We know that the Diophantine equation: $ax + by = c$, has infinitely many integer solutions if $\gcd(a,b)|c$. Now, I am asking about the case where this equation has infinitely many positive integers solutions.
Assuming $a>0$ and $b>0$
If $(x, y)$ is a solution, then the other solutions have the form $(x + kv, y − ku)$, where $k$ is an arbitrary integer, and $u$ and $v$ are the quotients of $a$ and $b$ (respectively) by the greatest common divisor of $a$ and $b$.
Now, $$x+kv≥0$$ and $$y-ku≥0$$ if $$-x/v≤k≤y/u$$
However, this is not able to give infinitely many integers $k$.