The ultimate sparseness occurs when $Ax^*(\lambda)=0$, which is the case when the minimizer $x^*$ is the projection of $b$ onto $\ker A$. For this to happen, $\lambda$ must be small enough so that the restriction of $A$ to the orthogonal complement of its kernel is bounded from below by a constant greater than $2\lambda \mathrm{dist}(b,\ker A)$. Here the lower bound for operator is understood in the $\ell^2\to\ell^1$ norm.