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January  2020, 19(1): 311-328. doi: 10.3934/cpaa.2020017

Multi-spikes solutions for a system of coupled elliptic equations with quadratic nonlinearity

Zhongwei Tang and  Huafei Xie ,

School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

* Corresponding author

Received  December 2018 Revised  March 2019 Published  July 2019

Fund Project: The first author is supported by NSFC (11571040, 11671331).

This paper is devoted to study the following systems of coupled elliptic equations with quadratic nonlinearity
$ \begin{equation*} \begin{cases} -\varepsilon^{2}\Delta v+P(x)v = \mu vw, &x\in{\mathbb{R}}^{N},\\ -\varepsilon^{2}\Delta w+Q(x)w = \frac{\mu}{2} v^{2}+\gamma w^{2}, &x\in{\mathbb{R}}^{N}, \end{cases} \end{equation*} $
which arises from second- harmonic generation in quadratic optical media. We assume that the potential functions
$ P(x) $
 and
$ Q(x) $
 are positive functions and have a strict local maxima at
$ x_{0} $
. Applying the finite dimensional reduction method, for any integer
$ 1\leq k\leq N+1 $
, we prove the existence of positive solutions which have
$ k $
 local maximum points that concentrate at
$ x_{0} $
 simultaneously when
$ \varepsilon $
 is small.
Keywords: Quadratic nonlinearity, concentrating solutions, finite dimensional reduction method.
Mathematics Subject Classification: Primary: 35J60; Secondary: 35J20.
Citation: Zhongwei Tang, Huafei Xie. Multi-spikes solutions for a system of coupled elliptic equations with quadratic nonlinearity. Communications on Pure and Applied Analysis, 2020, 19 (1) : 311-328. doi: 10.3934/cpaa.2020017
References:
[1]

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.

[2]

A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Math. Acad. Sci. Paris, 342 (2006), 453-458.  doi: 10.1016/j.crma.2006.01.024.

[3]

A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc., 75 (2007), 67-82.  doi: 10.1112/jlms/jdl020.

[4]

A. Ambrosetti, E. Colorado and D. Ruiz, Multi-bump solitons to linearly coupled systems of nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations, 30 (2007), 85-112.  doi: 10.1007/s00526-006-0079-0.

[5]

T. Bartsch, N. Dancer and Z. Q. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var. Partial Differential Equations, 37 (2010), 345-361.  doi: 10.1007/s00526-009-0265-y.

[6]

A. V. Buryak and Y. S. Kivshar, Solitons due to second harmonic generation, Phys. Lett. A, 197 (1995), 407-412.  doi: 10.1016/0375-9601(94)00989-3.

[7]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations. Ⅰ. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555.

[8]

A. V. Buryak, P. Di Trapani, D. V. Skryabin and S. Trillo, Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications, Phys. Rep., 370 (2002), 63-235.  doi: 10.1016/S0370-1573(02)00196-5.

[9]

T. Bartsch and Z. Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Partial Differential Equations, 19 (2006), 200-207. 

[10]

T. Bartsch, Z. Q. Wang and J. Wei, Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367.  doi: 10.1007/s11784-007-0033-6.

[11]

T. Cazenave, Semilinear Schrödinger Equations, vol.10 of Courant Lecture Notes in Mathematics, New York University Courant Insitute of Mathematical Sciences, New York, 2003. doi: 10.1090/cln/010.

[12]

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.

[13]

M. Conti, S. Terracini and G. Verzini, Nehari's problem and competing species systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 871-888.  doi: 10.1016/S0294-1449(02)00104-X.

[14]

M. Del Pino and P. Felmer, Multi-peak bound states for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 127-149.  doi: 10.1016/S0294-1449(97)89296-7.

[15]

E. N. Dancer and S. Yan, Multipeak solutions for a singularly perturbed Neumann problem, Pacific J. Math., 189 (1999), 241-262.  doi: 10.2140/pjm.1999.189.241.

[16]

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. Poincaré Anal. Non Linéaire, 27 (2010), 953-969.  doi: 10.1016/j.anihpc.2010.01.009.

[17]

C. Gui and J. Wei, Multiple interior peak solutions for some singularly perturbed Neumann problems, J. Differential Equations, 158 (1999), 1-27.  doi: 10.1016/S0022-0396(99)80016-3.

