Calculus of Variations and Geometric Measure Theory
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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.

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