On a semilinear variational problem

Schmidt, Bernd

ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 1, pp. 86-101.

We provide a detailed analysis of the minimizers of the functional $u\mapsto {\int }_{{ℝ}^n}{\left|\nabla u\right|}^2+D{\int }_{{ℝ}^n}{\left|u\right|}^{\gamma }$, $\gamma \in (0,2)$, subject to the constraint ${\parallel u\parallel }_{L^2}=1$. 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.

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