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

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.

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: −𝜔+𝐴₀²𝜙−𝐷₁𝐷₁𝜙−𝐷₂𝐷₂𝜙+𝑉^′𝜙²𝜕𝜙=0,₁𝐴₀=−Im𝜙𝐷₂𝜙,𝜕₂𝐴₀=Im𝜙𝐷₁𝜙,𝜕₁𝐴₂−𝜕₂𝐴₁=−𝜔+𝐴₀𝜙², (1)(2)(3)(4) which is derived from the system (5) with stationary solution ansatz 𝜓(𝑡,𝑥)=𝑒^𝑖𝜔𝑡𝜙(𝑥), 𝜙∈ℂ, and 𝐴_𝜇(𝑡,𝑥)=𝐴_𝜇(𝑥) for 𝜇=0,1,2. Consider 𝐷₀𝐷₀𝜓−𝐷₁𝐷₁+𝐷₂𝐷₂𝜓+𝑉^′𝜓²𝜕𝜓=0,₀𝐴₁−𝜕₁𝐴₀=Im𝜓𝐷₂𝜓,𝜕₀𝐴₂−𝜕₂𝐴₀=−Im𝜓𝐷₁𝜓,𝜕₁𝐴₂−𝜕₂𝐴₁=−Im𝜓𝐷₀𝜓, (5) where 𝜕₀=𝜕/𝜕𝑡, 𝜕₁=𝜕/𝜕𝑥₁, 𝜕₂=𝜕/𝜕𝑥₂ for (𝑡,𝑥₁,𝑥₂)∈ℝ¹⁺², 𝜓∶ℝ¹⁺²→ℂ is the complex scalar field, 𝐴_𝜇∶ℝ¹⁺²→ℝ is the gauge field, 𝐷_𝜇=𝜕_𝜇+𝑖𝐴_𝜇 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 𝐸(𝑡)=_ℝ2²_𝛼=0𝐷_𝛼𝜓(𝑡,𝑥)²+𝑉𝜓(𝑡,𝑥)²𝑑𝑥=𝐸(0). (6)

The special case with a self-dual potential 𝑉(|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]) 𝐷₁𝜙−𝑖𝐷₂𝜕𝜙=0,₁𝐴₂−𝜕₂𝐴₁+12𝜙²𝜙²−1=0,𝜔+𝐴₀−𝑖2𝜙²−1𝜙=0. (7) 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∈𝐿^𝑝(ℝ²), 𝐴₁,𝐴₂∈𝐿^𝑞(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 𝑑𝑉^′(𝑠)𝑠−𝑉(𝑠)≥0∀𝑠≥0, (8) 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): _ℝ2𝑑𝐷₁𝜙²+𝐷₂𝜙²+(1−𝑑)𝜔+𝐴₀²𝜙²+𝑑𝑉^′𝜙²𝜙²−𝑉𝜙²𝑑𝑥=0. (9) 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)For 𝛼>0, 𝛽>0, we take 1/2<𝑑<1.(2)For 𝛼>0, 𝛽<0, we take 1/3<𝑑<1/2.(3)For 𝛼<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 𝑑𝑉^′(𝑠)𝑠−𝑉(𝑠)=3𝑑−14𝑠³−2𝑑−12𝑠²+𝑑−14𝑠 (10) 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 𝑖𝐷₀𝜓+𝐷₁𝐷₁+𝐷₂𝐷₂𝜓−𝑉^′𝜙²𝜕𝜙=0,₀𝐴₁−𝜕₁𝐴₀=−Im𝜓𝐷₂𝜓,𝜕₀𝐴₂−𝜕₂𝐴₀=Im𝜓𝐷₁𝜓,𝜕₁𝐴₂−𝜕₂𝐴₁1=−2𝜓². (11) With the stationary solution ansatz 𝜓(𝑡,𝑥)=𝑒^𝑖𝜔𝑡𝜙(𝑥), 𝜙∈ℂ and 𝐴_𝜇(𝑡,𝑥)=𝐴_𝜇(𝑥) for 𝜇=0,1,2, we arrive at 𝜔+𝐴₀𝜙−𝐷₁𝐷₁𝜙−𝐷₂𝐷₂𝜙+𝑉^′𝜙²𝜕𝜙=0,₁𝐴₀=Im𝜙𝐷₂𝜙,𝜕₂𝐴₀=−Im𝜙𝐷₁𝜙,𝜕₁𝐴₂−𝜕₂𝐴₁1=−2𝜙². (12)(13)(14)(15)

