Asymptotic behavior of solutions to Schrödinger equations near an isolated singularity of the electromagnetic potential

Abstract

Abstract. Asymptotics of solutions to Schrödinger equations with singular magnetic and electric potentials is investigated. By using an Almgren type monotonicity formula, separation of variables, and an iterative Brezis-Kato type procedure, we describe the exact behavior near the singularity of solutions to linear and semilinear (critical and subcritical) elliptic equations with an inverse square electric potential and a singular magnetic potential with homogeneity of order −1.

Unique Continuation Property and Local Asymptotics of Solutions to Fractional Elliptic Equations

“…in the case s = 1) and Hardy-type potentials for different kinds of problems: in [12,14] for Schrödinger equations with electromagnetic potentials and in [13] for Schrödinger equations with inverse square many-particle potentials.…”

“…In the present paper we will follow the latter approach. Furthermore, in the spirit of [12,13,14], the combination of monotonicity methods with blow-up analysis will enable us to prove not only unique continuation but also the precise asymptotics of solutions stated in Theorem 1.1.…”

“…In particular, semilinear fractional elliptic equations involving the Hardy potential were treated in [9], where some existence and nonexistence results were obtained from lower bounds of positive solutions. Adapting the ideas in [12] in the nonlocal case of the present paper, we provide the behavior (therefore regularity) of solutions at the singularity. In particular, the asymptotics we have here show that the lower bound of positive solutions proved in [9] is sharp.…”

Ground State Solutions for Schrödinger Problems with Magnetic Fields and Hardy-Sobolev Critical Exponents

“…Then, we may assume that ( , ) ̸ = (0, 0) for large enough. Recalling (56), (57), and N in (13), there exists > 0 with → 1 as → ∞ such that ( , ) ∈ M. Then, by (60), we obtain that…”

“…Motivated by the seminal work [1], some papers dealt with the Schrödinger equations with nonsingular magnetic field, for example, [2][3][4][5][6][7][8][9][10][11] and references therein. For the singular magnetic potential, we refer to [12,13]. There are a few works about the nonsingular magnetic problems with critical exponents, such as [9,14,15] for Sobolev critical exponent and [16] for Hardy critical exponent.…”