Critical-eigenvalue nonexistence conjecture for strong-KPP reactions

Let $(\Omega,\varrho)$ be a domain and boundary parameter for the problem

, let $f$ satisfy the strong-KPP hypothesis, and let $\lambda(\Omega,\varrho)$ denote the relevant principal eigenvalue. **Critical strong-KPP nonexistence conjecture.** If

\lambda(\Omega,\varrho)=f'(0),

$$then$$

has no positive bounded solution. This extends the established conclusion for bounded or periodic domains to unbounded critical domains, where the critical case is described as subtle and largely open.