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Boundary value problems for harmonic functions on domains in Sierpinski gaskets
Multiple solutions for a nonlinear Schrödinger systems
Department of Mathematics, Tsinghua University, Beijing, 100084, China |
$ \left\{\begin{aligned}-\Delta u+a(x)u& = |u|^{p-2}u+\beta |u|^{\frac{p}{2}-2}u|v|^{\frac{p}{2}} \ \ \hbox{in } \mathbb{R}^{N}\\-\Delta v+a(x)v& = |v|^{p-2}v+\beta |v|^{\frac{p}{2}-2}v|u|^{\frac{p}{2}}\ \ \ \hbox{in } \mathbb{R}^{N}\\(u,v)&\in (H^1(\mathbb{R}^N))^2,\end{aligned}\right. \ \ \ \ \ \ \ {(P)} $ |
$ N\geq3 $ |
$ 2<p<\frac{2N}{N-2} = 2^{\ast} $ |
$ \beta\in \mathbb{R} $ |
$ a(x) $ |
$ \mathcal{C}^1 $ |
$ \beta<0 $ |
$ a(x) $ |
$ (P). $ |
References:
[1] |
R. Adams, Sobolev Spaces, Academic press, New York-London, 1975.
![]() ![]() |
[2] |
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. |
[3] |
A. Bahri and P. Lions,
On the existence of a positive solution of semilinear elliptic equations in unbounded domains, Ann. Ins. H. Poincaré Anayse Nonlinéaire, 14 (1997), 285-300.
doi: 10.1016/S0294-1449(97)80142-4. |
[4] |
T. Bartsch and Z. Wang,
Sign changing solutions of nonlinear Schrödinger equations, Top. Meth. Nonlinear Anal., 13 (1999), 191-198.
doi: 10.12775/TMNA.1999.010. |
[5] |
T. Bartsch, Z. Wang and J. Wei,
Bound states for a coupled Schr$\ddot{o}$dinger system, J. Fixed Point Theory Appl., 2 (2007), 67-82.
doi: 10.1007/s11784-007-0033-6. |
[6] |
T. Bartsch and M. Willelm,
Infinitely many nonradial solutions of an Euclidean scalar field equation, J. Funct. Anal., 117 (1993), 447-460.
doi: 10.1006/jfan.1993.1133. |
[7] |
V. Benci and G. Cerami,
Positive solutions of some nonlinear elliptic problems in exterior domains, Arch. Rat. Mech. Anal., 99 (1987), 283-300.
doi: 10.1007/BF00282048. |
[8] |
H. Berestycki and P. Lions, Nonlinear scalar field equations, Ⅰ Existence of a ground state. Ⅱ Existence of infinitely many solutions, Arch. Rat. Mech. Anal., 82 (1983), 313-346,347-376.
doi: 10.1007/BF00250556. |
[9] |
G. Cerami, D. Devillanova and S. Solimini,
Infinitely many bound states for some nonlinear scalar fileld equation, Calc. Var. Partial Differential Equations, 23 (2005), 139-168.
doi: 10.1007/s00526-004-0293-6. |
[10] |
G. Cerami, D. Passaseo and S. Solimini,
Infinitely many positive solutions to some scalar fileld equation, Comm. Pure Appl. math., 66 (2013), 372-413.
doi: 10.1002/cpa.21410. |
[11] |
M. Conti, S. Terracini and G. Verzini,
Neharis problem and competing species systems, Ann. Inst. H. Poincar Ana Non Linnaire, 19 (2002), 871-888.
doi: 10.1016/S0294-1449(02)00104-X. |
[12] |
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. Poincar Ana. Non Linaire, 27 (2010), 953-869.
doi: 10.1016/j.anihpc.2010.01.009. |
[13] |
G. Devillanova and S. Solimini,
Concentrations estimates and multiple solutions to elliptic problems at critical growth, Advances in Differential Equations, 7 (2002), 1257-1280.
|
[14] |
W. Ding and W. Ni,
On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Rat. Mech. Anal., 91 (1986), 283-308.
doi: 10.1007/BF00282336. |
[15] |
B. Esry, C. Greene, J. Burke Jr. and J. Bohn,
Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597.
|
[16] |
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-Heidelberg-New York, 1983.
doi: 10.1007/978-3-642-61798-0. |
[17] |
Q. Han and F. Lin, Elliptic Paritial Differential Equations, $2^nd$ edition, AMS, 2011. |
[18] |
P. L. Lions, The concentration compactness principle in the calculus of variations Parts Ⅰ and Ⅱ, Ann. Ins. H. Poincaré, 1 (1984), 109-145, 223-283. |
[19] |
T. Lin and J. Wei,
Ground state of $N$ couple nonlinear Schrödinger equations in $\mathbb{R}^n, n\leq 3$, Comm. Math. Phys., 255 (2005), 629-653.
