一阶多项式微分方程组的研究
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摘要
多项式微分方程组是非线性微分方程组的重要情形。本篇学位论文首先讨论了一阶多项式常微分方程组光滑解的存在性。我们利用与一元函数有关的常系数线性方程组的理论,将一阶多项式常微分方程组转化为二次多项式方程组与一阶常系数线性常微分方程组的合成。接着我们利用傅里叶变换将一阶常系数线性常微分方程组转化成线性方程组,并且求出其通解,然后根据该通解,我们将二次方程组转化为与非线性Fredholm积分方程组等价的不动点问题。我们利用压缩映射不动点原理和绍德尔不动点定理以及勒雷-邵德尔拓扑度理论,可以得到一阶多项式常微分方程组的光滑解的存在性有如下三种可能：
(1)在有界闭区间[a,b]上,(?)a1>0,当b-a (2)(?)a2>a1>0,当b-a (3)在有界闭区间[a,b]上,光滑解存在。并且,我们可以写出光滑解的与非线性Fredholm积分方程组有关的显式表达式。
接着,我们将结论推广到一阶的多项式偏微分方程组。我们利用与多元函数有关的常系数线性方程组的理论,将一阶多项式偏微分方程组转化为二次多项式方程组与一阶常系数线性偏微分方程组的合成。接着我们利用傅里叶变换将一阶常系数线性偏微分方程组转化成线性方程组,并且求出其通解,然后根据该通解,我们将二次方程组转化为与非线性Fredholm积分方程组等价的不动点问题。我们利用压缩映射不动点原理和绍德尔不动点定理以及勒雷-邵德尔拓扑度理论,可以得到一阶多项式偏微分方程组的光滑解的存在性有如下三种可能：
(1)在有界闭区域Ω上,(?)a1>0,当m(Ω) (2)(?)a2>a1>0,当m(Ω) (3)在有界闭区域Ω上,光滑解存在。同样,我们可以写出光滑解的与非线性Fredholm积分方程组有关的显式表达式。
最后,我们讨论了纳维-斯托克斯方程组。我们利用与四元函数有关的常系数线性方程组的理论,将纳维-斯托克斯方程组转化为二次多项式方程组与一阶常系数线性偏微分方程组的合成。接着我们利用傅里叶变换将一阶常系数线性偏微分方程组转化成线性方程组,并且求出其通解,然后根据该通解,我们将二次方程组转化为与非线性Fredholm积分方程组等价的不动点问题。我们利用压缩映射不动点原理和绍德尔不动点定理以及勒雷-邵德尔拓扑度理论,可以得到纳维-斯托克斯方程组的光滑解的存在性有如下三种可能：
(1)在Ω×[0,T]上,(?)a1>0,当m(Ω) (2)(?)a2>a1>0,当m(Q) (3)在Ω×[0,T]上,光滑解存在。同样,我们可以写出光滑解的与非线性Fredholm积分方程组有关的显式表达式。
The polynomial differential equations are the important scenarios of the nonlinear differential equations. This dissertation discussed how to solve the first order polyno-mial ordinary differential equations. At first, we put forward the theory of the linear equations about the unknown one variable functions with constant coefficients. Sec-ondly, we use this theory to convert the first order polynomial ordinary differential equations into the simultaneous of the first order linear ordinary differential equation-s with constant coefficients and the quadratic equations. Thirdly, we use the Fourier transformation to convert the first order linear ordinary differential equations with con-stant coefficients into the linear equations, and we work out the explicit general solution of it. With this general solution, we convert the quadratic equations into the question of the fixed-point of a continuous mapping. And it is equivalent to the nonlinear Fred-holm integral equations. We use the theories about the contraction mapping principle, the Schauder fixed-point theorem and the Leray-Schauder degree to get the existence and the uniqueness of the fixed-point. Finally, we find that there exist three scenarios about the smoothing-solution of the first order polynomial ordinary differential equa-tions:
(l)In the region [a, b], where [a, b] is a bounded and closed set,(?)a1>0, if b-a (2)In the region [a, b], where [a, b] is a bounded and closed set,(?)a2> a1>0, if b-a (3)In the region [a,b], where [a, b] is a bounded and closed set, the smoothing-solution exists.
Moreover we can write out the explicit expressions of the smoothing-solutions which are related with the nonlinear Fredholm integral equations.
Consequently we discussed how to solve the first order polynomial partial differential equations. At first, we put forward the theory of the linear equations about the un-known multivariate functions with constant coefficients. Secondly, we use this theory to convert the first order polynomial partial differential equations into the simultane-ous of the first order linear partial differential equations with constant coefficients and the quadratic equations. Thirdly, we use the Fourier transformation to convert the first order linear partial differential equations with constant coefficients into the linear equa-tions, and we work out the explicit general solution of it. With this general solution, we convert the quadratic equations into the question of the fixed-point of a continuous mapping. And it is equivalent to the nonlinear Fredholm integral equations. We use the theories about the contraction mapping principle, the Schauder fixed-point theorem and the Leray-Schauder degree to get the existence and the uniqueness of the fixed-point. Finally, we find that there exist three scenarios about the smoothing-solution of the first order polynomial partial differential equations:
(1)In the region Ω, where Ω is a bounded and closed set in Rn,3a1>0, if m(Ω) (2)In the region Ω, where Ω is a bounded and closed set in Rn,(?)a2> a1>0, if m(Ω) (3)In the region Q, where Ω. is a bounded and closed set in Rn, the solution exists. Moreover we can write out the explicit expressions of the smoothing-solutions which are related with the nonlinear Fredholm integral equations.
Finally we discussed how to solve the Navier-Stokes equations. At first, we put forward the theory of the linear equations which is about the unknown multivariate functions with constant coefficients. Secondly, we use this theory to convert the Navier-Stokes equations into the simultaneous of the first order linear partial differential equation-s with constant coefficients and the quadratic equations. Thirdly, we use the Fourier transformation to convert the first order linear partial differential equations with con-stant coefficients into the linear equations, and we work out the explicit solution of it. With this general solution, we convert the quadratic equations into the question of the fixed-point of a continuous mapping. And it is equivalent to the nonlinear Fredholm integral equations. We use the theories about the contraction mapping principle, the Schauder fixed-point theorem and the Leray-Schauder degree to get the existence and the uniqueness of the fixed-point. Finally, we find that there exist three scenarios about the smoothing-solution of the Navier-Stokes equations:
(1)In the region Ω×[0, T], where T>0, Ω is a bounded and closed set in R3,3a1>0, if m(Ω) (2)In the region Ω×[0, T], where T>0, Ω is a bounded and closed set in R3,3a2> a1>0, if m(Ω) (3)In the region Ω x [0,T], where T>0, Ω is a bounded and closed set in R3, the smoothing-solution exists.
