The Study On Some Problems On Positive Solutions Of Elliptic Equations And Systems

In this thesis, we deal with Dirichlet and Robin boundary value problems ofsecond order semilinear elliptic equations and systems. Some results on existence,uniqueness, multiplicity and non-degeneracy for positive solutions are demonstratedby employing blow-up technique, variational theory,fixed point theorem, sup-subsolution theory, a priori estimate and asymptotic analysis.As an introduction, in chapter one, the background and main results are brie?yaddressed. The general notations, lemmas and outline of this work are also givenin this chapter.In chapter two, we study the existence and uniqueness of positive solutionsof the general second order elliptic equations with Dirichlet boundary condition.The generally used variational theory can not be applied to prove the existenceof solutions when the problems have no variational structure. In addition to asu?cient and necessary condition for the existence of positive solutions is givenby using blow up technique and fixed point theorem, we also prove that if theuniqueness and non-degeneracy results are valid for positive solutions of a classof semi-linear elliptic equations, then they are still valid when one perturbs theoperator a little bit. As a consequence, some uniqueness results of positive solutionsunder the domain perturbation are also obtained.As Robin boundary value problems are concerned, most of mathematiciansthink that they are similar to Dirichlet boundary value problems. In fact, somerecent research works imply that these two kinds of problems are di?erent in manyaspects. Though the existence and a priori estimate of solutions for Robin bound-ary value problem are easy to obtain, the results on uniqueness and symmetryof positive solutions are very di?cult to demonstrate. We study three models ofRobin problems in chapter three, chapter four and chapter five respectively for theuniqueness and multiplicity of positive solutions.In chapter three, the uniqueness of positive solution to a second order ordinarydi?erential equations with Robin boundary condition is studied. The most powerfultool used to prove uniqueness results for Dirichlet problem of second order ordinarydi?erential equations is the so-called time mapping method. But the time mappingis not valid for general Robin problem because of the loss of symmetry. At first we proved the non-degeneracy and uniqueness of positive solution for the homogeneousRobin boundary value problem. Then, as an application of the non-degeneracy anduniqueness results, we gave a necessary and su?cient conditions for the existenceof one or two positive solutions for model one with the non-homogeneousity changesign, and for model two with concave-convex nonlinearities.In chapter four, we devote to prove uniqueness results of positive solutionsto Robin boundary value problem of semilinear elliptic equations and systems ongeneral region in n-dimension space. Since the moving plane method, which isusually applied to prove the symmetry and uniqueness of solutions for Dirichletboundary value problems on some symmetry domain, can not be used, the methodwe use is a priori estimate combining with the asymptotic analysis of solutions.Compared with the uniqueness results for Dirichlet boundary value problems, themain feature of our results is that there is no special request on the domain.In the last chapter, we consider the Robin boundary value problems ??u =f(u) on annulus in n-dimension space. Under some assumptions, we prove thatthe problem has at most one solution whenβ,the parameter in the boundarycondition, is small enough, whereas, it has at least k non-radial solutions whenβis large enough. This result gives another example which implies that Robinboundary value problems may be much di?erent from Dirichlet boundary valueproblems.