Brezis and Lions (1979) studied the degenerate semilinear elliptic equation −x2(uxx+uyy)+2u=f(u)-x^2(u_{xx}+u_{yy})+2u=f(u), f(0)=f′(0)=0f(0)=f'(0)=0, on a half-strip and proved that any solution continuous up to the degenerate boundary segment x=0x=0 and vanishing there satisfies x−2u∈C∞x^{-2}u\in C^\infty up to the boundary. The motivating application is the Yang–Mills multimeron problem: the scalar reduction x2Δψ=ψ(ψ2−1)x^2\Delta\psi=\psi(\psi^2-1) with boundary values ψ(0,y)=±1\psi(0,y)=\pm1 changing sign at finitely many points, and it is exactly at these sign-change points (the meron poles) that the vanishing hypothesis of Brezis and Lions fails. Existence of solutions and the leading asymptotic profile near such isolated singularities — asymptoticity to the explicit one-dimensional meron profile with o(1)o(1) error — were established by Caffarelli, Gidas and Spruck (1983) and generalized to hyperbolic space in any dimension by Pisante and Ponsiglione (2010). The fine local theory at the sign-change points remains open: no sharp rate, no complete asymptotic expansion, no optimal regularity class for the remainder, and no classification of possible local singular structures appears in the literature; the most recent treatment (Kichenassamy 2026) re-proves the Brezis–Lions regularity only under a hypothesis that explicitly excludes the sign-change points.

Background

The physical problem. A meron is a classical solution of the Euclidean SU(2)SU(2) Yang–Mills equations — the field equations of the simplest non-abelian gauge theory of particle physics, here written in four-dimensional Euclidean space rather than physical spacetime — carrying topological charge 1/21/2 (a topological invariant of the gauge field), concentrated at a single singular point; it was found explicitly by de Alfaro, Fubini and Furlan (1976). Glimm and Jaffe (1978) reduced the construction of multimeron configurations — merons connected in pairs along an axis — to a scalar boundary value problem: on the half-plane {x>0}\{x>0\}, where xx is the distance to the axis, the gauge profile is determined by a function ψ\psi solving the semilinear equation

x2(ψxx+ψyy)=ψ(ψ2−1)x^2(\psi_{xx}+\psi_{yy})=\psi(\psi^2-1)

with boundary data ψ(0,y)=±1\psi(0,y)=\pm1 prescribed piecewise on the boundary axis, changing sign at finitely many points (the meron poles). Jonsson, McBryan, Zirilli and Hubbard (1979) proved existence for this singular boundary value problem, thereby constructing multimeron solutions of the Yang–Mills equations.

The degenerate elliptic equation of Brezis–Lions. The coefficient x2x^2 vanishes on the boundary x=0x=0, so the equation is degenerate elliptic: the second-order part loses ellipticity exactly at the boundary, and standard boundary Schauder theory does not apply. Brezis and Lions (1979) considered the shifted form: setting u=ψ∓1u=\psi\mp1 near a boundary component on which ψ≡±1\psi\equiv\pm1 gives

−x2(uxx+uyy)+2u=f(u),f(0)=f′(0)=0-x^2(u_{xx}+u_{yy})+2u=f(u),\qquad f(0)=f'(0)=0

(for the multimeron nonlinearity, f(u)=−u2(3+u)f(u)=-u^2(3+u) with u=ψ−1u=\psi-1 on the ψ≡+1\psi\equiv+1 component, and f(u)=u2(3−u)=−u2(u−3)f(u)=u^2(3-u)=-u^2(u-3) with u=ψ+1u=\psi+1 on the ψ≡−1\psi\equiv-1 component) on a rectangle Ωa={(x,y):0<x<a, ∣y∣<a}\Omega_a=\{(x,y):0<x<a,\ |y|<a\}, and proved: if u∈C(Ωˉa)u\in C(\bar\Omega_a) solves the equation with trace u(0,y)=0u(0,y)=0 for ∣y∣<a|y|<a, then x−2u∈C∞(Ωˉa/2)x^{-2}u\in C^\infty(\bar\Omega_{a/2}).

Why the sign-change points are different. The hypothesis u(0,y)=0u(0,y)=0 holds on boundary segments where ψ\psi is constantly +1+1 or −1-1, but at a sign-change point the datum of ψ\psi jumps between +1+1 and −1-1, so the trace of uu is not even a well-defined continuous function there; the Brezis–Lions analysis does not reach these points. The behavior there is genuinely singular rather than merely unknown: the one-meron profile is the explicit bounded solution ψmeron(x,y)=sin⁡θ=y/x2+y2\psi_{\rm meron}(x,y)=\sin\theta=y/\sqrt{x^2+y^2} (in polar coordinates (r,θ)(r,\theta) about the pole, with the half-plane corresponding to θ∈(−π/2,π/2)\theta\in(-\pi/2,\pi/2); direct substitution verifies it solves the equation), and ψmeron\psi_{\rm meron} has no limit at the pole — its leading behavior is angular, depending on the direction of approach.

