The Ao–Musso–Pacard–Wei gluing method produces infinitely many finite-energy solutions of the cubic nonlinear Klein–Gordon (stationary Schrödinger) equation Δu−u+u3=0\Delta u-u+u^3=0 in R2\mathbb{R}^2 whose symmetry group reduces to the identity, by arranging signed copies of the ground-state bubble along 'balanced, flexible, closable' planar networks and running a finite-dimensional (Lyapunov–Schmidt) reduction. The method, however, is worked out only for the cubic nonlinearity in the plane and relies on a technical upper bound m[p,q]≪eγℓm_{[p,q]}\ll e^{\gamma\ell} on the number of interstitial bubbles per edge. The open problem is to extend this constructive program as a whole: carry out the full finite-dimensional reduction for general (not necessarily even) nonlinearities, build explicit non-symmetric closable flexible networks of genuinely non-planar type in spatial dimensions N≥3N\ge 3 together with the corresponding solutions, and remove the exponential upper bound for nonlinearities other than the cubic.

Background

The stationary nonlinear Schrödinger equation (equivalently, nonlinear Klein–Gordon equation) Δu−u+f(u)=0\Delta u-u+f(u)=0 in RN\mathbb{R}^N seeks finite-energy solutions, i.e. decaying functions in the energy space H1(RN)H^1(\mathbb{R}^N). For the cubic case f(u)=u3f(u)=u^3 in the plane (N=2N=2), the equation has a unique positive radially symmetric 'bubble' or ground state u0>0u_0>0 (the least-energy solution), and −u0-u_0 is also a solution because u3u^3 is odd, i.e. the equation is sign-equivariant (f(−u)=−f(u)f(-u)=-f(u)). A long gluing tradition, going back to constructions of constant-mean-curvature surfaces and minimal surfaces, builds new solutions by superposing far-apart copies of u0u_0 and −u0-u_0 and correcting the error through a fixed-point argument; the crucial analytic device is the finite-dimensional (Lyapunov–Schmidt) reduction, in which the mutual placement of the bubbles is governed by a finite system of 'balance' equations derived from the interaction of each bubble with the nearly-null translational modes of the linearized operator. Musso, Pacard and Wei (J. Eur. Math. Soc. 14 (2012) 1923–1953) used this philosophy to construct finite-energy sign-changing solutions with prescribed dihedral symmetry, i.e. invariance under the rotations and reflections of a regular polygon. Ao, Musso, Pacard and Wei (J. Funct. Anal. 270 (2016) 884–956) then removed all symmetry: they organize the signed peaks along a planar weighted graph (a 'network') (N,a)(\mathcal{N},a) with vertices, edges, and integer weights, subject to three combinatorial-algebraic conditions — 'balanced' (the signed bump positions satisfy a finite balance system analogous to a Toda-type equilibrium), 'flexible' (the derivative of the balance map with respect to the edge weights has maximal rank, so the balance system can be solved for admissible weights), and 'closable' (the polygonal cycles formed by chains of edges close up consistently) — and prove (Theorem 9.1) that every closable, flexible network satisfying compatibility conditions yields a genuine solution of Δu−u+u3=0\Delta u-u+u^3=0 in R2\mathbb{R}^2 with finite energy and trivial symmetry group, giving infinitely many such solutions. Three features of the construction are explicitly flagged as incomplete in the paper. First, the number of interstitial bumps m[p,q]m_{[p,q]} placed along an edge must satisfy ℓ3≪m[p,q]≪eγℓ\ell^3\ll m_{[p,q]}\ll e^{\gamma\ell} for a length scale ℓ\ell (Remark 9.1), where the upper bound — a consequence of the interaction estimates behind Proposition 8.1 of that paper — is removable for the cubic but 'for other applications it is not clear that this constraint can be removed'. Second, the construction generalizes to nonlinearities that are not necessarily even: here the operative relaxation is that the oddness f(−u)=−f(u)f(-u)=-f(u) is dropped, so ff may take the more general form f(u)=f1(u+)−f2(u−)f(u)=f_1(u_+)-f_2(u_-), with u±=max⁡(±u,0)u_\pm=\max(\pm u,0) and each fi∈C1,μf_i\in C^{1,\mu} for some μ∈(0,1)\mu\in(0,1) satisfying fi(0)=fi′(0)=0f_i(0)=f_i'(0)=0 and admitting a unique positive radially symmetric solution wiw_i of Δwi−wi+fi(wi)=0\Delta w_i-w_i+f_i(w_i)=0 that decays exponentially and is nondegenerate, i.e. Ker(Δ−1+fi′(wi))∩L∞(RN)=Span{∂x1wi,…,∂xNwi}\mathrm{Ker}(\Delta-1+f_i'(w_i))\cap L^\infty(\mathbb{R}^N)=\mathrm{Span}\{\partial_{x_1}w_i,\dots,\partial_{x_N}w_i\}; this is the class treated in the source's dihedral-symmetry paper. This case requires four distinct types of interaction functions and a considerably more involved reduction which the authors chose not to develop. Third, while the network notions extend naturally to higher dimensions (planar networks can be handled equivariantly in RN\mathbb{R}^N), the construction of genuinely higher-dimensional examples is 'quite difficult' and none are given. Two further features are noted descriptively in the paper rather than posed as open problems: checking closability in concrete examples requires computer-algebra (Mathematica) computations, and the scaling parameters ℓ\ell and κ\kappa are 'somehow quantized' by the integer choices m[p,q]m_{[p,q]}. In a different direction, variational methods produced infinitely many nonradial sign-changing solutions for general nonlinearities: Mederski (Nonlinearity 33 (2020) 6349–6380), working under the Berestycki–Lions conditions (the standard weak structural assumptions on ff — oddness is not required — under which ground states exist), obtains nonradial solutions for dimensions N≥4N\ge 4 and infinitely many when N≠5N\neq 5, and Jeanjean–Lu (Nonlinear Anal. 190 (2020) 111604) treat general nonlinearity as well; those solutions, however, are obtained in symmetry-reducing subspaces (they retain partial symmetry), prescribe no geometry, and do not implement the network construction, so the constructive, prescribed-geometry, trivial-symmetry program remains separate from these existence-in-bulk results.

