A pointwise lower bound for positive solutions of a Schrödinger equation in \(\mathbb{R}^ N\)
It is first shown that \((-\Delta+ q(x))u= f\geq 0\), \(f\neq 0\), implies \(u\geq c \varphi_1\), where \(\varphi_1\) is the first eigenfunction and \(c>0\) is a constant. Here, the potential has to grow at infinity, typically like \(|x|^{z +\varepsilon}\) with \(\varepsilon>0\), and it has to be a perturbation of a radially symmetric function. For the proof, weak and strong maximum principles in exterior domains are used. The result is sharp in the sense that it does not extend to the harmonic oscillator, i.e. \(\varepsilon=0\), as is demonstrated, too. Finally, applications to semilinear Schrödinger equations are given.
scientific article; zbMATH DE number 1000154