One of the motivations for the elastic net was the following limitation of LASSO:

"In the p > n case, the lasso selects at most n variables before it saturates, because of the nature of the convex optimization problem. This seems to be a limiting feature for a variable selection method. Moreover, the lasso is not well defined unless the bound on the L1-norm of the coefficients is smaller than a certain value." (http://onlinelibrary.wiley.com/doi/10.1111/j.1467-9868.2005.00503.x/full)

I understand that LASSO is a quadratic programming problem but also can be solved via LARS or element-wise gradient descent. But I do not understand where in these algorithms I encounter a problem if p > n where p is the number of predictors and n is the sample size. And why is this problem solved using elastic net where I augment the problem to p+n variables which clearly exceeds p.