About the solutions of the equation $ay^p-bx^q=k$

Question

For every combination of integer numbers $a,b,k$ is it possible to find at least one solution of the equation: $$ay^p-bx^q=k?$$ I know that in 1976 Michel Langevin proved that the problem is decidable and he showed it exists an upper bound for the values of $x$ and $y$. Has any progress been made since then? Thanks.