In this paper, we study the existence of positive and negative solutions for a class of fractional Schrodinger equations. Firstly, we give the definition of fractional Laplace operator and the conditions satisfied by nonlinear terms. This paper introduces the previous progress in this field, and gives the definitions of space and energy functional and the positive and negative parts of function. Then we introduce the main results of this paper. Next, we give the embedding relationship between workspace and Lp space and give the definition of inner product and norm of space. In order to obtain the existence of positive and negative solutions of the equation, we give the definitions of functions u+, u− and functional weak solutions. This paper mainly uses mountain pass lemma to prove. Firstly, according to the embedding relationship of workspace and the condition of nonlinear term f, it is proved that functional I satisfies mountain road structure. Secondly, we need to prove that functional I satisfies the (Cc) condition, we first prove that the sequence un is bounded, then prove that UN has convergent subsequence by the definition of inner product and holder inequality. Therefore, we prove that functional I satisfies the (Cc) condition. Then, we define functional I± and its inner product form to verify that functional I± also has mountain path structure and satisfies (Cc) condition. Finally, taking u+ and u− as experimental functions respectively, it is verified that they are the solutions of functional I. It is obtained that both u+ and u− are the solutions of functional I. Therefore, we get the conclusion.
| Published in | International Journal of Theoretical and Applied Mathematics (Volume 7, Issue 6) |
| DOI | 10.11648/j.ijtam.20210706.11 |
| Page(s) | 85-91 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2021. Published by Science Publishing Group |
Fractional Schrödinger Equation, Mountain Pass Theorem, Positive and Negative Solutions
| [1] | Chang X. Ground state solutions of asymptotically linear fractional Schrödinger equations [J]. Journal of Mathematical Physics, 54 (2013), 1-10. |
| [2] |
Secchi S, Ground state solutions for nonlinear fractional Schrödinger equations in  N [J]. Mathematics, 1 (2013), 1-17.
|
| [3] |
Tang, C. Multiple solutions for Kirchhoff-type equations in N.
|
| [4] | Sofiane K. Multiplicity results for a fractional Schrödinger equation with potentials [J]. Mathematics, 4 (2021), 1-19. |
| [5] | Dong. W, Xu. J, Wei. Z, Existence of weak solutions for a fractional Schrödinger equation [J]. Communications in Nonlinear Science and Numerical Simulation, 22 (2015), 1215-1222. |
| [6] | Zhao L, Zhao F. Positive solutions for Schrödinger- Poisson equations with a critical exponent [J]. Nonlinear Analysis, 2009, 6 (70): 2150-2164. |
| [7] | Brézis H, Lieb E. A relation between pointwise convergence of functions and convergence of functionals [J]. Proceedings of the American Mathematical Society, 1983, 88 (3): 486-490. |
| [8] | Ambrosetti A, Rabinowitz P H. Dual variational methods in critical point theory and applications [J]. Journal of functional Analysis, 1973, 14 (4): 349-381. |
| [9] | Alama S, Tarantello G. On semilinear elliptic equations with indefinite nonlinearities [J]. Calculus of Variations and Partial Differential Equations, 1993, 1 (4): 439-475. |
| [10] | Aubin J P, Ekeland I. Applied nonlinear analysis [M]. Courier Corporation, 2006. |
| [11] | Chen Z, Ji C. Existence and concentration of ground state solutions for a class of fractional Schrödinger equations [J]. Asymptotic Analysis, 2021 (47). |
| [12] | Gan W, Liu S. Multiple positive solutions of a class of Schrdinger-Poisson equation involving indefinite nonlinearity in R3 [J]. Applied Mathematics Letters, 2019, 93. |
| [13] | Liu W, Gan L. Existence of positive solutions with peaks on a Clifford torus for a fractional nonlinear Schrdinger equation [J]. Science China Mathematics, 2019. Yao D U, Tang C L. Positive Ground State Radial Solutions for a Class of Critical Schrödinger Equation [J]. Journal of Southwest China Normal University (Natural Science Edition), 2019. |
| [14] | Severo U B, Germano D. Existence and Nonexistence of Solution for a Class of Quasilinear Schrödinger Equations with Critical Growth [J]. Acta Applicandae Mathematicae, 2021, 173 (1). |
| [15] | Fan H, Feng Z, Yan X. Positive solutions for the fractional Schrödinger equations with logarithmic and critical nonlinearities [J]. 2021. |
| [16] | Shen L. Ground state solutions for a class of gauged Schrödinger equations with subcritical and critical exponential growth [J]. Mathematical Methods in the Applied Sciences, 2020, 43 (2). |
| [17] | Liu Z. Existence of global solutions for the semilinear nonlocal fractional Cauchy problem of the Schrödinger equation [J]. Boundary Value Problems, 2020 (1). |
APA Style
