Normalized solutions of Schrödinger equations with potentially bounded measures
Let \(\mu\) be a signed Radon measure on a domain \(X\subset \mathbb{R}^n\), \(n\geq 1\), with the Green function \(G_X\). Let \(\mu\) be potentially bounded, i.e., the potential \(G_X^{1_B| \mu| }\) is bounded for every ball \(B\subset X\). In [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 28, 413--470 (1999; Zbl 0940.35063)], the present author answered the question of when positive finely continuous solutions of \(\Delta h-h\mu =0\) (\(\equiv \mu\)-harmonic functions) satisfy the Harnack inequality. In particular, it was shown that \(\mu\)-harmonic functions can be very discontinuous. The paper under review is a continuation of that work. Here the author shows that under some general assumptions on \(G\) relative \(\mu\)-harmonic functions \(h/h_0\), where \(h\) and \(h_0\) are \(\mu\)-harmonic and \(h_0\) is strictly positive, are \(\rho\)-Hölder continuous with respect to the natural quasimetric \[ \rho (x,y)=(G(x,y)^{-1}+G(y,x)^{-1})/2. \] The author also studies some perturbations leading to a Brelot space admitting the Green function to be comparable (locally or globally) with \(G_X\).
scientific article; zbMATH DE number 2093712
scientific article