2A1=-Im(ψ-D0ψ),
where ∂0=∂/∂t, ∂1=∂/∂x1, ∂2=∂/∂x2 for (t,x1,x2)∈ℝ1+2, ψ:ℝ1+2→ℂ is the complex scalar field, Aμ:ℝ1+2→ℝ is the gauge field, Dμ=∂μ+iAμ is the covariant derivative for μ=0,1,2, and i denotes the imaginary unit.The Chern-Simons-Higgs system in (5) was introduced in [1, 2] to deal with the electromagnetic phenomena in planar domain such as fractional quantum Hall effect or high temperature superconductivity. The system in (5) has the conservation of the total energy
(6)E(t)=∫ℝ2‍∑α=02‍|Dαψ(t,x)|2+V(|ψ(t,x)|2)dx=E(0).The special case with a self-dual potentialV(|ϕ|2)=(1/4)|ϕ|2(|ϕ|2-1)2 has received much attention and has been studied by several authors, where one can derive the following system of first-order equations called self-dual equations (see [1, 2])
(7)D1ϕ-iD2ϕ=0,∂1A2-∂2A1+12|ϕ|2(|ϕ|2-1)=0,ω+A0-i2(|ϕ|2-1)ϕ=0.
We note that solutions to the self-dual equations (7) provide solutions to (1)–(4). For the self-dual potential V(|ϕ|2)=(1/4)|ϕ|2(|ϕ|2-1)2, there are two possible boundary conditions to make the energy finite; either |ϕ|→1 or |ϕ|→0 as |x|→∞. The former boundary condition is called “topological” while the latter “non-topological.” A lot of works have been done for the existence of solutions to the self-dual system [3–7]. Some existence results for the nonself-dual Chern-Simons-Higgs equations with the topological boundary condtion have been proved in [8–10]. From the mathematical point of view, it is meaningful to study existence and nonexistence of nontrivial solutions under various conditions on V. In this paper, we are concerned with the nonexistence of the non-trivial solution to (1)–(4) with the non-topological boundary condtion. The following is our first result.Theorem 1.
Let(ϕ,A0,A1,A2) be a classical solution of (1)–(4) such that ϕ∈H1(ℝ2) and A0∈Lp(ℝ2), A1,A2∈Lq(ℝ2) for any 2<p,q≤∞. Let V:ℝ→ℝ be a C1 function such that V(0)=0, inf{x>0∣V(x)≠0}=0 and V(|ϕ|2), |ϕ|2V′(|ϕ|2)∈L1(ℝ2). Assume that
(8)dV′(s)s-V(s)≥0∀s≥0,
where 0≤d≤1 is a constant. Then, one has ϕ≡0.The proof is based on the following Derrick-Pohozaev type identity for (1)–(4):
(9)∫ℝ2‍d(|D1ϕ|2+|D2ϕ|2)+(1-d)(ω+A0)2|ϕ|2+dV′(|ϕ|2)|ϕ|2-V(|ϕ|2)dx=0.
As a typical example, we consider V(|ϕ|2)=α|ϕ|6+β|ϕ|4. Then it is easy to check that dV′(s)s-V(s)=α(3d-1)s3+β(2d-1)s2. If one of the following conditions is satisfied, then we have ϕ≡0. (1)
Forα>0, β>0, we take 1/2<d<1.(2)
Forα>0, β<0, we take 1/3<d<1/2.(3)
Forα<0, β<0, we take 0<d<1/3. Note that for the self-dual potential V(|ϕ|2)=(1/4)|ϕ|2(|ϕ|2-1)2, we have
(10)dV′(s)s-V(s)=3d-14s3-2d-12s2+d-14s
which is not nonnegative for s≥0.The following Chern-Simons gauged Schrödinger system was proposed in [11] when the second quantized N body anyon problem is considered
(11)iD0ψ+(D1D1+D2D2)ψ-V′(|ϕ|2)ϕ=0,∂0A1-∂1A0=-Im(ψ-D2ψ),∂0A2-∂2A0=Im(ψ-D1ψ),∂1A2-∂2A1=-12|ψ|2.
