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March  2013, 12(2): 831-850. doi: 10.3934/cpaa.2013.12.831

Standing waves of nonlinear Schrödinger equations with optimal conditions for potential and nonlinearity

1. 

Faculty of Liberal Arts and Sciences, Hanbat National University, Daejeon, 305-719

2. 

Department of Mathematics, Pohang University of Science and Technology, Pohang, Kyungbuk 790-784

Received  September 2011 Revised  January 2012 Published  September 2012

We consider the singularly perturbed nonlinear elliptic problem \begin{eqnarray*} \varepsilon^2 \Delta v - V(x)v + f(v) =0, v > 0, \lim_{|x|\to \infty} v(x) = 0. \end{eqnarray*} Under almost optimal conditions for the potential $V$ and the nonlinearity $f$, we establish the existence of single-peak solutions whose peak points converge to local minimum points of $V$ as $\varepsilon \to 0$. Moreover, we exhibit a threshold on the condition of $V$ at infinity between existence and nonexistence of solutions.
Citation: Soohyun Bae, Jaeyoung Byeon. Standing waves of nonlinear Schrödinger equations with optimal conditions for potential and nonlinearity. Communications on Pure and Applied Analysis, 2013, 12 (2) : 831-850. doi: 10.3934/cpaa.2013.12.831
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, A. Malchiodi and S. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials, Arch. Ration. Mech. Anal., 159 (2001), 253-271. doi: 10.1007/s002050100152.

[3]

A. Ambrosetti, V. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc., 7 (2005), 117-144. doi: 10.4171/JEMS/24.

[4]

A. Ambrosetti and A. Malchiodi, "Perturbation Methods and Semilinear Elliptic Problems on $R^N$," Progress in Mathematics 240, Birkhäuser Verlag, Basel, 2006. doi: 10.1007/3-7643-7396-2.

[5]

A. Ambrosetti, A. Malchiodi and D. Ruiz, Bound states of Nonlinear Schrödinger equations with potentials vanishing at infinity, J. d'Analyse Math., 98 (2006), 317-348. doi: 10.1007/BF02790279.

[6]

A. Ambrosetti and D. Ruiz, Radial solutions concentrating on spheres of NLS with vanishing potentials, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 889-907. doi: 10.1017/S0308210500004789.

[7]

H. Berestycki and P.L. Lions, Nonlinear scalar field equations I existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-346. doi: 10.1007/BF00250555.

[8]

M. Bidaut-Veron, Local and global behavior of solutions of quasilinear equations of Emden-Fowler type, Arch. Rational Mech. Anal., 107 (1989), 293-324. doi: 10.1007/BF00251552.

[9]

J. Byeon, and L. Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity, Arch. Ration. Mech. Anal., 185 (2007), 185-200. doi: 10.1007/s00205-006-0019-3.

[10]

J. Byeon, L. Jeanjean and K. Tanaka, Standing waves for nonlinear Schrödinger equations with a general nonlinearity: one and two dimensional cases, Comm. Partial Differential Equations, 33 (2008), 1113-1136. doi: 10.1080/03605300701518174.

[11]

J. Byeon and Z.-Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations, Arch. Ration. Mech. Anal., 165 (2002), 295-316. doi: 10.1007/s00205-002-0225-6.

[12]

J. Byeon and Z.-Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations II, Calculus of Variations and PDE, 18 (2003), 207-219. doi: 10.1007/s00526-002-0191-8.

[13]

E. N. Dancer, K. Y. Lam and S. Yan, The effect of the graph topology on the existence of multipeak solutions for nonlinear Schrödinger equations, Abstr. Appl. Anal., 3 (1998), 293-318. doi: 10.1155/S1085337501000276.

[14]

E. N. Dancer and S. Yan, On the existence of multipeak solutions for nonlinear field equations on $R^N$, Discrete Contin. Dynam. Systems, 6 (2000), 39-50. doi: 10.3934/dcds.2000.6.39.

[15]

M. Del Pino and P. L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calculus of Variations and PDE, 4 (1996), 121-137. doi: 10.1007/BF01189950.

[16]

M. Del Pino and P. L. Felmer, Semi-classical states for nonlinear Schrödinger equations, J. Functional Analysis, 149 (1997), 245-265. doi: 10.1006/jfan.1996.3085.

[17]

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

[18]

M. Del Pino and P. L. Felmer, Semi-classical states for nonlinear Schrödinger equations: a variational reduction method, Math. Ann., 324 (2002), 1-32. doi: 10.1007/s002080200327.