Moreover we can write out the explicit expressions of the smoothing-solutions which are related with the nonlinear Fredholm integral equations.
(1)在有界闭区间[a,b]上,(?)a1>0,当b-a
接着,我们将结论推广到一阶的多项式偏微分方程组。我们利用与多元函数有关的常系数线性方程组的理论,将一阶多项式偏微分方程组转化为二次多项式方程组与一阶常系数线性偏微分方程组的合成。接着我们利用傅里叶变换将一阶常系数线性偏微分方程组转化成线性方程组,并且求出其通解,然后根据该通解,我们将二次方程组转化为与非线性Fredholm积分方程组等价的不动点问题。我们利用压缩映射不动点原理和绍德尔不动点定理以及勒雷-邵德尔拓扑度理论,可以得到一阶多项式偏微分方程组的光滑解的存在性有如下三种可能：
(1)在有界闭区域Ω上,(?)a1>0,当m(Ω)
最后,我们讨论了纳维-斯托克斯方程组。我们利用与四元函数有关的常系数线性方程组的理论,将纳维-斯托克斯方程组转化为二次多项式方程组与一阶常系数线性偏微分方程组的合成。接着我们利用傅里叶变换将一阶常系数线性偏微分方程组转化成线性方程组,并且求出其通解,然后根据该通解,我们将二次方程组转化为与非线性Fredholm积分方程组等价的不动点问题。我们利用压缩映射不动点原理和绍德尔不动点定理以及勒雷-邵德尔拓扑度理论,可以得到纳维-斯托克斯方程组的光滑解的存在性有如下三种可能：
(1)在Ω×[0,T]上,(?)a1>0,当m(Ω)
The polynomial differential equations are the important scenarios of the nonlinear differential equations. This dissertation discussed how to solve the first order polyno-mial ordinary differential equations. At first, we put forward the theory of the linear equations about the unknown one variable functions with constant coefficients. Sec-ondly, we use this theory to convert the first order polynomial ordinary differential equations into the simultaneous of the first order linear ordinary differential equation-s with constant coefficients and the quadratic equations. Thirdly, we use the Fourier transformation to convert the first order linear ordinary differential equations with con-stant coefficients into the linear equations, and we work out the explicit general solution of it. With this general solution, we convert the quadratic equations into the question of the fixed-point of a continuous mapping. And it is equivalent to the nonlinear Fred-holm integral equations. We use the theories about the contraction mapping principle, the Schauder fixed-point theorem and the Leray-Schauder degree to get the existence and the uniqueness of the fixed-point. Finally, we find that there exist three scenarios about the smoothing-solution of the first order polynomial ordinary differential equa-tions:
(l)In the region [a, b], where [a, b] is a bounded and closed set,(?)a1>0, if b-a
Moreover we can write out the explicit expressions of the smoothing-solutions which are related with the nonlinear Fredholm integral equations.
Consequently we discussed how to solve the first order polynomial partial differential equations. At first, we put forward the theory of the linear equations about the un-known multivariate functions with constant coefficients. Secondly, we use this theory to convert the first order polynomial partial differential equations into the simultane-ous of the first order linear partial differential equations with constant coefficients and the quadratic equations. Thirdly, we use the Fourier transformation to convert the first order linear partial differential equations with constant coefficients into the linear equa-tions, and we work out the explicit general solution of it. With this general solution, we convert the quadratic equations into the question of the fixed-point of a continuous mapping. And it is equivalent to the nonlinear Fredholm integral equations. We use the theories about the contraction mapping principle, the Schauder fixed-point theorem and the Leray-Schauder degree to get the existence and the uniqueness of the fixed-point. Finally, we find that there exist three scenarios about the smoothing-solution of the first order polynomial partial differential equations:
(1)In the region Ω, where Ω is a bounded and closed set in Rn,3a1>0, if m(Ω)
Finally we discussed how to solve the Navier-Stokes equations. At first, we put forward the theory of the linear equations which is about the unknown multivariate functions with constant coefficients. Secondly, we use this theory to convert the Navier-Stokes equations into the simultaneous of the first order linear partial differential equation-s with constant coefficients and the quadratic equations. Thirdly, we use the Fourier transformation to convert the first order linear partial differential equations with con-stant coefficients into the linear equations, and we work out the explicit solution of it. With this general solution, we convert the quadratic equations into the question of the fixed-point of a continuous mapping. And it is equivalent to the nonlinear Fredholm integral equations. We use the theories about the contraction mapping principle, the Schauder fixed-point theorem and the Leray-Schauder degree to get the existence and the uniqueness of the fixed-point. Finally, we find that there exist three scenarios about the smoothing-solution of the Navier-Stokes equations:
(1)In the region Ω×[0, T], where T>0, Ω is a bounded and closed set in R3,3a1>0, if m(Ω)
Moreover we can write out the explicit expressions of the smoothing-solutions which are related with the nonlinear Fredholm integral equations.
引文
[1]Ren Yongtai, Shi Xifu, The partial differential equation, North-east normal univer-sity, Liaoning people's press,1984.
[2]Shi Rongchang, Matrix analysis, Beijing technology university press,1996.
[3]Chaohao Gu, Daqian Li, Shuxing Chen, Songmu Zheng, Yongji Tian, The mathematical-physics equations, (second edition), Higher education press,2005.
[4]Jianfeng Wang, How to solve the polynomial ordinary differential equations, Appl math comput,218(2011),2421-2438.
[5]Gongqing Zhang, Yuanqu Lin, Functional analysis, Peking university press,2010.
[6]Daqian Li, Tiehu Qing, Physics and partial differential equations, (second edition), Higher education press,2005.
[7]Shuxing Chen, The general theory of the partial differential equation, Higher edu-cation press,1981.
