Learn English – Is solve correct in “solve a dilemma”

The sentence presented calls for the use of the to-infinitive "to solve".

When you refer to the function or purpose of something, you use either a to-infinitive or for + -ing form. For example:

This heater is to keep/for keeping the plants warm.

When you say why you do something or, in other words, refer to your intention or purpose of doing something, you use the phrase for + -ing form. For example:

I turned on the heater to keep the plants warm.

In the sentence in question, the OP states why they need to do more recycling, They need to do more recycling in order to solve this problem. Moreover, the sentence indicates a specific action or situation, not an action in general.

So you should use the to-infinitive (to solve) in the sentence.

Using "Solved by" expects a description of the method or the person who can solve it.

This problem can be solved by using Pythagoras’ theorem.

This problem can be solved by a 5-year-old

If you are not describing the method "solved in" is better

This problem can be solved in 2 ways

This problem can be solved in 5 minutes