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Existence and multiplicity of solutions for fractional Schrödinger-p-Kirchhoff equations in ℝ N

  • Huo Tao ORCID logo , Lin Li ORCID logo EMAIL logo and Patrick Winkert ORCID logo
Published/Copyright: March 26, 2024

Abstract

This paper concerns the existence and multiplicity of solutions for a nonlinear Schrödinger–Kirchhoff-type equation involving the fractional p-Laplace operator in N . Precisely, we study the Kirchhoff-type problem

( a + b 2 N | u ( x ) - u ( y ) | p | x - y | N + s p d x d y ) ( - Δ ) p s u + V ( x ) | u | p - 2 u = f ( x , u ) in  N ,

where a , b > 0 , ( - Δ ) p s is the fractional p-Laplacian with 0 < s < 1 < p < N s , V : N and f : N × are continuous functions while V can have negative values and f fulfills suitable growth assumptions. According to the interaction between the attenuation of the potential at infinity and the behavior of the nonlinear term at the origin, using a penalization argument along with L -estimates and variational methods, we prove the existence of a positive solution. In addition, we also establish the existence of infinitely many solutions provided the nonlinear term is odd.


Communicated by Christopher D. Sogge


Award Identifier / Grant number: 12361024

Funding statement: This work is supported by Research Fund of National Natural Science Foundation of China (No. 12361024), the Team Building Project for Graduate Tutors in Chongqing (No. yds223010) and Innovative Project of Chongqing Technology and Business University (No. CYS23567).

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Received: 2023-10-31
Revised: 2024-02-27
Published Online: 2024-03-26
Published in Print: 2025-02-01

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