Getting a wrong output using arraylists

You can use the following:

static List<Integer> sumDigPow(int a, int b) {
    List<Integer> eureka = new ArrayList<Integer>(0);
    String num;
    int sum = 0, multi;

    for (int i = a; i <= b; i++) {
        num = String.valueOf(i);
        for (int j = 0; j < num.length(); j++) {
            multi = (int) Math.pow(Character.getNumericValue(num.charAt(j)), j + 1);
            sum += multi;
        }

        if (sum == i) {
            eureka.add(i);
        }
        sum = 0;
    }
    return eureka;
}

Explanation:

1. You were not checking the second digit of the number.
2. Loop over each character of the String num.
3. There is no need of the digits arraylist, you can just use the numeric value of the char.

The problem were those lines :

num = String.valueOf(i);
digits.add(num);

You did not split your number into digits. You were just putting your whole numbers into digits list. Look at this code :

static List<Integer> sumDigPow(int a, int b) {
        List<Integer> eureka = new ArrayList<Integer>();
        List<String> digits;
        String num;
        int sum = 0, multi;

        for (int i = a; i <= b; i++) {
            num = String.valueOf(i);
            digits = Arrays.asList(num.split(""));

            for (int j = 0; j < digits.size(); j++) {
                multi = (int) Math.pow(Integer.parseInt(digits.get(j)), j + 1);
                sum += multi;
            }

            if (sum == i) eureka.add(i);

            sum = 0;
        }

        return eureka;
    }

I simply split your string number into digits using Arrays.asList(num.split("")). It outputs for a=1, b=100 the list :

1
2
3
4
5
6
7
8
9
89