A Counterexample to the Nodal Domain Conjecture and a Related Semilinear Equation

DOI10.2307/2045874zbMath0652.35085OpenAlexW4233798124WikidataQ125044226 ScholiaQ125044226MaRDI QIDQ3797611

Wei-Ming Ni, Chang-Shou Lin

Publication date: 1988

Full work available at URL: https://doi.org/10.2307/2045874

zbMATH Keywords

eigenvalue problemNodal Domain Conjecture

Mathematics Subject Classification ID

Boundary value problems for second-order elliptic equations (35J25) General topics in linear spectral theory for PDEs (35P05) Nonlinear elliptic equations (35J60)

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Cites Work

• Symmetry and related properties via the maximum principle
• Eigenfunctions and nodal sets
• On two conjectures in the fixed membrane eigenvalue problem
• Uniqueness and nonuniqueness for positive radial solutions of Δu + f(u, r) = 0
• Unnamed Item
• Unnamed Item