- Email: [email protected]
On the stationary solutions of multi-dimensional bipolar hydrodynamic model of semiconductors
Applied Mathematics Letters 64 (2017) 108–112
Contents lists available at ScienceDirect
Applied Mathematics Letters www.elsevier.com/locate/aml
On the stationary solutions of multi-dimensional bipolar hydrodynamic model of semiconductors Huimin Yu Department of Mathematics, Shandong Normal University, Jinan, 250014, China
article
info
Article history: Received 11 July 2016 Received in revised form 12 August 2016 Accepted 12 August 2016 Available online 31 August 2016
abstract In this note, we are interested in the existence and uniqueness of the steady solutions to N -dimensional bipolar hydrodynamic model of semiconductors. Different from the unipolar case, the time-independent model is an elliptic equation with nonlocal terms. For N = 1, 2, we achieve our goal by the calculus of variations, without any additional assumptions on the doping profile and the initial densities. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Euler–Poisson systems Bipolar multi-dimensional semiconductor device Stationary solution
1. Introduction In this paper, the steady solutions of the following Euler–Poisson system for the bipolar hydrodynamical model of semiconductors ⎧ ∂t ρ1 + ∇ · m ⃗ 1 = 0, ⎪ ⎪ ⎪ (m ⎪ ⎪ ⃗1⊗m ⃗ 1) ⎪ ⎪ + ∇ρ1 = ρ1 ∇ϕ − m ⃗ 1, ∂ m ⃗ + ∇ · t 1 ⎪ ⎪ ρ1 ⎨ ∂t ρ2 + ∇ · m ⃗ 2 = 0, (1.1) ⎪ (m ) ⎪ ⎪ ⃗ ⊗ m ⃗ 2 2 ⎪ ⎪ ∂t m ⃗2+∇· + ∇ρ2 = −ρ2 ∇ϕ − m ⃗ 2, ⎪ ⎪ ρ ⎪ 2 ⎪ ⎩ ∆ϕ = ρ1 − ρ2 − D(⃗x), are considered. Here ⃗x ∈ Ω , a bounded domain in RN (N ≥ 1), is the space variable, t ∈ R+ = [0, ∞) is the time variable. ρ1 , ρ2 , m ⃗ 1, m ⃗ 2 , ϕ, ∇ϕ are the unknown functions of ⃗x and t, representing the electron density, the hole density, the electron current density, the hole current density, the potential function and the electric field respectively. The function D(⃗x) > 0, called the doping profile, stands for the density of E-mail address: [email protected]. http://dx.doi.org/10.1016/j.aml.2016.08.007 0893-9659/© 2016 Elsevier Ltd. All rights reserved.
H. Yu / Applied Mathematics Letters 64 (2017) 108–112
109
impurities in semiconductor devices. The symbols (∇·) and ∆ denote the divergence and Laplacian in RN , ⨂ the symbol denotes the Kronecker tensor product. Several physical constants have been set to unity for the simplicity of presentation. We refer to [1] for the derivation of this model. The corresponding studies on the unipolar hydrodynamic model (i.e., ρ2 = 0, m ⃗ 2 = ⃗0 in (1.1)) have been extensively carried out, see [2–5] etc. Due to the complexity of the system, the related research on the bipolar semiconductor is very limited so far. In [6], Tsuge proved the existence and uniqueness of stationary solutions to a one-dimensional bipolar hydrodynamic models with Ohmic contact boundary. In this note, we consider the same problem with insulating boundary conditions, that is, for system (1.1), the initial conditions are prescribed as ρi (⃗x, 0) = ρi0 (⃗x) > 0, and the boundary conditions are ⏐ ⏐ m ⃗ i (⃗x, t) · ⃗n⏐
