Journal of the
Korean Mathematical Society JKMS

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