Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 44, 2019, 1101-1110

EXISTENCE OF WEAK SOLUTIONS FOR THE SCHRÖDINGER EQUATION AND ITS APPLICATION

Jinjin Huang

Zhoukou Normal University, School of Mathematics and Statistics
Zhoukou 466001, P.R. China; huangjinjin 'at' zknu.edu.cn

Abstract. In this paper, we are concerned with the existence of weak solutions for the Schrödinger equation with sign-changing potential in a smooth cone. For solving the Dirichlet boundary-value problem with respect to the Schrödinger operator, we prove the existence of at least one weak solution using changes of Schrödingerean harmonic measure, the energy estimate method and refined inequality technique. Due to the fact that the nonlinearity is allowed to change sign in our formulation, and the novelty of the boundary conditions, these results are new for discrete and arbitrary time scales. As an application, concentration results are also investigated.

2010 Mathematics Subject Classification: Primary 35J05, 35J10, 35C15.

Key words: Schödinger equations, sign-changing potential, cone.

Reference to this article: J. Huang: Existence of weak solutions for the Schrödinger equation and its application. Ann. Acad. Sci. Fenn. Math. 44 (2019), 1101-1110.

Full document as PDF file

https://doi.org/10.5186/aasfm.2019.4452