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Workgroup Nonlinear Partial Differential Equations

Kollegiengebäude Mathematik (20.30)
Room 3.029

Karlsruher Institut für Technologie
Institut für Analysis
Englerstraße 2
76131 Karlsruhe

Office hours:
Mon, Wed, Thu 10-13 and by email

Tel.: +49 721 608 42064

Fax.: +49 721 608 46530

Nonlinear Schrödinger equations - stationary aspects (Winter Semester 2010/11)

Lecturer: Prof. Dr. Wolfgang Reichel
Classes: Lecture (1053)
Weekly hours: 2

Course starts: October 28th

This course is a continuation of the lecture of Prof. Schnaubelt in the previous summer
semester on "Nonlinear Schrödinger Equations - dynamical aspects". However, the contents
and methods are independent.

In this course I will study solitary waves of the nonlinear Schrödinger equation (NLS). They
are solutions of

$-\Delta u + V(x) u = \Gamma(x) |u|^{p-1}u \mbox{ in } I\!\!R^n$

I will mainly discuss variational methods for proving existence of solutions. In the case of
constant coefficients we will also study qualitative properties of positive solutions.


Familiarity with variational methods is helpful. Some of the basic facts of the calculus of
variations will be reviewed. Knowledge in Lebesgue integral, Sobolev spaces, and functional
analytical concepts like weak convergence is essential.

Lecture: Thursday 11:30-13:00 1C-04 Begin: 28.10.2010
Lecturer Prof. Dr. Wolfgang Reichel
Office hours: Monday, 11:30-13:00 Before you e-mail: call or come!
Room 3.035 Kollegiengebäude Mathematik (20.30)
Email: Wolfgang.Reichel@kit.edu


A preliminary list of topics:

  1. Motivation and examples
  2. Constant coefficient case
  3. Asymptotically constant coefficients
  4. Periodic coefficients


  1. H. Berestycki, P.L. Lions: Nonlinear scalar fi eld equations I. Arch. Rational Mech. Anal. 82, 313-345 (1983).
  2. B. Gidas, Wei-Ming Ni, L. Nirenberg: Symmetry of positive solutions of nonlinear elliptic equations in Rn. Math. Anal. Appl., Part A, 369-402. Adv. in Math. Suppl. Stud. 7a (1981).
  3. A. Pankov: Periodic nonlinear Schroedinger equation with application to photonic crystals. Milan J. Math. 73, 259-287 (2005).
  4. W. Strauss: Existence of solitary waves in higher dimensions. Comm. Math. Phys. 55, 149-162 (1977).
  5. M. Struwe: Variational Methods. Springer Verlag.
  6. C. A. Stuart: A variational approach to bifurcation in LP on an unbounded symmetrical domain. Math. Ann. 263, 51-59 (1983).
  7. M. Willem: Minimax Theorems. Birkhäuser Verlag.