Concentration Phenomena For Some Singularly Perturbed Elliptic Equations

This thesis is concerned with the concentration phenomena of the solutions for the following singularly perturbed elliptic equationwhere B₁ is the unit ball centered at the origin of R^N, v denotes the unit outward vector to (?)B₁ ,ε> 0 is a parameter, and the nonlinearity f will be assumed to be a superlinear with subcritical exponent in the sense of Sobolev embedding theorem. This equation relates to biology mathematical problems and has many other physical applications. It has been received extensive attention from mathematician in the recent twenty years and many interesting results have been obtained. Along with the research of this problem, one finds that there are rich and interesting structures in it, though it seems very simple. We will construct many solutions of this equation using the so-called "localized energy" method. At the same time, these solutions will concentrate at the prescribed singularities. There are four parts in this thesis:In the first part, we will present the background of the singularly perturbed elliptic problem and list some basic knowledge needed in this thesis.In the second part, we will construct multiple interior peak solutions of the equation with the peaks located on a symmetric axis of the ball B₁ in R^N . More precisely, we will prove that for N ≥ 2, and for any given positive integer K(K ≥ 1) there exists a solution u_ε(x) = u_ε(|x'|,x_n) which is axially symmetric and has exactly K local maximum points Q₁^ε,... Q_k^ε located on the axis of symmetry, when ε > 0 is sufficiently small. Moreover, we can work out the exact location of these K interior maximum points, when ε→ 0.The third part concerns the construction of the solutions of the same equation with the peaks located on a hyperplane. We establish the relationship between the number of the peaks and e. We show that when e is sufficiently small there exists a solution withK interior peaks located on a hyperplane Γ = {x = (x₁,x') ∈ B₁|x₁= 0}, where1 ≤ K≤C/(ε|lnε|)^N-1(N ≥ 3) with C a positive constant depending on N and fonly. As a consequence, we obtain that there exists at least [C/(ε|lnε|)^N-1] solutions for εsufficiently small.Finally, in the fourth part, we will introduce some related topics and open questions that we can study in the future. At the same time, some relative results will be presented if necessary.