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Stability and symmetry-breaking bifurcation for the ground states of a NLS with a $δ^\prime$ interaction

We determine and study the ground states of a focusing Schrödinger equation in dimension one with a power nonlinearity $|ψ|^{2μ} ψ$ and a strong inhomogeneity represented by a singular point perturbation, the so-called (attractive) $δ^\prime$ interaction, located at the origin. The time-dependent problem turns out to be globally well posed in the subcritical regime, and locally well posed in the supercritical and critical regime in the appropriate energy space. The set of the (nonlinear) ground states is completely determined. For any value of the nonlinearity power, it exhibits a symmetry breaking bifurcation structure as a function of the frequency (i.e., the nonlinear eigenvalue) $ω$. More precisely, there exists a critical value $\om^*$ of the nonlinear eigenvalue $\om$, such that: if $\om_0 < \om < \om^*$, then there is a single ground state and it is an odd function; if $\om > \om^*$ then there exist two non-symmetric ground states. We prove that before bifurcation (i.e., for $\om < \om^*$) and for any subcritical power, every ground state is orbitally stable. After bifurcation ($\om =\om^*+0$), ground states are stable if $μ$ does not exceed a value $μ^\star$ that lies between 2 and 2.5, and become unstable for $μ> μ^*$. Finally, for $μ> 2$ and $\om \gg \om^*$, all ground states are unstable. The branch of odd ground states for $\om < \om^*$ can be continued at any $\om > \om^*$, obtaining a family of orbitally unstable stationary states. Existence of ground states is proved by variational techniques, and the stability properties of stationary states are investigated by means of the Grillakis-Shatah-Strauss framework, where some non standard techniques have to be used to establish the needed properties of linearization operators.