On criticality and ground states of second order elliptic equations. II (Q750756)

[For part I see ibid. 80, No.2, 237-250 (1989; Zbl 0697.35036).] Let P be a linear second order elliptic operator defined in a domain \(\Omega \subseteq {\mathbb{R}}^ n\). Let \({\mathcal C}_ P(\Omega)\) be the convex cone of all classical positive solutions of the equation \(Pu=0\) in \(\Omega\). P is said to be a subcritical operator in \(\Omega\) if P admits a positive (minimal) Green function \(G^{\Omega}_ P(x,y)\) in \(\Omega\). P is said to be supercritical in \(\Omega\) if \({\mathcal C}_ P(\Omega)=\emptyset\). Finally, P is a critical operator in \(\Omega\) if P is neither subcritical nor supercritical in \(\Omega\). Recall that P is critical in \(\Omega\) if and only if P admits a ground state (in the sense of Agmon) with eigenvalue zero. In this article we prove the following Theorem. Let P be a critical operator in \(\Omega \subseteq {\mathbb{R}}^ n\), and let \(P^*\) be its formal adjoint. Denote by \(\phi\) (x) (\(\psi\) (x)) the ground state for the equation \(Pu=0\) \((P^*u=0)\) in \(\Omega\). Let \(W\in C^{\alpha}_ 0(\Omega)\), \(W\not\equiv 0\), where \(0<\alpha \leq 1\). Then there exists \(t_ 0>0\) such that \(P+tW\) is subcritical in \(\Omega\) for all \(0<t<t_ 0\) if and only if \(\int_{\Omega}\psi (x)W(x)\phi (x)dx>0.\) This theorem extends a well known theorem of B. Simon [Theorem XIII.110 in \textit{M. Reed} and \textit{B. Simon}, Methods of modern mathematical physics. IV: Analysis of operators (1978; Zbl 0401.47001)].