The Concentration Phenomena Of Solutions For Some Semilinear Elliptic Equations And Systems

This dissertation is concerned with the concentration phenomena of solutions for some semilinear elliptic equations and systems. There are four chapters in this thesis.In Chapter One, we present the background of the related problems, and state the main results of this thesis.In Chapter Two, we consider the following Henon-like equation where Ω={x∈RN:1<|x|<3} with N≥4,2*=2N/(N-2), r>0and ε>0is a small parameter. We show that for any given k∈Z+, there exists a bubble solution concentrating simultaneously at2k different points for ε sufficiently small, among which k points are near the interior boundary{x∈RN:|x|=1} and the other k points are near the outward boundary {x e RN:|x|=3}. Moreover, the2k points tend to the boundary of Ω as ε goes to0.In Chapter Three, we study the following nonlinear Schrodinger equation where ε>0is a small parameter, N≥3and1<p<(N+2)/(N-2), V(x) is a nonnegative potential with compact support. For arbitrary positive integer k>1, we construct higher energy solutions with exactly k peaks which interact with each other and cluster around a local maximum point of V when ε is sufficiently small. The main part of the solutions decays exponentially but the perturbation part of them decays algebraically at infinity.Finally, in Chapter Four, we consider the existence of spike vector solutions for the nonlinear Schrodinger system where ε>0is a small parameter, P(x) and Q(x) are positive potentials, μ and v are positive constants and β≠0is a coupling constant. We investigate the effect of potentials and the nonlinear coupling to the solution structure. For any positive integer k E Z+, we construct a solution with k interacting spikes concentrating near the local maximum point x0of P(x) and Q(x) when P(x0)=Q(x0) in the attractive case. In contrast, we construct a solution with k interacting spikes near the local maximum point xo and x0of P(x) and Q(x) respectively when x0≠x0, moreover, spikes of u and v repel each other. Meanwhile, we prove the attractive phenomenon for β<0and the repulsive phenomenon for β>0.