题目: Exact multiplicity of the positive solutions of semilinear elliptic equations
摘要: We survey the results about exact multiplicity of positive solutions of a semilinear elliptic equation with zero Dirichlet boundary condition, and the domain is the unit ball or the whole space of n-dimensional space. It is well known that if the nonlinearity is smooth, then any positive solution to the equation is radially symmetric, and the solution set consists of curves. There are numerous work on the positive radially symmetric solutions of the equation in the last two decades. I will describe a unified approach by Tiancheng Ouyang and myself (and also Philip Korman and Yi Li) which determines the exact multiplicity of solutions and classifies the global bifurcation diagrams of the equation according to the convexity conditions on the nonlinearities. Related recent results will also be discussed.
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