Given a closed immersed solution to an elliptic equation, when is it possible to perturb it through embedded solutions? I will discuss several constructions related to this general question, indicate the methods used, and describe a general conjecture about when this should be possible. A motivation for this type of question is a solution to the question of Yau: Which three manifolds contain infinitely many minimal surface? Recent work of Neves-Marques answers this question in the case of positive Ricci curvature. However, even in this case, one can refine Yau's question by restricting topology, volume, etc. Desingularization methods can shed light one these questions.