With the stationary solution ansatz ψ(t,x)=eiωtϕ(x), ϕ∈ℂ and Aμ(t,x)=Aμ(x) for μ=0,1,2, we arrive at
(12)(ω+A0)ϕ-D1D1ϕ-D2D2ϕ+V′(|ϕ|2)ϕ=0,(13)∂1A0=Im(ϕ-D2ϕ),(14)∂2A0=-Im(ϕ-D1ϕ),(15)∂1A2-∂2A1=-12|ϕ|2.In the special case with the potentialV(|ϕ|2)=-(1/2)|ϕ|4, we can derive the following self dual equations [11–13]
(16)D1ϕ+iD2ϕ=0,ω+A0-12|ϕ|2=0,∂1A2-∂2A1+12|ϕ|2=0.
Note that solutions to the self-dual system (16) provide solutions to (12)–(15). The self-dual equations (16) can be transformed into the Liouville equation, an integrable equation whose solutions are explicitly known.For the nonself-dual potential of the formV(|ϕ|2)=-λ|ϕ|p (λ>0, p>2), some existence and nonexistence results have been studied in [14, 15] under the condition of the radially symmetric solution ψ(t,x)=eiωtu(|x|)(u∈ℝ). We prove the following nonexistence result, under various conditions on V, for (12)–(15).Theorem 2.
Let(ϕ,A0,A1,A2) be a classical solution of (1)–(4) such that ϕ∈H1(ℝ2), A0∈Lp(ℝ2) and A1,A2∈Lq(ℝ2) for 1<p≤∞ and 2<q≤∞. One also assumes that V:ℝ→ℝ is a C1 function such that V(0)=0 and V(|ϕ|2), |ϕ|2V′(|ϕ|2)∈L1(ℝ2). (1)
If the potentialV satisfies
(17)V′(s)s-V(s)≥0∀s≥0,
then one has ϕ≡0.(2)
Suppose thatϕ is a real-valued function; that is, ϕ(x)=u(x)∈ℝ and 1/p+1/q=1 for 2<q<∞. If the potential V satisfies, for a constant h≥2/3,
(18)ω(h-1)s+hV′(s)s-V(s)≥0∀s≥0,
then one has u≡0.The proof is based on the Derrick-Pohozaev type identities (40) and (45) for (12)–(15).Example 3.
For the static solution(ω=0), we consider the potential V(s)=s3-s2-s. Then, taking h=2/3, we can check
(19)23V′(|u|2)|u|2-V(|u|2)=|u|6-13|u|4+13|u|2≥0.
However we have, for the complex solution,
(20)V′(|ϕ|2)|ϕ|2-V(|ϕ|2)=2|ϕ|6-|ϕ|4
which is not nonnegative.
The paper is organized as follows. In Section2, we prove Theorem 1 by deriving Derrick-Pohozaev type identity. Theorem 2 is proved in Section 3. We conclude this section by giving a few notations. (i)
H1(ℝ2) denotes the usual Sobolev space W1,2(ℝ2).(ii)
BR:={x∈ℝ2∣|x|≤R} and ∂BR:={x∈ℝ2∣|x|=R}.(iii)
dσR:= the  surface  measure  on  ∂BR.


