Sublinear Lane–Emden Ground-State Asymptotics

Definitions and mathematical statement

Let $\Omega\subset{\mathbb{R}}^N$ be a bounded domain and let $1<q<p$, so that $p>1$. The dimension $N$, further hypotheses on $\Omega$, and the admissible values and dependence on $q$ of the parameter $\lambda$ are not specified in the supplied record. The $p$-Laplacian is \[ \Delta_p u=\operatorname{div}(|\nabla u|^{p-2}\nabla u). \] The stated Lane--Emden boundary value problem is \[ \begin{cases} -\Delta_pu=\lambda |u|^{q-2}u & \text{in }\Omega,\\ u=0 & \text{on }\partial\Omega. \end{cases} \] A standard weak interpretation of the boundary condition is $u\in W^{1,p}_0(\Omega)$, with \[ \int_\Omega |\nabla u|^{p-2}\nabla u\cdot\nabla\varphi\,{\,\mathrm{d}} x =\lambda\int_\Omega |u|^{q-2}u\varphi\,{\,\mathrm{d}} x \qquad\text{for every }\varphi\in W^{1,p}_0(\Omega). \] Here $|u|^{q-2}u$ is interpreted as zero at $u=0$. This weak formulation is an explicit interpretive framework; the supplied record does not specify the intended solution regularity.

The associated variational functional is \[ J_{\lambda,q}(u)=\frac1p\int_\Omega |\nabla u|^p\,{\,\mathrm{d}} x -\frac{\lambda}{q}\int_\Omega |u|^q\,{\,\mathrm{d}} x. \] Under the usual least-energy convention, a ground state is a nonzero weak solution minimizing $J_{\lambda,q}$ among nonzero weak solutions, and a least-energy nodal solution minimizes it among weak solutions whose positive and negative parts $u^+=\max\{u,0\}$ and $u^-=\max\{-u,0\}$ are both nonzero. These conventions make the terminology explicit but are not definitions confirmed by the supplied source; existence or attainment of either minimum is not asserted.

Question as supplied. Do the asymptotic results for ground states and least-energy nodal solutions of this problem as $q\to p$ remain valid in the range $1<q<p$, hence as $q\uparrow p$?

The supplied record does not state the asymptotic results to be extended, including their hypotheses, normalization, convergence topology, or limiting objects. Consequently, it does not determine a precise asymptotic assertion to prove or disprove. Its dated literature assessment also refers to an unestablished full $q>p$ asymptotic picture, whereas the extracted question asks about extension to $q<p$; this relationship remains unresolved.

Short English statement

The question asks whether unspecified asymptotic results for ground states and least-energy sign-changing solutions of the Dirichlet Lane--Emden problem extend to $q\uparrow p$ with $1<q<p$. The exact conclusions, parameter assumptions, and solution conventions require clarification before a precise theorem can be formulated.

Sources

• Supplied local registry record: permanent UnsolvedMath ID $30001258$, title Sublinear Lane--Emden Ground-State Asymptotics, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The proposed year is 2009; no individual proposer is supplied.
• Topological and Variational Methods for Partial Differential Equations (2009), Oberwolfach Reports, pp. 1427--1486, DOI 10.4171/owr/2009/26. Supplied report citation and literature-source link: https://doi.org/10.4171/owr/2009/26. The background describes this report as documenting workshop discussions and open problems; no precise problem page or speaker attribution is supplied.
• Supplied literature-source label: "inverse iteration for sublinear p-Laplacian problems," https://arxiv.org/abs/1011.3172. No authors or more detailed bibliographic attribution are supplied.
• Literature assessment checked at 2026-08-21: the supplied curation labels the problem partially solved and states that, for $q<p$, existence and uniqueness results and $p$-Laplacian eigenpair limits are known, while the full $q>p$ asymptotic picture requested in the extracted question has not been established. This is supplied dated curation, not an independently verified literature assessment; its hypotheses, scope, and relationship to the extracted question require review.
• The registry labels the statement corrected_verified; its accompanying verification note concerns balanced mathematical delimiters only. It does not establish mathematical correctness or verify the literature claims.