数学物理学报(英文版) ›› 2018, Vol. 38 ›› Issue (4): 1296-1310.doi: 10.1016/S0252-9602(18)30815-4
Amin ESFAHANI
收稿日期:
2016-09-02
修回日期:
2017-11-11
出版日期:
2018-08-25
发布日期:
2018-08-25
作者简介:
Amin ESFAHANI,E-mail:esfahani@du.ac.ir,amin@impa.br
Amin ESFAHANI
Received:
2016-09-02
Revised:
2017-11-11
Online:
2018-08-25
Published:
2018-08-25
摘要:
In this article, we establish the existence of a sign-changing solution and two sign-constant solutions for nonlinear nonlocal problem involving the BO-ZK operator on bounded domain. Our main tool is constrained minimization on appropriate Nehari manifolds.
Amin ESFAHANI. THREE NONTRIVIAL SOLUTIONS FOR A NONLINEAR ANISOTROPIC NONLOCAL EQUATION[J]. 数学物理学报(英文版), 2018, 38(4): 1296-1310.
Amin ESFAHANI. THREE NONTRIVIAL SOLUTIONS FOR A NONLINEAR ANISOTROPIC NONLOCAL EQUATION[J]. Acta mathematica scientia,Series B, 2018, 38(4): 1296-1310.
[1] | Ambrosetti A, Rabinowitz P. Dual variational methods in critical points theory and applications. J Funct Anal, 1973, 14:349-381 |
[2] | Alves C O, Souto M A S. Existence of least energy nodal solution for a Schrödinger-Poisson system in bounded domains. Z Angew Math Phys, 2014, 65:1153-1166 |
[3] | Barles G, Chasseigne E, Imbert C. Hölder continuity of solutions of second-order non-linear elliptic integrodifferential equations. J Eur Math Soc, 2011, 13:1-26 |
[4] | Barles G, Chasseigne E, Ciomaga A, Imbert C. Lipschitz regularity of solutions for mixed integro-differential equations. J Differential Equations, 2012, 252:6012-6060 |
[5] | Barles G, Chasseigne E, Ciomaga A, Imbert C. Large time behavior of periodic viscosity solutions for uniformly parabolic integro-differential equations. Calc Var Partial Differential Equations, 2014, 50:283-304 |
[6] | Bartsch T, Liu Z L, Weth T. Sign changing solutions of superlinear Schrödinger equations. Comm Partial Differential Equations, 2004, 29:25-42 |
[7] | Bartsch T, Weth T. Three nodal solutions of singularly perturbed elliptic equations on domains without topology. Ann Inst Henri Poincaré (C) Nonlinear Anal, 2005, 22:259-281 |
[8] | Berestycki H, Lions P-L. Nonlinear scalar field equations, Ⅱ, Existence of infinitely many solutions. Arch Rational Mech Anal, 1983, 82:347-375 |
[9] | Barrios B, Colorado E, Servadei R, Soria F. A critical fractional equation with concave-convex power nonlinearities. Ann Inst Henri Poincaré (C) Nonlinear Anal, 2015, 32:875-900 |
[10] | Cabré X, Sire Y. Nonlinear equations for fractional Laplacians I:Regularity, maximum principles, and Hamiltonian estimates. Ann Inst Henri Poincaré (C) Nonlinear Anal, 2014, 31:23-53 |
[11] | Cabré X, Sire Y. Nonlinear equations for fractional Laplacians Ⅱ:Existence, uniqueness, and qualitative properties of solutions. Trans Amer Math Soc, 2015, 367:911-941 |
[12] | Chang X, Wang Z-Q. Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity. Nonlinearity, 2013, 26:479-494 |
[13] | Cheng M. Bound state for the fractional Schrödinger equation with unbounded potential. J Math Phys, 2012, 53:043507 |
[14] | Ciomaga A. On the strong maximum principle for second order nonlinear parabolic integro-differential equations. Adv Differential Equations, 2012, 17:635-671 |
[15] | Dávila J, del Pino M, Dipierro S, Valdinoci E. Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum. Anal PDE, 2015, 8:1165-1235 |
[16] | Dávila J, del Pino M, Wei J. Concentrating standing waves for the fractional nonlinear Schrödinger equation. J Differential Equations, 2014, 256:858-892 |
[17] | Di Nezza E, Palatucci G, Valdinoci E. Hitchhiker's guide to the fractional Sobolev spaces. Bull Sci Math, 2012, 136:521-573 |
[18] | Esfahani A. Anisotropic Gagliardo-Nirenberg inequality with fractional derivatives. Z Angew Math Phys, 2015, 66:3345-3356 |
