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Xing, Yifan; Zhao, Jinhui

Existence and quantum calculus of weak solutions for a class of two-dimensional Schrödinger equations in \(\mathbb{C}_+\). (English) Zbl 1499.35208

Bound. Value Probl. 2018, Paper No. 59, 10 p. (2018).
Summary: The aim of this paper is to investigate the existence of weak solutions for a two-dimensional Schrödinger equation with a singular potential in \(\mathbb{C}_+\). Under appropriate assumptions on the nonlinearity, we introduce a new type of quantum calculus via the Morse theory and variational methods. By applying Schrödinger type inequalities and the well-known Banach fixed point theorem in conjunction with the technique of measures of weak noncompactness, the new and more accurate estimations for boundary behavior of them are also deduced.

MSC:

35J10 Schrödinger operator, Schrödinger equation
35D30 Weak solutions to PDEs
35Q40 PDEs in connection with quantum mechanics

Keywords:

two-dimensional Schrödinger equation; Schrödinger type inequality; boundary behavior; singular solution
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\textit{Y. Xing} and \textit{J. Zhao}, Bound. Value Probl. 2018, Paper No. 59, 10 p. (2018; Zbl 1499.35208)
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