ENTIRE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS

By A. Gladkov, Nickolai Slepchenkov. 2004

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Dual variational methods and nonvanishing for the nonlinear Helmholtz equation (2014, background): “l-valued solutions uof (1). We note that real-valued solutions of (1) with the decay property u(x) → 0 as |x| → ∞ have been studied extensively in the case where ∂uf(·,0) ≡ 0 on RN and λ≥ 0, see e.g. [19, 25, 27] and the references therein. On the contrary, very little is known in the case λ<0, w”
Regularity of flat level sets in phase transitions (2009, background): “Is it true that all the level sets of u are hyperplanes, at least if n ≤ 8? The conjecture was proved for n = 2 by Ghoussoub and Gui in [12] and for n = 3 by Ambrosio and Cabre in [2].”