I don't know of a totally specific anti-Gram Positive agent, but bile salts have been used often.

Aminoglycosides

These antibiotics target the ribosome and are highly effective against both Gram-negative and Gram-positive pathogens

Colistin or polymyxin B could probably work

Yes, there are bacteriolytic antibiotics that are effective against gram-negative bacteria. These antibiotics work by disrupting the bacterial cell wall, leading to cell lysis and death. One of the main classes of bacteriolytic antibiotics effective against gram-negative bacteria is beta-lactams. Here are a few examples: Carbapenems, Penicillins, Cephalosporins, and Monobactams.

Hello! So, just complementing everyone's answer, specially the one from Kais, many antibiotics have effect in both , Gram positive and negative, however, as Kais said, I would suggest as follow:

1- Carbapenems: almost by excellence, Gram Negative bacteria are sensible to them; such a powerful group that even Pseudomona aeruginosa is more often sensible than to any other group, hence, widely used on it . It also covers Klebsiella, pneumoniae, or spp. finally also Methycilin Resistant Staph aureus but this is Gram positive.

2- Cephalosporines: they mainly cover Gram positive, however, the 3rd and 4th generation of them are excellent covering Gram negative as well (or even have an increase activity against t negative in contrast against positive), excluding intracellular pathogens such as Neisseria gonorrhoeae( the actual ALMOST only treatment against it) they are amazing covering pseudomona, acitenobacter baumanii, klebsiella, an proteus, aside of course of many Gram positive due to their Beta lactamic ring.

3- Amynoglucosides: complementing Sreemoy's answer, these would be the first line of treatment , sometimes by themselves, some others with a Beta lactamic in order to cause synergism between each other, when covering infections by mainly enterobacteiaceae, but also Moraxella (catharralis mainly) dfor example.

4- penicillins and fluoroquinolones : I left intentionally these massive groups at last since they are mainly active against Gram positive, However, they have also a great activity against Gram Negative.

I hope this is helpful; I would suggest to take more info if needed, from "Goodman and Gillman manual of pharmacology" ( a jewel) and / or "Murray Medical microbiology" , these are the two references I took for this answer.

Saludos!