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On some local-nonlocal elliptic equation involving nonlinear terms with exponential growth
September  2017, 16(5): 1785-1805. doi: 10.3934/cpaa.2017087

## Segregated vector Solutions for nonlinear Schrödinger systems with electromagnetic potentials

 School of science, Jiangsu University of Science and Technology, Zhenjiang 212003, China

Received  November 2016 Revised  March 2017 Published  May 2017

Fund Project: This work was partially supported by NSFC (No.11601194), Ph.d star-up funds of JUST (1052931601,1052921513), Natural Science Foundation for Outstanding Young Scholars of Jiangsu Province (BK20160063) and Natural Science Foundation of China (11571140)

In this paper, we study the following nonlinear Schrödinger system in
 $\mathbb{R}^3$
 $\left\{ \begin{array}{*{35}{l}} {{(\frac{\nabla }{i}-A(y))}^{2}}u+{{\lambda }_{1}}(|y|)u=|u{{|}^{2}}u+\beta |v{{|}^{2}}u,&x\in {{\mathbb{R}}^{3}}, \\ {{(\frac{\nabla }{i}-A(y))}^{2}}v+{{\lambda }_{2}}(|y|)v=|v{{|}^{2}}v+\beta |u{{|}^{2}}v,&x\in {{\mathbb{R}}^{3}}, \\\end{array} \right.$
where
 $A(y)=A(|y|)∈ C^1(\mathbb{R}^3,\mathbb{R})$
is bounded,
 $λ_1(|y|),λ_2(|y|)$
are continuous positive radial potentials, and
 $β∈ \mathbb{R}$
is a coupling constant. We proved that if
 $A(y),λ_1(y),λ_2(y)$
satisfy some suitable conditions, the above problem has infinitely many non-radial segregated solutions.
Citation: Jing Yang. Segregated vector Solutions for nonlinear Schrödinger systems with electromagnetic potentials. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1785-1805. doi: 10.3934/cpaa.2017087
##### References:
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Dancer and S. Peng, On multi-bump semi-classical bound states of nonlinear Schrödinger euqations with electromagnetic fields, Adv. Differential Equations, 11 (2006), 781-812. Google Scholar [7] T. Bartsch, E. N. Dancer and Z.-Q. Wang, A Liouville theorem, a priori bounds, and bifurcating branches of positive solutions for a linear elliptic system, Calc. Var. Partial Differential Equations, 37 (2010), 345-361. doi: 10.1007/s00526-009-0265-y. Google Scholar [8] T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Partial Differential Equations, 19 (2006), 200-207. Google Scholar [9] D. Cao, E. S. Noussair and S. Yan, Solutions with multiple peaks for nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 235-264. doi: 10.1017/S030821050002134X. Google Scholar [10] D. Cao and Z. Tang, Existence and uniqueness of multi-bump bound states of nonlinear Schrödinger equations with electromagnetic fields, J. Differential Equations, 222 (2006), 381-424. doi: 10.1016/j.jde.2005.06.027. Google Scholar [11] S. Cingolani and S. Secchi, Semiclassical limit for nonlinear Schrödinger equation with electromagenetic fields, J. Math. Anal. Appl., 275 (2002), 108-130. doi: 10.1016/S0022-247X(02)00278-0. Google Scholar [12] S. Cingolani and S. Secchi, Semiclassical states for NLS equations with magnetic potentials having polynomial growths J. Math. Phys. , 46 (2005), 053503, 19pp. doi: 10.1063/1.1874333. Google Scholar [13] M. Conti, S. Terracini and G. Verzini, Nehari's problem and competing species systems, Ann. Inst. H. Poicaré Anal. Non Linéaire, 19 (2002), 871-888. doi: 10.1016/S0294-1449(02)00104-X. Google Scholar [14] E. N. Dancer, J. Wei and T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system, Ann. Inst. H. Poicaré