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Least energy solutions for nonlinear Schrödinger equation involving the fractional Laplacian and critical growth

The second author was supported by National Science Foundation of China(11571040)

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  • In this paper, we study a class of nonlinear Schrödinger equations involving the fractional Laplacian and the nonlinearity term with critical Sobolev exponent. We assume that the potential of the equations includes a parameter $λ$. Moreover, the potential behaves like a potential well when the parameter λ is large. Using variational methods, combining Nehari methods, we prove that the equation has a least energy solution which, as the parameter λ large, localizes near the bottom of the potential well. Moreover, if the zero set int $V^{-1}(0)$ of $V(x)$ includes more than one isolated component, then $u_\lambda (x)$ will be trapped around all the isolated components. However, in Laplacian case when $s=1$, for $\lambda$ large, the corresponding least energy solution will be trapped around only one isolated component and will become arbitrary small in other components of int $V^{-1}(0)$. This is the essential difference with the Laplacian problems since the operator $(-Δ)^{s}$ is nonlocal.

    Mathematics Subject Classification: Primary: 35Q55; Secondary: 35J65.


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  •   A. Ambrosetti , M. Badiale  and  S. Cingolani , Semiclassical states of nonlinear Schrödinger equations, Arch. Ration. Mech. Anal., 140 (1997) , 285-300.  doi: 10.1007/s002050050067.
      D. Applebaum , Lévy processes-from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004) , 1336-1347. 
      B. Barrios , E. Colorado , A. de Pablo  and  U. Sánchez , On some critical problems for the fractional Laplacian operator, J. Diff. Equa., 252 (2012) , 6133-6162.  doi: 10.1016/j.jde.2012.02.023.
      T. Bartsch , A. Pankov  and  Z. Wang , Nonlinear Schrödinger equations with steep pontential well, Comm. Contemp. Math., 3 (2001) , 549-569.  doi: 10.1142/S0219199701000494.
      T. Bartsch  and  Z. Wang , Multiple positive solutions for a nonlinear Schrödinger equation, Z. Angew. Math. Phys., 51 (2000) , 366-384.  doi: 10.1007/PL00001511.
      V. Benci  and  G. Cerami , Existence of positive solutions of the equation $-\triangle u+a(x)u=u^{\frac{N+2}{N-2}} \text{in}{\Bbb R}^{N}$, J. Funct. Anal., 88 (1990) , 90-117.  doi: 10.1016/0022-1236(90)90120-A.
      J. L. Bona  and  Y. A. Li , Decay and analyticity of solitary waves, J. Math. Pures Appl., 76 (1997) , 377-430.  doi: 10.1016/S0021-7824(97)89957-6.
      A. de Bouard  and  J. C. Saut , Symmetries and decay of the generalized Kadomtsev-Petviashvili solitary waves, SIAM J. Math. Anal., 28 (1997) , 1064-1085.  doi: 10.1137/S0036141096297662.
      H. Brezis  and  E. Lieb , A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983) , 486-490.  doi: 10.2307/2044999.
      X. Cabré  and  Y. Sire , Nonlinear equations for fractional Laplacians Ⅰ: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014) , 23-53.  doi: 10.1016/j.anihpc.2013.02.001.
      X. Cabré  and  J. Tan , Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010) , 2052-2093.  doi: 10.1016/j.aim.2010.01.025.
      L. Caffarelli , S. Salsa  and  L. Silvestre , Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008) , 425-461.  doi: 10.1007/s00222-007-0086-6.
      L. Caffarelli  and  L. Silvestre , An extension problem related to the fractional Laplacian, Comm. in Part. Diff. Equa., 32 (2007) , 1245-1260.  doi: 10.1080/03605300600987306.
      M. Cheng, Bound state for the fractional Schrödinger equation with unbounded potential J. Math. Phys. 53 (2012), 043507, 7pp. doi: 10.1063/1.3701574.
      W. Choi , S. Kim  and  K. Lee , Asymptotic behavior of solutions for nonlinear elliptic problems with the fractional Laplacian, J. Funct. Anal., 266 (2014) , 6531-6598.  doi: 10.1016/j.jfa.2014.02.029.
      S. Dipierro , G. Palatucci  and  E. Valdinoci , Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Matematiche (Catania), 68 (2013) , 201-216. 
      P. Felmer , A. Quaas  and  J. Tan , Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh., 142 (2012) , 1237-1262.  doi: 10.1017/S0308210511000746.
      R. L. Frank  and  E. Lenzmann , Uniqueness of non-linear ground states for fractional Laplacians $\text{in} \Bbb R$, Acta Math., 210 (2013) , 261-318.  doi: 10.1007/s11511-013-0095-9.
      R. L. Frank , E. Lenzmann  and  L. Silvestre , Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math., 69 (2016) , 1671-1726.  doi: 10.1002/cpa.21591.
      M. Gonzalez  and  J. Qing , Fractional conformal Laplacians and fractional Yamabe problems, Anal. PDE, 6 (2013) , 1535-1576.  doi: 10.2140/apde.2013.6.1535.
      T. Jin , Y. Li  and  J. Xiong , On a fractional Nirenberg problem, part Ⅰ: Blow up analysis and compactness of solutions, J. Eur. Math. Soc., 16 (2014) , 1111-1171.  doi: 10.4171/JEMS/456.
      M. Maris , On the existence, regularity and decay of solitary waves to a generalized Benjamin-Ono equation, Nonlin. Anal., 51 (2002) , 1073-1085.  doi: 10.1016/S0362-546X(01)00880-X.
      M. Reed and B. Simon, Methods of Modern Mathematical Physics, Ⅳ. Analysis of Operators, Academic Press, New York, 1978.
      L. Silvestre , Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007) , 67-112.  doi: 10.1002/cpa.20153.
      Y. Sire  and  E. Valdinoci , Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009) , 1842-1864.  doi: 10.1016/j.jfa.2009.01.020.
      J. Tan , The Brezis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Part. Diff. Equa., 42 (2011) , 21-41.  doi: 10.1007/s00526-010-0378-3.
      J. Tan , Positive solutions for non local elliptic problems, Discrete Contin. Dyn. Syst., 33 (2013) , 837-589.  doi: 10.3934/dcds.2013.33.837.
      J. Tan  and  J. Xiong , A Harnack inequality for fractional Laplace equations with lower order terms, Discrete Contin. Dyn. Syst., 31 (2011) , 975-983.  doi: 10.3934/dcds.2011.31.975.
      Z. Tang , Least energy solutions for semilinear Schrödinger equations involving critical growth and indefinite potentials, Comm. Pure Appl. Anal., 13 (2014) , 237-248.  doi: 10.3934/cpaa.2014.13.237.
      M. Weinstein , Solitary waves of nonlinear dispersive evolution equations with critical power nonlinearities, J. Diff. Equa., 69 (1987) , 192-203.  doi: 10.1016/0022-0396(87)90117-3.
      M. Willem, Minimax theorems. Progr. Nonlinear Differential Equations and their Applications 24. Birkhäuser Boston, Inc. , Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.
      S. Yan , J. Yang  and  X. Yu , Equations involving fractional Laplacian operator: Compactness and application, J. Funct. Anal., 269 (2015) , 47-79.  doi: 10.1016/j.jfa.2015.04.012.
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