J. Korean Math. Soc. 2020; 57(6): 1347-1372
Online first article July 21, 2020 Printed November 1, 2020
https://doi.org/10.4134/JKMS.j190646
Copyright © The Korean Mathematical Society.
Imed Abid
Higher Institute of Medical Technologies of Tunis
We study bifurcation for the following fractional Schr\"{o}dinger equation \begin{eqnarray*}\left\{ \begin{array}{rlll} (-\Delta)^{s}u+V(x)u& = \lambda\,f(u)& \hbox{in}\,\Omega \\ u&>0& \hbox{in}\,\Omega\\ u &=0 &\hbox{in}\,\R^n\setminus\Omega \\ \end{array} \right. \end{eqnarray*} where $02s,\;\Omega$ is a bounded smooth domain of $\R^n,$ $(-\Delta)^s$ is the fractional Laplacian of order $s,$ $V$ is the potential energy satisfying suitable assumptions and $\lambda$ is a positive real parameter. The nonlinear term $f$ is a positive nondecreasing convex function, asymptotically linear that is $\lim\limits_{t\rightarrow+\infty}\frac{f(t)}{t}= a \in(0,+\infty).$ We discuss the existence, uniqueness and stability of a positive solution and we also prove the existence of critical value and the uniqueness of extremal solutions. We take into account the types of Bifurcation problem for a class of quasilinear fractional Schr\"{o}dinger equations, we also establish the asymptotic behavior of the solution around the bifurcation point.
Keywords: Bifurcation problems, fractional partial differential equations, fractional Schr\"{o}dinger equations, weak solution, stability
MSC numbers: Primary 37K50, 35R11, 35J10, 35D30, 35B35
2018; 55(3): 531-551
2024; 61(2): 227-253
2022; 59(4): 733-756
2022; 59(2): 279-298
© 2022. The Korean Mathematical Society. Powered by INFOrang Co., Ltd