Anal. Non Linéaire, 27 (2010), 953-969. doi: 10.1016/j.anihpc.2010.01.009. Google Scholar [15] M. Esteban and P. L. Lions, Stationary solutions of nonlinear Schr#246;dinger equations with an external magnetic field, Differential Equations and the Calculus of Variations, Vol. Ⅰ, 401–449, Progr. Nonlinear Differential Equations Appl. , 1, Birkhäuser Boston, Boston, MA, 1989. Google Scholar [16] B. Helffer, Semiclassical analysis for Schrödinger operator with magnetic wells, Quasiclassical Methods (Minneapolis, MN, 1995), 99–114, IMA Vol. Math. Appl. , 95, Springer, New York, 1997. doi: 10.1007/978-1-4612-1940-8_4. Google Scholar [17] B. Helffer, On spectral theory for Schrödinger operator with magnetic potentials, Adv. Stud. Pure Math., 23 (1994), 113-141. Google Scholar [18] K. Kurata, Existence and semi-classical limit of the least energy solution to a nonlinear Schrödinger equation with electromagenetic fields, Nonlinear Anal., 41 (2000), 763-778. doi: 10.1016/S0362-546X(98)00308-3. Google Scholar [19] M. K. Kwong, Uniqueness of positive solutions of $Δ u-u+u^p=0$ in $\mathbb{R}^N$, Arch. Rational Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502. Google Scholar [20] G. Li, S. Peng and C. Wang, Infinitely many solutions for nonlinear Schrödinger equation with electromagenetic fields, J. Differential Equations, 251 (2011), 3500-3521. doi: 10.1016/j.jde.2011.08.038. Google Scholar [21] T.-C. Lin and J. Wei, Ground state of $N$ coupled nonlinear Schrödinger equations in $\mathbb{R}^n$, $n≤q3$, Comm. Math. Phys., 255 (2005), 629-653. doi: 10.1007/s00220-005-1313-x. Google Scholar [22] T. -C. Lin and J. Wei, Half-skyrmions and spike-vortex solutions of two-component nonlinear Schrödinger systems J. Math. Phys. , 48 (2007), 053518, 21 pp. doi: 10.1063/1.2722559. Google Scholar [23] T. -C. Lin and J. Wei, Ground state of N coupled nonlinear Schrödinger equations in $\mathbb{R}^n$, $n≤q 3$, [Comm. Math. Phys. , 255 (2005), 629–653 ]. Comm. Math. Phys. , 277 (2008), 573–576. doi: 10.1007/s00220-007-0365-5. Google Scholar [24] L. A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differential Equations, 229 (2006), 743-767. doi: 10.1016/j.jde.2006.07.002. Google Scholar [25] B. Noris, H. Tavares, S. Terracini and G. Verzini, Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition, Comm. Pure Appl. Math., 63 (2010), 267-302. doi: 10.1002/cpa.20309. Google Scholar [26] O. Rey, The role of the Green's function in a non-linear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal., 89 (1990), 1-52. doi: 10.1016/0022-1236(90)90002-3. Google Scholar [27] S. Peng and Z.-Q. Wang, Segregated and synchronized vector solutions for nonlinear Schrödinger systems, Arch. Ration Mech. Anal., 208 (2013), 305-339. doi: 10.1007/s00205-012-0598-0. Google Scholar [28] B. Sirakov, Least energy solitary waves for a system of nonlinear Schördinger equations in $R^N$, Comm. Math. Phys., 271 (2007), 199-221. doi: 10.1007/s00220-006-0179-x. Google Scholar [29] H. Pi and C. Wang, Multi-bump solutions for nonlinear Schrödinger equations with electromagnetic fields, ESAIM Control Optim. Calc. Var., 19 (2013), 91-111. doi: 10.1051/cocv/2011207. Google Scholar [30] C. Sulem and P. -L. Sulem, The nonlinear Schrödinger equation. Self-focusing and wave collapse, in Applied Mathematical Sciences, 139. Springer-Verlag, New York, 1999. Google Scholar

