Interior-Layer Peaks in Singularly Perturbed Elliptic Problems

Definitions and mathematical statement

Let $\Omega\subset{\mathbb{R}}^m$ be a bounded smooth domain and consider, for $\varepsilon>0$ tending to zero, \[ -\varepsilon^2\Delta u=u(u-a(x))(1-u)\quad\text{in }\Omega, \qquad \lim_{x\to\partial\Omega}u(x)=+\infty. \] The latter is an explosive boundary condition, not finite Dirichlet data; it applies for each fixed positive $\varepsilon$. The space dimension and the regularity/range of $a:\Omega\to{\mathbb{R}}$ are not specified in the restored statement.

The construction requested is a family of solutions with localized peaks in addition to an interior-transition-layer background. Schematically, if $U_\varepsilon$ is a layer solution, one seeks a solution $u_\varepsilon$ for which $u_\varepsilon-U_\varepsilon$ has one or more localized peak profiles while the interior layer persists. This is a nonlinear gluing/asymptotic description, not the assertion that a sum of two separate solutions is automatically a solution. The supplied assessment calls the intended problem an off-layer peak construction: the relation and separation of peak centers from the transition set must be specified.

For orientation only, in the usual bistable regime $0<a(x)<1$, an interior layer separates regions approaching the two stable phases $0$ and $1$, with a thin transition near an interior set. The record does not actually supply that coefficient hypothesis, a layer geometry, the limiting phases/profile, peak number and amplitude, a scale/separation condition, or nondegeneracy conditions for the construction. None is silently imposed. A precise existence theorem must choose these data and state in what topology the peak/layer asymptotics hold while satisfying the explosive boundary condition.

The supplied literature triage checked 21 August 2026 reports extensive spike and layer work but says no construction settling this exact off-layer peak problem in the stated generality was located. Related layer analysis is partial context, not a claimed resolution of the combined construction.

Short English statement

Construct solutions of the stated singularly perturbed equation that retain an interior transition layer and also develop localized off-layer peaks, with boundary blowup. The coefficient hypotheses, layer and peak geometry and precise asymptotics are missing. The supplied assessment reports related work without a general construction.

Sources

• Local UnsolvedMath registry, permanent ID 30000254, exact source code catalogue entry; Interior-Layer Peaks in Singularly Perturbed Elliptic Problems. Proposed year 2005, status open. The corrected_verified statement restores the equation, explosive boundary condition and peaks-on-layer-background relation; its claimed report inspection is supplied metadata.
• Supplied report citation: Topological and Variational Methods for Differential Equations (2006), Oberwolfach Reports, DOI 10.4171/owr/2005/29, pp. 1601--1678, https://doi.org/10.4171/owr/2005/29. Title year differs from DOI/proposed year.
• Supplied literature assessment checked 2026-08-21: "original problem report", https://doi.org/10.4171/owr/2005/29; "related singularly perturbed layer analysis", https://doi.org/10.1016/S0022-0396(98)93619-1. This dated OWR-derived triage requests independent verification of time-sensitive status and cited claims.