$string = 'john  doe';

// split on 1+ whitespace & ignore empty (eg. trailing space)
$searchValues = preg_split('/s+/', $string, -1, PREG_SPLIT_NO_EMPTY); 

$users = User::where(function ($q) use ($searchValues) {
  foreach ($searchValues as $value) {
    $q->orWhere('name', 'like', "%{$value}%");
  }
})->get();