[8]Caisheng Chen, Gang Li, Jidong Chou, Wenchu Wang, The mathematical-physics equations, Science education press,2008.
[9]Tian Ma, The theory and method of partial differential equations, Science education press,2011.
[10]Yazhe, Chen, Elliptic partial differential equation and Elliptic partial differential equations of second order, Science education press,1991.
[11]Gilbarg D and Trudinger N S. Elliptic partial differential equations of second order [M].2nd ED. Berlin:Springer-Verlag,1983.
[12]Jidong Chou, Function of complex variable and integral transforms, Hohai uni-versity press,2001.
[13]Chongshi Wu, The mathematical-physics methods, Beijing University press, 2003.
[14]Lishang, Jiang, Lectures on the mathematical-physics equations, Higher educa-tion press,2007.
[15]Dajun Guo, Nonlinear functional analysis, (second edition), Sandong technology and science press,2002.
[16]Mingyou Qi, The introduction of linear partial differential operators, Science ed-ucation press,1986.
[17]Shuxing Chen, The modern method of partial differential equations, Higher edu-cation press,1992.
[18]L. Hormander, The analysis of linear partial differential operators II, Springer-Verlag,1983.
[19]Zhuxi Chen, Partial differential equations, Chinese science and technology uni-versity press,2002.
[20]C.O. Alves, S.H. M. Soares, Nodal solutions for singularly perturbed equations with critical exponential growth, J.D.E.234,464-484(2007).
[21]A. Ambrosetti, V. Felli, A. Malchiodi, Ground states of nonlinear Schrodinger equations with potentials vanishing at infinity, J. Eur. Math. Soc.7,117-144(2005).
[22]A. Ambrosetti, Z.Q. Wang, Nonlinear Schrodinger equations with vanishing and decaying potentials, Differential Integral Equations 18,1321-1332(2005).
[23]T. Bartsch, T. Weth, M. Willem, Partial symmetry of least energy nodal solutions to some variational problems, J.Anal.Math.96,1-18(2005).
[24]H. Berestycki, P.L. Lions, Nonlinear scalar field equations, I, Existence of ground state, Arch.Rat.Mech. Anal.82,313-345(1983).
[25]J. Byeon, Z.Q. Wang, Spherifical semiclassical states of a critical frequency for Schrodinger equations with decaying potentials, J. Eur. Math. Soc.8,217-228(2006).
[26]D.M. Cao, Nontrivial solutions of semilinear elliptic equation with critical expo-nent in R2, Comm.P.D.E.17,407-435(1992).
[27]M. Del Pino, P.L. Felmer, Local mountian passes for semilinear elliptic problems in unbounded domain, Calc. Var. P.D.E.4,121-137(1996).
[28]J.M. do'O, M.A.S. Souto, On a class of nonlinear Schrodinger equations in R2 involving critical growth, J.D.E.174,289-311(2001).
[29]M.W. Fei, H.C. Yin, On existence and concentration of bound states of nonlinear Schrodinger equations with compactly supported and competing potentials, Pacf-ic.J.M.244,261-296(2010).
[30]D.G. de Figueiredo, J.M. do'O, B. Ruf, On an inequality by N. Trudinger and J. Morse and related elliptic equations, Comm.Pure.Appl.Math.55,135-152(2002).
[31]D.G. de Figueiredo, O.H. Miyagaki, B. Ruf, Elliptic equations in R2 with nonlin-earities in the critical growth range, Calc.Var.P.D.E.3,139-153(1995).
[32]B. Gidas, W.M. Ni, L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in RN, Adv.Math.Suppl.Stud.7A,369-402(1981).
[33]Gilbarg D and Trudinger N S. Elliptic partial differential equations of second order. Second edition. Grundlehren der Mathematischen Wissenschaften,224, Springer, Berlin-New York,1998.
[34]Z.L. Liu, Z.Q. Wang, Sign-changing solutions of nonlinear elliptic equations, Front.Math.China 3,221-238(2008).
[35]E.S. Noussair, J. Wei, On the effect of domain geometry on the existence of n-odal solutions in singular perturbations problems, Indiana Univ. Math. J.46,1255-1271(1997).
[36]E.S. Noussair, J. Wei, On the location of spikes and profile of nodal. solutions for a singularly perturbed Neumann problem, Comm.P.D.E.23,793-816(1998).
[37]Y. Sato, Sign-changing multi-peak solutions for nonlinear Schrodinger equations with critical frequency, Comm.Pure.Appl.Anal.7,883-903(2008).
[38]X.F. Wang, B. Zeng, On concentration of positive bound states of nonlinear Schrodinger equations with competing potential functions, SIAM J.Math.Anal.28, 633-655(1997).
[39]X.F. Wang, On concentration of positive bound states of nonlinear Schrodinger equations, Comm.Math.Phys.153,229-244(1993).
[40]M. Willem, Minimax theorems, Progress in nonlinear differential equations and their applications,24. Birkhauser Boston, Inc. Boston, MA,1996.
[41]H.C. Yin, P.Z. Zhang, Bound states of nonlinear Schrodinger equations with po-tentials tending to zero at infinity, J.D.E.247,618-647(2009).
[42]A. Azzam, On Dirichlet's problem for elliptic equations in sectionally smooth n-dimensional domains, SIAM J.Math.Anal. Vol.11,248-253(1980).
[43]A. Azzam, Smoothness properties of mixed boundary value problems for elliptic equations in sectionally smooth n-dimensional domains, Ann. Polon. Math.40,81-93(1981).
[44]L. Bers, Mathematical aspects of subsonic and transonic gas dynamics, John Wil-ley Sons, Inc. New York, Chapman Hall, Ltd, London(1958).
[45]S. Canic, B.L. Keyltz, G.M. Lieberman, A proof of existence of perturbed steady transonic shocks via a free boundary problem, Comm.Pure Appl.Math, Vol.LⅢ, 484-511(2000).
[46]G.Q. Chen, J. Chen, K.W. Song, Transonic nozzle flows and free boundary prob-lems for the full Euler equations, J.D.E.229,92-120(2006).
[47]G.Q. Chen, J. Chen, M. Feldman, Transonic shocks and free boundary problem-s for the full Euler equations in infinite nozzles, J.Math. Pures Appl.(9)88,191-218(2007).
