Suppose that $(k,l,m) \in \mathbb{N_0}^3$.

If $(k,l,m)=(0,0,0)$ then for $n=1,2$ there is an infinite number of solutions and, by the theorem of of Wiles there are no solutions when $n \geq 3$.

Is it known can there be an infinite number of solutions of $a^{n+k}+b^{n+l}=c^{n+m}$ if $k,l,m$ are not all equal and $n \geq 3$?

What is known of when there is a finite non-zero number of solutions?

Can someone summarize all that is known about this generalized Fermat problem?

We see that a mapping $m: (k,l,m) \to a^{n+k}+b^{n+l}=c^{n+m}$ is a mapping that sends some element of $\mathbb{N_0}^3$ to a family of equations over the integers and because every such equation has its own set of solutions for every $n \in \mathbb N$ we can define some mapping $s$ that sends some element of $\mathbb{N_0}^3$ to a set of all solutions.

A question can then be phrased of when is $s(k,l,m)$ an empty set and when it is not an empty set.

We could probably somehow lower "dimensionality" of this problem by choosing some $(k,l,m)$ and fixing either $a$ or $b$ or $c$ but I am not sure how general results could be obtained in such a way.

Also, I do not know are techniques of Wiles suited for this more general problem, and, if they are not, I do not know why they are not?

It would be nice if someone could explain in non-strict technical sense all that is known about this generalized Fermat problem.

If this is off-topic and written in non-professional way then please vote to close this question of mine.