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

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.


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


[1] Gil’, A., Nogin, V.: Complex powers of a differential operator related to the Schrödinger operator. Vladikavkaz. Mat. Zh. 19(1), 18-25 (2017) · Zbl 07259885
[2] Nakao, M.L., Narazaki, T.: Existence and decay of solutions of some nonlinear wave equations in noncylindrical domains. Math. Rep. Coll. Gen. Educ. Kyushu Univ. 11(2), 117-125 (1978) · Zbl 0393.35042
[3] Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions, 10th printing. National Bureau of Standards, Washington (1972) · Zbl 0543.33001
[4] Bardos, C., Chen, G.: Control and stabilization for the wave equation. III: domain with moving boundary. SIAM J. Control Optim. 19, 123-138 (1981) · Zbl 0461.93038
[5] Bresters, D.W.: On the equation of Euler-Poisson-Darboux. SIAM J. Math. Anal. 1, 31-41 (1973) · Zbl 0248.35077
[6] Ferreira, J.: Nonlinear hyperbolic-parabolic partial differential equation in noncylindrical domain. Rend. Circ. Mat. Palermo 44(1), 135-146 (1995) · Zbl 0830.35020
[7] Medeiros, L.A.: Nonlinear wave equations in domains with variable boundary. Arch. Ration. Mech. Anal. 47, 47-58 (1972) · Zbl 0239.35066
[8] Li, Z.: Boundary behaviors of modified Green’s function with respect to the stationary Schrödinger operator and its applications. Bound. Value Probl. 2015, Article ID 242 (2015) · Zbl 1334.35023
[9] Schwartz, L.: Théorie des Distributions. Hermann, Paris (1978) · Zbl 0399.46028
[10] Marion, O.: Hilbert transform, Plemelj relation, and Fourier transform of distributions. SIAM J. Math. Anal. 4(4), 656-670 (1973) · Zbl 0236.46045
[11] Pandey, J.: The Hilbert Transform of Schwartz Distributions and Applications. Wiley, New York (1996) · Zbl 0844.46022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.