5 research outputs found
A mean field type flow
We consider a gradient flow related to the mean field type equation. First,
we show that this flow exists for all time. Next, we prove a compactness result
for this flow allowing us to get, under suitable hypothesis on its energy, the
convergence of the flow to a solution of the mean field type equation. We also
get a divergence result if the energy of the initial data is largely negative
Orbitally stable standing waves of a mixed dispersion nonlinear Schr\"odinger equation
We study the mixed dispersion fourth order nonlinear Schr\"odinger equation
\begin{equation*} %\tag{\protect{4NLS}}\label{4nls} i \partial_t \psi -\gamma
\Delta^2 \psi +\beta \Delta \psi +|\psi|^{2\sigma} \psi =0\ \text{in}\ \R
\times\R^N, \end{equation*} where and . We
focus on standing wave solutions, namely solutions of the form , for some . This ansatz yields the
fourth-order elliptic equation \begin{equation*}
%\tag{\protect{*}}\label{4nlsstar} \gamma \Delta^2 u -\beta \Delta u +\alpha u
=|u|^{2\sigma} u. \end{equation*} We consider two associated constrained
minimization problems: one with a constraint on the -norm and the other on
the -norm. Under suitable conditions, we establish existence of
minimizers and we investigate their qualitative properties, namely their sign,
symmetry and decay at infinity as well as their uniqueness, nondegeneracy and
orbital stability.Comment: 37 pages. To appear in SIAM J. Math. Ana