Concentration Phenomena Of Solutions For Some Elliptic Equations In 2-Dimension

The main pursuit of this dissertation is to study concentration phenomena of solutions for some elliptic equations in 2-dimension. Concentration phenomena arise from many fields, such as geometry, electrochemical, hydrodynamic turbulence, geophysical flows and so on. So, it is important for us to understand further this phenomena. The main research results are as follows:First, in chapter 2, we investigate the elliptic equation -div(a(x)â–½u) + a(x)u = 0 posed on a bounded smooth domain Ω in R² with nonlinear Neumann boundary condition , where ε > 0 is a small parameter. We show that if a family of solutions u_ε satisfy that is bounded, then it will develop up to subsequences a finite number of bubbles , in the sense that with m,k_i∈Z⁺. Location of blow-up points is characterized in terms of function a(x). Reciprocally, in chapter 3, we established that there exists a solution u_ε for which εe^u develops (up to subsequences) m boundary Dirac deltas with weight 2Ï€, which verifies the existence of large energy solutions to the equation in chapter 2.Next, we consider, in chapter 4, the semi-linear equation Δu + ε²(e^u — e^-u) = 0, which is a steady states of two-dimensional vortices in an inviscid fluid, posed on a bounded smooth domain Ω in R² with homogeneous Neumann boundary condition, where ε > 0 is a small parameter. We show that for any given positive integer k, there exists a family of solution u_ε for ε sufficiently small, for which with the propertynamely which change sign and concentrate positively and negatively at different points ζ₁,…,ζ_2k in Ω where (ζ₁, …, ζ_2k) are critical points of some functional defined explicitly in terms of the associated Green's function.Finally, we consider the following anisotropic sinh-Poisson equationdiv(a(x)â–½u) + ε²a(x)(e^u - e^-u) = 0 in Ω, u = 0 on ,where Ω(?)R² is a bounded smooth domain and a(x) is a positive smooth function. We investigate the effect of anisotropic coefficient a(x) on the existence of bubbling solutions.We show that there exists a family of solution u_ε concentrating positively and negatively at given local minimum points of a(x) for e sufficiently small, for which with the propertyε²a(x)(e^uε - e^-uε)→0.This result shows a striking difference with the isotropic case (a(x) ≡ Constant) in [15] and [73].