In the special case with the potential 𝑉(|𝜙|²)=−(1/2)|𝜙|⁴, we can derive the following self dual equations [11–13] 𝐷₁𝜙+𝑖𝐷₂𝜙=0,𝜔+𝐴₀−12𝜙²𝜕=0,₁𝐴₂−𝜕₂𝐴₁+12𝜙²=0. (16) 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 form 𝑉(|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)=𝑒^𝑖𝜔𝑡𝑢(|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∈𝐿^𝑝(ℝ²) and 𝐴₁,𝐴₂∈𝐿^𝑞(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 potential 𝑉 satisfies 𝑉^′(𝑠)𝑠−𝑉(𝑠)≥0∀𝑠≥0, (17)
  then one has 𝜙≡0.(2)  Suppose that 𝜙 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, 𝜔(ℎ−1)𝑠+ℎ𝑉^′(𝑠)𝑠−𝑉(𝑠)≥0∀𝑠≥0, (18)
  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 23𝑉^′|𝑢|²|𝑢|²−𝑉|𝑢|²=|𝑢|⁶−13|𝑢|⁴+13|𝑢|²≥0. (19) However we have, for the complex solution, 𝑉^′𝜙²𝜙²−𝑉𝜙²=2𝜙⁶−𝜙⁴ (20) which is not nonnegative.
The paper is organized as follows. In Section 2, 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 𝑊^1,2(ℝ²). (ii)𝐵_𝑅∶={𝑥∈ℝ²∣|𝑥|≤𝑅} and 𝜕𝐵_𝑅∶={𝑥∈ℝ²∣|𝑥|=𝑅}.(iii)𝑑𝜎_𝑅∶= the  surface  measure  on  𝜕𝐵_𝑅.

2. Proof of Theorem 1

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 𝑥_𝑘𝐷_𝑘𝜙 and integrating over 𝐵_𝑅, we obtain −_𝐵𝑅𝜔+𝐴₀²𝜙𝑥_𝑘𝐷_𝑘𝜙𝑑𝑥−_𝐵𝑅𝐷_𝑗𝐷_𝑗𝜙𝑥_𝑘𝐷_𝑘+𝜙𝑑𝑥_𝐵𝑅𝑉^′𝜙²𝜙𝑥_𝑘𝐷_𝑘𝜙𝑑𝑥=0. (21)

Now we set I=_𝐵𝑅𝜔+𝐴₀²𝜙𝑥_𝑘𝐷_𝑘𝜙𝑑𝑥,II=_𝐵𝑅𝐷_𝑗𝐷_𝑗𝜙𝑥_𝑘𝐷_𝑘𝜙𝑑𝑥,III=_𝐵𝑅𝑉^′𝜙²𝜙𝑥_𝑘𝐷_𝑘𝜙𝑑𝑥. (22)

Then, integrating by parts and taking real parts, we have Re{I}=_𝜕𝐵𝑅𝑅2𝜔+𝐴₀²𝜙²𝑑𝜎_𝑥−_𝐵𝑅𝜔+𝐴₀²𝜙²−𝑑𝑥_𝐵𝑅𝜔𝜕_𝑗𝐴₀𝑥_𝑗𝜙²−𝑑𝑥_𝐵𝑅12𝑥_𝑗𝜕_𝑗𝐴²₀𝜙²{}=𝑑𝑥,ReIII_𝜕𝐵𝑅𝑅2𝑉𝜙²𝑑𝜎_𝑥−_𝐵𝑅𝑉𝜙²𝑑𝑥. (23)

For  II, we have II=_𝐵𝑅𝜕_𝑗𝑥_𝑘𝐷_𝑗𝜙𝐷_𝑘𝜙−𝐷_𝑗𝜙²−𝑥_𝑘𝐷_𝑗𝜙𝐷_𝑗𝐷_𝑘=𝜙𝑑𝑥_𝜕𝐵𝑅𝑥_𝑗𝑥_𝑘𝑅𝐷_𝑗𝜙𝐷_𝑘𝜙𝑑𝜎_𝑅−_𝐵𝑅𝐷_𝑗𝜙²𝑑𝑥−_𝐵𝑅𝑥_𝑘𝐷_𝑗𝜙𝐷_𝑘𝐷_𝑗𝜙+𝑖𝐹_𝑗𝑘𝜙𝑑𝑥, (24) where we used the notation 𝐹_𝑗𝑘=𝜕_𝑗𝐴_𝑘−𝜕_𝑘𝐴_𝑗 and the following identity: 𝐷_𝑗𝐷_𝑘𝜙=𝐷_𝑘𝐷_𝑗𝜙+𝑖𝐹_𝑗𝑘𝜙. (25)