[48]S.X. Chen, Stability on transonic shock fronts in two-dimensional Euler systems, Trans.Amer.Math.Soc.357,287-308(2005).
[49]S.X. Chen, Transonic shocks in 3-D compressible folow passing a duct with gen-eral section for Euler systems, Trans.Amer.Math.Soc.360,5265-5289(2008).
[50]R. Courant, K.O. Friedrichs, Supersonic flow and shock waves, Interscience Pub-lishers Inc, New York,1948.
[51]V. Elling, T.P. Liu, Supersonic flow onto a solid wedge, Comm. Pure Appl.Math. Vol. XI,1347-1448(2008).
[52]P. Embid, J. Goodman, A. Majda, Multiple steady states for 1-D transonic flow, SIAM I. Sci.Statist.Comput.5,21-41(1984).
[53]D. Gilbarg, L. Hormander, Intermediate Schauder estimates, Arch. Rational. Mech. Anal.74,297-318(1980).
[54]H.M. Glaz, T.P. Liu, The asymptotic analysis of wave interactions and numerical calculations of transonic nozzle flow, Adv. in Appl.Math.5,111-146(1984).
[55]F. John, Formation of singularities in one-dimensional nonlinear wave propaga-tion, Comm. Pure Appl.Math.27,377-405(1974).
[56]A.G. Kuz'min, Boundary-Value problems for transonic flow, John Wiley Sons, Ltd(2002).
[57]J. Li, Z.P. Xin, H.C. Yin, On transonic shocks in a nozzle with variable end pres-sures, Comm.Math.Phys.291,111-150(2009).
[58]J. Li, Z.P. Xin, H.C. Yin, A free boundary value problem for the full steady com-pressible Euler system and two dimensional transonic shocks in a large variable nozzle, Math.Res.Lett. Vol.16,777-786(2009).
[59]D.Q. Li, S.M. Zheng, Y.J. Tan, W.X. Shen, Boundary value problems with e-quivalued surface resistivity well-logging, Pitman Research Notes in Mathematics Series,382, Longman, Harlow(1998).
[60]G.M. Lieberman, Mixed boundary value problems for elliptic and parabolic dif-ferential equation of second order, J.Math.Anal.Appl.113,442-440(1986).
[61]L. Liu, H.R. Yuan, Stability of cylindrical transonic shocks for the two-dimensional steady compressible Euler system, J. Hyperbolic Differ.Equ.5,347-379(2008).
[62]T.P. Liu, Nonlinear stability abd instability of transonic flows through a nozzle, Comm.Math.Phys.83,243-260(1982).
[63]T.P. Liu, Transonic gas flow in a duct of varying area, Arch.Rational Mech.Anal. 80,1-18(1982).
[64]C.S. Morawetz, Potential theory for regular and Mach reflection of a shock at wedge, Comm,Pure Appl.Math.47,593-624(1994).
[65]D.H. Wagner, Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions, J.D.E.68,118-136(1987).
[66]S.K. Wen, Multi-dimensional conservation laws and transonic shock problem, Master philosophy thesis of Chinese University of Hong Kong(2009).
[67]Z.P. Xin, H,C, Yin, W. Yan, Transonic shock problem for the Euler system in a nozzle, Arch.Rat.Mech.Anal.194,1-47(2009).
[68]Z.P. Xin, H,C, Yin, Transonic shock in a nozzle I,2-D case, Comm. Pure Ap-pl.Math. Vol. Ⅷ,999-1050(2005).
[69]Z.P. Xin, H,C, Yin,3-dimensional transonic shock in a nozzle, Pacific J.Math. 236,No.1,139-193(2008).
[70]Z.P. Xin, H,C, Yin, Transonic shock in a curved nozzle,2-D and 3-D complete Euler systems, J.D.E.245,1014-1085(2008).
[71]G. Xu, H.C. Yin, Global transonic conic shock wave for the symmetrically per-turbed supersonic flow past a cone, J.D.E.245,3389-3432(2008).
[72]G. Xu, H.C. Yin, Global multidimensional transonic conic shock wave for the symmetrically perturbed supersonic flow past a cone, SIAM J.Math.Anal. Vol.41, 178-218(2009).
[73]H.C. Yin, C.H. Zhou, On global transonic shocks for the steady supersonic Euler flows past sharp 2-D wedges, J.D.E.246,4466-4496(2009).
[74]H.R. Yuan, On transonic shocks in two-dimensional variable-area ducts for steady Euler system, SIAM J.Math.Anal.38,1343-1370(2006).
[75]Y.X. Zheng, Two-dimensional regular shock reflection for the pressure gradient system of conservation laws, Arch Math.Appl.Sin.Engl.Ser.22,177-210(2006).
[76]R. Adams, J.F. fournier, Sobolev Space, Second edition, Academic Press, New York,2003.
[77]A. Ambrosetti, A Malchiodi, D. Ruiz, Bound states of nonlinear Schrodinger e-quations with potentials vanishing at invity, J.Anal.Math.98,317-348(2006).
[78]D. Bonheure, J.V. Schaftingen, Nonlinear Schrodinger equations with potentials vanishing at invity, C.R.Acag.Sci.Paris 342,903-908(2006).
[79]D. Bonheure, J.V. Schaftingen, Bound state solutions for a class of nonlinear Schrodinger equations, Rev.Mat.Iberoam.24,297-351(2008).
[80]J. Byeon, Z.Q. Wang, Standing waves with a critical frequency for nonlinear Schrodinger equations, II, Calc.Var.P.D.E.18,207-219(2003).
[81]S. Cingolani, M. Lazzo, Multiple positive solutions to nonlinear Schrodinger e-quations with competing potential functions, J.D.E.160,118-132(2000).
[82]D.G. de Figueiredo, B. Ruf, Existence and non-existence of radial solutions for elliptic equations with critical exponent in R2, Comm. Pure Appl.Math.48,639-655(1995).
[83]A. Floer, A. Weinstein, Nonspreading wave packets for the cubic Schrodinger equations with a bounded potential, J.Funct.Anal.69,397-408(1986).
[84]B. Gidas, W.M. Ni, L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in RN, Adv.Math.Suppl.Stud.7A,369-402(1981).