[19] | Esfahani A, Pastor A. Instability of solitary wave solutions for the generalized BO-ZK equation. J Differential Equations, 2009, 247:3181-3201 |
[20] | Esfahani A, Pastor A, Bona J L. Stability and decay properties of solitary wave solutions for the generalized BOZK equation. Adv Differential Equations, 2015, 20:801-834 |
[21] | Fall M M, Valdinoci E. Uniqueness and nondegeneracy of positive solutions of (-△)su + u=up in Rn when s is close to 1. Comm Math Phys, 2014, 329:383-404 |
[22] | Farina A, Valdinoci E. Regularity and rigidity theorems for a class of anisotropic nonlocal operators. Manuscripta Math, 2017, 153:53-70 |
[23] | Felmer P, Quaas A, Tan J. Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian. Proc Roy Soc Edinburgh Sect A, 2012, 142:1237-1262 |
[24] | Furtado M F, Maia L A, Medeiros E S. Positive and nodal solutions for a nonlinear Schrdinger equation with indefinite potential. Adv Nonlinear Stud, 2008, 8:353-373 |
[25] | Frank R L, Lenzmann E. Uniqueness of non-linear ground states for fractional Laplacians in R. Acta Math, 2013, 210:261-318 |
[26] | Frank R L, Lenzmann E, Silvestre L. Uniqueness of radial solutions for the fractional Laplacian. Comm Pure Appl Math, 2016, 69:1671-1726 |
[27] | Frank R L, Lieb E H, Seiringer R. Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators. J Amer Math Soc, 2008, 21:925-950 |
[28] | Haškovec J, Schmeiser C. A note on the anisotropic generalizations of the Sobolev and Morrey embedding theorems. Monatsh Math, 2009, 158:71-79 |
[29] | Jones C., Küpper T. On the infinitely many solutions of a semilinear elliptic equation. SIAM J Math Anal, 1986, 17:803-835 |
[30] | Laskin N. Fractional quantum mechanics and Lévy path integrals. Phys Lett A, 2000, 268:298-305 |
[31] | Laskin N. Fractional Schrödinger equation. Phys Rev E, 2002, 66:056108 |
[32] | Latorre J C, Minzoni A A, Smyth N F, Vargas C A. Evolution of Benjamin-Ono solitons in the presence of weak Zakharov-Kuznetsov lateral dispersion. Chaos, 2006, 16:043103-1-043103-10 |
[33] | Lions P-L. The concentration-compactness principle in the calculus of variations. The locally compact case. I. Ann Inst H Poincaré, Anal Non linéaire, 1984, 1:109-145; The locally compact case. Ⅱ. Ann Inst H Poincaré, Anal Non linéaire, 1984, 4:223-283 |
[34] | Miranda C. Un'osservazione su un teorema di Brouwer. Bol Un Mat Ital, 1940, 3:5-7 |
[35] | Noussair E S, Wei J C. On the effect of domain geometry on the existence and profile of nodal solution of some singularly perturbed semilinear Dirichlet problem. Indiana Univ Math J, 1997, 46:1321-1332 |
[36] | Ribaud F, Vento S. Local and global well-posedness results for the Benjamin-Ono-Zakharov-Kuznetsov equation. Discrete Contin Dyn Syst, 2017, 37:449-483 |
[37] | Pucci P, Serrin J. A general variational inequality. Indiana Univ Math J, 1986, 35:681-703 |
[38] | Ros-Oton X, Serra J. Nonexistence results for nonlocal equations with critical and supercritical nonlinearities. Comm Partial Differential Equations, 2015, 40:115-133 |
[39] | Secchi S. Ground state solutions for nonlinear fractional Schrödinger equations in RN. J Math Phys, 2013, 54:031501 |
[40] | Servadei R, Valdinoci E. Mountain Pass solutions for non-local elliptic operators. J Math Anal Appl, 2012, 389:887-898 |
[41] | Servadei R, Valdinoci E. The Brezis-Nirenberg result for the fractional Laplacian. Trans Amer Math Soc, 2015, 367:67-102 |
[42] | Struwe M. Variational Methods:Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems. Ergeb Math Grenzgeb (3). Berlin, Heidelberg:Springer-Verlag, 1990 |
[43] | Wang Z, Zhou H-S. Sign-changing solutions for the nonlinear Schrödinger-Poisson system in R3. Calc Var, 2015, 52:927-943 |
[44] | Wang Z, Zhou H-S. Radial sign-changing solution for fractional Schrödinger equation. Discrete Contin Dyn Syst, 2016, 36:499-508 |