Geometric uniformity of minimizing tiled deformations for non-symmetric elliptic equations: extending the Bahri–Lions multiplicity theorem to arbitrary bounded smooth domains

For the Dirichlet problem $-\Delta u = |u|^{p-1}u + w$ on a bounded smooth domain $\Omega\subset\mathbb{R}^n$ with a non-symmetric forcing term $w\in L^2(\Omega)$, the Bahri–Lions conjecture predicts infinitely many solutions for every $w$ throughout the subcritical range $1<p<(n+2)/(n-2)$. Molle and Passaseo (2022) proved this for $n=1$ and, for $n>1$, reduced it to a geometric uniformity condition on the bi-Lipschitz distortion of deformations of a tiling pattern (conditions (2.53)/(3.7)), which they could verify only in dimension one; for $n>1$ they proposed the localized variant (3.7) but did not verify it. The open problem is to establish the conjecture for $n>1$ on an arbitrary bounded smooth domain $\Omega$. The 2023 follow-up settles the conjecture only for cubes and special tessellable domains, by a method that bypasses rather than verifies the condition; the general-domain case remains open.

Background

Consider the model boundary-value problem $-\Delta u = |u|^{p-1}u + w$ in $\Omega$, $u = 0$ on $\partial\Omega$, where $\Omega$ is a bounded smooth domain of $\mathbb{R}^n$, $w\in L^2(\Omega)$ is a fixed forcing term, and $1<p<(n+2)/(n-2)$ is the subcritical range below the critical Sobolev exponent $(n+2)/(n-2)$, the threshold for the compact embedding of the energy space $H_0^1(\Omega)$ into $L^{p+1}(\Omega)$ (for $n=2$ the critical exponent is infinite and the range is all finite $p>1$). When $w=0$ the nonlinearity $|u|^{p-1}u$ is odd, the energy functional is even, and the symmetric mountain-pass theorem of Ambrosetti–Rabinowitz yields infinitely many solutions. The forcing term $w$ breaks the symmetry: solutions need not come in pairs, and infinite multiplicity is no longer automatic. (The Morse index of a solution is the maximal dimension of a subspace on which the second variation of the energy is negative definite; Bahri and Lions used lower bounds on it, combined with min–max constructions, to force the existence of many critical points.) Bahri (1981) proved that for a generic (open dense set of) $w$ there are infinitely many solutions in the full subcritical range; Bahri and Lions (1988), via Morse-index estimates on min–max critical points of the associated energy functional, proved infinite multiplicity for every $w\in L^2(\Omega)$ but only in the smaller range $1<p<n/(n-2)$, $n\ge 3$. The Bahri–Lions conjecture asserts that infinite multiplicity for every $w$ persists throughout the full subcritical range $1<p<(n+2)/(n-2)$; Ramos–Tavares–Zou (2009) revisited the 1988 theorem but the general full-range statement remained out of reach for general domains. Molle and Passaseo (2022) introduced a direct construction: for each $k$ they build a test configuration made of $k$ mutually interacting nodal regions (subdomains where the test function alternates sign) arranged by deforming a reference cubic tiling $P_k$ of $\Omega$ (in dimension one, a subdivision of the interval), and minimize over a class $C^{\overline{L}}$ of Lipschitz deformations $T$ of the tiling whose minimal bi-Lipschitz distortion constant $\mathcal{L}(T)$ — the least $L\ge 1$ such that both $T$ and $T^{-1}$ are $L$-Lipschitz — is at most $\overline{L}$. Their Theorem 2.7 states that if there exists a distortion cap $\overline{L}\ge L'_\Omega$ — where $L'_\Omega\ge 1$ is a domain-dependent constant, defined in their (2.49) from the smoothness of $\Omega$, which guarantees that the class of surjective bi-Lipschitz deformations $T$ with $T(P_k)=\overline{\Omega}$ is nonempty for all large $k$ — such that the minimizing deformations $T_k^{\overline{L}}$ — defined by the two-step minimization of their (2.50)/(2.52), the strict inequality leaving room for the small bi-Lipschitz perturbations used in the proof of Proposition 2.8 — satisfy the uniformity condition

$$\limsup_{k\to\infty}\mathcal{L}\bigl(T_k^{\overline{L}}\bigr) < \overline{L},$$