[85]G.B. Li, X.P. Zhu, Nontrivial solution of nonlinear scalar old equations with strong nonlinearity, Acta Math.Sci.8,431-448(1988).
[86]Y.Q. Li, Z.Q. Wang, J. Zeng, Ground states of nonlinear Schrodinger equations with potentials, Ann.Inst.H.Poincar Anal.Non Linaire 23,829-837(2006).
[87]P.L. Lions, The concentration-compactness principle in the calculus of variation-s:the locally compact case, Part I, Ann.Inst.H.Poincar Anal.Non Linaire 1,109-145(1984).
[88]C.Y. Liu, Z.P. Wang, H.S. Zhou, Asymptotically linear Schrodinger equations with potential vanishing at invity, J.D.E.245,201-222(2008).
[89]W.M. Ni, On the elliptic equation its generalizations and applicatons in geometry, Indina Univ.Math.J.31,493-529(1982).
[90]X.M. Wang, On concentration of positive bound states of nonlinear Schrodinger equations, Comm.Math.Phys.153,229-244(1993).
[91]X.F. Wang, Z. Bin, On concentration of positive bound states of nonlinear Schrodinger equations with competing potential functions, SIAM J.Math.Anal.28, 633-655(1997).
[92]M. Williem, Minimax Theorems, PNLDE 24, Birkh Muser,1996.
[93]H.C. Yin, P.Z. Zhang, Bound states of nonlinear Schrodinger equations with po-tentials tending to zero at invity, J.D.E.247,618-647(2009).
[94]R.K. Agarwal, D.W. Halt, A modified CUSP scheme in wave particle split for-m for unstructured grid Euler Dews, Frontiers of Computational Fluid Dynamic-s(D.A.Caughey and M.M.Hafez, eds.)(1994).
[95]S. Alinhac, Blow-up of small data solutions for a quasilinear wave equations in two space dimensions, Ann. of Math. (2) 149,97-127(1999).
[96]S. Alinhac, The null condition for quasilinear wave equations in two space dimen-sions I, Invent.Math.145,597-618(2001).
[97]S. Alinhac, An example of blow-up at invity for quasilinear wave equations, As-terisque 284,1-91(2003).
[98]D. Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data, Comm.Pure Appl.Math.39,267-282(1986).
[99]R.T. Glassey, J.K. Hunter, Y.X. Zheng, Singularities of a variational wave equa-tion, J.D.E.129,49-78(1996).
[100]P. Godin, Lifespan of solutions of semilinear wave equations in two space di-mensions, Comm.P.D.E.18,No.5-6,895-916(1993).
[101]L. Hormander, The lifespan of classical solutions of nonlinear hyperbolic equa-tions, Mittag-Leur report(1985).
[102]L. Hormander, Lectures on nonlinear hyperbolic equations, Mathematiques Ap-plications 26, Spring Verlag, Heidelberg,1997.
[103]A. Hoshiga, The asymptotic behaviour of the radially symmetric solutions to quasilinear wave equations in two space dimensions, Hokkaido Math.J.24,575-615(1995).
[104]A. Hoshiga, The existence of the global solutions to semilinear wave equations with a class of cubic nonlinearities in 2-dimensional space, Hokkaido Math.J.37, 669-688(2008).
[105]J.K. Hunter, R.A. Saxton, Dynamics of director olds, SIAM. J.Appl.Math.51, 1498-1521(1991).
[106]F. John, Blow-up of radial solutions of utt= c2(ut) in three space dimensions, Mat.Appl.Comput.4,3-18(1985).
[107]S. Klainerman, G. Ponce, Global, small amplitude solutions to nonlinear evolu-tion equations, Comm. Pure Appl.Math.36,133-141(1983).
[108]S. Klainerman, The null condition and global existence to nonlinear wave equa-tions, Lectures in Applied Mathematics 23,193-236(1986).
[109]S. Klainerman, Remarks on the global Sobolev inequalities in the Minkowski space Rn+1, Comm. Pure Appl.Math.40,111-117(1987).
[110]Li, Ta-tsien, Y.M. Chen, Initial value problems for nonlinear wave equations, Comm.P.D.E.13,383-422(1988).
[111]H. Lindblad, On the lifespan of solutions of nonlinear wave equations with small initial data, Comm. Pure Appl.Math.43(1990),445-472(1990).
[112]H. Lindblad, Global solutions of nonlinear wave equations, Comm. Pure Ap-pl.Math.45(9),1063-1096(1992).
[113]H. Lindblad, Global solutions of quasilinear wave equations, Amer.J.Math.130, 115-157(2008).
[114]J. Metcalfe, C.D. Sogge, Global existence for high dimensional quasilinear wave equations exterior to star-shaped obstacles, Discrete Contin. Dyn.Syst.28,1589-1601(2010).
[115]R.A. Saxton, Dynamic instablity of the liquid crystal director, Current progress in hyperbolic system:Riemann problems and computations(Brunswick, ME, 1988),325-330, Contemp.Math.100, Amer.Math.Soc. Providence, RI,1989.
[116]H.C. Yin, Q.J. Qiu, The blow-up of solutions for 3-D axisymmetric compressible Euler equations, Nagoya MathJ. Vol.154,157-169(1999).
[117]H.C. Yin, Formation and construction of a shock wave for 3-D compressible Euler equations with the spherical initial data, Nagoya Math.J.175,125-164(2004).
[118]P. Zhang, Y.X. Zheng, On the second order asymptotic equation of a variational wave equation, Proc.Roy.Soc. Edinburgh SectA 132,483-509(2002).
[119]P. Zhang, Y.X. Zheng, Energy conservative solutions to a one-dimensional full variational wave system, Comm. Pure Appl.Math. DOI 10.1002/cpa.20380(on-line)(2011).
[120]Y.X. Zheng, Existence of solutions to the transonic pressur-gradient equations of the compressible Euler equations in elliptic regions, Comm.P.D.E.22,1849-1868(1997).
[2]Shi Rongchang, Matrix analysis, Beijing technology university press,1996.
[3]Chaohao Gu, Daqian Li, Shuxing Chen, Songmu Zheng, Yongji Tian, The mathematical-physics equations, (second edition), Higher education press,2005.
[4]Jianfeng Wang, How to solve the polynomial ordinary differential equations, Appl math comput,218(2011),2421-2438.
