Multi-solitons for the nonlinear Schrödinger equation with repulsive Dirac delta potential

We prove the existence of multi-soliton solutions for the nonlinear Schrödinger equation with repulsive Dirac delta potential and $L^2$-supercritical focusing nonlinear term. Our main contribution is to treat the unmoving part of the multi-solitons, which is the ground state of the equation. The linearized operator around it has two unstable eigenvalues. This is the main difference from NLS without potential, whose existence of multi-solitons is investigated by Côte, Martel, and Merle (2011).