RESEARCH

RESEARCH INTERESTS

Semipositone Problems
Superlinear problems on exterior domains.
• $$\left\{ \begin{array}{cl} - \Delta u = \lambda f(u), & x \in \Omega^c, \\ u=0, & x \in \partial \Omega, \\ u \to 0, & \|x\| \to \infty. \end{array} \right.$$
• $f(0)<0$ (semipositone)
• $\displaystyle \lim_{s \to \infty}\frac{f(s)}{s} = \infty$ (superlinear)
• $\Omega$ is a bounded domain in $\mathbb{R}^n$.
• ? existence, uniqueness, multiplicity of solutions
Nonlinear Boundary Conditions
Semipositone problems with nonlinear boundary conditions.
• $$\left\{ \begin{array}{cl} - \Delta u = \lambda f(u), & x \in \Omega^c, \\ \frac{\partial u}{\partial \eta} + c(u) u =0, & x \in \partial \Omega, \\ u \to 0, & \|x\| \to \infty. \end{array} \right.$$
• $f$ is semipositone and superlinear.
• $c:[0,\infty) \to (0,\infty)$ is continuous
• $\Omega$ is a bounded domain in $\mathbb{R}^n$.
• ? existence, uniqueness, multiplicity of solutions
Math Biology
Density dependent dispersal on the boundary
• Modeling habitat surrounded by hostile matrix with nonlinear density dependent dispersal on the boundary using reaction-diffusion equations with nonlinear boundary conditions. (Single PDE)
• Modeling competing species with nonlinear dispersal on the boundary based on density of competitor. (PDE systems)
• ? existence, uniqueness, multiplicity, and stablity of steady states

Dr. Jim Cronin

Louisiana State University

Dr. Jerome Goddard

Auburn University at Montgomery

Dr. Catherine Payne

Winston-Salem State University

Dr. Stephen Robinson

Wake Forest University

Dr. Ratnasingham Shivaji

University of North Carolina at Greensboro

Dr. Inbo Sim

University of Ulsan

Dr. Byungjae Son

Wayne State University

Noam Chomsky