How to grow vegetables on a patio garden?

We live in an apartment complex, and I have always wanted to grow vegetables. The problem is, the fence is tall and we have a huge sycamore overhead, so we barley get any light. The only place we have light is on one fence railing (that's where I have been growing a couple of radishes in a small pot) but it isn't very big and can't hold much weight. We also have a dirt patch surrounding the patio (about 12 feet long and 2 feet wide) but it gets no sun. Other than radishes, I also have two fava bean plants growing and they are now taller than me, but combined, they only have 4 bean sprouts. Is there any way I can grow some veggies on my patio? I have thought about upside down tomato planters, but I don't know if they will work when you ony have flickers of shade throughout the day in only some places that are higher up. Oh yeah, and I'm also starting carrots in a place that has about 4 hours of sun... That's the most we get and it's tiny. So yeah, is there any way I can grow veggies on my patio??? HELP PLEASE.