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

Wayne State University

University of North Carolina at Greensboro

Universidad de Concepcion

Auburn University at Montgomery

Louisiana State University

Wake Forest University

University of Ulsan

Winston-Salem State University

*Noam Chomsky*