m ⃗ i (⃗x, 0) = m ⃗ i0 (⃗x),
⏐ ⏐ ∇ϕ(⃗x, t) · ⃗n⏐
= ⃗0, ∂Ω
= ⃗0,
i = 1, 2,
for i = 1, 2, t ≥ 0,
(1.2)
(1.3)
∂Ω
⏐ ⏐ where ⃗n is the outer normal vector of Ω in ⃗x. Moreover, the compatibility condition m ⃗ i0 ·⃗n⏐ = ⃗0, (i = 1, 2) ∂Ω satisfies. Throughout this paper, we assume the doping profile D(⃗x) satisfies D∗ = sup D(⃗x) ≥ inf D(⃗x) = D∗ . x
x
(1.4)
To get the existence and uniqueness of the stationary solutions, we first rewrite the time-independent equation into an elliptic equation with nonlocal terms. By the calculus of variations, we achieve our aim. It is worthy to point out that the mapping T (Φ) = eΦ (H 1 (Ω ) → L1 (Ω )) is compact for spacial dimension N = 1, 2, we get a unique steady solution to problem (1.1)–(1.3). Yet, the compactness of the above mapping is lost for N ≥ 3, and the existence of the stationary solution is still open. Compared with the corresponding result in [7], where the existence is obtained by Schauder’s fixed point theorem, the same result is obtained in this note without any assumption on the doping profile and the initial densities. 2. Main results In this part, we will state the existence and uniqueness of the stationary solutions to problem (1.1)–(1.3) for N = 1, 2. Moreover, some important estimates will be given. For conciseness, we denote ⃗x as x. We consider the steady equation as follows ⎧ ρ1 = ρ¯1 ∇Φ, ⎨∇¯ ∇¯ ρ2 = −¯ ρ2 ∇Φ, (2.1) ⎩ ∆Φ = ρ¯1 − ρ¯2 − D(⃗x) with the boundary condition ⏐ ⏐ ∇Φ(x, t) · ⃗n⏐
= ⃗0.
(2.2)
∂Ω
Here, we are only concerned with the classical solutions in the region where the density inf ρ¯1 > 0
and
x
hold. To eliminate the additive constant, we set ∇Φ =
∫ Ω
inf ρ¯2 > 0 x
(2.3)
Φ(x)dx = 0 . Evidently, (2.1)1 and (2.1)2 indicate
∇¯ ρ2 ∇¯ ρ1 = ∇(ln ρ¯1 ) = − = −∇(ln ρ¯2 ). ρ¯1 ρ¯2
(2.4)
H. Yu / Applied Mathematics Letters 64 (2017) 108–112
110
Then ρ¯2 (x) = C2 e−Φ ,
ρ¯1 (x) = C1 eΦ ,
(2.5)
where C1 and C2 are two unknown positive constants. To determine the two constants C1 and C2 , we suppose1 ) ∫ ( ρi (x, 0) − ρ¯i (x) dx = 0 for i = 1, 2. (2.6) Ω
Then ∫
eΦ(x) dx =
C1 ∫Ω C2
∫ ρ1 (x, 0)dx =: A > 0,
e−Φ(x) dx =
Ω ∫
(2.7) ρ2 (x, 0)dx =: B > 0.
Ω
Ω
Substituting (2.5)(2.7) into (2.1)3 , we get ⎧ e−Φ eΦ ⎪ ⎪ − B ∫ −Φ − D(x), ∆Φ = A ∫ Φ ⎪ ⎪ ⎨ e dx e dx Ω Ω
(2.8)
⎪∫ ⎪ ⎪ ⎪ ⎩ Φdx = 0. Ω
Set ∫ J(Φ) = Ω
2
|∇Φ| dx + A log 2
(∫
) (∫ ) ∫ −Φ e dx + B log e dx − D(x)Φdx, Φ
Ω
Ω
(2.9)
Ω
we have the following claim. ∫ Claim: J admits a minimizer in A = {Φ|Φ ∈ H 1 (Ω ), Ω Φdx = 0}. To see this, we suppose {Φn } is a minimizing sequence. Then (∫ ) (∫ ) ∫ ∫ 2 |∇Φn | dx + A log eΦn dx + B log e−Φn dx − D(x)Φn dx ≤ C1 2 Ω Ω Ω Ω
(2.10)
holds for some constant C1 . Jensen’s inequality implies that ∫ ∫ 1 Φn dx Φn |Ω| Ω e dx ≥ |Ω |e = |Ω |, ∫ Ω ∫ − 1 Φ dx e−Φn dx ≥ |Ω |e |Ω| Ω n = |Ω |, Ω
and (2.10) turns into 2
∫ Ω
|∇Φn | dx − 2
∫ D(x)Φn dx ≤ C2 .
