On a semilinear variational problem
Zentrum Mathematik, Technische Universität München, Boltzmannstr. 3, 85747 Garching, Germany. firstname.lastname@example.org
Revised: 27 July 2009
We provide a detailed analysis of the minimizers of the functional , , subject to the constraint . This problem, e.g., describes the long-time behavior of the parabolic Anderson in probability theory or ground state solutions of a nonlinear Schrödinger equation. While existence can be proved with standard methods, we show that the usual uniqueness results obtained with PDE-methods can be considerably simplified by additional variational arguments. In addition, we investigate qualitative properties of the minimizers and also study their behavior near the critical exponent 2.
Mathematics Subject Classification: 35J20 / 49J45 / 35Q55
Key words: Nonlinear minimum problem / parabolic Anderson model / variational methods / Gamma-convergence / ground state solutions
© EDP Sciences, SMAI, 2009