# A mountain pass theorem (existence and bifurcation)

##
Hans-Jӧrg Ruppen

created by malchiodi on 21 Nov 2016

modified on 28 Nov 2016

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.