I am not sure how is $\ell_q$-LASSO called, but here I am talking about LASSO regression, with $\| \beta \|_{\ell_q}$ regularization, $q< 1$. In popular literature, such as Elements of Statistical Learning or Statistical Learning with Sparsity by Hastie et. al., I am able to find only their rough definitions with some graphs, and comments about how their non-convexity makes estimating it troublesome and computationally inefficient.
However, I am interested in a deeper discussion of the approach, such as its Oracle Properties, asymptotic behaviour regarding variable selection and parameter estimation, and various cases when the method doesn't work. So far I've tested simple cases by hand in Monte Carlo experiments and find that it is able to outperform $\ell_1$ LASSO in various cases by a significant margin, hence I assume there should be strong properties at work there. That is, I'd like to understand why it works well for me.
My questions:
Does this method have an official name for easier literature search?
If you have any links to papers with proved properties, they would be very useful for further read.
I am also very interested in counterexamples, that would be able to break the method. (Can't construct them by myself, as I wasn't able to find assumptions, under which, say, $\ell_q$-lasso would guarantee consistent variable selection). For example, with $\ell_1$-lasso, it is known that significant multicorrelation can easily break consistent variable selection. At the same time, I find $\ell_q$-lasso being able to deal with same examples without problems.
Note: I've mentioned variable selection in the questions only as an example, other properties are also of interest.