(2.11)
Ω
Using H¨ older inequality, we get ∫ Ω 1
2
|∇Φn | dx ≤ C(ε) + ε 2
∫
Φn2 dx
Ω
Noticing the conservation of the total charge: integrating (1.1)1 and (1.1)3 in Ω gives
(∫
)
∫ =−
ρi dx Ω
we see this assumption is natural.
t
∇·m ⃗ i dx = 0, Ω
for i = 1, 2,
(2.12)
H. Yu / Applied Mathematics Letters 64 (2017) 108–112
111
satisfies for some small enough constant ε > 0. Thus, ∥Φn ∥H 1 ≤ C3 follows Poinc´are inequality. Now, we suppose Φn → Φ in H 1 (Ω ) weakly. Since the map: Φ −→ eΦ (H 1 (Ω ) −→ L1 (Ω )) is compact for N = 1, 2, then (∫ ) ( ) (∫ ) ∫ eΦ dx , (2.13) lim log eΦn dx = log eΦn dx = log lim n→∞
n→∞
Ω
(∫ lim log
n→∞
e−Φn dx
)
( = log
Ω
) (∫ ) e−Φn dx = log e−Φ dx .
∫ lim
n→∞
Ω
Ω
Ω
(2.14)
Ω
Moreover, by the lower semi-continuous of weak convergence, we have limn→∞ J(Φn ) ≥ J(Φ). So, Φ is a ∫ minimizer. We will be done, once we show Ω Φdx = 0. This is true for H 1 (Ω ) is compactly embedded in L1 (Ω ). ˜ u To give the uniqueness, we suppose u and u ˜ are two minimizers of J[·] over A. Then, v = u+ 2 ∈ A. Now, u] we claim J[v] ≤ J[u]+J[˜ with a strict inequality, unless u = u ˜ holds almost everywhere. To see this, we 2
compute 2
˜ u |∇( u+ 2 )| dx + A log 2
) ∫ u+u ˜ e dx + B log e dx − D(x) dx J[v] = 2 Ω Ω Ω Ω [ ] (∫ ) 21 (∫ ) 21 ∫ 1 2 2 2 u ≤ (2|∇u| + 2|∇˜ u| − |∇u − ∇˜ u| )dx + A log eu dx e˜ dx Ω 8 Ω Ω [(∫ ) 12 ] ) 12 (∫ ∫ ∫ 1 1 u e−˜ dx − + B log e−u dx D(x)udx − D(x)˜ udx 2 Ω 2 Ω Ω Ω 1 1 u). ≤ J(u) + J(˜ 2 2 ∫
(∫
u+u 2
)
(∫
˜
˜ u − u+ 2
(2.15)
The strict inequality holds since u ̸= u ˜. This is a contradiction, since u and u ˜ are minimizers. 1 1 For ∥Φ∥L∞ ≤ C, if we suppose 0 < m ≤ eΦ ≤ M , then M ≤ e−Φ ≤ m . Noticing (2.5) and (2.7) we have Am AM ≤ ρ¯1 ≤ |Ω |M |Ω |m
and
Bm BM ≤ ρ¯2 ≤ . |Ω |M |Ω |m
(2.16)
Moreover, for the unique solution Φ, the monotone properties of ρ¯1 , ρ¯2 and the maximum principle give D∗ ≤ ρ¯1 − ρ¯2 ≤ D∗ .
(2.17)
As the result, we have: Theorem 2.1. Problem (2.1)–(2.2) has a unique smooth solution (¯ ρ1 , ρ¯2 , Φ) for N = 1, 2 satisfying ∫ (i) D∗ ≤ ρ¯1 − ρ¯2 ≤ D∗ and Ω Φdx = 0, (ii) ρ¯1 and ρ¯2 have positive bounds from bottom and top, that is, there exist positive constants C1 and C2 such that 0 < C1 ≤ ρ¯1 , ρ¯2 ≤ C2 .
3. Conclusion remarks 1. Guo and Strauss considered the same problem on the unipolar hydrodynamic model of semiconductors in [3] section 2, where the leading equation is elliptic. However, in the bipolar case, the corresponding equation is an elliptic one with nonlocal terms.
