# 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. Catherine Payne

Winston-Salem State University

#### Dr. Ratnasingham Shivaji

University of North Carolina at Greensboro

#### Dr. Inbo Sim

University of Ulsan

#### Dr. Byungjae Son

Wayne State University

Noam Chomsky