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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 $0<s<1,\;n>2s,\;\Omega$ 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.
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참고문헌 (28)
참고문헌 신청
A. Ambrosetti / 1973 / Dual variational methods in critical point theory and applications / J. Functional Analysis 14:349~381
G. Autuori / 2013 / Elliptic problems involving the fractional Laplacian in RN / J. Dierential Equations 255(8):2340~2362
T. Bartsch / 1995 / Existence and multiplicity results for some superlinear elliptic problems on RN / Comm. Partial Dierential Equations 20(9-10):1725~1741
G. M. Bisci / 2015 / Ground state solutions of scalar eld fractional Schrodinger equations / Calc. Var. Partial Dierential Equations 54(3):2985~3008
C. Brandle / 2013 / A concave-convex elliptic problem involving the fractional Laplacian / Proc. Roy. Soc. Edinburgh Sect. A 143(1):39~71
G. Autuori / 2013 / Elliptic problems involving the fractional Laplacian in RN / J. Dierential Equations 255(8):2340~2362
T. Bartsch / 1995 / Existence and multiplicity results for some superlinear elliptic problems on RN / Comm. Partial Dierential Equations 20(9-10):1725~1741
G. M. Bisci / 2015 / Ground state solutions of scalar eld fractional Schrodinger equations / Calc. Var. Partial Dierential Equations 54(3):2985~3008
C. Brandle / 2013 / A concave-convex elliptic problem involving the fractional Laplacian / Proc. Roy. Soc. Edinburgh Sect. A 143(1):39~71
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