112
H. Yu / Applied Mathematics Letters 64 (2017) 108–112
2. Schauder’s fixed point theorem is used in [7], in which the authors need ∫2 n20 dx D∗ > − 1 N −1 . 2
(3.1)
In this note, we delete the assumption (3.1). Furthermore, the uniqueness is a direct conclusion by the convexity of J. 3. Since the mapping T (Φ) = eΦ (H 1 (Ω ) → L1 (Ω )) is compact for spacial dimension N = 1, 2, we get our main result. However, the compactness of the above mapping is lost for N ≥ 3. The existence of the stationary solutions is still open. Acknowledgments The author would like to thank Dr. Marcello Lucia for his thoughtful discussion and valuable suggestions. This project is supported in part by the National Natural Science Foundation of China (Grant Nos. 11541005, 11671237), Shandong Provincial Natural Science Foundation (Grant No. ZR2015AM001), and Domestic Visiting Scholars Program of Excellent Young Teachers in Shandong Province. References
[1] P. Markowich, C. Ringhofer, C. Schmeiser, Semiconductor Equations, Springer-Verlag, Vienna, 1990. [2] P. Degond, P. Markowich, On a one-dimensional steady-state hydrodynamic model, Appl. Math. Lett. 3 (1990) 25–29. [3] Y. Guo, W. Strauss, Stability of semiconductor states with insulating and contact boundary conditions, Arch. Ration. Mech. Anal. 179 (2006) 1–30. [4] H. Yu, Large time behavior of Euler-Poisson equations for isothermal fluids with spherical symmetry, J. Math. Anal. Appl. 363 (2010) 302–309. [5] H. Yu, Large time behavior of entropy solutions to a unipolar hydrodynamic model of semiconductors, Commun. Math. Sci. 14 (2016) 69–82. [6] N. Tsuge, Existence and uniqueness of stationary solutions to a one-dimensional bipolar hydrodynamic models of semiconductors, Nonlinear Anal. 73 (2010) 779–787. [7] H. Yu, Y. Zhan, Large time behavior of solutions to multi-dimensional bipolar hydrodynamic model of semiconductors with vacuum, J. Math. Anal. Appl. 438 (2016) 856–874.
Contents lists available at ScienceDirect
Applied Mathematics Letters www.elsevier.com/locate/aml
On the stationary solutions of multi-dimensional bipolar hydrodynamic model of semiconductors Huimin Yu Department of Mathematics, Shandong Normal University, Jinan, 250014, China
article
info
Article history: Received 11 July 2016 Received in revised form 12 August 2016 Accepted 12 August 2016 Available online 31 August 2016
abstract In this note, we are interested in the existence and uniqueness of the steady solutions to N -dimensional bipolar hydrodynamic model of semiconductors. Different from the unipolar case, the time-independent model is an elliptic equation with nonlocal terms. For N = 1, 2, we achieve our goal by the calculus of variations, without any additional assumptions on the doping profile and the initial densities. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Euler–Poisson systems Bipolar multi-dimensional semiconductor device Stationary solution
1. Introduction In this paper, the steady solutions of the following Euler–Poisson system for the bipolar hydrodynamical model of semiconductors ⎧ ∂t ρ1 + ∇ · m ⃗ 1 = 0, ⎪ ⎪ ⎪ (m ⎪ ⎪ ⃗1⊗m ⃗ 1) ⎪ ⎪ + ∇ρ1 = ρ1 ∇ϕ − m ⃗ 1, ∂ m ⃗ + ∇ · t 1 ⎪ ⎪ ρ1 ⎨ ∂t ρ2 + ∇ · m ⃗ 2 = 0, (1.1) ⎪ (m ) ⎪ ⎪ ⃗ ⊗ m ⃗ 2 2 ⎪ ⎪ ∂t m ⃗2+∇· + ∇ρ2 = −ρ2 ∇ϕ − m ⃗ 2, ⎪ ⎪ ρ ⎪ 2 ⎪ ⎩ ∆ϕ = ρ1 − ρ2 − D(⃗x), are considered. Here ⃗x ∈ Ω , a bounded domain in RN (N ≥ 1), is the space variable, t ∈ R+ = [0, ∞) is the time variable. ρ1 , ρ2 , m ⃗ 1, m ⃗ 2 , ϕ, ∇ϕ are the unknown functions of ⃗x and t, representing the electron density, the hole density, the electron current density, the hole current density, the potential function and the electric field respectively. The function D(⃗x) > 0, called the doping profile, stands for the density of E-mail address: [email protected]. http://dx.doi.org/10.1016/j.aml.2016.08.007 0893-9659/© 2016 Elsevier Ltd. All rights reserved.
