Sharp counterexamples related to the De Giorgi conjecture in dimensions 4≤𝑛≤8

Moradifam, Amir

doi:10.1090/S0002-9939-2013-12040-X

Sharp counterexamples related to the De Giorgi conjecture in dimensions $4\leq n \leq 8$
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Proc. Amer. Math. Soc. 142 (2014), 199-203 Request permission

Abstract:

In this note, we show that in dimensions $n\geq 4$ there exists a smooth bounded potential $V$ such that $(\Delta +V)w=0$ has a positive solution $u$ as well as a bounded sign-changing solution $v$ satisfying \begin{equation*} \int _{B_{R}}v^2\leq CR^3 \ \ \ \ \forall R>0, \end{equation*} for some $C>0$ independent of $R$. This in particular implies that the Ambrosio-Cabré proof of the De Giorgi conjecture in dimension $n=3$ cannot be extended to dimensions $4\leq n \leq 8$. We also answer an open question of L. Moschini [L. Moschini, New Liouville theorems for linear second order degenerate elliptic equations in divergence form, Ann. Inst. H. Poincarè Anal. Non Linéaire 22 (2005), 11-23].

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