28 nov 2016 -- 10:00 [open in google calendar]
Scuola Normale Superiore, Aula Tonelli
NOTE THE CHANGE OF TIME AND VENUE
Abstract.
We present a new variational characterization of multiple critical
points for even energy functionals functionals corresponding to
nonlinear Schrödinger equations of the following type:
$
\left\{
\begin{array}{l}
-\Delta u + V(x) u - q(x)
u
^\sigma u = \lambda u, \quad (x\in\mathbf{R}^N)\\
u\in H^1(\mathbf{R}^N)\setminus\{0\}.
\end{array}
\right.
$
We assume $N\geq 3$, $q(x)\in L^\infty(\mathbf{R}^N)$, $q(x)>0$ a.e. with $\lim_{
x
\to\infty}q(x)=0$ and $0<\sigma <\frac{4}{N-2}$.
Our results cover the following 3 cases in a uniform way:
(i) $V(x)\equiv 0$;
(ii) $V(x)$ is a Coulomb potential and
(iii) $V(x)\in L^\infty(\mathbf{R}^N)$ with $V(x+k)\equiv V(x)$ for all $k\in \mathbf{Z}^N$.
The eigenvalue $\lambda$ thereby may or may not lie inside a spectral gap.
Our variational characterization is ``simple'' and well suited for discussing multiple bifurcation of solutions.