Concentration Phenomena For Equations With Supercritical Nonlinearity

In this thesis, we study the concentration phenomena for semilinear elliptic problems with supercritical nonlinearity. In the first part, we construct positive solutions of the semilinear elliptic problem Δu + λu + u^p = 0 with homogeneous Dirichlet boundary conditions in a bounded smooth domain Ω C R^N (N ≥ 4), when the exponent p is supercritical and close enough to N+2/N-2 and the parameter λ∈R is small enough. As p →N+2/N-2 and λ→0 at some appropriated rate, these solutions have "multiple bubbles" at finitely many of points which are the critical points of a function whose definition involves the Green's function of —Δ with homogeneous Dirichlet boundary condition. Our result in this part extends the result of Del Pino, Dolbeault and Musso in [DDMl] when is a ball and the solutions are radially symmetric.In the second part, we construct positive solutions to the same supercritical equation for N = 3, that is with p = 5 + ε. When the parameter A is a positive constant in some range and ε→0⁺, we also construct a sequence of positive solutions with "multiple bubbles" at a finite number of points.Finally, we consider the problem Δu + /u/^p1u +ε^1/2f(x) = 0 in a bounded domain Ω? C R^N (N ≥ 3), and u = ε^1/2 g on the boundary of Ω, where ε> 0 and p = (N+2/N-2) + ε. Here the functions f ∈ C^0,a(?) and g∈C^2,a(αΩ) for some a ∈ (0,1). We will conclude that the problem admits "multiply bubbles" solutions with finite singular points whose positions are related to the Green's function and the functions f and g. We give some applications of the result in the symmetric domains .