Bifurcation problem for a class of quasilinear fractional Schr\"{o}dinger equations
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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.
Abstract
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.
참고문헌 (28)
- A. Ambrosetti, "Dual variational methods in critical point theory and applications", J. Functional Analysis, 14, pp.349~381 (1973)[DOI]
- G. Autuori, "Elliptic problems involving the fractional Laplacian in RN", J. Dierential Equations, 255(8), pp.2340~2362 (2013)[DOI]
- T. Bartsch, "Existence and multiplicity results for some superlinear elliptic problems on RN", Comm. Partial Dierential Equations, 20(9-10), pp.1725~1741 (1995)[DOI]
- G. M. Bisci, "Ground state solutions of scalar eld fractional Schrodinger equations", Calc. Var. Partial Dierential Equations, 54(3), pp.2985~3008 (2015)[DOI]
- C. Brandle, "A concave-convex elliptic problem involving the fractional Laplacian", Proc. Roy. Soc. Edinburgh Sect. A, 143(1), pp.39~71 (2013)[DOI]
- H. Brezis, "Collection Mathematiques Appliquees pour la Matrise" (1983), Masson
- L. Caarelli, "An extension problem related to the fractional Laplacian", Comm. Partial Dierential Equations, 32(7-9), pp.1245~1260 (2007)[DOI]
- X. Chang, "Ground state solutions of asymptotically linear fractional Schrodinger equations", J. Math. Phys, 54(6), pp.10~ (2013)[DOI]
- X. Chang, "Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian", J. Dierential Equations, 256(8), pp.2965~2992 (2014)[DOI]
- M. Cheng, "Bound state for the fractional Schrodinger equation with unbounded potential", J. Math. Phys, 53(4), pp.7~ (2012)[DOI]
- S. Dipierro, "Existence and symmetry results for a Schrodinger type problem involving the fractional Laplacian", Matematiche (Catania), 68(1), pp.201~216 (2013)[DOI]
- E. Di Nezza, "Hitchhiker's guide to the fractional Sobolev spaces", Bull. Sci. Math, 136(5), pp.521~573 (2012)[DOI]
- M. M. Fall, "Ground states and concentration phenomena for the fractional Schrodinger equation", Nonlinearity, 28(6), pp.1937~1961 (2015)[DOI]
- P. Felmer, "Positive solutions of the nonlinear Schrodinger equation with the fractional Laplacian", Proc. Roy. Soc. Edinburgh Sect. A, 142(6), pp.1237~1262 (2012)[DOI]
- H. Jin, "Ground state solutions for nonlinear fractional Schrodinger equations involving critical growth", Electron. J. Dierential Equations, 2017, pp.19~ (2017)
- J. Korvenpaa, "Fractional superharmonic functions and the Perron method for nonlinear integro-dierential equations", Math. Ann, 369(3-4), pp.1443~1489 (2017)[DOI]
- N. Laskin, "Fractional quantum mechanics and Levy path integrals", Phys. Lett. A, 268(4-6), pp.298~305 (2000)[DOI]
- N. Laskin, "Fractional Schrodinger equation", Phys. Rev. E, 66(5), pp.7~ (2002)[DOI]
- N. Laskin, "Fractional quantum mechanics" (2018), World Scientic Publishing Co. Pte. Ltd
- S. Secchi, "Ground state solutions for nonlinear fractional Schrodinger equations in RN", J. Math. Phys, 54(3), pp.17~ (2013)[DOI]
- S. Secchi, "Perturbation results for some nonlinear equations involving fractional operators", Dier. Equ. Appl, 5(2), pp.221~236 (2013)[DOI]
- S. Secchi, "On fractional Schrodinger equations in RN without the Ambrosetti-Rabinowitz condition", Topol. Methods Nonlinear Anal, 47(1), pp.19~41 (2016)
- R. Servadei, "Variational methods for non-local operators of elliptic type", Discrete Contin. Dyn. Syst, 33(5), pp.2105~2137 (2013)[DOI]
- J. Tan, "The Brezis-Nirenberg type problem involving the square root of the Laplacian", Calc. Var. Partial Dierential Equations, 42(1-2), pp.21~41 (2011)[DOI]
- J. Tan, "Nonlinear fractional eld equations", Nonlinear Anal, 75(4), pp.2098~2110 (2012)[DOI]
- K. Teng, "Multiple solutions for a class of fractional Schrodinger equations in RN", Nonlinear Anal. Real World Appl, 21, pp.76~86 (2015)[DOI]
- W. Zhang, "Innitely many radial and non-radial solutions for a fractional Schrodinger equation", Comput. Math. Appl, 71(3), pp.737~747 (2016)[DOI]
- H. Zhang, "Existence and multiplicity of solutions for superlinear fractional Schrodinger equations in RN", J. Math. Phys, 56(9), pp.13~ (2015)[DOI]