Non-stop "Upper Respiratory Infection" for 5 Months - HELP

We got Gulliver when he was about 10 weeks old back in November. We believe he is some sort of a Maine-coon cross since he so fluffy and big, with a HUGE tail. Shortly after we got him he began to have eye discharge started sneezing a lot. We took him to the vet for vaccinations and they told us he had an upper respiratory tract infection. They gave us antibiotics and eye cream. It seemed to help a little bit at first, but shortly after the symptoms came back. We took him back to the vet again only to be given the same eye cream and to be told that they could not do anything about it. Apparently sometimes these infections are chronic. It is worth noting that we have another cat, Ellie, who contracted the infection from Gulliver in late November and was better within a week. It just doesn't make sense. 

Gulliver's eye discharge is brown and crusty, I clean his eyes with a warm cloth everyday but it doesn't seem to help. He sneezes, but he has no nasal discharge other than that and does not seem to wheeze or snort. He is also neutered and is vaccinated for everything other than upper respiratory tract infection because they won't do it when he's sick and he's always sick. He is active, playful and has a very healthy appetite. However, if he runs around too much with his sister Ellie he seems to have trouble breathing and pants like a dog. This worries me. 

I cannot afford any more fruitless vet trips. It's obvious that what the vet is giving us is not working for Gulliver. My boyfriend and I are both students and as such have very limited funds. I need advice on what I can do to help our Gully, what medicines I should request from the vet, and if there could be an underlying problem. Should I go to a different vet for a second opinion? Or is Gulliver just going to have to live with this cold for the rest of his life? 

Any advice and tips on dealing with upper respiratory tract infections are welcome. 

It really sounds like he might have airborne allergies at this point in time.  I don't read anything that indicates an upper respiratory infection.  I have one who gets brown crusties (I call them eye boogers) from time to time and I just pick them off with my fingernail.  As long as the eye discharge isn't yellow or green, I think you're okay. Is the discharge actively seeping out brown or is it dried on and brown? 

I'm actually thinking Feline Herpes is the likely case here, especially since your other cat contracted it. The virus can present different ways in different cats depending on that particular cat's immune system.

Antibiotics will help with secondary infection, but will not treat the virus itself.

Famciclovir (an antiviral) and L-lysine can help. As well as the eye drops Idoxuridine and Cidofovir.

I would guess the eye ointment your vet prescribed was Terramycin?

Has your vet ever done an eye stain on Gulliver?