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

Louisiana State University

Auburn University at Montgomery

Swarthmore College

Winston-Salem State University

Universidad de Concepcion

Wake Forest University

University of North Carolina at Greensboro

University of Ulsan

Wayne State University

*Noam Chomsky*