Chee Meng Tan

Harvey Mudd College Mathematics 2007

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Thesis Proposal: Radial Solutions to Superlinear Dirichlet Problem
Thesis Advisor: Prof. Alfonso Castro
Second Reader: Prof. Jon Jacobsen
Thesis Report: Thesis

Radial Solutions to Superlinear Dirichlet Problem

My study of the semilinear Dirichlet problem has its beginnings in the summer of 2005, as a three person research problem, which included Professor Alfonso Castro of the Harvey Mudd College mathematics department, graduate student John Kwon of the Claremont Graduate University and myself. The partial differential equation that we have studied is of the type,

$\displaystyle \left \{\begin{array}{ll} \Delta u+g(u(x))=h(\Vert x\Vert) \:, &... ... u(x)=0\:, & \mbox{$x\in B(0,1)\subseteq \mathbb{R}^N$}, \end{array} \right.$

whereby $ g(u)=u^p$ for $ u<0$, and $ g(u)=u^q$ for $ u>0$ with $ p<\frac{N+2}{N-2}$ and $ q>\frac{N+2}{N-2}$. It must be mentioned that our results pertain to radial solutions in $ B(0,1)$, and that there may well be radial solutions that we have not found.

In previous years similar studies of the semi-linear Dirichlet problem have been made, but the non-linearity $ g(u)$ is either $ u^p$, $ u^q$ or $ u^\frac{N+2}{N-2}$. Our studies confirm that even with different non-linearities for $ u<0$ and $ u>0$, infinitely many solutions exist to this problem.