$$\begin{array}{ll} \text{maximize} & \alpha x_1 + x_2\\ \text{subject to} & p_1 x_1 + p_2 x_2 = y\end{array}$$ where $\alpha\geq 0$
I've derived the maxiimizers
$$x_1^* = {(\frac{{\alpha} {p_2}}{2p_1}})^2 $$
and
$$x_2^* = {\frac{y}{p_2}}-{\frac{p_2}{p_1}}{(\frac{{\alpha} }{2}})^2 $$
However, from the graph I sketched I can see that the utility function can potentially touch the axes so there could be corner solutions, but I'm not sure under what circumstances they exist.
Can someone please explain it to me.