you haven't initialized the variable $user so in the controller and before using it you've to add $user = new User; assuming that you already have use AppUser;
or $user = new AppUser; if you don't.
Update if you're trying to update an existing user record you've to initialize the $user variable by selecting based on it's primary key which should be sent in the request and then doing the initialization like:
$user_id = $request->input('user_id');
$user = User::find($user_id);