I was wondering if the equation
$x^p+y^p=2z^p$
has been studied. For small cases it is seen that the only solutions are trivial: $x=y=z$. There are probably methods to solve this for regular primes, but in the general, maybe we shall consider an elliptic curve as in FLT?
On
This equation was solved in a paper of Henri Darmon and Loic Merel in Crelle's journal in 1997. The techniques involved are similar to those in Wiles' proof, with additional complications due to the fact that the trivial solution $x=y=z=1$ actually corresponds to a modular form of weight $2$ and level $32$.
The Diophantine equation $Ax^n+By^n=Cz^n$ for coprime integers $(A,B,C)$ is closely related to FLT, and has been studied, too. There are choices for $A,B,C$ and $n=p$, such that there are indeed some nontrivial solutions, e.g., $$ x^5+13y^5=8z^5 $$ has the nontrivial solution $(x,y,z)=(3,1,2)$. But usually there are very few solutions. For an overview see the article On the number of solutions to the generalised Fermat equation by Andrew Granville.
In addition, even a more generalized Fermat has been studied, i.e., $$ Ax^p+Bx^q=Cx^r. $$ Here several cases are still not known. Even for $A=B=C=1$,the hyperbolic case, i.e. $\frac{1}{p}+\frac{1}{p}+\frac{1}{p}<1$ is only known conjecturally. For a very nice overview see the article The ABC's of Number Theory by Noam D. Elkies.