doi: 10.1007/s00220-005-1313-x. |
[20] |
T. C. Lin and J. Wei,
Spike in two coupled nonlinear Schrödinger equations, Ann. Inst. H. Poincar Ana. Non Linaire, 22 (2005), 403-439.
doi: 10.1016/j.anihpc.2004.03.004. |
[21] |
C. Maniscalco,
Multiple solutions for semilinear elliptic problems in $\mathbb{R}^N$, Ann. Univ. Ferrara., 37 (1991), 95-110.
|
[22] |
Z. Nehari,
On a nonlinear differential equation arising in nuclear physics, Proc. Roy. Irish. Acad., 62 (1963), 117-235.
|
[23] |
E. Noris, H. Tavares, S. Terracini and G. Verzini,
Uniform holder bounds for nonlinear Schrödinger system with strong competition, Comm. Pure Appl. Math., 63 (2010), 267-302.
doi: 10.1002/cpa.20309. |
[24] |
R. Palais,
Lusternik-Schnirelmann theory on Banach spaces, Topology, 5 (1996), 115-132.
doi: 10.1016/0040-9383(66)90013-9. |
[25] |
S. Peng and Z. Wang,
Segregated and synchronized vector solutions for nonlinear Schrödinger system, Arch. Rat. Mech. Anal., 208 (2013), 305-339.
doi: 10.1007/s00205-012-0598-0. |
[26] |
P. Rabinowitz,
On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.
doi: 10.1007/BF00946631. |
[27] |
Y. Rudyak and F. Schlenk,
Lusternik-Schnirelmann theory for fixed points of maps, Topological Methods in Nonlinear Analysis, 21 (2003), 171-194.
doi: 10.12775/TMNA.2003.011. |
[28] |
S. Solimini,
Morse index estimates in min-max theorems, Manuscripta Math., 63 (1989), 421-453.
doi: 10.1007/BF01171757. |
[29] |
W. Strauss,
Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.
|
[30] |
S. Terracini and G. Verzini,
Multiple phase in $k-$mixture of Bose-Einstein condensates, Arch. Rat. Mech. Anal., 194 (2009), 717-741.
doi: 10.1007/s00205-008-0172-y. |
[31] |
J. Wei and T. Weth,
Nonradial symmetric bound states for a system of coupled Schrödinger equation, Rend. Lincei Mat. Appl., 18 (2007), 279-293.
doi: 10.4171/RLM/495. |
[32] |
J. Wei and T. Weth,
Radial solutions and phase sepration in a system of two coupled systems of nonlinear Schrödinger equations, Arch. Rat. Mech. Anal., 190 (2008), 83-106.
doi: 10.1007/s00205-008-0121-9. |
[33] |
J. Wei and W. Yao,
Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, Comm. Pure Appl. Anal., 11 (2012), 1003-1011.
doi: 10.3934/cpaa.2012.11.1003. |
[34] |
C. Zelati and P. Robinowitz,
Homoclinic type solutions for a semilinear elliptic PDE on $\mathbb{R}^N$, Comm. Pure. Appl. Math., 10 (1992), 1217-1269.
doi: 10.1002/cpa.3160451002. |
show all references
References:
[1] |
R. Adams, Sobolev Spaces, Academic press, New York-London, 1975.
![]() ![]() |
[2] |
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. |
[3] |
A. Bahri and P. Lions,
On the existence of a positive solution of semilinear elliptic equations in unbounded domains, Ann. Ins. H. Poincaré Anayse Nonlinéaire, 14 (1997), 285-300.
doi: 10.1016/S0294-1449(97)80142-4. |
[4] |
T. Bartsch and Z. Wang,
Sign changing solutions of nonlinear Schrödinger equations, Top. Meth. Nonlinear Anal., 13 (1999), 191-198.
doi: 10.12775/TMNA.1999.010. |
[5] |
T. Bartsch, Z. Wang and J. Wei,
Bound states for a coupled Schr$\ddot{o}$dinger system, J. Fixed Point Theory Appl., 2 (2007), 67-82.
doi: 10.1007/s11784-007-0033-6. |
[6] |
T. Bartsch and M. Willelm,
Infinitely many nonradial solutions of an Euclidean scalar field equation, J. Funct. Anal., 117 (1993), 447-460.
doi: 10.1006/jfan.1993.1133. |
[7] |
V. Benci and G. Cerami,
Positive solutions of some nonlinear elliptic problems in exterior domains, Arch. Rat. Mech. Anal., 99 (1987), 283-300.
doi: 10.1007/BF00282048. |
[8] |
H. Berestycki and P. Lions, Nonlinear scalar field equations, Ⅰ Existence of a ground state. Ⅱ Existence of infinitely many solutions, Arch. Rat. Mech. Anal., 82 (1983), 313-346,347-376.
doi: 10.1007/BF00250556. |
[9] |
G. Cerami, D. Devillanova and S. Solimini,
Infinitely many bound states for some nonlinear scalar fileld equation, Calc. Var. Partial Differential Equations, 23 (2005), 139-168.