[18]

C. Gui and J. Wei, On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems, Canad. J. Math., 52 (2000), 522-538.  doi: 10.4153/CJM-2000-024-x.

[19]

Y. Li, On a singularly perturbed elliptic equation, Adv. Differential Equations, 2 (1997), 955-980. 

[20]

Y. Li and L. Nirenberg, The Dirichlet problem for singularly perturbed elliptic equations, Comm. Pure Appl. Math., 51 (1998), 1445-1490.  doi: 10.1002/(SICI)1097-0312(199811/12)51:11/12<1445::AID-CPA9>3.3.CO;2-Q.

[21]

C. Lin, W. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27.  doi: 10.1016/0022-0396(88)90147-7.

[22]

Z. Liu and Z. Q. Wang, Ground states and bound states of a nonlinear Schrödinger system, Adv. Nonlinear Stud., 10 (2010), 175-193.  doi: 10.1515/ans-2010-0109.

[23]

E. Montefusco, B. Pellacci and M. Squassina, Semiclassical states for weakly coupled nonlinear Schrödinger systems, J. Eur. Math. Soc., 10 (2008), 47-71.  doi: 10.4171/JEMS/103.

[24]

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.

[25]

E. S. Noussair and S. Yan, On positive multipeak solutions of a nonlinear elliptic problem, J. Lond. Math. Soc., 62 (2000), 213-227.  doi: 10.1112/S002461070000898X.

[26]

W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162. 

[27]

B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbb{R}^{N}$, Comm. Math. Phys., 271 (2007), 199-221.  doi: 10.1007/s00220-006-0179-x.

[28]

C. Sulem and P. L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse, vol.39 of Applied Mathematical Sciences. Springer-Verlag, New York, 1999.

[29]

R. Tian and Z. Q. Wang, Multiple solitary wave solutions of nonlinear Schrödinger systems, Topol. Methods Nonlinear Anal., 37 (2011), 203-223. 

[30]

J. Wei and T. Weth, Radial solutions and phase separation in a system of two coupled Schrödinger equations, Arch. Ration. Mech. Anal., 190 (2008), 83-106.  doi: 10.1007/s00205-008-0121-9.

[31]

C. Wang and J. Zhou, Infinitely many solitary waves due to the second-harmonic generation in quadratic media, bo be appeared in Acta Math. Sci. Ser. B (Engl. Ed.) (2020, no.1).

[32]

A. C. Yew, Multipulses of nonlinearly coupled Schrödinger equations, J. Differential Equations, 173 (2001), 92-137.  doi: 10.1006/jdeq.2000.3922.

[33]

A. C. Yew, A. R. Champneys and P. J. McKenna, Multiple solitary waves due to second-harmonic generation in quadratic media, J. Nonlinear Sci., 9 (1999), 33-52.  doi: 10.1007/s003329900063.

[34]

L. Zhao, F. Zhao and J. Shi, Higher dimensional solitary waves generated by second-harmonic generation in quadratic media, Calc. Var. Partial Differential Equations, 54 (2015), 2657-2691.  doi: 10.1007/s00526-015-0879-1.

show all references

References:
[1]

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.

[2]

A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Math. Acad. Sci. Paris, 342 (2006), 453-458.  doi: 10.1016/j.crma.2006.01.024.

[3]

A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc., 75 (2007), 67-82.  doi: 10.1112/jlms/jdl020.

[4]

A. Ambrosetti, E. Colorado and D. Ruiz, Multi-bump solitons to linearly coupled systems of nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations, 30 (2007), 85-112.  doi: 10.1007/s00526-006-0079-0.

[5]

T. Bartsch, N. Dancer and Z. Q. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var. Partial Differential Equations, 37 (2010), 345-361.  doi: 10.1007/s00526-009-0265-y.

[6]

A. V. Buryak and Y. S. Kivshar, Solitons due to second harmonic generation, Phys. Lett. A, 197 (1995), 407-412.  doi: 10.1016/0375-9601(94)00989-3.

[7]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations. Ⅰ. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555.

[8]

A. V. Buryak, P. Di Trapani, D. V. Skryabin and S. Trillo, Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications, Phys. Rep., 370 (2002), 63-235.  doi: 10.1016/S0370-1573(02)00196-5.

[9]

T. Bartsch and Z. Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Partial Differential Equations, 19 (2006), 200-207. 

