On a compact spin manifold (M,g,σ)(M,g,\sigma) of dimension m≥2m\ge2, the nonlinear Dirac equation Dψ=λψ+f(∣ψ∣)ψ+∣ψ∣2/(m−1)ψD\psi=\lambda\psi+f(|\psi|)\psi+|\psi|^{2/(m-1)}\psi couples a strongly indefinite leading operator with a Sobolev-critical nonlinearity. Bartsch and Xu (2021) proved that a least-energy solution exists for every λ>0\lambda>0 under mild assumptions on ff, and for every real λ\lambda outside the nonpositive spectrum of the Dirac operator DD under stronger assumptions. They explicitly posed as a challenging open problem the remaining range: λ≤0\lambda\le0 in the mild setting, and λ=λk≤0\lambda=\lambda_k\le0 (a nonpositive eigenvalue of DD) in the stronger setting, warning that the answer may depend on an intricate combination of the geometry of (M,g,σ)(M,g,\sigma) and properties of ff. This is the spinorial counterpart of the classical threshold phenomenon of Brezis and Nirenberg for the scalar critical-exponent equation. No resolution is known in the literature to date: subsequent work either restricts to positive parameters, excludes spectral values of DD, or works in perturbative regimes.

Background

A compact spin manifold (M,g,σ)(M,g,\sigma) of dimension m≥2m\ge2 is a closed Riemannian manifold MM with metric gg together with a spin structure σ\sigma, that is, a lift of the oriented orthonormal frame bundle to a principal Spin(m)\mathrm{Spin}(m)-bundle. On such a manifold the Atiyah-Singer Dirac operator DD is a first-order, self-adjoint, elliptic operator acting on sections ψ\psi (spinors) of the spinor bundle S(M)\mathbb{S}(M). Its spectrum spec⁡(D)={λk:k∈Z}\operatorname{spec}(D)=\{\lambda_k:k\in\mathbb{Z}\} consists of real eigenvalues of finite multiplicity, ordered as …<λ−1<λ0≤0<λ1<…\ldots<\lambda_{-1}<\lambda_0\le0<\lambda_1<\ldots with ∣λk∣→∞|\lambda_k|\to\infty; note that λ0\lambda_0 may equal 00 and is never positive.

The equation under study is the nonlinear Dirac equation with critical Sobolev exponent

Dψ=λψ+f(∣ψ∣)ψ+∣ψ∣2m−1ψD\psi=\lambda\psi+f(|\psi|)\psi+|\psi|^{\frac{2}{m-1}}\psi

(denoted (NLD) below), where λ∈R\lambda\in\mathbb{R} and f:[0,∞)→[0,∞)f:[0,\infty)\to[0,\infty) is a subcritical perturbation in the sense that f(s)=o(s2m−1)f(s)=o\big(s^{\frac{2}{m-1}}\big) as s→∞s\to\infty; write F(s)=∫0sf(t)t dtF(s)=\int_0^s f(t)t\,dt. The exponent is critical because the natural form domain of DD is the fractional Sobolev space H1/2(M,S(M))H^{1/2}(M,\mathbb{S}(M)), whose embedding into LpL^p is critical precisely at 2∗=2mm−12^*=\frac{2m}{m-1}; the model case f≡0f\equiv0 is the spinorial Brezis-Nirenberg equation Dψ=λψ+∣ψ∣2∗−2ψD\psi=\lambda\psi+|\psi|^{2^*-2}\psi. Weak solutions are critical points of the energy functional

Lλ(ψ)=12∫M(Dψ,ψ) dvolg−λ2∫M∣ψ∣2 dvolg−∫MF(∣ψ∣) dvolg−12∗∫M∣ψ∣2∗ dvolg.L_\lambda(\psi)=\tfrac12\int_M(D\psi,\psi)\,dvol_g-\tfrac{\lambda}{2}\int_M|\psi|^2\,dvol_g-\int_MF(|\psi|)\,dvol_g-\tfrac{1}{2^*}\int_M|\psi|^{2^*}\,dvol_g.

A least-energy solution (also called a ground state) is a nonzero weak solution at which LλL_\lambda attains its minimum over all nonzero weak solutions.

Two structural difficulties govern the problem. First, LλL_\lambda is strongly indefinite: because spec⁡(D)\operatorname{spec}(D) is an infinite two-sided sequence, both the positive and negative spectral subspaces of D−λD-\lambda are infinite-dimensional, so critical points have infinite Morse index and co-index, and the classical Nehari-manifold minimization is unavailable. Second, the critical exponent destroys compactness: a Palais-Smale condition holds only below an explicit critical energy level γcrit\gamma_{\mathrm{crit}} associated with concentration (bubbling) of spinors, mirroring the compactness loss in the scalar Brezis-Nirenberg problem.

The scalar analogue is the classical result of Brezis and Nirenberg (1983) for −Δu=λu+u2∗−1-\Delta u=\lambda u+u^{2^*-1} on a bounded domain Ω⊂Rn\Omega\subset\mathbb{R}^n with Dirichlet boundary conditions: for n≥4n\ge4 a solution exists precisely when 0<λ<λ1(Ω)0<\lambda<\lambda_1(\Omega), while in dimension n=3n=3 the threshold also depends on the geometry of the domain; in particular, solvability fails for nonpositive λ\lambda in dimensions n≥4n\ge4. Whether the spinorial problem exhibits an analogous breakdown of solvability for nonpositive λ\lambda is exactly what is at stake here.

