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

Abstract We prove the existence of multi-soliton solutions for the nonlinear Schrödinger equation with repulsive Dirac delta potential and $$L^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -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 et al. (Rev Mat Iberoam 27:273–302, 2011).