Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 43, 2018, 991-1021

THE EXISTENCE AND CONCENTRATION BEHAVIOR OF GROUND STATE SOLUTIONS FOR FRACTIONAL SCHRÖDINGER–KIRCHHOFF TYPE EQUATIONS

Huifang Jia and Gongbao Li

Central China Normal University, School of Mathematics and Statistics
Wuhan, 430079, P.R. China; hf_jia 'at' mails.ccnu.edu.cn

Normal University, School of Mathematics and Statistics
Wuhan, 430079, P.R. China; ligb 'at' mail.ccnu.edu.cn

Abstract. In this paper, we study the following fractional Schrödinger Kirchhoff type problem

(Q_ε) L_ε^su = K(x)f(u) in R³, u ∈ H^s(R³),

where L_ε^s is a nonlocal operator defined by

L_ε^su = M(1/ε^3–2s ∫∫_{R3 × R3}|u(x) – u(y)|²/|x – y|^3+2s dx dy + 1/ε³ ∫_R3V(x)u² dx)[ε^2s(–Δ)^su + V(x)u],

ε is a small positive parameter, 3/4 < s < 1 is a fixed constant, the operator (–Δ)^s is the fractional Laplacian of order s, M, V, K and f are continuous functions. Under proper assumptions on M, V, K and f, we prove the existence and concentration phenomena of solutions of the problem (Q_ε). With minimax theorems and the Ljusternik–Schnirelmann theory, we also obtain multiple solutions of problem (Q_ε) by employing the topology of the set where the potentials V(x) attains its minimum and K(x) attains its maximum.

2010 Mathematics Subject Classification: Primary 35J20, 35J25, 35J60.

Key words: Schrödinger–Kirchhoff type equations, concentration behavior, ground state solution, variational methods.

Reference to this article: H. Jia and G. Li: The existence and concentration behavior of ground state solutions for fractional Schrödinger–Kirchhoff type equations. Ann. Acad. Sci. Fenn. Math. 43 (2018), 991-1021.

Full document as PDF file

https://doi.org/10.5186/aasfm.2018.4361