Abstract: In this paper, we focus on semilinear elliptic equations and systems. Using the critical point theory and the modern PDE methods, we obtained a series of interesting results on the existence, multiplicity and the asymptotic behavior of solutions for nonlinear elliptic equations. In particular, we solved or partially solved two open problems in semilinear elliptic equations.

Key words: semilinear elliptic equation, critical point, Hardy singluar term, Palais-Smale sequence