Existence and orbital stability/instability of standing waves with prescribed mass for the L2-supercritical NLS in bounded domains and exterior domains

Relying on bifurcation type arguments, we obtain threshold results for the existence, nonexistence and multiplicity of positive solutions with prescribed L-2 norm for the semi-linear elliptic equation in bounded domains: {-Delta u - f (x, u) = lambda u, x epsilon Omega, u vertical bar partial derivative Omega = 0, and we do not assume that f is autonomous. Moreover, we provide a lower bound for the threshold. For almost every L-2 mass in the existence range, there exist one orbitally stable standing wave and one unstable standing wave associated to these positive solutions. We also study the existence of prescribed norm solutions in exterior domains and orbitally unstable standing waves for almost every L-2 mass in the existence range.

Automated summary

By using bifurcation type arguments, we establish threshold results for the existence, nonexistence, and multiplicity of positive solutions with prescribed L-2 norm for a semi-linear elliptic equation in bounded domains. We do not assume f to be autonomous. Furthermore, we provide a lower bound for the threshold. For almost every L-2 mass in the existence range, there exist one orbitally stable standing wave and one unstable standing wave associated with these positive solutions. We also study the existence of prescribed norm solutions in exterior domains and orbitally unstable standing waves for almost every L-2 mass in the existence range.