  A fundamental example of reaction-diffusion system exhibiting Turing type
pattern formation is the Gierer-Meinhardt system, which reduces to the shadow
Gierer-Meinhardt problem in a suitable singular limit. Thanks to its
applicability in a large range of biological applications, this singularly
perturbed problem has been widely studied in the last few decades via rigorous,
asymptotic, and numerical methods. However, standard matched asymptotics
methods do not apply (Ni 1998, Wei 1998), and therefore analytical expressions
for single spike solutions are generally lacking.
  By introducing an ansatz based on generalized hyperbolic functions, we
determine exact radially symmetric solutions to the one-dimensional shadow
Gierer-Meinhardt problem for any $1 < p < \infty$, representing both inner and
boundary spike solutions depending on the location of the peak. Our approach
not only confirms numerical results existing in literature, but also provides
guidance for tackling extensions of the shadow Gierer-Meinhardt problem based
on different boundary conditions (e.g. mixed) and/or $n$-dimensional domains.
