Convexity domain of x cubed plus y cubed plus Axy
= Convexity domain of x cubed plus y cubed plus Axy
The <Hessian matrix> of
$$
f(x,y)=x^3+y^3+Axy
$$
is positive semidefinite exactly when
$$
x\geq0,\qquad y\geq0,\qquad 36xy\geq A^2.
$$
This region is convex: for $A\ne0$ it is the epigraph of $A^2/(36x)$ in the positive quadrant, and for $A=0$ it is the closed first quadrant.