The Dirichlet Problem for Singularly Perturbed Elliptic Equations

There has been much work on various singularly perturbed partial differential equations or systems. Such equations or systems depend on some small parameters ε > 0, solutions denoted as uε. There are at least two types of questions being investigated. The first type is to study possible behavior of uε as ε tends to zero. The second is to actually construct, by various methods, such solutions. In this paper we mainly present some results of the second type. We will study some specific singularly perturbed partial differential equations. However, the methods we used are useful in studying other problems as well. Let Ω ⊂ Rn be a bounded domain with smooth boundary. We consider { −ε2∆ũ+ ũ = ũq , ũ > 0 , in Ω , ũ ∣∣ ∂Ω = 0 , (0.1)