Infinitely Many Small Energy Solutions to Schrödinger-Kirchhoff Type Problems Involving the Fractional <inline-formula><math display="inline"><semantics><mrow><mi>r</mi><mo>(</mo><mo>·</mo><mo>)</mo></mrow></semantics></math></inline-formula>-Laplacian in <inline-formula><math display="inline"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mi>N</mi></msup></mrow></semantics></math></inline-formula>

This paper is concerned with the existence result of a sequence of infinitely many small energy solutions to the fractional <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo>(...

Full description

Bibliographic Details
Published in:	Fractal and Fractional
Main Author:	Yun-Ho Kim
Format:	Article
Language:	English
Published:	MDPI AG 2023-02-01
Subjects:	fractional <i>r</i>(<i>x</i>)-Laplacian variable exponent fractional Sobolev spaces weak solution dual fountain theorem
Online Access:	https://www.mdpi.com/2504-3110/7/3/207

Similar Items

General New Results on <inline-formula><math display="inline"><semantics><mrow><mrow><mo>(</mo><mrow><mi>ϕ</mi><mo>,</mo><mi mathvariant="script">F</mi></mrow><mo>)</mo></mrow><mo>−</mo></mrow></semantics></math></inline-formula>Contractions in <inline-formula><math display="inline"><semantics><mrow><mi>b</mi><mo>−</mo></mrow></semantics></math></inline-formula>Metric-like-Spaces
by: Kastriot Zoto, et al.
Published: (2023-07-01)

Riemann–Hilbert Problems and Soliton Solutions of Type (<inline-formula><math display="inline"><semantics><mrow><msup><mi>λ</mi><mo>∗</mo></msup></mrow></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><mrow><mo>−</mo><msup><mi>λ</mi><mo>∗</mo></msup></mrow></semantics></math></inline-formula>) Reduced Nonlocal Integrable mKdV Hierarchies
by: Wen-Xiu Ma
Published: (2022-03-01)

The Dynamics of <inline-formula><math display="inline"><semantics><mrow><msubsup><mrow><mi>NO</mi></mrow><mn>3</mn><mo>−</mo></msubsup></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><msubsup><mrow><mi>NH</mi></mrow><mn>4</mn><mo>+</mo></msubsup></mrow></semantics></math></inline-formula> Uptake in Duckweed Are Coordinated with the Expression of Major Nitrogen Assimilation Genes
by: Yuzhen Zhou, et al.
Published: (2021-12-01)

Finiteness of <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="script">N</mi><mo>=</mo><mn>4</mn></mrow></semantics></math></inline-formula> Super-Yang–Mills Effective Action in Terms of Dressed <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="script">N</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula> Superfields
by: Igor Kondrashuk, et al.
Published: (2023-11-01)

On Modified Interpolative Almost <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="script">E</mi><mo>−</mo></mrow></semantics></math></inline-formula>Type Contraction in Partial Modular <inline-formula><math display="inline"><semantics><mrow><mi>b</mi><mo>−</mo></mrow></semantics></math></inline-formula>Metric Spaces
by: Dilek Kesik, et al.
Published: (2023-07-01)

Multicorn Sets of <inline-formula><math display="inline"><semantics><mrow><msup><mrow><mover accent="true"><mi>z</mi><mo>¯</mo></mover></mrow><mi>k</mi></msup><mo>+</mo><msup><mi>c</mi><mi>m</mi></msup></mrow></semantics></math></inline-formula> via S-Iteration with <i>h</i>-Convexity
by: Asifa Tassaddiq, et al.
Published: (2023-06-01)

Asymptotic Behaviours for an Index Whittaker Transform over <inline-formula><math display="inline"><semantics><mrow><msup><mi mathvariant="script">E</mi><mo>′</mo></msup><mrow><mo>(</mo><msub><mi mathvariant="double-struck">R</mi><mo>+</mo></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula>
by: Emilio R. Negrín, et al.
Published: (2025-02-01)

<inline-formula><math display="inline"><semantics><mi mathvariant="script">O</mi></semantics></math></inline-formula>-Regular Mappings on <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="double-struck">C</mi><mrow><mo>(</mo><mi mathvariant="double-struck">C</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>: A Structured Operator–Theoretic Framework
by: Ji Eun Kim
Published: (2025-10-01)

Scattering in the Energy Space for Solutions of the Damped Nonlinear Schrödinger Equation on <inline-formula><math display="inline"><semantics><mrow><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mi>d</mi></msup><mo>×</mo><mi mathvariant="double-struck">T</mi></mrow></semantics></math></inline-formula>
by: Taim Saker, et al.
Published: (2025-06-01)

Conversion of Unweighted Graphs to Weighted Graphs Satisfying Properties <inline-formula><math display="inline"><semantics><mi mathvariant="monospace">R</mi></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><mo>−</mo><mi mathvariant="monospace">SR</mi></mrow></semantics></math></inline-formula>
by: Xiaolong Shi, et al.
Published: (2023-11-01)

