Concentration Phenomena Of Solutions To Singular Perturbed Elliptic Equations

This dissertation is devoted to the study of concentration phenomena of solutions to some singular perturbed problems. These phenomena often arise in the research of geometry, physics, chemistry, biology and so on, which are very interesting.In Chapter 1, we will first introduce the background and the method of this dissertation, and then state the main result.In Chapter 2, we will consider the following Liouville-type equation with singular sourceswhereδ_pi is the Dirac measure at pi. The singular itemsδ_pi lead to different characters of solutions at pi. Using "localized energy method", we construct a family of solutions, which concentrate at every pi and some other points ofΩ, forming lots of bubbles. Note that all the bubbles here are isolated, and the concentration phenomena at pi are different from those at other points.Chapter 3 is a generalization of Chapter 2. We consider the following anisotropic Emden-Fowler equation with a singular sourcewhere a(x) > 0 is a smooth function overΩ. By construction, we prove that if the singular source p is a local strict maximum point of a(x), then there exists a family of solutions with concentration phenomena at p. But now along withε→0, all the bubbles are close to each other and the corresponding points are concentrating to p.In Chapter 4, we consider the following non-autonomous singularly perturbedNeumann problemwhere the exponent p is sub-critical, a(x) is a positive smooth function onΩ. We prove that there exists a family of solutions, which shows interior concentration phenomena at a local minimum point of a(x). Moreover, we establish the relationship between the number of the interior peaks and the parameter e.Finally, we try to rise some questions to the related problems.