H. Yu / Applied Mathematics Letters 64 (2017) 108–112
109
impurities in semiconductor devices. The symbols (∇·) and ∆ denote the divergence and Laplacian in RN , ⨂ the symbol denotes the Kronecker tensor product. Several physical constants have been set to unity for the simplicity of presentation. We refer to [1] for the derivation of this model. The corresponding studies on the unipolar hydrodynamic model (i.e., ρ2 = 0, m ⃗ 2 = ⃗0 in (1.1)) have been extensively carried out, see [2–5] etc. Due to the complexity of the system, the related research on the bipolar semiconductor is very limited so far. In [6], Tsuge proved the existence and uniqueness of stationary solutions to a one-dimensional bipolar hydrodynamic models with Ohmic contact boundary. In this note, we consider the same problem with insulating boundary conditions, that is, for system (1.1), the initial conditions are prescribed as ρi (⃗x, 0) = ρi0 (⃗x) > 0, and the boundary conditions are ⏐ ⏐ m ⃗ i (⃗x, t) · ⃗n⏐
m ⃗ i (⃗x, 0) = m ⃗ i0 (⃗x),
⏐ ⏐ ∇ϕ(⃗x, t) · ⃗n⏐
= ⃗0, ∂Ω
= ⃗0,
i = 1, 2,
for i = 1, 2, t ≥ 0,
(1.2)
(1.3)
∂Ω
⏐ ⏐ where ⃗n is the outer normal vector of Ω in ⃗x. Moreover, the compatibility condition m ⃗ i0 ·⃗n⏐ = ⃗0, (i = 1, 2) ∂Ω satisfies. Throughout this paper, we assume the doping profile D(⃗x) satisfies D∗ = sup D(⃗x) ≥ inf D(⃗x) = D∗ . x
x
(1.4)
To get the existence and uniqueness of the stationary solutions, we first rewrite the time-independent equation into an elliptic equation with nonlocal terms. By the calculus of variations, we achieve our aim. It is worthy to point out that the mapping T (Φ) = eΦ (H 1 (Ω ) → L1 (Ω )) is compact for spacial dimension N = 1, 2, we get a unique steady solution to problem (1.1)–(1.3). Yet, the compactness of the above mapping is lost for N ≥ 3, and the existence of the stationary solution is still open. Compared with the corresponding result in [7], where the existence is obtained by Schauder’s fixed point theorem, the same result is obtained in this note without any assumption on the doping profile and the initial densities. 2. Main results In this part, we will state the existence and uniqueness of the stationary solutions to problem (1.1)–(1.3) for N = 1, 2. Moreover, some important estimates will be given. For conciseness, we denote ⃗x as x. We consider the steady equation as follows ⎧ ρ1 = ρ¯1 ∇Φ, ⎨∇¯ ∇¯ ρ2 = −¯ ρ2 ∇Φ, (2.1) ⎩ ∆Φ = ρ¯1 − ρ¯2 − D(⃗x) with the boundary condition ⏐ ⏐ ∇Φ(x, t) · ⃗n⏐
= ⃗0.
(2.2)
∂Ω
Here, we are only concerned with the classical solutions in the region where the density inf ρ¯1 > 0
and
x
hold. To eliminate the additive constant, we set ∇Φ =
∫ Ω
inf ρ¯2 > 0 x
(2.3)
Φ(x)dx = 0 . Evidently, (2.1)1 and (2.1)2 indicate
∇¯ ρ2 ∇¯ ρ1 = ∇(ln ρ¯1 ) = − = −∇(ln ρ¯2 ). ρ¯1 ρ¯2
(2.4)
H. Yu / Applied Mathematics Letters 64 (2017) 108–112
110
Then ρ¯2 (x) = C2 e−Φ ,
ρ¯1 (x) = C1 eΦ ,
(2.5)
where C1 and C2 are two unknown positive constants. To determine the two constants C1 and C2 , we suppose1 ) ∫ ( ρi (x, 0) − ρ¯i (x) dx = 0 for i = 1, 2. (2.6) Ω
Then ∫
eΦ(x) dx =
C1 ∫Ω C2
∫ ρ1 (x, 0)dx =: A > 0,
e−Φ(x) dx =
Ω ∫
(2.7) ρ2 (x, 0)dx =: B > 0.