[5]Gongqing Zhang, Yuanqu Lin, Functional analysis, Peking university press,2010.
[6]Daqian Li, Tiehu Qing, Physics and partial differential equations, (second edition), Higher education press,2005.
[7]Shuxing Chen, The general theory of the partial differential equation, Higher edu-cation press,1981.
[8]Caisheng Chen, Gang Li, Jidong Chou, Wenchu Wang, The mathematical-physics equations, Science education press,2008.
[9]Tian Ma, The theory and method of partial differential equations, Science education press,2011.
[10]Yazhe, Chen, Elliptic partial differential equation and Elliptic partial differential equations of second order, Science education press,1991.
[11]Gilbarg D and Trudinger N S. Elliptic partial differential equations of second order [M].2nd ED. Berlin:Springer-Verlag,1983.
[12]Jidong Chou, Function of complex variable and integral transforms, Hohai uni-versity press,2001.
[13]Chongshi Wu, The mathematical-physics methods, Beijing University press, 2003.
[14]Lishang, Jiang, Lectures on the mathematical-physics equations, Higher educa-tion press,2007.
[15]Dajun Guo, Nonlinear functional analysis, (second edition), Sandong technology and science press,2002.
[16]Mingyou Qi, The introduction of linear partial differential operators, Science ed-ucation press,1986.
[17]Shuxing Chen, The modern method of partial differential equations, Higher edu-cation press,1992.
[18]L. Hormander, The analysis of linear partial differential operators II, Springer-Verlag,1983.
[19]Zhuxi Chen, Partial differential equations, Chinese science and technology uni-versity press,2002.
[20]C.O. Alves, S.H. M. Soares, Nodal solutions for singularly perturbed equations with critical exponential growth, J.D.E.234,464-484(2007).
[21]A. Ambrosetti, V. Felli, A. Malchiodi, Ground states of nonlinear Schrodinger equations with potentials vanishing at infinity, J. Eur. Math. Soc.7,117-144(2005).
[22]A. Ambrosetti, Z.Q. Wang, Nonlinear Schrodinger equations with vanishing and decaying potentials, Differential Integral Equations 18,1321-1332(2005).
[23]T. Bartsch, T. Weth, M. Willem, Partial symmetry of least energy nodal solutions to some variational problems, J.Anal.Math.96,1-18(2005).
[24]H. Berestycki, P.L. Lions, Nonlinear scalar field equations, I, Existence of ground state, Arch.Rat.Mech. Anal.82,313-345(1983).
[25]J. Byeon, Z.Q. Wang, Spherifical semiclassical states of a critical frequency for Schrodinger equations with decaying potentials, J. Eur. Math. Soc.8,217-228(2006).
[26]D.M. Cao, Nontrivial solutions of semilinear elliptic equation with critical expo-nent in R2, Comm.P.D.E.17,407-435(1992).
[27]M. Del Pino, P.L. Felmer, Local mountian passes for semilinear elliptic problems in unbounded domain, Calc. Var. P.D.E.4,121-137(1996).
[28]J.M. do'O, M.A.S. Souto, On a class of nonlinear Schrodinger equations in R2 involving critical growth, J.D.E.174,289-311(2001).
[29]M.W. Fei, H.C. Yin, On existence and concentration of bound states of nonlinear Schrodinger equations with compactly supported and competing potentials, Pacf-ic.J.M.244,261-296(2010).
[30]D.G. de Figueiredo, J.M. do'O, B. Ruf, On an inequality by N. Trudinger and J. Morse and related elliptic equations, Comm.Pure.Appl.Math.55,135-152(2002).
[31]D.G. de Figueiredo, O.H. Miyagaki, B. Ruf, Elliptic equations in R2 with nonlin-earities in the critical growth range, Calc.Var.P.D.E.3,139-153(1995).
[32]B. Gidas, W.M. Ni, L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in RN, Adv.Math.Suppl.Stud.7A,369-402(1981).
[33]Gilbarg D and Trudinger N S. Elliptic partial differential equations of second order. Second edition. Grundlehren der Mathematischen Wissenschaften,224, Springer, Berlin-New York,1998.
[34]Z.L. Liu, Z.Q. Wang, Sign-changing solutions of nonlinear elliptic equations, Front.Math.China 3,221-238(2008).
[35]E.S. Noussair, J. Wei, On the effect of domain geometry on the existence of n-odal solutions in singular perturbations problems, Indiana Univ. Math. J.46,1255-1271(1997).
[36]E.S. Noussair, J. Wei, On the location of spikes and profile of nodal. solutions for a singularly perturbed Neumann problem, Comm.P.D.E.23,793-816(1998).
[37]Y. Sato, Sign-changing multi-peak solutions for nonlinear Schrodinger equations with critical frequency, Comm.Pure.Appl.Anal.7,883-903(2008).
[38]X.F. Wang, B. Zeng, On concentration of positive bound states of nonlinear Schrodinger equations with competing potential functions, SIAM J.Math.Anal.28, 633-655(1997).
[39]X.F. Wang, On concentration of positive bound states of nonlinear Schrodinger equations, Comm.Math.Phys.153,229-244(1993).
[40]M. Willem, Minimax theorems, Progress in nonlinear differential equations and their applications,24. Birkhauser Boston, Inc. Boston, MA,1996.
[41]H.C. Yin, P.Z. Zhang, Bound states of nonlinear Schrodinger equations with po-tentials tending to zero at infinity, J.D.E.247,618-647(2009).
[42]A. Azzam, On Dirichlet's problem for elliptic equations in sectionally smooth n-dimensional domains, SIAM J.Math.Anal. Vol.11,248-253(1980).
[43]A. Azzam, Smoothness properties of mixed boundary value problems for elliptic equations in sectionally smooth n-dimensional domains, Ann. Polon. Math.40,81-93(1981).
[44]L. Bers, Mathematical aspects of subsonic and transonic gas dynamics, John Wil-ley Sons, Inc. New York, Chapman Hall, Ltd, London(1958).
[45]S. Canic, B.L. Keyltz, G.M. Lieberman, A proof of existence of perturbed steady transonic shocks via a free boundary problem, Comm.Pure Appl.Math, Vol.LⅢ, 484-511(2000).