[19]

A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equations with a bounded potential, J. Functional Analysis, 69 (1986), 397-408. doi: 10.1016/0022-1236(86)90096-0.

[20]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," 2nd edition, Grundlehren 224, Springer, Berlin, Heidelberg, New York and Tokyo, 1983.

[21]

M. Guedda and L. Veron, Local and global properties of solutions of quasilinear elliptic equations, J. Differential Equations, 76 (1988), 159-189. doi: 10.1016/0022-0396(88)90068-X.

[22]

C. Gui, Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method, Comm. Partial Differential Equations, 21 (1996), 787-820. doi: 10.1080/03605309608821208.

[23]

L. Jeanjean and K. Tanaka, A remark on least energy solutions in $R^N$, Proc. Amer. Math. Soc., 131 (2003), 2399-2408. doi: 10.1090/S0002-9939-02-06821-1.

[24]

L. Jeanjean and K. Tanaka, Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities, Calculus of Variations and PDE, 21 (2004), 287-318. doi: 10.1007/s00526-003-0261-6.

[25]

O. Kwon, Existence of multi-bump standing waves with a critical frequency for nonlinear Schrödingerinfinity, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 833-852. doi: 10.1017/S0308210508000309.

[26]

V. Kondratiev, V. Liskevich and Z. Sobol, Second-order semilinear elliptic inequalities in exterior domains, J. Differential Equations, 187 (2003), 429-455 doi: 10.1016/S0022-0396(02)00036-0.

[27]

X. Kang and J. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations, Adv. Differential Equations, 5 (2000), 899-928.

[28]

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

[29]

P. L. Lions, The concentration -compactness principle in the calculus of variations. The locally compact case, part II , Ann. Inst. Henri Poincaré, 1 (1984), 223-283.

[30]

V. Liskevich, S. Lyakhova and V. Moroz, Positive solutions to singular semilinear elliptic equations with critical potential on cone-like domains, Adv. Differential Equations, 4 (2006), 361-398.

[31]

V. Moroz and J. Van Schaftingen, Semiclassical stationary states for nonlinear Schrodinger equations with fast decaying potentials, Calculus of Variations and PDE, 37 (2010), 1-27. doi: 10.1007/s00526-009-0249-y.

[32]

W. M. Ni and J. Serrin, Nonexistence theorems for quasilinear partial differential equations, Proceedings of the Conference Commemorating the 1st Centennial of the Circolo Matematico di Palermo(Palermo, 1984), Rend. Circ. Mat. Palermo (2) Suppl. (1985), 171-185.

[33]

Y. G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_a$, Comm. Partial Differential Equations, 13 (1988), 1499-1519. doi: 10.1080/03605308808820585.

[34]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631.

[35]

X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244. doi: 10.1007/BF02096642.

[36]

H. Yin and P. Zhang, Bound states of nonlinear Schrodinger equations with potentials tending to zero at infinity, J. of Differential Equations, 247 (2009), 618-647. doi: 10.1016/j.jde.2009.03.002.

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, A. Malchiodi and S. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials, Arch. Ration. Mech. Anal., 159 (2001), 253-271. doi: 10.1007/s002050100152.

[3]

A. Ambrosetti, V. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc., 7 (2005), 117-144. doi: 10.4171/JEMS/24.

[4]

A. Ambrosetti and A. Malchiodi, "Perturbation Methods and Semilinear Elliptic Problems on $R^N$," Progress in Mathematics 240, Birkhäuser Verlag, Basel, 2006. doi: 10.1007/3-7643-7396-2.

[5]

A. Ambrosetti, A. Malchiodi and D. Ruiz, Bound states of Nonlinear Schrödinger equations with potentials vanishing at infinity, J. d'Analyse Math., 98 (2006), 317-348. doi: 10.1007/BF02790279.

[6]

A. Ambrosetti and D. Ruiz, Radial solutions concentrating on spheres of NLS with vanishing potentials, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 889-907. doi: 10.1017/S0308210500004789.

[7]

H. Berestycki and P.L. Lions, Nonlinear scalar field equations I existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-346. doi: 10.1007/BF00250555.

[8]

M. Bidaut-Veron, Local and global behavior of solutions of quasilinear equations of Emden-Fowler type, Arch. Rational Mech. Anal., 107 (1989), 293-324. doi: 10.1007/BF00251552.

[9]

J. Byeon, and L. Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity, Arch. Ration. Mech. Anal., 185 (2007), 185-200. doi: 10.1007/s00205-006-0019-3.