The hyperbolic Allen–Cahn reformulation. Dividing by x2x^2 and endowing the half-plane with the hyperbolic metric (dx2+dy2)/x2(dx^2+dy^2)/x^2 (constant curvature −1-1, for which x2Δx^2\Delta is the hyperbolic Laplacian), the equation becomes the Allen–Cahn equation ΔHψ=W′(ψ)\Delta_{\mathbb H}\psi=W'(\psi) with double-well potential W(u)=14(1−u2)2W(u)=\frac14(1-u^2)^2 — the standard model of phase transitions — on the hyperbolic plane; the boundary axis together with the point at infinity forms its conformal boundary (circle at infinity). This reformulation, developed by Birindelli–Mazzeo (2009), Pisante–Ponsiglione (2010) and Mazzeo–Saez (2014), connects the 1979 question to the active study of Allen–Cahn solutions with prescribed phase regions Ω±\Omega^\pm on the sphere at infinity: in two dimensions, prescribing a sign pattern on the boundary axis with finitely many change points is exactly the multimeron data.

Method context. Analyses of such equations use comparison and barrier arguments adapted to degenerate operators (Caffarelli–Gidas–Spruck), variational methods (Pisante–Ponsiglione), and Fuchsian reduction — the systematic expansion of solutions of equations whose principal coefficient vanishes on a hypersurface, in powers and possible logarithms (Kichenassamy 2007). The natural regularity scales near x=0x=0 are weighted spaces in which the lifted operator D=x∂xD=x\partial_x has controlled coefficients, such as xαx^\alpha-weighted Hölder classes.

Problem Statement

Determine the precise local behavior, near a boundary sign-change point, of bounded solutions of the degenerate semilinear elliptic multimeron equation

x2(ψxx+ψyy)=ψ(ψ2−1)in {(x,y):0<x<a, ∣y∣<a}x^2(\psi_{xx}+\psi_{yy})=\psi(\psi^2-1)\quad\text{in }\{(x,y):0<x<a,\ |y|<a\}

with boundary data ψ(0,y)=±1\psi(0,y)=\pm1 prescribed piecewise on the axis x=0x=0 and changing sign at the point (0,0)(0,0) (the equation, the domain shape, and the class of data are invariant under translations in yy and under the dilations (x,y)↦(λx,λy)(x,y)\mapsto(\lambda x,\lambda y), so every isolated sign-change point is equivalent to this one) — equivalently, after the shift u=ψ∓1u=\psi\mp1, of solutions of −x2(uxx+uyy)+2u=f(u)-x^2(u_{xx}+u_{yy})+2u=f(u) with f(0)=f′(0)=0f(0)=f'(0)=0, exactly at the locus where the Brezis–Lions hypothesis u(0,y)=0u(0,y)=0 fails. The solutions considered are bounded and classical in the interior, continuous up to the boundary away from the sign-change point. The target is a complete local theory at (0,0)(0,0), consisting of three mandatory parts: (1) the complete asymptotic expansion of the remainder ψ−ψmeron\psi-\psi_{\rm meron}, including the sharp rate, where ψmeron(x,y)=±y/x2+y2\psi_{\rm meron}(x,y)=\pm y/\sqrt{x^2+y^2} (sign according to the orientation of the local data) is the explicit one-dimensional meron profile — including the structure of the successive terms and whether the scale invariance of the equation forces logarithmic terms; (2) the optimal regularity class of the remainder in scales adapted to the degenerate operator x2Δx^2\Delta, with matching upper and lower bounds establishing optimality; (3) the classification of possible local singular structures: whether every bounded solution with the given local sign data is asymptotic to the meron profile, or whether other local behaviors occur, and if so, which. The question may be posed for either the specific multimeron nonlinearity or the general Brezis–Lions class f(0)=f′(0)=0f(0)=f'(0)=0, with the sign-change datum and the degeneracy of the operator retained and, in the general class, ψmeron\psi_{\rm meron} replaced by the corresponding one-dimensional profile of that equation. Already established, and not to be re-proved as the answer: existence for general pole configurations (Jonsson–McBryan–Zirilli–Hubbard 1979; Caffarelli–Gidas–Spruck 1983; Pisante–Ponsiglione 2010) and the o(1)o(1) leading asymptoticity to the one-dimensional profile (Caffarelli–Gidas–Spruck 1983; Pisante–Ponsiglione 2010).

References

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