一类超线性Dirichlet问题无穷多个径向对称解的存在性

应用常微分方程的能量分析法和相平面分析法证明了球上一类超线性Dirichlet问题存在无穷多个径向对称解.首先将所研究的问题转化为常微分方程,进而利用压缩映射原理证明常微分方程问题存在解,从而得到原问题存在无穷多个径向对称解.这一结果对某些不满足PS序列紧性条件和超出Sobolev嵌入定理临界指数的非线性增长条件仍然成立,并给出具体实例,说明了采用这种方法研究问题的优势.

Existence of Infinite Radially Symmetric Solutions for Superlinear Dirichlet Problem

A radially symmetric superlinear Dirichlet problem was studied by using the energy analysis and the phase plane analysis.The problem was turned into a boundary value problem of ordinary differential equation.The contraction mapping principle was used to verify the existence of solutions of the differential equation and thereby to prove the existence of infinite radially symmetric solutions.The same result can be still obtained when the growth of the nonlinearity surpasses the critical exponent of the Sobolev embedding theorem or does not satisfy the Palais Smale condition.Some examples were given to show the advantage of these methods.