show all references

##### References:
 [1] L. Abatangelo and S. Terracini, Solutions to nonlinear Schrödinger equation with singular electromagnetic potential and critical exponent, J. Fixed Point Theory Appl., 10 (2011), 147-180. doi: 10.1007/s11784-011-0053-0. Google Scholar [2] N. Akhmediev and A. Ankiewicz, Partially coherent solitons on a finite background, Phys. Rev. Lett. , 82 (1999), 2661.Google Scholar [3] A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equation, C. R. Math. Acad. Sci. Paris, 342 (2006), 453-458. doi: 10.1016/j.crma.2006.01.024. Google Scholar [4] A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equation, J. Lond. Math. Soc., 75 (2007), 67-82. doi: 10.1112/jlms/jdl020. Google Scholar [5] W. Ao and J. Wei, Infinitely many positive solutions for nonlinear equations with non-symmetric potential, Calc. Var. Partial Differential Equations, 51 (2014), 761-798. doi: 10.1007/s00526-013-0694-5. Google Scholar [6] T. Bartsch, E. N. Dancer and S. Peng, On multi-bump semi-classical bound states of nonlinear Schrödinger euqations with electromagnetic fields, Adv. Differential Equations, 11 (2006), 781-812. Google Scholar [7] T. Bartsch, E. N. Dancer and Z.-Q. Wang, A Liouville theorem, a priori bounds, and bifurcating branches of positive solutions for a linear elliptic system, Calc. Var. Partial Differential Equations, 37 (2010), 345-361. doi: 10.1007/s00526-009-0265-y. Google Scholar [8] T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Partial Differential Equations, 19 (2006), 200-207. Google Scholar [9] D. Cao, E. S. Noussair and S. Yan, Solutions with multiple peaks for nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 235-264. doi: 10.1017/S030821050002134X. Google Scholar [10] D. Cao and Z. Tang, Existence and uniqueness of multi-bump bound states of nonlinear Schrödinger equations with electromagnetic fields, J. Differential Equations, 222 (2006), 381-424. doi: 10.1016/j.jde.2005.06.027. Google Scholar [11] S. Cingolani and S. Secchi, Semiclassical limit for nonlinear Schrödinger equation with electromagenetic fields, J. Math. Anal. Appl., 275 (2002), 108-130. doi: 10.1016/S0022-247X(02)00278-0. Google Scholar [12] S. Cingolani and S. Secchi, Semiclassical states for NLS equations with magnetic potentials having polynomial growths J. Math. Phys. , 46 (2005), 053503, 19pp. doi: 10.1063/1.1874333. Google Scholar [13] M. Conti, S. Terracini and G. Verzini, Nehari's problem and competing species systems, Ann. Inst. H. Poicaré Anal. Non Linéaire, 19 (2002), 871-888. doi: 10.1016/S0294-1449(02)00104-X. Google Scholar [14] E. N. Dancer, J. Wei and T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system, Ann. Inst. H. Poicaré Anal. Non Linéaire, 27 (2010), 953-969. doi: 10.1016/j.anihpc.2010.01.009. Google Scholar [15] M. Esteban and P. L. Lions, Stationary solutions of nonlinear Schr#246;dinger equations with an external magnetic field, Differential Equations and the Calculus of Variations, Vol. Ⅰ, 401–449, Progr. Nonlinear Differential Equations Appl. , 1, Birkhäuser Boston, Boston, MA, 1989. Google Scholar [16] B. Helffer, Semiclassical analysis for Schrödinger operator with magnetic wells, Quasiclassical Methods (Minneapolis, MN, 1995), 99–114, IMA Vol. Math. Appl. , 95, Springer, New York, 1997. doi: 10.1007/978-1-4612-1940-8_4. Google Scholar [17] B. Helffer, On spectral theory for Schrödinger operator with magnetic potentials, Adv. Stud. Pure Math., 23 (1994), 113-141. Google Scholar [18] K. Kurata, Existence and semi-classical limit of the least energy solution to a nonlinear Schrödinger equation with electromagenetic fields, Nonlinear Anal., 41 (2000), 763-778. doi: 10.1016/S0362-546X(98)00308-3. Google Scholar [19] M. K. Kwong, Uniqueness of positive solutions of $Δ u-u+u^p=0$ in $\mathbb{R}^N$, Arch. Rational Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502. Google Scholar [20] G. Li, S. Peng and C. Wang, Infinitely many solutions for nonlinear Schrödinger equation with electromagenetic fields, J. Differential Equations, 251 (2011), 3500-3521. doi: 10.1016/j.jde.2011.08.038. Google Scholar [21] T.-C. Lin and J. Wei, Ground state of $N$ coupled nonlinear Schrödinger equations in $\mathbb{R}^n$, $n≤q3$, Comm. Math. Phys., 255 (2005), 629-653. doi: 10.1007/s00220-005-1313-x. Google Scholar [22] T. -C. Lin and J. Wei, Half-skyrmions and spike-vortex solutions of two-component nonlinear Schrödinger systems J. Math. Phys. , 48 (2007), 053518, 21 pp. doi: 10.1063/1.2722559. Google Scholar [23] T. -C. Lin and J. Wei, Ground state of N coupled nonlinear Schrödinger equations in $\mathbb{R}^n$, $n≤q 3$, [Comm. Math. Phys. , 255 (2005), 629–653 ]. Comm. Math. Phys. , 277 (2008), 573–576. doi: 10.1007/s00220-007-0365-5. Google Scholar [24] L. A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differential Equations, 229 (2006), 743-767. doi: 10.1016/j.jde.2006.07.002. Google Scholar [25] B. Noris, H. Tavares, S. Terracini and G. Verzini, Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition, Comm. Pure Appl. Math., 63 (2010), 267-302. doi: 10.1002/cpa.20309. Google Scholar [26] O. Rey, The role of the Green's function in a non-linear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal., 89 (1990), 1-52. doi: 10.1016/0022-1236(90)90002-3. Google Scholar [27] S. Peng and Z.-Q. Wang, Segregated and synchronized vector solutions for nonlinear Schrödinger systems, Arch. Ration Mech. Anal., 208 (2013), 305-339. doi: 10.1007/s00205-012-0598-0. Google Scholar [28] B. Sirakov, Least energy solitary waves for a system of nonlinear Schördinger equations in $R^N$, Comm. Math. Phys., 271 (2007), 199-221. doi: 10.1007/s00220-006-0179-x. Google Scholar [29] H. Pi and C. Wang, Multi-bump solutions for nonlinear Schrödinger equations with electromagnetic fields, ESAIM Control Optim. Calc. Var., 19 (2013), 91-111. doi: 10.1051/cocv/2011207. Google Scholar [30] C. Sulem and P. -L. Sulem, The nonlinear Schrödinger equation. Self-focusing and wave collapse, in Applied Mathematical Sciences, 139. Springer-Verlag, New York, 1999. Google Scholar
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