[46]G.Q. Chen, J. Chen, K.W. Song, Transonic nozzle flows and free boundary prob-lems for the full Euler equations, J.D.E.229,92-120(2006).
[47]G.Q. Chen, J. Chen, M. Feldman, Transonic shocks and free boundary problem-s for the full Euler equations in infinite nozzles, J.Math. Pures Appl.(9)88,191-218(2007).
[48]S.X. Chen, Stability on transonic shock fronts in two-dimensional Euler systems, Trans.Amer.Math.Soc.357,287-308(2005).
[49]S.X. Chen, Transonic shocks in 3-D compressible folow passing a duct with gen-eral section for Euler systems, Trans.Amer.Math.Soc.360,5265-5289(2008).
[50]R. Courant, K.O. Friedrichs, Supersonic flow and shock waves, Interscience Pub-lishers Inc, New York,1948.
[51]V. Elling, T.P. Liu, Supersonic flow onto a solid wedge, Comm. Pure Appl.Math. Vol. XI,1347-1448(2008).
[52]P. Embid, J. Goodman, A. Majda, Multiple steady states for 1-D transonic flow, SIAM I. Sci.Statist.Comput.5,21-41(1984).
[53]D. Gilbarg, L. Hormander, Intermediate Schauder estimates, Arch. Rational. Mech. Anal.74,297-318(1980).
[54]H.M. Glaz, T.P. Liu, The asymptotic analysis of wave interactions and numerical calculations of transonic nozzle flow, Adv. in Appl.Math.5,111-146(1984).
[55]F. John, Formation of singularities in one-dimensional nonlinear wave propaga-tion, Comm. Pure Appl.Math.27,377-405(1974).
[56]A.G. Kuz'min, Boundary-Value problems for transonic flow, John Wiley Sons, Ltd(2002).
[57]J. Li, Z.P. Xin, H.C. Yin, On transonic shocks in a nozzle with variable end pres-sures, Comm.Math.Phys.291,111-150(2009).
[58]J. Li, Z.P. Xin, H.C. Yin, A free boundary value problem for the full steady com-pressible Euler system and two dimensional transonic shocks in a large variable nozzle, Math.Res.Lett. Vol.16,777-786(2009).
[59]D.Q. Li, S.M. Zheng, Y.J. Tan, W.X. Shen, Boundary value problems with e-quivalued surface resistivity well-logging, Pitman Research Notes in Mathematics Series,382, Longman, Harlow(1998).
[60]G.M. Lieberman, Mixed boundary value problems for elliptic and parabolic dif-ferential equation of second order, J.Math.Anal.Appl.113,442-440(1986).
[61]L. Liu, H.R. Yuan, Stability of cylindrical transonic shocks for the two-dimensional steady compressible Euler system, J. Hyperbolic Differ.Equ.5,347-379(2008).
[62]T.P. Liu, Nonlinear stability abd instability of transonic flows through a nozzle, Comm.Math.Phys.83,243-260(1982).
[63]T.P. Liu, Transonic gas flow in a duct of varying area, Arch.Rational Mech.Anal. 80,1-18(1982).
[64]C.S. Morawetz, Potential theory for regular and Mach reflection of a shock at wedge, Comm,Pure Appl.Math.47,593-624(1994).
[65]D.H. Wagner, Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions, J.D.E.68,118-136(1987).
[66]S.K. Wen, Multi-dimensional conservation laws and transonic shock problem, Master philosophy thesis of Chinese University of Hong Kong(2009).
[67]Z.P. Xin, H,C, Yin, W. Yan, Transonic shock problem for the Euler system in a nozzle, Arch.Rat.Mech.Anal.194,1-47(2009).
[68]Z.P. Xin, H,C, Yin, Transonic shock in a nozzle I,2-D case, Comm. Pure Ap-pl.Math. Vol. Ⅷ,999-1050(2005).
[69]Z.P. Xin, H,C, Yin,3-dimensional transonic shock in a nozzle, Pacific J.Math. 236,No.1,139-193(2008).
[70]Z.P. Xin, H,C, Yin, Transonic shock in a curved nozzle,2-D and 3-D complete Euler systems, J.D.E.245,1014-1085(2008).
[71]G. Xu, H.C. Yin, Global transonic conic shock wave for the symmetrically per-turbed supersonic flow past a cone, J.D.E.245,3389-3432(2008).
[72]G. Xu, H.C. Yin, Global multidimensional transonic conic shock wave for the symmetrically perturbed supersonic flow past a cone, SIAM J.Math.Anal. Vol.41, 178-218(2009).
[73]H.C. Yin, C.H. Zhou, On global transonic shocks for the steady supersonic Euler flows past sharp 2-D wedges, J.D.E.246,4466-4496(2009).
[74]H.R. Yuan, On transonic shocks in two-dimensional variable-area ducts for steady Euler system, SIAM J.Math.Anal.38,1343-1370(2006).
[75]Y.X. Zheng, Two-dimensional regular shock reflection for the pressure gradient system of conservation laws, Arch Math.Appl.Sin.Engl.Ser.22,177-210(2006).
[76]R. Adams, J.F. fournier, Sobolev Space, Second edition, Academic Press, New York,2003.
[77]A. Ambrosetti, A Malchiodi, D. Ruiz, Bound states of nonlinear Schrodinger e-quations with potentials vanishing at invity, J.Anal.Math.98,317-348(2006).
[78]D. Bonheure, J.V. Schaftingen, Nonlinear Schrodinger equations with potentials vanishing at invity, C.R.Acag.Sci.Paris 342,903-908(2006).
[79]D. Bonheure, J.V. Schaftingen, Bound state solutions for a class of nonlinear Schrodinger equations, Rev.Mat.Iberoam.24,297-351(2008).
[80]J. Byeon, Z.Q. Wang, Standing waves with a critical frequency for nonlinear Schrodinger equations, II, Calc.Var.P.D.E.18,207-219(2003).
[81]S. Cingolani, M. Lazzo, Multiple positive solutions to nonlinear Schrodinger e-quations with competing potential functions, J.D.E.160,118-132(2000).
[82]D.G. de Figueiredo, B. Ruf, Existence and non-existence of radial solutions for elliptic equations with critical exponent in R2, Comm. Pure Appl.Math.48,639-655(1995).