Ω
Ω
Substituting (2.5)(2.7) into (2.1)3 , we get ⎧ e−Φ eΦ ⎪ ⎪ − B ∫ −Φ − D(x), ∆Φ = A ∫ Φ ⎪ ⎪ ⎨ e dx e dx Ω Ω
(2.8)
⎪∫ ⎪ ⎪ ⎪ ⎩ Φdx = 0. Ω
Set ∫ J(Φ) = Ω
2
|∇Φ| dx + A log 2
(∫
) (∫ ) ∫ −Φ e dx + B log e dx − D(x)Φdx, Φ
Ω
Ω
(2.9)
Ω
we have the following claim. ∫ Claim: J admits a minimizer in A = {Φ|Φ ∈ H 1 (Ω ), Ω Φdx = 0}. To see this, we suppose {Φn } is a minimizing sequence. Then (∫ ) (∫ ) ∫ ∫ 2 |∇Φn | dx + A log eΦn dx + B log e−Φn dx − D(x)Φn dx ≤ C1 2 Ω Ω Ω Ω
(2.10)
holds for some constant C1 . Jensen’s inequality implies that ∫ ∫ 1 Φn dx Φn |Ω| Ω e dx ≥ |Ω |e = |Ω |, ∫ Ω ∫ − 1 Φ dx e−Φn dx ≥ |Ω |e |Ω| Ω n = |Ω |, Ω
and (2.10) turns into 2
∫ Ω
|∇Φn | dx − 2
∫ D(x)Φn dx ≤ C2 .
(2.11)
Ω
Using H¨ older inequality, we get ∫ Ω 1
2
|∇Φn | dx ≤ C(ε) + ε 2
∫
Φn2 dx
Ω
Noticing the conservation of the total charge: integrating (1.1)1 and (1.1)3 in Ω gives
(∫
)
∫ =−
ρi dx Ω
we see this assumption is natural.
t
∇·m ⃗ i dx = 0, Ω
for i = 1, 2,
(2.12)
H. Yu / Applied Mathematics Letters 64 (2017) 108–112
111
satisfies for some small enough constant ε > 0. Thus, ∥Φn ∥H 1 ≤ C3 follows Poinc´are inequality. Now, we suppose Φn → Φ in H 1 (Ω ) weakly. Since the map: Φ −→ eΦ (H 1 (Ω ) −→ L1 (Ω )) is compact for N = 1, 2, then (∫ ) ( ) (∫ ) ∫ eΦ dx , (2.13) lim log eΦn dx = log eΦn dx = log lim n→∞
n→∞
Ω
(∫ lim log
n→∞
e−Φn dx
)
( = log
Ω
) (∫ ) e−Φn dx = log e−Φ dx .