Problem Statement

Extend the Ao–Musso–Pacard–Wei constructive program for the equation Δu−u+f(u)=0\Delta u-u+f(u)=0 with finite energy beyond the planar cubic case, as one coherent objective — a constructive existence theory of sign-changing finite-energy solutions whose symmetry group is trivial, valid for general nonlinearities and higher spatial dimensions. Concretely, as formulated in the source paper's closing Section 11 and Remark 9.1: (1) carry out the complete finite-dimensional (Lyapunov–Schmidt) reduction for nonlinearities ff that are not necessarily even (that is, for the class f(u)=f1(u+)−f2(u−)f(u)=f_1(u_+)-f_2(u_-) defined in the Background, in which oddness of ff is dropped) — introducing the four required types of interaction functions, proving the analogues of the balance/flexibility/closability framework, and producing explicit worked examples of solutions with trivial symmetry group; (2) construct, for every dimension N≥3N\ge 3, explicit non-symmetric balanced, flexible, closable networks in RN\mathbb{R}^N that are genuinely non-planar (not contained in an affine plane, so the construction is not reducible to the planar one composed with an equivariant embedding) and derive the corresponding solutions; (3) remove the technical upper bound m[p,q]≪eγℓm_{[p,q]}\ll e^{\gamma\ell} on the interstitial bump counts for nonlinearities and applications other than the cubic, where its removal is not known. Item (1) must hold for the whole admissible class f(u)=f1(u+)−f2(u−)f(u)=f_1(u_+)-f_2(u_-) (generality over ff, not a single nonlinearity); item (2) must supply at least one explicit genuinely non-planar example for every dimension N≥3N\ge 3, together with its solution; and item (3) must remove the bound for the whole class of admissible nonlinearities and applications other than the cubic. No item requires an infinite family of solutions: it suffices to exhibit one explicit construction covering each regime the item claims. The single scientific objective uniting these three items is the network-gluing construction itself: each item is an explicitly stated open limitation of the same reduction, and the objective comprises all three enumerated items (1)–(3): the construction must apply, with proofs and explicit examples, outside the cubic planar case, and the technical upper bound of item (3) must be removed for nonlinearities other than the cubic. The nonlinearity extension and the higher-dimensional extension may be established independently: it is enough to handle the non-odd case in the plane and the higher-dimensional case for the cubic separately, and the combined case of a non-odd ff on a genuinely non-planar network in RN\mathbb{R}^N, N≥3N\ge 3, need not be handled separately. The objective's requirement of validity 'for general nonlinearities and higher spatial dimensions' is accordingly the union of these two separately established extensions, not their joint case.

References

  1. Weiwei Ao, Monica Musso, Frank Pacard, Juncheng Wei, Solutions without any symmetry for semilinear elliptic problems, Journal of Functional Analysis 270 (2016), 884–956. DOI: 10.1016/j.jfa.2015.10.015; arXiv:1401.0271. https://doi.org/10.1016/j.jfa.2015.10.015
  2. Monica Musso, Frank Pacard, Juncheng Wei, Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation, Journal of the European Mathematical Society 14 (2012), 1923–1953. DOI: 10.4171/jems/351. https://doi.org/10.4171/jems/351
  3. Weiwei Ao, Monica Musso, Juncheng Wei, Nondegeneracy of nonradial sign-changing solutions to the nonlinear Schrödinger equation, Bulletin de la Société Mathématique de France 147 (2019), 1–48. DOI: 10.24033/bsmf.2774; arXiv:1505.07488. https://doi.org/10.24033/bsmf.2774
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