Taking the real parts and integrating by parts, we obtain Re{II}=_𝜕𝐵𝑅𝑥_𝑗𝑥_𝑘𝑅𝐷_𝑗𝜙𝐷_𝑘𝜙𝑑𝜎_𝑅−_𝐵𝑅𝐷_𝑗𝜙²−𝑑𝑥_𝐵𝑅12𝑥_𝑘𝜕_𝑘𝐷_𝑗𝜙²−𝑑𝑥_𝐵𝑅𝑥_𝑘𝐹_𝑗𝑘Im𝜙𝐷_𝑗𝜙=𝑑𝑥_𝜕𝐵𝑅𝑥_𝑗𝑥_𝑘𝑅𝐷_𝑗𝜙𝐷_𝑘𝜙𝑑𝜎_𝑅−_𝜕𝐵𝑅𝑅2𝐷_𝑗𝜙²𝑑𝜎_𝑅+_𝐵𝑅12𝜙²𝑥_𝑗𝜕_𝑗𝐴²₀𝑑𝑥+_𝐵𝑅𝜔𝜙²𝑥_𝑗𝜕_𝑗𝐴₀𝑑𝑥, (26) where we used (2)–(4) in the following way: −_𝐵𝑅𝑥₁𝐹₂₁Im𝜙𝐷₂𝜙+𝑥₂𝐹₁₂Im𝜙𝐷₁𝜙=𝑑𝑥_𝐵𝑅𝑥_𝑗𝜙²𝜔+𝐴₀𝜕_𝑗𝐴₀𝑑𝑥. (27) Combining (23) and (26), we have from the identity (21) _𝐵𝑅𝜔+𝐴₀²𝜙²𝑑𝑥−_𝐵𝑅𝑉𝜙²=𝑑𝑥_𝜕𝐵𝑅𝑥_𝑗𝑥_𝑘𝑅Re𝐷_𝑗𝜙𝐷_𝑘𝜙−𝑅2𝐷_𝑗𝜙²−𝑅2𝑉𝜙²+𝑅2𝜔+𝐴₀²𝜙²𝑑𝜎_𝑅. (28) Thus we have _𝐵𝑅𝜔+𝐴₀²𝜙²−𝑉𝜙²𝑑𝑥≤𝐶_𝜕𝐵𝑅𝑅𝐷_𝑗𝜙²+𝜔²𝜙²+𝐴₀²𝜙²+𝑉𝜙²𝑑𝜎_𝑅, (29) where 𝐶 is a positive constant. Considering the Sobolev embedding and the condition of Theorem 1, we know that |𝐷_𝑗𝜙|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 {𝑅_𝑛}→∞ such that _{𝜕𝐵𝑅𝑛}𝑅_𝑛𝐷_𝑗𝜙²+𝜔²𝜙²+𝐴₀²𝜙²+𝑉𝜙²𝑑𝜎_𝑅𝑛⟶0, (30) and consequently _ℝ2𝜔+𝐴₀²𝜙²−𝑉𝜙²𝑑𝑥=lim_{𝑛→∞𝐵𝑅𝑛}𝜔+𝐴₀²𝜙²−𝑉𝜙²𝑑𝑥=0. (31) On the other hand, we know from (1) that 0=_ℝ2𝜙−𝜔+𝐴₀²𝜙−𝐷_𝑗𝐷_𝑗𝜙+𝑉^′𝜙²𝜙𝑑𝑥=_ℝ2−𝜔+𝐴₀²𝜙²+𝐷_𝑗𝜙²+𝑉^′𝜙²𝜙²𝑑𝑥, (32) by taking care of the boundary integral terms as before. Combining (31) and (32), we obtain _ℝ2𝑑𝐷_𝑗𝜙²+(1−𝑑)𝜔+𝐴₀²𝜙²+𝑑𝑉^′𝜙²𝜙²−𝑉𝜙²𝑑𝑥=0, (33) where 0≤𝑑≤1 is a constant. We are ready to prove Theorem 1.

(1) For 0<𝑑≤1, we have 𝐷_𝑗𝜙≡0. If 𝜙(𝑥₀)≠0, then there exists 𝛿>0 such that 𝜙(𝑥)≠0 for 𝐵_𝑥0(𝛿)={𝑥||𝑥−𝑥₀|<𝛿}. In the region 𝐵_𝑥0(𝛿), we have 𝐴_𝑗=𝑖(𝜕_𝑗𝜙/𝜙). Using (4) we have 𝐴₀(𝑥)+𝜔=0 in 𝐵_𝑥0(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 Theorem 2

Repeating the similar argument to the proof of Theorem 1, we derive Derrick-Pohozaev type identities for (12)–(15). Suppose that (𝜙,𝐴₀,𝐴₁,𝐴₂) is a solution of (12)–(15). Multiplying (12) by 𝑥_𝑘𝐷_𝑘𝜙 and integrating over 𝐵_𝑅, we obtain _𝐵𝑅𝜔+𝐴₀𝜙𝑥_𝑘𝐷_𝑘𝜙𝑑𝑥−_𝐵𝑅𝐷_𝑗𝐷_𝑗𝜙𝑥_𝑘𝐷_𝑘+𝜙𝑑𝑥_𝐵𝑅𝑉^′𝜙²𝜙𝑥_𝑘𝐷_𝑘𝜙𝑑𝑥=0. (34)

