Any interior solution for $u(x,y) = min\left \{ x,y \right \}^{2} + max\left \{ x,y \right \}$?
Because the two segments of the IC bend away from the origin, the only possible solutions are at the corners or at $x=y$.

Write $p_x$ and $p_y$ for the prices, and $M$ for the budget.

Suppose we try to set up a corner solution in which only one of the two goods is consumed. Since utility is symmetric, it makes sense to spend all of your money on whichever good is cheaper. Suppose this is good $x$. We then have $x=\frac{M}{p_x}&gt;0=y$ and $u=x$.

The best $x=y$ bundle that can be afforded is such that $x p_x+x p_y=M\iff x=y=\frac{M}{p_x+p_y}$. Utility is then
$$\frac{M^2}{(p_x+p_y)^2}+\frac{M}{p_x+p_y}.$$

You should easily be able to compute a condition on $M$ such that the bundle with $x=y$ is optimal: $$M&gt;\frac{p_y(p_x+p_y)}{p_x}.$$

You can see this dependence on M if we look at a contour plot of the indifference curves: