I am a lawyer, and I have, in my own startup, contended with a company attempting to sue me/my startup for allegedly infringing on their trademark.
I'm sympathetic to your cause, but playing with intellectual property issues is serious stuff and the reality of the situation is that better-funded players will not infrequently attempt to bully you into getting what they want, as appears to have occurred here. Obviously hindsight is 20/20, but you should have consulted an attorney immediately on receiving a cease-and-desist letter, even if you thought that their claim was entirely frivolous (which, given the facts presented here, is hard to say). Had this occurred, you almost certainly would either not be in this situation, or saved yourself a ton of trouble.
Given that you did not and this situation has evolved to what it is, your only choice is to retain an attorney if you’d like to continue operate the app and not pay whatever damages they alleged. And, frankly, you're in a poor position because, since the court has already entered a default judgment against you, you will have to show cause as to why the court should set aside this judgment, which will require more time and effort than if you have initially opposed the lawsuit, even if what you say regarding a piece of the document missing is true.
If you do retain an attorney and fight this, realize that, if you lose, the opposing side might be able to recoup the entirety of their attorney's fees (which would be very expensive for you, particularly in the unlikely event you went to trial or the like, and it appears that they have already alleged this right now), and, possibly, your profits (depending on a number of factors). That's very serious stuff.
I wish I could give more direct advice, and I strongly agree with commenters here in other posts lamenting how elitist law is and so on, but you have absolutely no alternative but to retain an attorney. I don't know where you live and where this litigation is taking place, but there are tons of lawyer referral services out there (your local county bar will have such a service, you can find out on their website), and you can speak to various attorneys who might understand your financial situation and be able to work with whatever budget you have. Just because an attorney says that they charged $X does not mean that you can’t find another, competent attorney, who might charge less. If I were in your position, and wanted to fight this, I would spend my time either finding/raising money, or talking to attorneys about how you can structure a fee schedule that would work with your finances (lawyers are open to negotiating: you can say "I can only pay you this or that" or "I can pay you this or that in the future"—see what they say and how flexible they will be). I am skeptical that someone would take your case pro bono.
As a general piece of advice based on personal experience to others: don't ignore intellectual property issues. Getting a trademark is not particularly expensive (and, unlike this situation, something that you can, actually, do without consulting an attorney, or at least for little $ should you need to consult an attorney) and can save you headaches in the future. If you receive a cease and desist-type letter, take it seriously, regardless of how frivolous you think their claim is. If whoever sent you the letter went to the trouble to have their lawyer draft a letter to you--which means that they probably paid the lawyer a non-insignificant fee to do that--they are certainly taking it seriously, and so should you. Even well-funded companies don't throw money away to, of all people, lawyers, unless they think they can get something they want from you, and/or they think they actually have a meritorious claim. Do not attempt to respond to the cease and desist letter, unless you have experience in how to do this. If you want to avoid paying an enormously onerous legal fee, you can try your hand at drafting a response—but absolutely get a lawyer to look at it prior to sending it. This may be more affordable than you think (~1-2 hour of work, or the like, for a lawyer). If you're worrying that you may be infringing someone else's trademark, or someone will try to sue you in the future (as apparently happened here), you can get a declaratory judgment (for a fee) from a court stating that you're not infringing-again, something to talk to a lawyer about.
Good post though I will say even without the power of hindsight consulting a lawyer should've been step one. There's enough big trademark cases to go around in the public eye to know that these things shouldn't be taken lightly.
I'm sympathetic to your cause, but playing with intellectual property issues is serious stuff and the reality of the situation is that better-funded players will not infrequently attempt to bully you into getting what they want, as appears to have occurred here. Obviously hindsight is 20/20, but you should have consulted an attorney immediately on receiving a cease-and-desist letter, even if you thought that their claim was entirely frivolous (which, given the facts presented here, is hard to say). Had this occurred, you almost certainly would either not be in this situation, or saved yourself a ton of trouble.
Given that you did not and this situation has evolved to what it is, your only choice is to retain an attorney if you’d like to continue operate the app and not pay whatever damages they alleged. And, frankly, you're in a poor position because, since the court has already entered a default judgment against you, you will have to show cause as to why the court should set aside this judgment, which will require more time and effort than if you have initially opposed the lawsuit, even if what you say regarding a piece of the document missing is true.
If you do retain an attorney and fight this, realize that, if you lose, the opposing side might be able to recoup the entirety of their attorney's fees (which would be very expensive for you, particularly in the unlikely event you went to trial or the like, and it appears that they have already alleged this right now), and, possibly, your profits (depending on a number of factors). That's very serious stuff.
I wish I could give more direct advice, and I strongly agree with commenters here in other posts lamenting how elitist law is and so on, but you have absolutely no alternative but to retain an attorney. I don't know where you live and where this litigation is taking place, but there are tons of lawyer referral services out there (your local county bar will have such a service, you can find out on their website), and you can speak to various attorneys who might understand your financial situation and be able to work with whatever budget you have. Just because an attorney says that they charged $X does not mean that you can’t find another, competent attorney, who might charge less. If I were in your position, and wanted to fight this, I would spend my time either finding/raising money, or talking to attorneys about how you can structure a fee schedule that would work with your finances (lawyers are open to negotiating: you can say "I can only pay you this or that" or "I can pay you this or that in the future"—see what they say and how flexible they will be). I am skeptical that someone would take your case pro bono.
As a general piece of advice based on personal experience to others: don't ignore intellectual property issues. Getting a trademark is not particularly expensive (and, unlike this situation, something that you can, actually, do without consulting an attorney, or at least for little $ should you need to consult an attorney) and can save you headaches in the future. If you receive a cease and desist-type letter, take it seriously, regardless of how frivolous you think their claim is. If whoever sent you the letter went to the trouble to have their lawyer draft a letter to you--which means that they probably paid the lawyer a non-insignificant fee to do that--they are certainly taking it seriously, and so should you. Even well-funded companies don't throw money away to, of all people, lawyers, unless they think they can get something they want from you, and/or they think they actually have a meritorious claim. Do not attempt to respond to the cease and desist letter, unless you have experience in how to do this. If you want to avoid paying an enormously onerous legal fee, you can try your hand at drafting a response—but absolutely get a lawyer to look at it prior to sending it. This may be more affordable than you think (~1-2 hour of work, or the like, for a lawyer). If you're worrying that you may be infringing someone else's trademark, or someone will try to sue you in the future (as apparently happened here), you can get a declaratory judgment (for a fee) from a court stating that you're not infringing-again, something to talk to a lawyer about.