Concentrating Solutions For Some Planar Elliptic Problems With Weighted Nolinearity

Over the past decades, some planar elliptic problems with weighted nonlin-earity have aroused wide concern among the people, such as the singular Liouville equation, the Heron equation and the weighted sinh-Poisson equation. These equations not only have practical background, but also bring some interesting questions to mathematics itself. This thesis will be devoted to studying the existence of concentrating solutions for these equations through the Lyapunov-Schmidt finite dimensional reduction method.First of all, we will briefly introduce the question background, the research method and main results of this thesis in Chapter1.In Chapter2, we use some assumptions of stability of critical points to con-struct multiple interior concentrating solutions for the singular Liouville equation.In Chapter3, we prove the existence of mixed interior and boundary concen-trating solutions for the planar elliptic Neumann problem involving exponential nonlinearity with mixed interior and boundary singular sources.In Chapter4, we use the assumption of C0-stable critical points to give the existence of multiple interior concentrating solutions (or spike solutions) for a two-dimensional elliptic problem with large exponent in weighted nonlinearityIn Chapter5, we prove the existence of mixed interior and boundary con-centrating solutions (or spike solutions) for a two-dimensional elliptic Neumann problem with large exponent in weighted nonlinearity.In the last chapter, we consider the weighted sinh-Poisson equation on the unite disk, and prove the existence of nodal bubbling solutions concentrating positively and negatively at the origin and outside the origin.In order to obtain the above results, we need to carry out the whole re-duction procedure, especially the analysis of the bounded invertibility of the linearized operator L of the problem at the suitable approximate solution. Based on this point, many difficulties in technique need to be overcome. Here our main innovation points are as follows:Firstly, using del Pino et al’s nondegeneracy result of entire solutions of the Liouville equation with a singular source of integer multiplicity, we adopt a new cut-off function concerning with separate polar variables on the whole disk in analysis of invertibility of L with sources of integer multiplicity, which plays a very important role in some related orthogonality integrations and helps us regain the needed strictly diagonal dominant linear system.Secondly, we introduce a suitable weighted L∞-norm related to a "gap" interval to overcome the difficulty produced by the singularities of L at singular sources with coefficients located in the interval (-1,0).Lastly, the presence of boundary singular sources in Neumann problems also brings some technical difficulties. A valid approach exactly helps us carry out the reduction on the half disk by adopting an idea of del Pino and Wei to strengthen the boundary near each boundary source.