*Published Paper*

**Inserted:** 14 jun 2002

**Last Updated:** 17 dec 2003

**Journal:** Comm. Pure Appl. Anal.

**Volume:** 2

**Pages:** 323-353

**Year:** 2003

**Abstract:**

We investigate the existence of global solutions for the two-body problem, when the particles interact with a potential of the form ${1/r^\alpha}$, for $\alpha >0$. Our solutions are pointwise limits of approximate solutions $u_\alpha(\epsilon_k,\nu_k)$ which solve the equation of motion with the regularized potential ${1/(r^2+\epsilon_k^2)^{\alpha/2}}$, and with an initial condition $\nu_k$; ${(\epsilon_k,\nu_k)}_k$ is a sequence converging to $(0,\nu ^*)$ as $k$ tends to infinity, where $\nu^*$ is an initial condition leading to collision in the non-regularized problem. We classify all the possible limits and we compare them with the already known solutions, in particular with those obtained in a paper by McGehee using branch regularization and block regularization. It turns out that when $\alpha > 2$ the double limit exist, therefore in this case the problem can be regularized according to a suitable definition.

**Keywords:**
two-body problem, binary collisions, regularization