  In this paper, we study the bound state analysis of a one dimensional
nonlinear version of the Schr\"{o}dinger equation for the harmonic oscillator
potential perturbed by a $\delta$ potential, where the nonlinear term is taken
to be proportional to $\delta(x) |\psi(x)|^2 \psi(x)$. The bound state wave
functions are explicitly found and the bound state energy of the system is
algebraically determined by the solution of an implicit equation. Then, we
apply this model to the Bose-Einstein condensation of a Bose gas in a harmonic
trap with a dimple potential. We propose that the many-body interactions of the
Bose gas can be effectively described by the nonlinear term in the
Schr\"{o}dinger equation. Then, we investigate the critical temperature, the
condensate fraction, and the density profile of this system numerically.