Curvature Pinching Problems for Compact Pseudo-Umbilical PMC Submanifolds in <inline-formula><math display="inline"><semantics><mrow><msup><mi mathvariant="double-struck">S</mi><mi>m</mi></msup><mo>(</mo><mi>c</mi><mo>)</mo><mo>×</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>
by: Wang-Hua Qiu, et al.
Published: (2023-12-01)

Understanding the Functional Expression of <inline-formula><math display="inline"><semantics><mrow><msup><mrow><mi>Na</mi></mrow><mo>+</mo></msup></mrow></semantics></math></inline-formula>-Coupled SLC4 Transporters in the Renal and Nervous Systems: A Review
by: Le Du, et al.
Published: (2021-09-01)

A Pinching Theorem for Compact Minimal Submanifolds in Warped Products <inline-formula><math display="inline"><semantics><mrow><mi>I</mi><msub><mo>×</mo><mi>f</mi></msub><msup><mi mathvariant="double-struck">S</mi><mi>m</mi></msup><mrow><mo>(</mo><mi>c</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>
by: Xin Zhan, et al.
Published: (2020-08-01)

Inextensible Flows of Null Cartan Curves in Minkowski Space <inline-formula><math display="inline"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mrow><mn>2</mn><mo>,</mo><mn>1</mn></mrow></msup></mrow></semantics></math></inline-formula>
by: Samah Gaber, et al.
Published: (2023-02-01)

Local Convergence of the Continuous and Semi-Discrete Wavelet Transform in <inline-formula><math display="inline"><mrow><msup><mi>L</mi><mi>p</mi></msup><mo>(</mo><mi mathvariant="double-struck">R</mi><mo>)</mo></mrow></math></inline-formula>
by: Jaime Navarro-Fuentes, et al.
Published: (2021-03-01)

Best Proximity Point for <inline-formula><math display="inline"><semantics><mrow><msub><mo>Γ</mo><mi>τ</mi></msub><mi mathvariant="script">F</mi></mrow></semantics></math></inline-formula>-Fuzzy Proximal Contraction
by: Uma Devi Patel, et al.
Published: (2023-02-01)

On the Truncated Multidimensional Moment Problems in <inline-formula><math display="inline"><semantics><mrow><msup><mi mathvariant="double-struck">C</mi><mi>n</mi></msup></mrow></semantics></math></inline-formula>
by: Sergey Zagorodnyuk
Published: (2022-01-01)

Is the Non-Pointness of the Electron Observable in <inline-formula><math display="inline"><semantics><mrow><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup></mrow></semantics></math></inline-formula> Annihilation at Center-of-Mass Energies 55–207 GeV?
by: Yutao Chen, et al.
Published: (2023-07-01)

Application of the Exp<inline-formula><math display="inline"><semantics><mrow><mfenced><mrow><mo>−</mo><mi>φ</mi><mfenced><mi>ξ</mi></mfenced></mrow></mfenced></mrow></semantics></math></inline-formula>-Expansion Method to Find the Soliton Solutions in Biomembranes and Nerves
by: Attia Rani, et al.
Published: (2022-09-01)

A Classification of Elements of Function Space <i>F</i>(<inline-formula><math display="inline"><semantics><mrow><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>,<inline-formula><math display="inline"><semantics><mrow><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>)
by: Mohsen Soltanifar
Published: (2023-08-01)

On (<inline-formula><math display="inline"><semantics><mrow><mover><mi>m</mi><mo>¯</mo></mover></mrow></semantics></math></inline-formula>, <i>m</i>)-Conformal Mappings
by: Branislav M. Randjelović, et al.
Published: (2025-08-01)

Conceptual Neighborhood Graphs of Topological Relations in <inline-formula><math display="inline"><semantics><mrow><msup><mrow><mi mathvariant="double-struck">Z</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></semantics></math></inline-formula>
by: Brendan Patrick Hall, et al.
Published: (2025-03-01)

Hořava–Lifshitz <inline-formula><math display="inline"><semantics><mrow><mi>F</mi><mo>(</mo><mover accent="true"><mi>R</mi><mo>¯</mo></mover><mo>)</mo></mrow></semantics></math></inline-formula> Theories and the Swampland
by: Hugo García-Compeán, et al.
Published: (2023-10-01)

Ar(c)tivism and Policing: Unveiling the Theatrics of Justice and Resistance in Nigeria’s S<inline-formula><math display="inline"><semantics><mrow><mover><mi mathvariant="bold">ọ</mi><mo mathvariant="bold">̀</mo></mover></mrow></semantics></math></inline-formula>r<inline-formula><math display="inline"><semantics><mrow><mover><mi mathvariant="bold">ọ</mi><mo mathvariant="bold">̀</mo></mover></mrow></semantics></math></inline-formula>-Sókè Movement
by: Friday Gabriel, et al.
Published: (2025-04-01)

Generalized Finslerian Wormhole Models in <inline-formula><math display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi mathvariant="script">R</mi><mo>,</mo><mi mathvariant="script">T</mi><mo>)</mo></mrow></semantics></math></inline-formula> Gravity
by: B. R. Yashwanth, et al.
Published: (2024-08-01)

