Broken-symmetry multiplicity for superlinear elliptic Dirichlet problems: the Bahri–Lions conjecture on general domains

When the nonlinearity of a superlinear elliptic Dirichlet problem is odd in the unknown, the energy functional is even, and equivariant minimax theory yields infinitely many solutions. Whether infinite multiplicity persists when this symmetry is broken by a perturbation is a classical question highlighted in the early 1980s and sharpened by Bahri and Lions: for the model problem $-\Delta u=|u|^{p-1}u+\psi$, $u=0$ on $\partial\Omega$, they proved infinitely many solutions for every $\psi\in L^2(\Omega)$ whenever $1<p<N/(N-2)$, and conjectured the same throughout the subcritical range $1<p<(N+2)/(N-2)$. Ramos–Tavares–Zou (Adv. Math. 2009) revisited this theorem: on general bounded domains they constructed infinitely many sign-changing solutions with two-sided energy growth for a broad class of odd superlinear nonlinearities plus a lower-growth non-odd perturbation, but only under a growth restriction coupling the exponents, and they could bound the number of nodal domains of the constructed solutions only from above. Molle–Passaseo (2022, 2023) verified the conjecture on special domains (domains satisfying a geometric condition, in particular radial ones, and cubes, with solutions of prescribed nodal structure), leaving general domains open. This record asks whether infinite multiplicity persists throughout the full subcritical range when the symmetry is broken: in the model class, the Bahri–Lions conjecture on arbitrary bounded smooth domains for every $\psi\in L^2(\Omega)$, and, in the general odd-plus-perturbation framework of Ramos–Tavares–Zou, whether infinite sign-changing multiplicity holds without their coupling growth restriction. The core question is open in the literature; the related question of lower bounds on nodal counts is open context rather than a completion target.

Background

For a bounded smooth domain $\Omega\subset\mathbb{R}^N$, $N\ge 3$, consider the semilinear Dirichlet problem $-\Delta u = g(x,u)+f(x,u)\ \text{in }\Omega,\qquad u=0\ \text{on }\partial\Omega,$ where weak solutions are sought in the Sobolev space $H^1_0(\Omega)$ (functions with square-integrable gradient and zero trace). A solution is called sign-changing (or nodal) if it takes both positive and negative values; its nodal domains are the connected components of the sets $\{u>0\}$ and $\{u<0\}$. Solutions are critical points of the energy functional $J(u)=\frac12\int_\Omega|\nabla u|^2-\int_\Omega G(x,u)-\int_\Omega F(x,u)$, where $G,F$ are the primitives of $g,f$ in the second variable.

If $f\equiv0$ and $g$ is odd in $u$ (so $J$ is even), equivariant minimax theory in the spirit of Lusternik–Schnirelmann category (a topological invariant counting how many contractible closed sets are needed to cover a given set) yields infinitely many solutions; the min–max construction is organized by an index $k$ running over eigenfunction truncations of $H^1_0(\Omega)$. When $f\not\equiv0$ is not odd in $u$, this symmetric structure breaks down, and the multiplicity question becomes genuinely difficult: it is a classical problem from the early 1980s (Ambrosetti, Bahri, Berestycki, Rabinowitz, Struwe and others; see the historical accounts in the Molle–Passaseo and Bonheure–Moreira dos Santos–Tavares references below) to determine whether infinitely many solutions survive. A standard hypothesis on the odd part $g$ is superlinear growth $2<p<2^*$ with $2^*=2N/(N-2)$ the critical Sobolev exponent, where $p$ is the exponent convention in which $|g(x,u)|\le C(1+|u|^{p-1})$, so that the primitive $G$ grows like $|u|^{p}$; this is paired with an Ambrosetti–Rabinowitz condition $g(x,s)s\ge\mu G(x,s)-C$ for some $\mu>2$, which forces unbounded energy levels. The perturbation $f$ is allowed strictly weaker growth, imposed pointwise on $f$ itself: $|f(x,u)|\le C(1+|u|^{\nu})$ with $\nu+1<\mu$.

For the model problem $-\Delta u=|u|^{p-1}u+\psi$ with $\psi\in L^2(\Omega)$, Bahri (1981) proved that infinitely many solutions exist for an open dense set of $\psi$, and Bahri–Lions (1988), combining min–max levels with Morse index information, proved that infinitely many solutions exist for every $\psi\in L^2(\Omega)$ whenever $1<p<N/(N-2)$; they conjectured the same conclusion throughout the subcritical range $1<p<(N+2)/(N-2)$. Here the Morse index of a critical point is the maximal dimension of a subspace on which the second variation of $J$ is negative definite, and it controls multiplicity: for classical solutions the number of nodal domains is at most one more than the Morse index.