Jianing Wang. (2021). Positive and Negative Solutions of a Class of Fractional Schrödinger Equation. International Journal of Theoretical and Applied Mathematics, 7(6), 85-91. https://doi.org/10.11648/j.ijtam.20210706.11
ACS Style
Jianing Wang. Positive and Negative Solutions of a Class of Fractional Schrödinger Equation. Int. J. Theor. Appl. Math. 2021, 7(6), 85-91. doi: 10.11648/j.ijtam.20210706.11
AMA Style
Jianing Wang. Positive and Negative Solutions of a Class of Fractional Schrödinger Equation. Int J Theor Appl Math. 2021;7(6):85-91. doi: 10.11648/j.ijtam.20210706.11
Share This Article
Twitter
Linked In
Facebook
Copy
Download@article{10.11648/j.ijtam.20210706.11,
author = {Jianing Wang},
title = {Positive and Negative Solutions of a Class of Fractional Schrödinger Equation},
journal = {International Journal of Theoretical and Applied Mathematics},
volume = {7},
number = {6},
pages = {85-91},
doi = {10.11648/j.ijtam.20210706.11},
url = {https://doi.org/10.11648/j.ijtam.20210706.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20210706.11},
abstract = {In this paper, we study the existence of positive and negative solutions for a class of fractional Schrodinger equations. Firstly, we give the definition of fractional Laplace operator and the conditions satisfied by nonlinear terms. This paper introduces the previous progress in this field, and gives the definitions of space and energy functional and the positive and negative parts of function. Then we introduce the main results of this paper. Next, we give the embedding relationship between workspace and Lp space and give the definition of inner product and norm of space. In order to obtain the existence of positive and negative solutions of the equation, we give the definitions of functions u+, u− and functional weak solutions. This paper mainly uses mountain pass lemma to prove. Firstly, according to the embedding relationship of workspace and the condition of nonlinear term f, it is proved that functional I satisfies mountain road structure. Secondly, we need to prove that functional I satisfies the (Cc) condition, we first prove that the sequence un is bounded, then prove that UN has convergent subsequence by the definition of inner product and holder inequality. Therefore, we prove that functional I satisfies the (Cc) condition. Then, we define functional I± and its inner product form to verify that functional I± also has mountain path structure and satisfies (Cc) condition. Finally, taking u+ and u− as experimental functions respectively, it is verified that they are the solutions of functional I. It is obtained that both u+ and u− are the solutions of functional I. Therefore, we get the conclusion.},
year = {2021}
}
TY - JOUR T1 - Positive and Negative Solutions of a Class of Fractional Schrödinger Equation AU - Jianing Wang Y1 - 2021/12/29 PY - 2021 N1 - https://doi.org/10.11648/j.ijtam.20210706.11 DO - 10.11648/j.ijtam.20210706.11 T2 - International Journal of Theoretical and Applied Mathematics JF - International Journal of Theoretical and Applied Mathematics JO - International Journal of Theoretical and Applied Mathematics SP - 85 EP - 91 PB - Science Publishing Group SN - 2575-5080 UR - https://doi.org/10.11648/j.ijtam.20210706.11 AB - In this paper, we study the existence of positive and negative solutions for a class of fractional Schrodinger equations. Firstly, we give the definition of fractional Laplace operator and the conditions satisfied by nonlinear terms. This paper introduces the previous progress in this field, and gives the definitions of space and energy functional and the positive and negative parts of function. Then we introduce the main results of this paper. Next, we give the embedding relationship between workspace and Lp space and give the definition of inner product and norm of space. In order to obtain the existence of positive and negative solutions of the equation, we give the definitions of functions u+, u− and functional weak solutions. This paper mainly uses mountain pass lemma to prove. Firstly, according to the embedding relationship of workspace and the condition of nonlinear term f, it is proved that functional I satisfies mountain road structure. Secondly, we need to prove that functional I satisfies the (Cc) condition, we first prove that the sequence un is bounded, then prove that UN has convergent subsequence by the definition of inner product and holder inequality. Therefore, we prove that functional I satisfies the (Cc) condition. Then, we define functional I± and its inner product form to verify that functional I± also has mountain path structure and satisfies (Cc) condition. Finally, taking u+ and u− as experimental functions respectively, it is verified that they are the solutions of functional I. It is obtained that both u+ and u− are the solutions of functional I. Therefore, we get the conclusion. VL - 7 IS - 6 ER -
Information
Email: