You should be perfectly fine.
A 3.5 GPA isn't that bad to begin with, though it is cutting it close for some grad schools that want 3.5 as the minimum. The upward trend and extenuating medical circumstances though should be more than enough to get any school to overlook a less than optimal GPA, just make sure to explain in your essays. If you have good GRE scores to prove the upward trend, I think you might even be better off than a consistent but slightly higher GPA. The grad schools I've applied to all have an "overcoming adversity" component they want you to address, you have something to talk about for this. Go ahead and apply wherever you want, 3.5 is fine for most grad schools and your publications, research experience, and letters of recommendation will be more important for the more competitive programs.