[10]

J. Byeon, L. Jeanjean and K. Tanaka, Standing waves for nonlinear Schrödinger equations with a general nonlinearity: one and two dimensional cases, Comm. Partial Differential Equations, 33 (2008), 1113-1136. doi: 10.1080/03605300701518174.

[11]

J. Byeon and Z.-Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations, Arch. Ration. Mech. Anal., 165 (2002), 295-316. doi: 10.1007/s00205-002-0225-6.

[12]

J. Byeon and Z.-Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations II, Calculus of Variations and PDE, 18 (2003), 207-219. doi: 10.1007/s00526-002-0191-8.

[13]

E. N. Dancer, K. Y. Lam and S. Yan, The effect of the graph topology on the existence of multipeak solutions for nonlinear Schrödinger equations, Abstr. Appl. Anal., 3 (1998), 293-318. doi: 10.1155/S1085337501000276.

[14]

E. N. Dancer and S. Yan, On the existence of multipeak solutions for nonlinear field equations on $R^N$, Discrete Contin. Dynam. Systems, 6 (2000), 39-50. doi: 10.3934/dcds.2000.6.39.

[15]

M. Del Pino and P. L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calculus of Variations and PDE, 4 (1996), 121-137. doi: 10.1007/BF01189950.

[16]

M. Del Pino and P. L. Felmer, Semi-classical states for nonlinear Schrödinger equations, J. Functional Analysis, 149 (1997), 245-265. doi: 10.1006/jfan.1996.3085.

[17]

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

[18]

M. Del Pino and P. L. Felmer, Semi-classical states for nonlinear Schrödinger equations: a variational reduction method, Math. Ann., 324 (2002), 1-32. doi: 10.1007/s002080200327.

[19]

A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equations with a bounded potential, J. Functional Analysis, 69 (1986), 397-408. doi: 10.1016/0022-1236(86)90096-0.

[20]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," 2nd edition, Grundlehren 224, Springer, Berlin, Heidelberg, New York and Tokyo, 1983.

[21]

M. Guedda and L. Veron, Local and global properties of solutions of quasilinear elliptic equations, J. Differential Equations, 76 (1988), 159-189. doi: 10.1016/0022-0396(88)90068-X.

[22]

C. Gui, Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method, Comm. Partial Differential Equations, 21 (1996), 787-820. doi: 10.1080/03605309608821208.

[23]

L. Jeanjean and K. Tanaka, A remark on least energy solutions in $R^N$, Proc. Amer. Math. Soc., 131 (2003), 2399-2408. doi: 10.1090/S0002-9939-02-06821-1.

[24]

L. Jeanjean and K. Tanaka, Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities, Calculus of Variations and PDE, 21 (2004), 287-318. doi: 10.1007/s00526-003-0261-6.

[25]

O. Kwon, Existence of multi-bump standing waves with a critical frequency for nonlinear Schrödingerinfinity, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 833-852. doi: 10.1017/S0308210508000309.

[26]

V. Kondratiev, V. Liskevich and Z. Sobol, Second-order semilinear elliptic inequalities in exterior domains, J. Differential Equations, 187 (2003), 429-455 doi: 10.1016/S0022-0396(02)00036-0.

[27]

X. Kang and J. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations, Adv. Differential Equations, 5 (2000), 899-928.

[28]

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

[29]

P. L. Lions, The concentration -compactness principle in the calculus of variations. The locally compact case, part II , Ann. Inst. Henri Poincaré, 1 (1984), 223-283.

[30]

V. Liskevich, S. Lyakhova and V. Moroz, Positive solutions to singular semilinear elliptic equations with critical potential on cone-like domains, Adv. Differential Equations, 4 (2006), 361-398.

[31]

V. Moroz and J. Van Schaftingen, Semiclassical stationary states for nonlinear Schrodinger equations with fast decaying potentials, Calculus of Variations and PDE, 37 (2010), 1-27. doi: 10.1007/s00526-009-0249-y.

[32]

W. M. Ni and J. Serrin, Nonexistence theorems for quasilinear partial differential equations, Proceedings of the Conference Commemorating the 1st Centennial of the Circolo Matematico di Palermo(Palermo, 1984), Rend. Circ. Mat. Palermo (2) Suppl. (1985), 171-185.

[33]

Y. G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_a$, Comm. Partial Differential Equations, 13 (1988), 1499-1519. doi: 10.1080/03605308808820585.

[34]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631.

[35]

X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244. doi: 10.1007/BF02096642.

[36]

H. Yin and P. Zhang, Bound states of nonlinear Schrodinger equations with potentials tending to zero at infinity, J. of Differential Equations, 247 (2009), 618-647. doi: 10.1016/j.jde.2009.03.002.

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