我們仔細的研讀了由Thierry Cazenave教授所編寫的書籍"Semilinear Schrödinger equations"中的第八章，其內容是討論一個半線性Schrödinger方程的駐波解存在性與穩定性。

作者先給出駐波解的型式，將駐波解代入方程後得到橢圓方程。對於這個橢圓方程作者給出Pohozaev恆等式和一些恆等式，以及一些相關的泛函，並且在給定限制條件下，證明能量泛函為最小值的函數就是橢圓方程的解。這裡的極小化序列函數只要求弱收斂在能量空間裡，還有全部的基態可以被唯一的好函數藉由平移以及旋轉所生成。

其中也證明了維度等於 1 的時候，束縛態就是基態，但是在維度大於等於 2 時，束縛態就不只是基態!

接著，作者給出一些引理及推論去證明當非線性作用強的時候，則薛丁格方程的駐波解是不穩定的，作者的策略是證明駐波解附近有爆炸解，從而得到駐波解的不穩定性。

另外，當非線性作用弱的時候，作者使用了 P.L.Lions 的集中緊緻性原理，證明極小化序列函數會強收斂在能量空間裡，再去證明基態解存在性等價於一個適當的極小值問題的解，然後用反證法得到駐波解是軌道穩定的。

最後的部分則是證明在特定非線性次數下，基態跟Gagliardo-Nirenberg不等式的最佳常數是相關的。

我們除了研讀這些章節的內容外，也補充了作者省略的證明細節，以及修正一些打字方面的小錯誤。

We study the chapter eight of the book ”Semilinear Schrödinger equations” written by Professor Thierry Cazenave, which is to discuss the existence and stability of standing wave solutions for a semilinear Schrödinger equation.
First, giving the expression of the standing wave and substituting the standing wave into the semilinear Schrödinger equation, the author got an elliptic equation. Then derived the Pohozaev identity, some other identities, and some related functionals to prove that the minimizer of the functional under a suitable constraint condition, which is the solution of the elliptic equation. In addition, the argument of the convergence of the minimizing sequence only requires weak convergence !
The family of ground states can be generated by the unique good function, through the modulo space transformation and Rotation. In the case of the dimension 1, every bound state is a ground state. This is not true when the dimension is greater than 1 !
Second, when the equation has strong non-linear interaction, the author gives some lemmas and corollaries to show that the standing wave is unstable. The strategies for the proof is to show that if the initial data is close to the ground state, then the corresponding maximal solution shall blow up in finite time.
Furthermore, when the equation has weak non-linear interaction, the author use the Concentration Compactness principle to claim that the minimizing sequence strong converges in the energy space, and the ground state is the minimizer of an appropriate minimization problem.
Therefore, the standing wave solution is then proved by contradiction to be orbitally stable.
The last part is to prove that under a certain non-linear exponent, the mass of the ground state is related to the best constant of the Gagliardo-Nirenberg inequality.
In addition to studying the contents of this chapter, we elaborate the proof in details which omitted by the author and we corrected some minor errors and typos.

[1] Adams R. A. ”Sobolev Spaces”. Academic Press, New York, 1975.
[2] Antoni Zygmund and Richard L. Wheeden. ”Measure and Integral, An Introduction to Real Analysis”.
[3] B. Gidas, Ni, W. M., and L. Nirenberg. ”Symmetry of positive solutions of Part A, 369-402. Advances in Mathematics Supplementary Studies”, 7. Academic, New York- London, 1981.
[4] D. Gilbarg and N.S. Trudinger. ”Elliptic partial differential equation of second order”.
2nd ed. Grundlehren der Mathematischen Wissenschaften, 224. springer, Berlin, 1983.
[5] Enrico Giusti ”Direct methods in the calculus of variations”, Singapore ; River Edge, NJ : World Scientific, 2003.
[6] Haim Brezis. ”Functional Analysis, Sobolev space and Partial Differential Equation”.
[7] H. Berestycki and P.L.Lions. ”Nonlinear Scalar Field Equation”. Arch. Ration. Mech. Anal. 82: 313-375, 1983.
[8] H. Royden and P. Fitzpatrick. ”Real Analysis”. 4ed.
[9] J. Bergh and J. Löfström ”Interpolation spaces. Springer”, New York, 1976.
[10] Jindrich Necas ”Direct methods in the theory of elliptic equations”, Berlin, Heidelberg : Springer-Verlag Berlin Heidelberg, 2012.
[11] Lawrence C. Evans. ”Partial Differential Equation”, 1ed.
[12] Lawrence C. Evans. ”Weak Convergence Methods for Nonlinear Partial Differential Equations ”, GBMS Vol.74, American. Society, 1990.
[13] M. K. Kwong. ”Uniqueness of positive solutions of ”. Arch. Ration. Mech. Anal. 105(3):243-266, 1989.
[14] M. I. Weinstein. ”Nonlinear Schrödinger equation and sharp interpolation estimates”. Comm. Math. Phys. 87:567-576, 1985.
[15] M. Struwe. ”Variational methods : applications to nonlinear partial differential equations and Hamiltonian systems”, 2006.
[16] Mihlin S.G. ”Variational methods in Mathematical Physics”. State Publishing House for Technical Literature, 1957.
[17] Philppe G. Ciarlet. ”Linear and Nonlinear Functional Analysis with Application”.
[18] P. L. Lions. ”The concentration-compactness principle in the calculus of variations”. The locally compact case. Ann. Inst. Poincaré Anal. Nonlineaire, 1, 1984.
[19] P. L. Lions. ”The concentration-compactness principle in the calculus of variations” The limit case, Rev. Mat. Iberoamericano, 1, 1985.
[20] S. Laurent ”Aspects of Sobolev-Type Inequalities”, Cambridge University Press, 2002.
[21] T. Cazenave and P.L.Lions. ”Orbital stability of standing waves for some nonlinear Schrödinger equations”. Comm. Math. Phys. 85(4):549-561, 1982.
[22] T. Cazenave. ”Semilinear Schrödinger Equations ”.
[23] T. Cazenave. ”An introduction to nonlinear Schrödinger equations”.
[24] Talenti G. ”Best constants in Sobolev inequality”, Annali di Mat., 110, 353-372, 1976.
[25] Walter A. Strauss ”Partial Differential Equations: An Introduction”, 2ed.
[26] Zeidler, Eberhard ”Applied functional analysis: Variational Methods and Optimization”. Applied Mathematical Sciences 109. New York, NY: Springer-Verlag. ISBN 978-1-4612-9529-7, 1995.
[27] Yosida K. ”Functional analysis”. Springer-Verlag, 1978.