[10]

T. Bartsch, Z. Q. Wang and J. Wei, Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367.  doi: 10.1007/s11784-007-0033-6.

[11]

T. Cazenave, Semilinear Schrödinger Equations, vol.10 of Courant Lecture Notes in Mathematics, New York University Courant Insitute of Mathematical Sciences, New York, 2003. doi: 10.1090/cln/010.

[12]

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.

[13]

M. Conti, S. Terracini and G. Verzini, Nehari's problem and competing species systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 871-888.  doi: 10.1016/S0294-1449(02)00104-X.

[14]

M. Del Pino and P. Felmer, Multi-peak bound states for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 127-149.  doi: 10.1016/S0294-1449(97)89296-7.

[15]

E. N. Dancer and S. Yan, Multipeak solutions for a singularly perturbed Neumann problem, Pacific J. Math., 189 (1999), 241-262.  doi: 10.2140/pjm.1999.189.241.

[16]

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. Poincaré Anal. Non Linéaire, 27 (2010), 953-969.  doi: 10.1016/j.anihpc.2010.01.009.

[17]

C. Gui and J. Wei, Multiple interior peak solutions for some singularly perturbed Neumann problems, J. Differential Equations, 158 (1999), 1-27.  doi: 10.1016/S0022-0396(99)80016-3.

[18]

C. Gui and J. Wei, On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems, Canad. J. Math., 52 (2000), 522-538.  doi: 10.4153/CJM-2000-024-x.

[19]

Y. Li, On a singularly perturbed elliptic equation, Adv. Differential Equations, 2 (1997), 955-980. 

[20]

Y. Li and L. Nirenberg, The Dirichlet problem for singularly perturbed elliptic equations, Comm. Pure Appl. Math., 51 (1998), 1445-1490.  doi: 10.1002/(SICI)1097-0312(199811/12)51:11/12<1445::AID-CPA9>3.3.CO;2-Q.

[21]

C. Lin, W. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27.  doi: 10.1016/0022-0396(88)90147-7.

[22]

Z. Liu and Z. Q. Wang, Ground states and bound states of a nonlinear Schrödinger system, Adv. Nonlinear Stud., 10 (2010), 175-193.  doi: 10.1515/ans-2010-0109.

[23]

E. Montefusco, B. Pellacci and M. Squassina, Semiclassical states for weakly coupled nonlinear Schrödinger systems, J. Eur. Math. Soc., 10 (2008), 47-71.  doi: 10.4171/JEMS/103.

[24]

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.

[25]

E. S. Noussair and S. Yan, On positive multipeak solutions of a nonlinear elliptic problem, J. Lond. Math. Soc., 62 (2000), 213-227.  doi: 10.1112/S002461070000898X.

[26]

W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162. 

[27]

B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbb{R}^{N}$, Comm. Math. Phys., 271 (2007), 199-221.  doi: 10.1007/s00220-006-0179-x.

[28]

C. Sulem and P. L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse, vol.39 of Applied Mathematical Sciences. Springer-Verlag, New York, 1999.

[29]

R. Tian and Z. Q. Wang, Multiple solitary wave solutions of nonlinear Schrödinger systems, Topol. Methods Nonlinear Anal., 37 (2011), 203-223. 

[30]

J. Wei and T. Weth, Radial solutions and phase separation in a system of two coupled Schrödinger equations, Arch. Ration. Mech. Anal., 190 (2008), 83-106.  doi: 10.1007/s00205-008-0121-9.

[31]

C. Wang and J. Zhou, Infinitely many solitary waves due to the second-harmonic generation in quadratic media, bo be appeared in Acta Math. Sci. Ser. B (Engl. Ed.) (2020, no.1).

[32]

A. C. Yew, Multipulses of nonlinearly coupled Schrödinger equations, J. Differential Equations, 173 (2001), 92-137.  doi: 10.1006/jdeq.2000.3922.

[33]

A. C. Yew, A. R. Champneys and P. J. McKenna, Multiple solitary waves due to second-harmonic generation in quadratic media, J. Nonlinear Sci., 9 (1999), 33-52.  doi: 10.1007/s003329900063.

[34]

L. Zhao, F. Zhao and J. Shi, Higher dimensional solitary waves generated by second-harmonic generation in quadratic media, Calc. Var. Partial Differential Equations, 54 (2015), 2657-2691.  doi: 10.1007/s00526-015-0879-1.

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