For the spinorial problem, Isobe (2011) obtained ground states of the Brezis-Nirenberg equation via a dual variational principle, requiring dimension m≥4m\ge4, λ>0\lambda>0, and λ∉spec⁡(D)\lambda\notin\operatorname{spec}(D). Bartsch and Xu (2021) developed a direct variational approach for the general equation (NLD): the Nehari-Pankov manifold (a natural constraint adapted to strongly indefinite functionals) combined with a modified nonlinear projection onto ker⁡(D−λ)\ker(D-\lambda) and a generalized linking argument. Their Theorem 2.1 yields least-energy solutions for every λ>0\lambda>0 under mild assumptions on ff, and for every λ∈R∖{λk:k≤0}\lambda\in\mathbb{R}\setminus\{\lambda_k:k\le0\} under stronger assumptions; Corollary 2.3 covers the Brezis-Nirenberg equation for every λ>0\lambda>0 in all dimensions m≥2m\ge2, removing Isobe's restrictions. The range that these methods do not reach is precisely the nonpositive spectral range, which the authors themselves state as an open problem (their Remark 2.2 c)): whether (NLD) has a least-energy solution for λ≤0\lambda\le0 under the mild hypotheses, or for λ=λk≤0\lambda=\lambda_k\le0 under the stronger hypotheses — a question they warn may depend on an intricate combination of conditions on the geometry of (M,g,σ)(M,g,\sigma) and properties of ff.

Problem Statement

Let (M,g,σ)(M,g,\sigma) be a compact spin manifold of dimension m≥2m\ge2 without boundary, with Dirac operator DD acting on the spinor bundle S(M)\mathbb{S}(M), and let the spectrum of DD be ordered as …<λ−1<λ0≤0<λ1<…\ldots<\lambda_{-1}<\lambda_0\le0<\lambda_1<\ldots. Consider the nonlinear Dirac equation with critical Sobolev exponent

Dψ=λψ+f(∣ψ∣)ψ+∣ψ∣2m−1ψD\psi=\lambda\psi+f(|\psi|)\psi+|\psi|^{\frac{2}{m-1}}\psi

where λ∈R\lambda\in\mathbb{R} and f:[0,∞)→[0,∞)f:[0,\infty)\to[0,\infty) is continuous with f(0)=0f(0)=0; write F(s)=∫0sf(t)t dtF(s)=\int_0^s f(t)t\,dt. A least-energy solution is a nonzero weak solution ψ∈H1/2(M,S(M))\psi\in H^{1/2}(M,\mathbb{S}(M)) at which the associated energy functional attains its minimum over all nonzero weak solutions.

Question A (mild hypotheses). Assume: (f1) f:[0,∞)→[0,∞)f:[0,\infty)\to[0,\infty) is continuous and f(0)=0f(0)=0; (f2) f(s)/s2m−1→0f(s)/s^{\frac{2}{m-1}}\to0 as s→+∞s\to+\infty; (f3) the function s↦f(s)+s2m−1s\mapsto f(s)+s^{\frac{2}{m-1}} is strictly increasing. Does the equation have a least-energy solution for λ≤0\lambda\le0?

Question B (stronger hypotheses). Assume f∈C1f\in C^1 with f′(s)>0f'(s)>0, together with (f1) and: (f4) f(s)sF(s)m+12m→0\frac{f(s)s}{F(s)^{\frac{m+1}{2m}}}\to0 as s→+∞s\to+\infty; (f5) lim⁡ε→0+εm−1∣ln⁡ε∣max⁡{3−m,0}∫01/εF(ε−m−12(1+r2)−m−12)rm−1 dr=∞\lim_{\varepsilon\to0^+}\frac{\varepsilon^{m-1}}{|\ln\varepsilon|^{\max\{3-m,0\}}}\int_0^{1/\varepsilon}F\left(\varepsilon^{-\frac{m-1}{2}}(1+r^2)^{-\frac{m-1}{2}}\right)r^{m-1}\,dr=\infty. Does the equation have a least-energy solution for λ=λk≤0\lambda=\lambda_k\le0?

The posing authors warn that the answer may depend on an intricate combination of conditions on the geometry of (M,g,σ)(M,g,\sigma) and properties of ff.

References

  1. Haïm Brezis, Louis Nirenberg. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Communications on Pure and Applied Mathematics 36(4):437-477, 1983. DOI: 10.1002/cpa.3160360405
  2. Takeshi Isobe. Nonlinear Dirac equations with critical nonlinearities on compact spin manifolds. Journal of Functional Analysis 260(1):253-307, 2011. DOI: 10.1016/j.jfa.2010.09.008
  3. Thomas Bartsch, Tian Xu. A spinorial analogue of the Brezis-Nirenberg theorem involving the critical Sobolev exponent. Journal of Functional Analysis 280(12):108991, 2021. Preprint arXiv:1810.05548 (2018). DOI: 10.1016/j.jfa.2021.108991; https://arxiv.org/abs/1810.05548
  4. Yanheng Ding, Tian Xu. The Spinorial Brezis-Nirenberg Problem. In: Localization Approaches in Strongly Indefinite Problems, pp. 155-190. Springer, 2025 (online December 2024). DOI: 10.1007/978-981-97-9539-0_6
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