(the paper's condition (2.53)), then the subcritical problem has infinitely many solutions; Theorem 3.8 gives the analogous high-multiplicity statement, under the same condition (2.53) applied to the critical functional $\mathcal{F}$, for the critical problem $-\Delta u = |u|^{4/(n-2)}u + \lambda u + w$ with $\lambda>0$ and $n\ge 4$. Lemma 2.9 verifies condition (2.53) for $n=1$ (where $\mathcal{L}(T_k^{\overline{L}})\to 1$), but for $n>1$ the class of all Lipschitz deformations is too large — equal-volume subdomains can still change shape, letting the distortion blow up — and the authors only introduce condition (3.7), a localized variant in which the deformations stay in a neighborhood of a fixed reference map $T_0$ (Remark 3.1 of the paper). They verify (3.7) only for $n=1$ (with $T_0$ the identity); for $n>1$ they note that (3.7) "seems to have more chances" than (2.53) to be satisfied, that it holds or fails depending on the choice of the cap $\tilde{L}$, the reference map $T_0$ and the neighborhood radius $r$, and they conjecture that by a suitable choice one should obtain infinitely many nodal solutions for every bounded domain. Thus for $n>1$ the multiplicity theorems are conditional: their uniformity hypothesis is verified only in dimension one, and not for a general bounded smooth domain $\Omega$.

Problem Statement

For the Dirichlet problem $-\Delta u = |u|^{p-1}u + w$ in a bounded domain $\Omega\subset\mathbb{R}^n$ ($n>1$) with $w\in L^2(\Omega)$ arbitrary and $1<p<(n+2)/(n-2)$ (for $n=2$: every finite $p>1$): prove that for every bounded smooth domain $\Omega$ and every $w\in L^2(\Omega)$ the problem admits infinitely many solutions throughout the full subcritical range — that is, settle the Bahri–Lions conjecture on arbitrary bounded smooth domains.

The 2023 cube result, which bypasses the condition via the cube's unique shape-factor minimization among polyhedra, does not answer this question: the general bounded smooth domain case remains open, whether it is reached by extending such a uniformity mechanism beyond cubes and the tessellable special classes or by any other proof of the conjecture. The analogous critical-exponent question — the critical problem $-\Delta u = |u|^{4/(n-2)}u + \lambda u + w$ with $\lambda>0$ and $n\ge 4$ (the exponent $|u|^{4/(n-2)}u$ corresponding to $p=(n+2)/(n-2)$) on general domains — is a related question beyond this target.

References

1. Riccardo Molle and Donato Passaseo, "Infinitely many solutions for elliptic equations with non-symmetric nonlinearities", Calculus of Variations and Partial Differential Equations 61(3), art. 115 (2022). DOI: 10.1007/s00526-022-02223-6; arXiv:2112.02298; https://doi.org/10.1007/s00526-022-02223-6
2. Riccardo Molle and Donato Passaseo, "On the Bahri–Lions conjecture for elliptic equations with non-symmetric nonlinearities", Calculus of Variations and Partial Differential Equations 62(6), art. 177 (2023). DOI: 10.1007/s00526-023-02507-5; https://doi.org/10.1007/s00526-023-02507-5
3. A. Bahri and P.-L. Lions, "Morse index of some min-max critical points. I. Application to multiplicity results", Communications on Pure and Applied Mathematics 41(8), 1027–1037 (1988). DOI: 10.1002/cpa.3160410803; https://doi.org/10.1002/cpa.3160410803
4. M. Ramos, H. Tavares, and W. Zou, "A Bahri–Lions theorem revisited", Advances in Mathematics 222(6), 2173–2195 (2009). DOI: 10.1016/j.aim.2009.07.013; https://doi.org/10.1016/j.aim.2009.07.013