## 2. Proof of Theorem1

We apply Derrick-Pohozaev argument to derive the identity (9) which prove Theorem 1. From now on, we adopt the summation convention for repeated indices.Suppose that(ϕ,A0,A1,A2) is a solution of (1)–(4). Multiplying (1) by xkDkϕ¯ and integrating over BR, we obtain
(21)-∫BR‍(ω+A0)2ϕxkDkϕ¯dx-∫BR‍DjDjϕxkDkϕ¯dx+∫BR‍V′(|ϕ|2)ϕxkDkϕ¯dx=0.Now we set(22)I=∫BR‍(ω+A0)2ϕxkDkϕ¯dx,II=∫BR‍DjDjϕxkDkϕ¯dx,III=∫BR‍V′(|ϕ|2)ϕxkDkϕ¯dx.Then, integrating by parts and taking real parts, we have(23)Re{I}=∫∂BR‍R2(ω+A0)2|ϕ|2dσx-∫BR‍(ω+A0)2|ϕ|2dx-∫BR‍ω∂jA0xj|ϕ|2dx-∫BR‍12xj∂j(A02)|ϕ|2dx,Re{III}=∫∂BR‍R2V(|ϕ|2)dσx-∫BR‍V(|ϕ|2)dx.ForII, we have
(24)II=∫BR‍∂j(xkDjϕDkϕ¯)-|Djϕ|2-xkDjϕDjDkϕ¯dx=∫∂BR‍xjxkRDjϕDkϕ¯dσR-∫BR‍|Djϕ|2dx-∫BR‍xkDjϕ(DkDjϕ+iFjkϕ)¯dx,
where we used the notation Fjk=∂jAk-∂kAj and the following identity:
(25)DjDkϕ=DkDjϕ+iFjkϕ.Taking the real parts and integrating by parts, we obtain(26)Re{II}=∫∂BR‍xjxkRDjϕDkϕ¯dσR-∫BR‍|Djϕ|2dx-∫BR‍12xk∂k(|Djϕ|2)dx-∫BR‍xkFjkIm(ϕ-Djϕ)dx=∫∂BR‍xjxkRDjϕDkϕ¯dσR-∫∂BR‍R2|Djϕ|2dσR+∫BR‍12|ϕ|2xj∂j(A02)dx+∫BR‍ω|ϕ|2xj∂jA0dx,
where we used (2)–(4) in the following way:
(27)-∫BR‍x1F21Im(ϕ-D2ϕ)+x2F12Im(ϕ-D1ϕ)dx=∫BR‍xj|ϕ|2(ω+A0)∂jA0dx.
Combining (23) and (26), we have from the identity (21)
(28)∫BR‍(ω+A0)2|ϕ|2dx-∫BR‍V(|ϕ|2)dx=∫∂BR‍xjxkRRe(DjϕDkϕ¯)-R2|Djϕ|2-R2V(|ϕ|2)+R2(ω+A0)2|ϕ|2dσR.
Thus we have
(29)|∫BR‍(ω+A0)2|ϕ|2-V(|ϕ|2)dx|≤C∫∂BR‍R(|Djϕ|2+ω2|ϕ|2+|A0|2|ϕ|2+V(|ϕ|2))dσR,
where C is a positive constant. Considering the Sobolev embedding and the condition of Theorem 1, we know that |Djϕ|2, ω2|ϕ|2, |A0|2|ϕ|2, V(|ϕ|2)∈L1(ℝ2). Applying the idea in [16], we know that there exists a sequence {Rn}→∞ such that
(30)∫∂BRn‍Rn(|Djϕ|2+ω2|ϕ|2+|A0|2|ϕ|2+V(|ϕ|2))dσRn⟶0,
and consequently
(31)∫ℝ2‍(ω+A0)2|ϕ|2-V(|ϕ|2)dx=limn→∞∫BRn‍(ω+A0)2|ϕ|2-V(|ϕ|2)dx=0.
On the other hand, we know from (1) that
(32)0=∫ℝ2‍ϕ-(-(ω+A0)2ϕ-DjDjϕ+V′(|ϕ|2)ϕ)dx=∫ℝ2‍-(ω+A0)2|ϕ|2+|Djϕ|2+V′(|ϕ|2)|ϕ|2dx,
by taking care of the boundary integral terms as before. Combining (31) and (32), we obtain
(33)∫ℝ2‍d|Djϕ|2+(1-d)(ω+A0)2|ϕ|2+dV′(|ϕ|2)|ϕ|2-V(|ϕ|2)dx=0,
where 0≤d≤1 is a constant. We are ready to prove Theorem 1.(
1
) For 0<d≤1, we have Djϕ≡0. If ϕ(x0)≠0, then there exists δ>0 such that ϕ(x)≠0 for Bx0(δ)={x||x-x0|<δ}. In the region Bx0(δ), we have Aj=i(∂jϕ/ϕ). Using (4) we have A0(x)+ω=0 in Bx0(δ). On the other hand, from (2) and (3), we deduce that A0(x)=constant=-ω. By (1), we obtain V′(|ϕ|2)ϕ=0 for all x∈ℝ2. By the condition of V′ and ϕ∈L2, we conclude that ϕ≡0.(
2
) For d=0, we have V(|ϕ|2)=0. By the condition of V and ϕ∈L2, we have ϕ≡0.


