Sharp counterexamples related to the De Giorgi conjecture in dimensions \(4\leq n \leq 8\) (Q2862183)

The following problem concerning Liouville comparison principle is known to be closely related to the celebrated De Giorgi conjecture:NEWLINENEWLINELet \(\varphi\) be a strictly positive function on \(\mathbb{R}^n\), and consider the divergence form operator \(L=- \nabla\cdot (\varphi^2\nabla)\) on \(\mathbb{R}^n\). Suppose that the equation \(L u=0\) in \(\mathbb{R}^n\) admits a nonzero (sub)solution \(\psi\) with \(\psi_+\neq 0\). Find a natural growth condition on \(\varphi\psi\) that insures that \(\psi\) is the constant function.NEWLINENEWLINEFor example, by a result of the reviewer [Commun. Math. Phys. 272, No. 1, 75--84 (2007; Zbl 1135.35021)] it follows that for \(n\geq 2\) if \(|(\varphi\psi)(x)|= O(|x|^{(2-n)/2})\) as \(x\to \infty\), then \(\psi\) is the constant function. \textit{L.~Moschini} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 22, No. 1, 11--23 (2005; Zbl 1130.35070)] improved a result of \textit{L. Ambrosio} and \textit{X. Cabré} [J. Am. Math. Soc. 13, No. 4, 725--739 (2000; Zbl 0968.35041)], and proved that \(\psi=\mathrm{constant}\) if \(\int_{B_R} (\varphi\psi)^2 \,\mathrm{d}x\leq CR^2\log R\) as \(R\to \infty\).NEWLINENEWLINEThe author shows that Moschini condition is sharp. He constructs for \(n\geq 3\) a nonconstant positive (sub)solution \(\psi\) of the equation \(L u=0\) in \(\mathbb{R}^n\) that satisfies the growth condition \(\int_{B_R} (\varphi\psi)^2 \,\mathrm{d}x\leq CR^2(\log R)^2\) as \(R\to \infty\).NEWLINENEWLINESimilarly, the author constructs in dimensions \(n\geq 4\) a smooth bounded potential \(V\) such that the equation \((-\Delta + V )w = 0\) in \(\mathbb{R}^n\) admits a positive solution \(u\) and a bounded sign-changing solution \(v\) satisfying \(\int_{B_R} v^2(x)\,\mathrm{d}x\leq CR^3\), where \(C > 0\) is a constant independent of \(R\).