## Unstable normalized standing waves for the space periodic NLS.(English)Zbl 1405.35191

Summary: For the stationary nonlinear Schr\`‘odinger equation $$-\Delta u+ V(x)u- f(u) = \lambda u$$ with periodic potential $$V$$ we study the existence and stability properties of multibump solutions with prescribed $$L^2$$-norm. To this end we introduce a new nondegeneracy condition and develop new superposition techniques which allow us to match the $$L^2$$-constraint. In this way we obtain the existence of infinitely many geometrically distinct solutions to the stationary problem. We then calculate the Morse index of these solutions with respect to the restriction of the underlying energy functional to the associated $$L^2$$-sphere, and we show their orbital instability with respect to the Schrödinger flow. Our results apply in both, the mass-subcritical and the mass-supercritical regime.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian 35J20 Variational methods for second-order elliptic equations
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