I know that equations of the form
$$\displaystyle ax^d + by^d = h$$
with $a,b,h \in \mathbb{Z}$ have been thoroughly investigated as a special (and interesting) case of the Thue-Mahler equation, for instance by Evertse.


What about the case when the powers of $x$ and $y$ are unequal? It seems a rather natural question to ask. In particular, can one obtain a similar bound on the number of solutions as in the homogeneous case?


More specifically, in the homogeneous case it was obtained by Evertse in a 1984 paper that the number of solutions is $O\_\epsilon(h^\epsilon)$ for any $\epsilon > 0$.



Reference request: on sums of the form $ax^m + by^n = h$
I assume that you're asking about integer solutions, but of course, if the genus of the associated curve is at least 2, you can also ask for bounds for the number of rational solutions. Anyway, Evertse and I have a paper that deals with $S$-integral points on these curves (and somewhat more general ones):


Evertse, J.-H.; Silverman, J. H., 
Uniform bounds for the number of solutions to $Y^n=f(X)$,
*Math. Proc. Cambridge Philos. Soc.* **100** (1986), no. 2, 237–248. MR0848850


In this paper, the degree of $f(X)$ can be arbitrary, subject to the curve $Y^n=f(X)$ have genus at least one.


There's also the following paper of mine that gives an upper bound in terms of the rank of the Jacobian of the curve. The Catalan curve part is relevant to your question.


Silverman, Joseph H. A quantitative version of Siegel's theorem: integral points on elliptic curves and Catalan curves. *J. Reine Angew. Math.* **378** (1987), 60–100. MR0895285


You might also find the following paper to be of interest. It gives (somewhat) uniform upper bounds for the number of *rational points* on families of curves of genus at least two:


de Diego, Teresa,
Théorème de Faltings (conjecture de Mordell) pour les familles algébriques de courbes. 
*C. R. Acad. Sci. Paris* Sér. I Math. **323** (1996), no. 2, 175–178. MR1402538


