Dynamical and variational properties of the NLS-$\delta'_s$ equation on the star graph

We study the nonlinear Schr\"odinger equation with $\delta'_s$ coupling of intensity $\beta\in\mathbb{R}\setminus\{0\}$ on the star graph $\Gamma$ consisting of $N$ half-lines. The nonlinearity has the form $g(u)=|u|^{p-1}u, p>1.$ In the first part of the paper, under certain restriction on $\beta$, we prove the existence of the ground state solution as a minimizer of the action functional $S_\omega$ on the Nehari manifold. It appears that the family of critical points which contains a ground state consists of $N$ profiles (one symmetric and $N-1$ asymmetric). In particular, for the attractive $\delta'_s$ coupling ($\beta<0$) and the frequency $\omega$ above a certain threshold, we managed to specify the ground state. The second part is devoted to the study of orbital instability of the critical points. We prove spectral instability of the critical points using Grillakis/Jones Instability Theorem. Then orbital instability for $p>2$ follows from the fact that data-solution mapping associated with the equation is of class $C^2$ in $H^1(\Gamma)$. Moreover, for $p>5$ we complete and concertize instability results showing strong instability (by blow up in finite time) for the particular critical points.