On locally finite $p$-groups satisfying an Engel condition

For a given positive integer $n$ and a given prime number $p$, let $r=r(n,p)$ be the integer satisfying $p^{r-1}<n\leq p^{r}$. We show that every locally finite $p$-group, satisfying the $n$-Engel identity, is (nilpotent of $n$-bounded class)-by-(finite exponent) where the best upper bound for the exponent is either $p^{r}$ or $p^{r-1}$ if $p$ is odd. When $p=2$ the best upper bound is $p^{r-1},p^{r}$ or $p^{r+1}$. In the second part of the paper we focus our attention on $4$-Engel groups. With the aid of the results of the first part we show that every $4$-Engel $3$-group is soluble and the derived length is bounded by some constant.