Strong Quantization for Fourth-Order Navier Problems

Definitions and mathematical statement

Let $\Omega\subset{\mathbb{R}}^4$ be a bounded domain and let $u_k>0$ be solutions with the regularity required for the stated Navier problem \[ \Delta^2u_k=\lambda_k u_k e^{2u_k^2}\quad\text{in }\Omega, \qquad u_k=\Delta u_k=0\quad\text{on }\partial\Omega, \qquad\lambda_k\to0. \] Impose the exact energy condition from the corrected statement: \[ \int_\Omega|\Delta u_k|^2\,{\,\mathrm{d}} x =\lambda_k\int_\Omega u_k^2e^{2u_k^2}\,{\,\mathrm{d}} x \longrightarrow\Lambda\in(0,\infty). \] Thus the PDE has $u_k e^{2u_k^2}$ on its right side; the energy density has the additional factor $u_k$.

Suppose the sequence has $I$ distinct blow-up points and its total limiting energy is quantized as $\Lambda=L\Lambda_1$, where $\Lambda_1$ is the source's elementary bubble energy and $L$ its total bubble multiplicity. The question is whether \[ L=I, \] that is, whether every blow-up point carries only one bubble and is simple in the source's sense. The question does not ask for total multiplicity $L=1$ when several distinct blow-up points exist.

To clarify the claimed equivalence, if the complete concentration decomposition is \[ \lambda_k u_k^2e^{2u_k^2}\,{\,\mathrm{d}} x \ \rightharpoonup\ \sum_{j=1}^{I}\ell_j\Lambda_1\delta_{x_j}, \qquad\ell_j\in{\mathbb{Z}}_{\geq1},\qquad L=\sum_{j=1}^{I}\ell_j, \] then $L=I$ is equivalent to every $\ell_j=1$. This decomposition is explanatory and conditional, not an additional theorem asserted from the extract. The value/normalization of $\Lambda_1$, the underlying quantization result, absence of residual energy, boundary blow-up treatment and precise analytic meaning of simplicity are not supplied.

The original extracted equation $\lambda_k u_k^2e^{2u_k}$ differs from the corrected verified equation and is not combined with it. The literature-assessment phrase $\lambda_k u_k^2e^{2u_k^2}$ matches the energy density rather than the corrected PDE right-hand side. The explicitly restored clean equation controls this draft.

Short English statement

For the corrected fourth-order Navier equation and bounded positive energy limit, must the total bubble count equal the number of concentration points? This asks for one bubble at each point, while preserving the distinctions between the corrected PDE, its energy density and the earlier inconsistent extraction.

Sources

• Supplied report: Partielle Differentialgleichungen (2008), Oberwolfach Reports, DOI 10.4171/owr/2007/35, pp. 2035--2078; proposed year 2007. https://doi.org/10.4171/owr/2007/35.
• Supplied reference: Robert, fourth-order quantization under additional hypotheses, https://doi.org/10.1017/S0308210506000096.
• Local UnsolvedMath registry, permanent ID 30000801, catalogue entry; created 25 August 2026, updated 16 September 2026. The corrected_verified note says the official report restored the equation, Navier data, energy limit and $L=I$ question. The supplied 21 August 2026 literature triage labels the exact simplicity question open and says related fourth-order equations do not settle it. Its equation description differs from the clean PDE as explained above; no independent literature assessment is asserted.