∫ lim
n→∞
Ω
Ω
Ω
(2.14)
Ω
Moreover, by the lower semi-continuous of weak convergence, we have limn→∞ J(Φn ) ≥ J(Φ). So, Φ is a ∫ minimizer. We will be done, once we show Ω Φdx = 0. This is true for H 1 (Ω ) is compactly embedded in L1 (Ω ). ˜ u To give the uniqueness, we suppose u and u ˜ are two minimizers of J[·] over A. Then, v = u+ 2 ∈ A. Now, u] we claim J[v] ≤ J[u]+J[˜ with a strict inequality, unless u = u ˜ holds almost everywhere. To see this, we 2
compute 2
˜ u |∇( u+ 2 )| dx + A log 2
) ∫ u+u ˜ e dx + B log e dx − D(x) dx J[v] = 2 Ω Ω Ω Ω [ ] (∫ ) 21 (∫ ) 21 ∫ 1 2 2 2 u ≤ (2|∇u| + 2|∇˜ u| − |∇u − ∇˜ u| )dx + A log eu dx e˜ dx Ω 8 Ω Ω [(∫ ) 12 ] ) 12 (∫ ∫ ∫ 1 1 u e−˜ dx − + B log e−u dx D(x)udx − D(x)˜ udx 2 Ω 2 Ω Ω Ω 1 1 u). ≤ J(u) + J(˜ 2 2 ∫
(∫
u+u 2
)
(∫
˜
˜ u − u+ 2
(2.15)
The strict inequality holds since u ̸= u ˜. This is a contradiction, since u and u ˜ are minimizers. 1 1 For ∥Φ∥L∞ ≤ C, if we suppose 0 < m ≤ eΦ ≤ M , then M ≤ e−Φ ≤ m . Noticing (2.5) and (2.7) we have Am AM ≤ ρ¯1 ≤ |Ω |M |Ω |m
and
Bm BM ≤ ρ¯2 ≤ . |Ω |M |Ω |m
(2.16)
Moreover, for the unique solution Φ, the monotone properties of ρ¯1 , ρ¯2 and the maximum principle give D∗ ≤ ρ¯1 − ρ¯2 ≤ D∗ .
(2.17)
As the result, we have: Theorem 2.1. Problem (2.1)–(2.2) has a unique smooth solution (¯ ρ1 , ρ¯2 , Φ) for N = 1, 2 satisfying ∫ (i) D∗ ≤ ρ¯1 − ρ¯2 ≤ D∗ and Ω Φdx = 0, (ii) ρ¯1 and ρ¯2 have positive bounds from bottom and top, that is, there exist positive constants C1 and C2 such that 0 < C1 ≤ ρ¯1 , ρ¯2 ≤ C2 .
3. Conclusion remarks 1. Guo and Strauss considered the same problem on the unipolar hydrodynamic model of semiconductors in [3] section 2, where the leading equation is elliptic. However, in the bipolar case, the corresponding equation is an elliptic one with nonlocal terms.
112
H. Yu / Applied Mathematics Letters 64 (2017) 108–112
2. Schauder’s fixed point theorem is used in [7], in which the authors need ∫2 n20 dx D∗ > − 1 N −1 . 2
(3.1)
In this note, we delete the assumption (3.1). Furthermore, the uniqueness is a direct conclusion by the convexity of J. 3. Since the mapping T (Φ) = eΦ (H 1 (Ω ) → L1 (Ω )) is compact for spacial dimension N = 1, 2, we get our main result. However, the compactness of the above mapping is lost for N ≥ 3. The existence of the stationary solutions is still open. Acknowledgments The author would like to thank Dr. Marcello Lucia for his thoughtful discussion and valuable suggestions. This project is supported in part by the National Natural Science Foundation of China (Grant Nos. 11541005, 11671237), Shandong Provincial Natural Science Foundation (Grant No. ZR2015AM001), and Domestic Visiting Scholars Program of Excellent Young Teachers in Shandong Province. References
[1] P. Markowich, C. Ringhofer, C. Schmeiser, Semiconductor Equations, Springer-Verlag, Vienna, 1990. [2] P. Degond, P. Markowich, On a one-dimensional steady-state hydrodynamic model, Appl. Math. Lett. 3 (1990) 25–29. [3] Y. Guo, W. Strauss, Stability of semiconductor states with insulating and contact boundary conditions, Arch. Ration. Mech. Anal. 179 (2006) 1–30. [4] H. Yu, Large time behavior of Euler-Poisson equations for isothermal fluids with spherical symmetry, J. Math. Anal. Appl. 363 (2010) 302–309. [5] H. Yu, Large time behavior of entropy solutions to a unipolar hydrodynamic model of semiconductors, Commun. Math. Sci. 14 (2016) 69–82. [6] N. Tsuge, Existence and uniqueness of stationary solutions to a one-dimensional bipolar hydrodynamic models of semiconductors, Nonlinear Anal. 73 (2010) 779–787. [7] H. Yu, Y. Zhan, Large time behavior of solutions to multi-dimensional bipolar hydrodynamic model of semiconductors with vacuum, J. Math. Anal. Appl. 438 (2016) 856–874.