## 3. Proof of Theorem2

Repeating the similar argument to the proof of Theorem1, we derive Derrick-Pohozaev type identities for (12)–(15). Suppose that (ϕ,A0,A1,A2) is a solution of (12)–(15). Multiplying (12) by xkDkϕ¯ and integrating over BR, we obtain
(34)∫BR‍(ω+A0)ϕxkDkϕ¯dx-∫BR‍DjDjϕxkDkϕ¯dx+∫BR‍V′(|ϕ|2)ϕxkDkϕ¯dx=0.Now we set(35)I=∫BR‍(ω+A0)ϕxkDkϕ¯dx,II=∫BR‍DjDjϕxkDkϕ¯dx,III=∫BR‍V′(|ϕ|2)ϕxkDkϕ¯dx.Then, integrating by parts and taking real parts, we have(36)Re{I}=∫∂BR‍R2(ω+A0)|ϕ|2dσx-∫BR‍(ω+A0)|ϕ|2dx-∫BR‍12xj∂jA0|ϕ|2dx,Re{II}=∫∂BR‍xjxkRDjϕDkϕ¯dσR-∫∂BR‍R2|Djϕ|2dσR-∫BR‍12xj∂jA0|ϕ|2dx,Re{III}=∫∂BR‍R2V(|ϕ|2)dσx-∫BR‍V(|ϕ|2)dx.Then we have from the identity (34)
(37)∫BR‍V(|ϕ|2)+(ω+A0)|ϕ|2dx=∫∂BR‍-xjxkRRe(DjϕDkϕ¯)+R2|Djϕ|2+R2V(|ϕ|2)+R2(ω+A0)|ϕ|2dσR.
Applying the same argument in Section 2, the right hand side of the above equality vanishes. Then we conclude that
(38)∫ℝ2‍V(|ϕ|2)+(ω+A0)|ϕ|2dx=0.
On the other hand, we know from (12)
(39)∫ℝ2‍(ω+A0)|ϕ|2+|Djϕ|2+V′(|ϕ|2)|ϕ|2dx=0.
Combining (38) and (39), we end up with
(40)∫ℝ2‍|Djϕ|2+V′(|ϕ|2)|ϕ|2-V(|ϕ|2)dx=0.
Following the reasoning in Theorem 1, we deduce ϕ≡0 from the fact Djϕ≡0.For the proof of the second result in Theorem2, we assume ϕ(x)=u(x)∈ℝ. Then (13)–(15) can be rewritten by
(41)∂1A0=A2u2,∂2A0=-A1u2,∂1A2-∂2A1=-12u2.
It is easy to check the following identity:
(42)∂1(A2A0)-∂2(A1A0)=(A12+A22-12A0)u2,
from which we derive, with the condition A0∈Lp, A1,A2∈Lq for 1/p+1/q=1, 2<q<∞,
(43)∫ℝ2‍12A0u2dx=∫R2‍(A12+A22)u2dx.
Considering |Dju|2=|∇u|2+(A12+A22)u2, we have from (38) and (39)
(44)∫ℝ2‍V(|u|2)+ω|u|2+2(A12+A22)u2dx=0,∫ℝ2‍|∇u|2+3(A12+A22)u2+ω|u|2+V′(|u|2)|u|2dx=0.
Then we obtain, for a constant h≥2/3,
(45)∫ℝ2‍h|∇u|2+(3h-2)(A12+A22)u2+ω(h-1)|u|2+hV′(|u|2)|u|2-V(|u|2)dx=0,
which proves the second result in Theorem 2.



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