[83]A. Floer, A. Weinstein, Nonspreading wave packets for the cubic Schrodinger equations with a bounded potential, J.Funct.Anal.69,397-408(1986).
[84]B. Gidas, W.M. Ni, L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in RN, Adv.Math.Suppl.Stud.7A,369-402(1981).
[85]G.B. Li, X.P. Zhu, Nontrivial solution of nonlinear scalar old equations with strong nonlinearity, Acta Math.Sci.8,431-448(1988).
[86]Y.Q. Li, Z.Q. Wang, J. Zeng, Ground states of nonlinear Schrodinger equations with potentials, Ann.Inst.H.Poincar Anal.Non Linaire 23,829-837(2006).
[87]P.L. Lions, The concentration-compactness principle in the calculus of variation-s:the locally compact case, Part I, Ann.Inst.H.Poincar Anal.Non Linaire 1,109-145(1984).
[88]C.Y. Liu, Z.P. Wang, H.S. Zhou, Asymptotically linear Schrodinger equations with potential vanishing at invity, J.D.E.245,201-222(2008).
[89]W.M. Ni, On the elliptic equation its generalizations and applicatons in geometry, Indina Univ.Math.J.31,493-529(1982).
[90]X.M. Wang, On concentration of positive bound states of nonlinear Schrodinger equations, Comm.Math.Phys.153,229-244(1993).
[91]X.F. Wang, Z. Bin, On concentration of positive bound states of nonlinear Schrodinger equations with competing potential functions, SIAM J.Math.Anal.28, 633-655(1997).
[92]M. Williem, Minimax Theorems, PNLDE 24, Birkh Muser,1996.
[93]H.C. Yin, P.Z. Zhang, Bound states of nonlinear Schrodinger equations with po-tentials tending to zero at invity, J.D.E.247,618-647(2009).
[94]R.K. Agarwal, D.W. Halt, A modified CUSP scheme in wave particle split for-m for unstructured grid Euler Dews, Frontiers of Computational Fluid Dynamic-s(D.A.Caughey and M.M.Hafez, eds.)(1994).
[95]S. Alinhac, Blow-up of small data solutions for a quasilinear wave equations in two space dimensions, Ann. of Math. (2) 149,97-127(1999).
[96]S. Alinhac, The null condition for quasilinear wave equations in two space dimen-sions I, Invent.Math.145,597-618(2001).
[97]S. Alinhac, An example of blow-up at invity for quasilinear wave equations, As-terisque 284,1-91(2003).
[98]D. Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data, Comm.Pure Appl.Math.39,267-282(1986).
[99]R.T. Glassey, J.K. Hunter, Y.X. Zheng, Singularities of a variational wave equa-tion, J.D.E.129,49-78(1996).
[100]P. Godin, Lifespan of solutions of semilinear wave equations in two space di-mensions, Comm.P.D.E.18,No.5-6,895-916(1993).
[101]L. Hormander, The lifespan of classical solutions of nonlinear hyperbolic equa-tions, Mittag-Leur report(1985).
[102]L. Hormander, Lectures on nonlinear hyperbolic equations, Mathematiques Ap-plications 26, Spring Verlag, Heidelberg,1997.
[103]A. Hoshiga, The asymptotic behaviour of the radially symmetric solutions to quasilinear wave equations in two space dimensions, Hokkaido Math.J.24,575-615(1995).
[104]A. Hoshiga, The existence of the global solutions to semilinear wave equations with a class of cubic nonlinearities in 2-dimensional space, Hokkaido Math.J.37, 669-688(2008).
[105]J.K. Hunter, R.A. Saxton, Dynamics of director olds, SIAM. J.Appl.Math.51, 1498-1521(1991).
[106]F. John, Blow-up of radial solutions of utt= c2(ut) in three space dimensions, Mat.Appl.Comput.4,3-18(1985).
[107]S. Klainerman, G. Ponce, Global, small amplitude solutions to nonlinear evolu-tion equations, Comm. Pure Appl.Math.36,133-141(1983).
[108]S. Klainerman, The null condition and global existence to nonlinear wave equa-tions, Lectures in Applied Mathematics 23,193-236(1986).
[109]S. Klainerman, Remarks on the global Sobolev inequalities in the Minkowski space Rn+1, Comm. Pure Appl.Math.40,111-117(1987).
[110]Li, Ta-tsien, Y.M. Chen, Initial value problems for nonlinear wave equations, Comm.P.D.E.13,383-422(1988).
[111]H. Lindblad, On the lifespan of solutions of nonlinear wave equations with small initial data, Comm. Pure Appl.Math.43(1990),445-472(1990).
[112]H. Lindblad, Global solutions of nonlinear wave equations, Comm. Pure Ap-pl.Math.45(9),1063-1096(1992).
[113]H. Lindblad, Global solutions of quasilinear wave equations, Amer.J.Math.130, 115-157(2008).
[114]J. Metcalfe, C.D. Sogge, Global existence for high dimensional quasilinear wave equations exterior to star-shaped obstacles, Discrete Contin. Dyn.Syst.28,1589-1601(2010).
[115]R.A. Saxton, Dynamic instablity of the liquid crystal director, Current progress in hyperbolic system:Riemann problems and computations(Brunswick, ME, 1988),325-330, Contemp.Math.100, Amer.Math.Soc. Providence, RI,1989.
[116]H.C. Yin, Q.J. Qiu, The blow-up of solutions for 3-D axisymmetric compressible Euler equations, Nagoya MathJ. Vol.154,157-169(1999).
[117]H.C. Yin, Formation and construction of a shock wave for 3-D compressible Euler equations with the spherical initial data, Nagoya Math.J.175,125-164(2004).
[118]P. Zhang, Y.X. Zheng, On the second order asymptotic equation of a variational wave equation, Proc.Roy.Soc. Edinburgh SectA 132,483-509(2002).
[119]P. Zhang, Y.X. Zheng, Energy conservative solutions to a one-dimensional full variational wave system, Comm. Pure Appl.Math. DOI 10.1002/cpa.20380(on-line)(2011).
[120]Y.X. Zheng, Existence of solutions to the transonic pressur-gradient equations of the compressible Euler equations in elliptic regions, Comm.P.D.E.22,1849-1868(1997).