Ramos–Tavares–Zou (2009) revisited this theorem. On arbitrary bounded smooth domains, for a general pair $(g,f)$ of Carathéodory functions (functions measurable in $x$ for each fixed $u$ and continuous in $u$ for each fixed $x$) as above, they proved under the coupling growth restriction $\frac{2p}{N(p-2)}>\frac{\mu}{\mu-\nu-1}$ that the problem admits a sequence of sign-changing solutions $u_{k_n}$ whose energies grow polynomially in $k_n$ from both sides; under an additional differentiability hypothesis each $u_{k_n}$ has at most $k_n+1$ nodal domains; in the purely even case they obtain sign-changing solutions whose Morse index satisfies the sandwich $m(u_k)\le k\le m^*(u_k)$ with $m^*$ the augmented (Morse index plus nullity) index. They also proved parallel results for a fourth-order biharmonic problem under Navier boundary conditions $u=\Delta u=0$ — where the reduction relies on positivity of the associated Green function — and for the strongly coupled Hamiltonian system $-\Delta u=|v|^{q-1}v$, $-\Delta v=|u|^{p-1}u$ in the subcritical-hyperbola regime (documented in the Bonheure–Moreira dos Santos–Tavares survey). Beyond what the theorem establishes, three directions remain: the growth restriction itself (how far the multiplicity persists toward the critical exponent on general domains), any lower bound on the nodal counts of the constructed solutions, and the two extensions (fourth-order clamped boundary conditions and a priori bounds for the system).

Molle–Passaseo developed a different, non-perturbative gluing method: piecing together Dirichlet problems on Lipschitz-deformed cube partitions, they first (2022) obtained, for domains satisfying a geometric condition on such deformations (verified in dimension one and adaptable to radial solutions in radially symmetric domains), infinitely many nodal solutions with prescribed 'check' nodal structure whose nodal counts and energies tend to infinity; then (2023) they gave a complete positive answer to the Bahri–Lions conjecture when $\Omega$ is a cube, for every $\psi\in L^2(\Omega)$ and the full range $1<p<(N+2)/(N-2)$, again with prescribed nodal structure. Their method depends essentially on the cube-like geometry, and both the conjecture on general domains and the nodal-count question in the perturbation-from-symmetry framework remain unresolved. The Ramos–Tavares–Zou approach remains active: Chen–Chen–Li–Liao (2025) extended it to subelliptic operators, where they show the sufficient conditions it yields differ from the Euclidean case.

Problem Statement

Work on the class of semilinear Dirichlet problems $-\Delta u = g(x,u)+f(x,u)\ \text{in }\Omega,\qquad u=0\ \text{on }\partial\Omega,$ where $\Omega\subset\mathbb{R}^N$ ($N\ge3$) is an arbitrary bounded smooth domain, $g$ is a Carathéodory function (measurable in $x$ for each fixed $u$, continuous in $u$ for each fixed $x$), odd and superlinear in $u$ with growth bound $|g(x,u)|\le C(1+|u|^{p-1})$ for some $2<p<2^*=\frac{2N}{N-2}$ (so that the primitive $G$ grows like $|u|^{p}$) and satisfying an Ambrosetti–Rabinowitz condition with constant $\mu>2$, and $f$ is a Carathéodory perturbation (same joint measurability/continuity) that is not odd in $u$ and satisfies the pointwise growth bound $|f(x,u)|\le C(1+|u|^{\nu})$ with $\nu+1<\mu$, where $\mu$ is the Ambrosetti–Rabinowitz constant of $g$.

Part A (general class, full-range multiplicity). Decide whether, for every bounded smooth domain $\Omega$ and every admissible pair $(g,f)$ in the full subcritical range $2<p<2^*$, the problem admits infinitely many sign-changing weak solutions — that is, whether the coupling growth restriction $\frac{2p}{N(p-2)}>\frac{\mu}{\mu-\nu-1}$ used by Ramos–Tavares–Zou can be removed.

Part B (model case, the Bahri–Lions conjecture on general domains). Decide whether, for every bounded smooth domain $\Omega$, every $\psi\in L^2(\Omega)$ and every $1<p<\frac{N+2}{N-2}$, the model problem $-\Delta u=|u|^{p-1}u+\psi$, $u=0$ on $\partial\Omega$, admits infinitely many solutions; this is currently verified only when $\Omega$ is a cube (Molle–Passaseo 2023) or satisfies the Molle–Passaseo geometric condition (2022). Note the exponent conventions: in the model problem the primitive of $|u|^{p-1}u$ grows like $|u|^{p+1}$, so the subcritical bound $p<\frac{N+2}{N-2}$ corresponds to $p+1<2^*$ in the primitive-exponent convention of the general class. The conjecture ranges over every $\psi\in L^2(\Omega)$, including perturbations unbounded in $x$, and is therefore stated in its own $L^2$ class rather than as a literal special case of the pointwise-bounded framework of Part A.

Parts A and B address one scientific objective: to decide to what extent odd symmetry of the nonlinearity is essential for infinite multiplicity of superlinear elliptic Dirichlet problems under symmetry breaking, up to the critical exponent. The two Parts are independent sub-claims, neither implying the other: resolving either Part answers its own question, and both must be decided for a complete resolution of the record. Two further issues in the same lineage — extension to fourth-order clamped-plate boundary conditions $u=\partial_\nu u=0$ without assuming positivity of the biharmonic Green function, and uniform a priori bounds for positive solutions of the strongly coupled system $-\Delta u=|v|^{q-1}v$, $-\Delta v=|u|^{p-1}u$ in the full subcritical-hyperbola regime — are separate adjacent open problems and are not part of the completion boundary of this problem. Likewise, lower bounds on the number of nodal domains of solutions of the perturbed problem are not part of the completion boundary of this problem.

References

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