How to handle push notifications if the application is already running?

You can implement application:didReceiveRemoteNotification:

Here is a possible sample code:

- (void)application:(UIApplication *)application
   didReceiveRemoteNotification:(NSDictionary *)userInfo
{
  NSString *message = nil;
  id alert = [userInfo objectForKey:@"alert"];
  if ([alert isKindOfClass:[NSString class]]) {
    message = alert;
  } else if ([alert isKindOfClass:[NSDictionary class]]) {
    message = [alert objectForKey:@"body"];
  }
  if (alert) {
    UIAlertView *alertView = [[UIAlertView alloc] initWithTitle:@"Title"
                                       message:@"AThe message."  delegate:self
                             cancelButtonTitle:@"button 1"
                             otherButtonTitles:@"button", nil];
    [alertView show];
    [alertView release];
  }