give up 后可以加不定式吗？

可以啊

give up 的各种用法
To surrender:
1 投降：
The suspects gave themselves up.
这些嫌疑犯投降了
2 To devote (oneself) completely:
将（自己）完全奉献：
gave herself up to her work.
献身于她自己的工作
3 To cease to do or perform:
停止做或执行：
gave up their search.
停止他们的搜查
4 To desist from; stop:
停止；结束：
gave up smoking.
戒烟
5 To part with; relinquish:
让出；放弃：
gave up the apartment; gave up all hope.
让出公寓；放弃所有的希望
6 To lose hope for:
对…失去希望：
We had given the dog up as lost.
最后我们当那只狗已经走失了而不再存有指望
7 To lose hope of seeing:
没有希望见到：
We'd given you up an hour ago.
一个小时前我们就以为见不到你了
8 To admit defeat.
承认失败
9 To abandon what one is doing or planning to do. Often used with on :
中止：放弃某人正在做的或打算做的。常与on 一起用：
gave up on writing the novel.
放弃写小说了

give up doing sth~(是最常用的）

不行，要+doing sth