Existence of Ground State Solutions for a Class of Non-Autonomous Fractional Kirchhoff Equations
Abstract
:1. Introduction
- with .
- and on a positive measure set.
- and on a positive measure set.
2. Preliminary Results
- is a regular manifold diffeomorphic to the sphere of .
- I is bounded from below on .
- u is a critical point of I if and only if u is a critical point of I constrained on .
3. Proof of Theorem 1
- (i)
- (ii)
- There exists a solution , with , a number , and with for each , and for (as ), nontrivial solutions of the problem (7), such that
- (a.1)
- ;
- (a.2)
- ;
- (a.3)
- (b.1)
- ,
- (b.2)
- ,
- (b.3)
- (c.1)
- ;
- (c.2)
- ;
- (c.3)
- .
4. Proof of Theorem 2
- For all , there is a unique such that for and for , where . Moreover, and .
- There is such that , where .
- For each compact subset W of , there exists such that for all .
- If , then for any .
- (i)
- is a sequence of I if is a sequence of Φ; is a sequence of Φ if is a bounded sequence of .
- (ii)
- w is a critical point of Φ if and only if is a nontrivial point of . Moreover, the corresponding values of Φ and coincide and .
- (iii)
- A minimizer of on is a ground state of Equation (1).
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Gu, G.; Mu, C.; Yang, Z. Existence of Ground State Solutions for a Class of Non-Autonomous Fractional Kirchhoff Equations. Fractal Fract. 2024, 8, 113. https://doi.org/10.3390/fractalfract8020113
Gu G, Mu C, Yang Z. Existence of Ground State Solutions for a Class of Non-Autonomous Fractional Kirchhoff Equations. Fractal and Fractional. 2024; 8(2):113. https://doi.org/10.3390/fractalfract8020113
Chicago/Turabian StyleGu, Guangze, Changyang Mu, and Zhipeng Yang. 2024. "Existence of Ground State Solutions for a Class of Non-Autonomous Fractional Kirchhoff Equations" Fractal and Fractional 8, no. 2: 113. https://doi.org/10.3390/fractalfract8020113
APA StyleGu, G., Mu, C., & Yang, Z. (2024). Existence of Ground State Solutions for a Class of Non-Autonomous Fractional Kirchhoff Equations. Fractal and Fractional, 8(2), 113. https://doi.org/10.3390/fractalfract8020113