Pith Number

pith:JVZA4GOJ

pith:2026:JVZA4GOJQQYUUEEPBVLEXTGCIE

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On the Minimax Bifurcation Formula

Y. Sh. Il'yasov

A variational minimax method identifies the critical parameter for saddle-node bifurcations directly as an extremal value of an extended Rayleigh quotient.

arxiv:2605.17331 v1 · 2026-05-17 · math.AP · math-ph · math.MP

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Claims

C1strongest claim

We prove an abstract minimax bifurcation formula, establish the existence and characterization of weak saddle-node bifurcation points, and justify finite-dimensional Galerkin approximations.

C2weakest assumption

The critical parameter for saddle-node bifurcations can be characterized variationally as an extremal value of an extended Rayleigh quotient in abstract nonlinear equations, including those without classical variational structure.

C3one line summary

Proves an abstract minimax bifurcation formula for detecting and characterizing weak saddle-node bifurcation points in nonlinear equations, with Galerkin approximations and applications to non-variational elliptic systems.

References

40 extracted · 40 resolved · 0 Pith anchors

[1] A. Ambrosetti, H. Brezis, G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994) 519–543 1994

[2] I. Babuška, J. Osborn, Eigenvalue problems, in: Handbook of Numerical Analysis, Vol. II, North-Holland, Amsterdam, 1991, pp. 641–787 1991

[3] A. Bagirov, N. Karmitsa, M.M. Mäkelä, Introduction to Nonsmooth Opti- mization: Theory, Practice and Software, Springer, Cham, 2014 2014

[4] H. Berestycki, I. Capuzzo-Dolcetta, L. Nirenberg, Variational methods for indefinite superlinear homogeneous elliptic problems, NoDEA Nonlinear Differential Equations Appl. 2 (1995) 553–572 1995

[5] H. Berestycki, L. Nirenberg, S.R.S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math. 47 (1994) 47–92 1994

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:03:52.472538Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

4d720e19c984314a108f0d564bccc24126222bb05611c5c32e87faaa2cd16dab

Aliases

arxiv: 2605.17331 · arxiv_version: 2605.17331v1 · doi: 10.48550/arxiv.2605.17331 · pith_short_12: JVZA4GOJQQYU · pith_short_16: JVZA4GOJQQYUUEEP · pith_short_8: JVZA4GOJ

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/JVZA4GOJQQYUUEEPBVLEXTGCIE \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 4d720e19c984314a108f0d564bccc24126222bb05611c5c32e87faaa2cd16dab

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "8cc59704ae0febe5bdbea85bcb096e03d4f8dbaf2987cf3ad768596b5c4dc255",
    "cross_cats_sorted": [
      "math-ph",
      "math.MP"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.AP",
    "submitted_at": "2026-05-17T08:48:09Z",
    "title_canon_sha256": "aff6c80d4696c618afe31bba3f46e6ce945b2addea3e28b9b21fba67802eaebf"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.17331",
    "kind": "arxiv",
    "version": 1
  }
}