doi: 10.1007/s00526-004-0293-6. |
[10] |
G. Cerami, D. Passaseo and S. Solimini,
Infinitely many positive solutions to some scalar fileld equation, Comm. Pure Appl. math., 66 (2013), 372-413.
doi: 10.1002/cpa.21410. |
[11] |
M. Conti, S. Terracini and G. Verzini,
Neharis problem and competing species systems, Ann. Inst. H. Poincar Ana Non Linnaire, 19 (2002), 871-888.
doi: 10.1016/S0294-1449(02)00104-X. |
[12] |
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. Poincar Ana. Non Linaire, 27 (2010), 953-869.
doi: 10.1016/j.anihpc.2010.01.009. |
[13] |
G. Devillanova and S. Solimini,
Concentrations estimates and multiple solutions to elliptic problems at critical growth, Advances in Differential Equations, 7 (2002), 1257-1280.
|
[14] |
W. Ding and W. Ni,
On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Rat. Mech. Anal., 91 (1986), 283-308.
doi: 10.1007/BF00282336. |
[15] |
B. Esry, C. Greene, J. Burke Jr. and J. Bohn,
Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597.
|
[16] |
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-Heidelberg-New York, 1983.
doi: 10.1007/978-3-642-61798-0. |
[17] |
Q. Han and F. Lin, Elliptic Paritial Differential Equations, $2^nd$ edition, AMS, 2011. |
[18] |
P. L. Lions, The concentration compactness principle in the calculus of variations Parts Ⅰ and Ⅱ, Ann. Ins. H. Poincaré, 1 (1984), 109-145, 223-283. |
[19] |
T. Lin and J. Wei,
Ground state of $N$ couple nonlinear Schrödinger equations in $\mathbb{R}^n, n\leq 3$, Comm. Math. Phys., 255 (2005), 629-653.
doi: 10.1007/s00220-005-1313-x. |
[20] |
T. C. Lin and J. Wei,
Spike in two coupled nonlinear Schrödinger equations, Ann. Inst. H. Poincar Ana. Non Linaire, 22 (2005), 403-439.
doi: 10.1016/j.anihpc.2004.03.004. |
[21] |
C. Maniscalco,
Multiple solutions for semilinear elliptic problems in $\mathbb{R}^N$, Ann. Univ. Ferrara., 37 (1991), 95-110.
|
[22] |
Z. Nehari,
On a nonlinear differential equation arising in nuclear physics, Proc. Roy. Irish. Acad., 62 (1963), 117-235.
|
[23] |
E. Noris, H. Tavares, S. Terracini and G. Verzini,
Uniform holder bounds for nonlinear Schrödinger system with strong competition, Comm. Pure Appl. Math., 63 (2010), 267-302.
doi: 10.1002/cpa.20309. |
[24] |
R. Palais,
Lusternik-Schnirelmann theory on Banach spaces, Topology, 5 (1996), 115-132.
doi: 10.1016/0040-9383(66)90013-9. |
[25] |
S. Peng and Z. Wang,
Segregated and synchronized vector solutions for nonlinear Schrödinger system, Arch. Rat. Mech. Anal., 208 (2013), 305-339.
doi: 10.1007/s00205-012-0598-0. |
[26] |
P. Rabinowitz,
On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.
doi: 10.1007/BF00946631. |
[27] |
Y. Rudyak and F. Schlenk,
Lusternik-Schnirelmann theory for fixed points of maps, Topological Methods in Nonlinear Analysis, 21 (2003), 171-194.
doi: 10.12775/TMNA.2003.011. |
[28] |
S. Solimini,
Morse index estimates in min-max theorems, Manuscripta Math., 63 (1989), 421-453.
doi: 10.1007/BF01171757. |
[29] |
W. Strauss,
Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.
|
[30] |
S. Terracini and G. Verzini,
Multiple phase in $k-$mixture of Bose-Einstein condensates, Arch. Rat. Mech. Anal., 194 (2009), 717-741.
doi: 10.1007/s00205-008-0172-y. |
[31] |
J. Wei and T. Weth,
Nonradial symmetric bound states for a system of coupled Schrödinger equation, Rend. Lincei Mat. Appl., 18 (2007), 279-293.
doi: 10.4171/RLM/495. |
[32] |
J. Wei and T. Weth,
Radial solutions and phase sepration in a system of two coupled systems of nonlinear Schrödinger equations, Arch. Rat. Mech. Anal., 190 (2008), 83-106.
doi: 10.1007/s00205-008-0121-9. |
[33] |
J. Wei and W. Yao,
Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, Comm. Pure Appl. Anal., 11 (2012), 1003-1011.
doi: 10.3934/cpaa.2012.11.1003. |
[34] |
C. Zelati and P. Robinowitz,
Homoclinic type solutions for a semilinear elliptic PDE on $\mathbb{R}^N$, Comm. Pure. Appl. Math., 10 (1992), 1217-1269.
doi: 10.1002/cpa.3160451002. |
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