Radially symmetric internal layers in a semilinear elliptic system

Abstract:

Let $B$ denote the unit ball in ${R^N},\quad N \geqslant 1$. We consider the problem of finding nonconstant solutions to a class of elliptic systems including the Gierer and Meinhardt model of biological pattern formation, \[ (1.1)\qquad {\varepsilon ^2}\Delta u - u + \frac {{{u^2}}} {{1 + k{u^2}}} + p = 0\quad {\text {in}}B,\] \[ (1.2)\quad D\Delta v - v + {u^2} = 0\quad {\text {in}}B,\] \[ (1.3)\quad \frac {{\partial u}} {{\partial n}} = 0 = \frac {{\partial v}} {{\partial n}}\quad {\text {on}}\partial B,\] where $\varepsilon$, $D$, $k$ and $\rho$ denote positive constants and $n$ the unit outer normal to $\partial B$. Assuming that the parameters $\rho$, $k$ are small and $D$ large, we construct a family of radially symmetric solutions to (1.1)-(1.3) indexed by the parameter $\varepsilon$, which exhibits an internal layer in $B$, as $\varepsilon \to 0$.