## 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.

### MSC:

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

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