Instability of standing waves for fractional NLS with combined nonlinearities

We study the existence and strong instability of standing wave solutions for the fractional nonlinear Schrödinger equation \begin{equation*} \begin{cases} \mathrm{i} \psi_{t}=\left(-\Delta\right)^s \psi-\left(|\psi|^{p-2}\psi+\mu|\psi|^{q-2}\psi\right), & \quad(t,x)\in(0,\infty)\times\mathbb{R}^N,\ \psi(0,x)=\psi_0(x),&\quad x\in\mathbb{R}^N, \end{cases} \end{equation*} where $N\geq2$ , $0 \lt s \lt 1$ , $2 \lt q \lt p \lt 2_s^*=2N/(N-2s)$ , and $\mu\in\mathbb{R}$ . The primary challenge lies in the inhomogeneity of the nonlinearity. We deal with the following three cases: (i) for $2 \lt q \lt p \lt 2+4s/N$ and µ < 0, there exists a threshold mass a 0 for the existence of the least energy normalized solution; (ii) for $2+4s/N \lt q \lt p \lt 2_s^*$ and µ > 0, we reveal the existence of the ground state solution, explore the strong instability of standing waves, and provide a blow-up criterion; (iii) for $2 \lt q\leq2+4s/N \lt p \lt 2_s^*$ and µ < 0, the strong instability of standing wave solutions is demonstrated. These findings are illuminated through variational characterizations, the profile decomposition, and the virial estimate.