I have a nonlinear bi-variate optimization problem like $\max \: f(x,y)$ where $f(x,y)$ is a nonlinear and differentiable function of both variables, and $0\le x\le 1$, $\:0\le y \le ub$. In order to find the optimal solution, can I find the stationary points, $\nabla f(x,y)=0$, and boundary points and evaluate the objective function at the corners, such as $(0,0), (0,ub), (1,0),(1,ub)$ points and stationary points $\nabla f(x,y)=0$ (KKT points)? The highest order in the function is 3 and in form of $xy^2$ or $yx^2$?
The KKT conditions are necessary conditions for an optimum to your problem, so if you can find all feasible points satisfying them, the one with the best objective function will be your optimum. There is no need to consider the corners of the feasible region explicitly. If any of them is optimal, it will also be a KKT point.