Now we set I=_𝐵𝑅𝜔+𝐴₀𝜙𝑥_𝑘𝐷_𝑘𝜙𝑑𝑥,II=_𝐵𝑅𝐷_𝑗𝐷_𝑗𝜙𝑥_𝑘𝐷_𝑘𝜙𝑑𝑥,III=_𝐵𝑅𝑉^′𝜙²𝜙𝑥_𝑘𝐷_𝑘𝜙𝑑𝑥. (35)

Then, integrating by parts and taking real parts, we have Re{I}=_𝜕𝐵𝑅𝑅2𝜔+𝐴₀𝜙²𝑑𝜎_𝑥−_𝐵𝑅𝜔+𝐴₀𝜙²−𝑑𝑥_𝐵𝑅12𝑥_𝑗𝜕_𝑗𝐴₀𝜙²{}=𝑑𝑥,ReII_𝜕𝐵𝑅𝑥_𝑗𝑥_𝑘𝑅𝐷_𝑗𝜙𝐷_𝑘𝜙𝑑𝜎_𝑅−_𝜕𝐵𝑅𝑅2𝐷_𝑗𝜙²𝑑𝜎_𝑅−_𝐵𝑅12𝑥_𝑗𝜕_𝑗𝐴₀𝜙²{}=𝑑𝑥,ReIII_𝜕𝐵𝑅𝑅2𝑉𝜙²𝑑𝜎_𝑥−_𝐵𝑅𝑉𝜙²𝑑𝑥. (36)

Then we have from the identity (34) _𝐵𝑅𝑉𝜙²+𝜔+𝐴₀𝜙²=𝑑𝑥_𝜕𝐵𝑅−𝑥_𝑗𝑥_𝑘𝑅Re𝐷_𝑗𝜙𝐷_𝑘𝜙+𝑅2𝐷_𝑗𝜙²+𝑅2𝑉𝜙²+𝑅2𝜔+𝐴₀𝜙²𝑑𝜎_𝑅. (37) Applying the same argument in Section 2, the right hand side of the above equality vanishes. Then we conclude that _ℝ2𝑉𝜙²+𝜔+𝐴₀𝜙²𝑑𝑥=0. (38) On the other hand, we know from (12) _ℝ2𝜔+𝐴₀𝜙²+𝐷_𝑗𝜙²+𝑉^′𝜙²𝜙²𝑑𝑥=0. (39) Combining (38) and (39), we end up with _ℝ2𝐷_𝑗𝜙²+𝑉^′𝜙²𝜙²−𝑉𝜙²𝑑𝑥=0. (40) Following the reasoning in Theorem 1, we deduce 𝜙≡0 from the fact 𝐷_𝑗𝜙≡0.

For the proof of the second result in Theorem 2, we assume 𝜙(𝑥)=𝑢(𝑥)∈ℝ. Then (13)–(15) can be rewritten by 𝜕₁𝐴₀=𝐴₂𝑢²,𝜕₂𝐴₀=−𝐴₁𝑢²,𝜕₁𝐴₂−𝜕₂𝐴₁1=−2𝑢². (41) It is easy to check the following identity: 𝜕₁𝐴₂𝐴₀−𝜕₂𝐴₁𝐴₀=𝐴²₁+𝐴²₂−12𝐴₀𝑢², (42) from which we derive, with the condition 𝐴₀∈𝐿^𝑝, 𝐴₁,𝐴₂∈𝐿^𝑞 for 1/𝑝+1/𝑞=1, 2<𝑞<∞, _ℝ212𝐴₀𝑢²𝑑𝑥=_𝑅2𝐴²₁+𝐴²₂𝑢²𝑑𝑥. (43) Considering |𝐷_𝑗𝑢|²=|∇𝑢|²+(𝐴²₁+𝐴²₂)𝑢², we have from (38) and (39) _ℝ2𝑉|𝑢|²+𝜔|𝑢|²+2𝐴²₁+𝐴²₂𝑢²𝑑𝑥=0,_ℝ2|∇𝑢|²+3𝐴²₁+𝐴²₂𝑢²+𝜔|𝑢|²+𝑉^′|𝑢|²|𝑢|²𝑑𝑥=0. (44) Then we obtain, for a constant ℎ≥2/3, _ℝ2ℎ|∇𝑢|²+(3ℎ−2)𝐴²₁+𝐴²₂𝑢²+𝜔(ℎ−1)|𝑢|²+ℎ𝑉^′|𝑢|²|𝑢|²−𝑉|𝑢|²𝑑𝑥=0, (45) which proves the second result in Theorem 2.  

Acknowledgments

The author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011-0015866) and also partially supported by the TJ Park Junior Faculty Fellowship.