DNA Code from Cyclic and Skew Cyclic Codes over <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">F</mi><mn>4</mn></msub><mrow><mo>[</mo><mi>v</mi><mo>]</mo></mrow><mo>/</mo><mrow><mo>〈</mo><msup><mi>v</mi><mn>3</mn></msup><mo>〉</mo></mrow></mrow></semantics></math></inline-formula>
by: Om Prakash, et al.
Published: (2023-01-01)

G-Fractional Diffusion on Bounded Domains in <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup></semantics></math></inline-formula>
by: Luca Angelani, et al.
Published: (2023-03-01)

Multiplicity Results of Solutions to Non-Local Magnetic Schrödinger–Kirchhoff Type Equations in <inline-formula><math display="inline"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mi>N</mi></msup></mrow></semantics></math></inline-formula>
by: Kisoeb Park
Published: (2022-01-01)

Neutron Stars in the Context of <i>f</i>(<inline-formula><math display="inline"><semantics><mrow><mi mathvariant="double-struck">T</mi><mo>,</mo><mi mathvariant="script">T</mi></mrow></semantics></math></inline-formula>) Gravity
by: Clésio E. Mota, et al.
Published: (2023-05-01)

Effects on Iron Metabolism and System <inline-formula><math display="inline"><semantics><mrow><msup><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi mathvariant="bold-italic">c</mi></msub></mrow><mo mathvariant="bold">−</mo></msup></mrow></semantics></math></inline-formula> /GPX4 Pathway from Hydroquinone Suggest Ferroptosis of Jurkat Cells
by: Nana Liu, et al.
Published: (2024-08-01)

<inline-formula><math display="inline"><semantics><mrow><mi>S</mi><mi>L</mi><mo>(</mo><mn>2</mn><mo>,</mo><mi mathvariant="double-struck">C</mi><mo>)</mo></mrow></semantics></math></inline-formula> Scheme Processing of Singularities in Quantum Computing and Genetics
by: Michel Planat, et al.
Published: (2023-02-01)

A Novel Formulation of the Fractional Derivative with the Order <inline-formula><math display="inline"><semantics><mrow><mi>α</mi><mo>≥</mo><mn>0</mn></mrow></semantics></math></inline-formula> and without the Singular Kernel
by: Hassan Kamil Jassim, et al.
Published: (2022-11-01)

Structural Properties of The Clifford–Weyl Algebra <inline-formula><math display="inline"><semantics><msubsup><mi>𝒜</mi><mrow><mi>q</mi></mrow><mo>±</mo></msubsup></semantics></math></inline-formula>
by: Jia Zhang, et al.
Published: (2025-09-01)

Digital Relations in <inline-formula><math display="inline"><semantics><mrow><msup><mrow><mi mathvariant="double-struck">Z</mi></mrow><mrow><mn>1</mn></mrow></msup></mrow></semantics></math></inline-formula>: Discretized Time and Rasterized Lines
by: Matthew P. Dube
Published: (2025-08-01)

On the Generalized Fractional Convection–Diffusion Equation with an Initial Condition in <inline-formula><math display="inline"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mi mathvariant="bold-italic">n</mi></msup></mrow></semantics></math></inline-formula>
by: Chenkuan Li, et al.
Published: (2025-05-01)

<inline-formula><math display="inline"><semantics><mrow><mi>U</mi><mo>(</mo><mi mathvariant="fraktur">h</mi><mo>)</mo></mrow></semantics></math></inline-formula>-Free Modules over the Lie Algebras of Differential Operators
by: Munayim Dilxat, et al.
Published: (2022-05-01)

On Several Parameters of Super Line Graph <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="script">L</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>
by: Jiawei Meng, et al.
Published: (2023-03-01)

The Connection between Der(<inline-formula><math display="inline"><semantics><mrow><msubsup><mi mathvariant="bold">U</mi><mi mathvariant="bold">q</mi><mo mathvariant="bold">+</mo></msubsup><mrow><mo mathvariant="bold">(</mo><mi mathvariant="bold">g</mi><mo mathvariant="bold">)</mo></mrow></mrow></semantics></math></inline-formula>) and Der(<inline-formula><math display="inline"><semantics><mrow><msubsup><mi mathvariant="bold">U</mi><mrow><mi mathvariant="bold">r</mi><mo mathvariant="bold">,</mo><mi mathvariant="bold">s</mi></mrow><mo mathvariant="bold">+</mo></msubsup><mrow><mo mathvariant="bold">(</mo><mi mathvariant="bold">g</mi><mo mathvariant="bold">)</mo></mrow></mrow></semantics></math></inline-formula>)
by: Yongyue Zhong, et al.
Published: (2024-08-01)

On the Vacuum Structure of the <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="script">N</mi><mo>=</mo><mn>4</mn></mrow></semantics></math></inline-formula> Conformal Supergravity
by: Ioannis Dalianis, et al.
Published: (2021-10-01)

<inline-formula><math display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi mathvariant="script">R</mi><mo>,</mo><mi mathvariant="script">T</mi><mo>)</mo></mrow></semantics></math></inline-formula>-Gravity Model with Perfect Fluid Admitting Einstein Solitons
by: Mohd Danish Siddiqi, et al.
Published: (2021-12-01)