We establish asymptotic formulas for all the eigenvalues of the linearization problem of the Neumann problem for the scalar field equation in a finite interval

\[ \begin{cases} \varepsilon^2u_{xx}-u+u^3=0, & 0< x<1,\\ u_x(0)=u_x(1)=0. \end{cases} \]

$\varepsilon ^2u_{xx}+u-u^3=0$

$-3$

$0$

$-3$

$0$

We construct countably infinitely many non-radial singular solutions of the problem

of the form

where v(σ) depends only on σ ∈𝕊N−1. To this end we construct countably infinitely many solutions of

using ordinary differential equation techniques.

We consider the activator-inhibitor Gierer–Meinhardt reaction-diffusion system of biological pattern formation in a closed bounded domain. The existence and stability of a boundary apike-layer solution to the Gierer–Meinhardt model, and it, so-called shadow limit, is analysed. In the limit of small activator diffusivity, together with a large inhibitor diffusivity, an equilibrium boundary spike-layer solution is constructed that concentrates at a non-degenerate critical point P of the boundary. By non-degenerate we mean that every principal curvature of the boundary has a local maximum at P, and hence the mean curvature at the boundary has a local maximum at P. Rigorous results for the stability of such a boundary spike-layer solution are given.