Uniqueness of minimum norm solution

Question

Let's state a regularized least squares problem $min_{x\in R^N}\ \ \Vert y-Ax \Vert _2 ^2\ \ +\ \delta\Vert x \Vert _2 ^2$ has a unique solution for every $\delta > 0$, and when $\delta\to0$, the regularized solution goes to the minimum norm solution of $min_{x\in R^N}\ \ \Vert x \Vert _2 ^2\ \ subject\ to\ A^TAx = A^Ty$. $A$ is a $M * N $ matrix with $rank(A) < N$. How do we prove that the minimum norm solution is always unique?