- Never give up on your hope, whatever hardships you face.
- Never give up your hope whatever hardships you face.
What's the difference between the two sentences in meaning? Are both 2 grammatically correct?
Here are some more idiomatic expressions:
While the original sentences you mentioned make sense, they don't sound like the most standard English.
To give up X (or give X up) is to surrender or abandon it. (X can be a habit: “I gave up sugary drinks long ago.”)
To give up on X is to abandon expectation that X will have a good result. “I gave up on that project; it was going nowhere.” “My parents gave up on me after my second time in prison.”