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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 & 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. doi: 10.1007/s002050050067. Google Scholar [2] A. Ambrosetti, A. Malchiodi and S. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials,, Arch. Ration. Mech. Anal., 159 (2001), 253. doi: 10.1007/s002050100152. Google Scholar [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. doi: 10.4171/JEMS/24. Google Scholar [4] A. Ambrosetti and A. Malchiodi, "Perturbation Methods and Semilinear Elliptic Problems on $R^N$,", Progress in Mathematics 240, (2006). doi: 10.1007/3-7643-7396-2. Google Scholar [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. doi: 10.1007/BF02790279. Google Scholar [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. doi: 10.1017/S0308210500004789. Google Scholar [7] H. Berestycki and P.L. Lions, Nonlinear scalar field equations I existence of a ground state,, Arch. Ration. Mech. Anal., 82 (1983), 313. doi: 10.1007/BF00250555. Google Scholar [8] M. Bidaut-Veron, Local and global behavior of solutions of quasilinear equations of Emden-Fowler type,, Arch. Rational Mech. Anal., 107 (1989), 293. doi: 10.1007/BF00251552. Google Scholar [9] J. Byeon, and L. Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity,, Arch. Ration. Mech. Anal., 185 (2007), 185. doi: 10.1007/s00205-006-0019-3. Google Scholar [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. doi: 10.1080/03605300701518174. Google Scholar [11] J. Byeon and Z.-Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations,, Arch. Ration. Mech. Anal., 165 (2002), 295. doi: 10.1007/s00205-002-0225-6. Google Scholar [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. doi: 10.1007/s00526-002-0191-8. Google Scholar [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. doi: 10.1155/S1085337501000276. Google Scholar [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. doi: 10.3934/dcds.2000.6.39. Google Scholar [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. doi: 10.1007/BF01189950. Google Scholar [16] M. Del Pino and P. L. Felmer, Semi-classical states for nonlinear Schrödinger equations,, J. Functional Analysis, 149 (1997), 245. doi: 10.1006/jfan.1996.3085. Google Scholar [17] M. Del Pino and P. L. Felmer, Multi-peak bound states for nonlinear Schrödinger equations,, Ann. Inst. Henri Poincar\'e, 15 (1998), 127. doi: 10.1016/S0294-1449(97)89296-7. Google Scholar [18] M. Del Pino and P. L. Felmer, Semi-classical states for nonlinear Schrödinger equations: a variational reduction method,, Math. Ann., 324 (2002), 1. doi: 10.1007/s002080200327. Google Scholar [19] A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equations with a bounded potential,, J. Functional Analysis, 69 (1986), 397. doi: 10.1016/0022-1236(86)90096-0. Google Scholar [20] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", 2$^{nd}$ edition, (1983). Google Scholar [21] M. Guedda and L. Veron, Local and global properties of solutions of quasilinear elliptic equations,, J. Differential Equations, 76 (1988), 159. doi: 10.1016/0022-0396(88)90068-X. Google Scholar [22] C. Gui, Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method,, Comm. Partial Differential Equations, 21 (1996), 787. doi: 10.1080/03605309608821208. Google Scholar [23] L. Jeanjean and K. Tanaka, A remark on least energy solutions in $R^N$,, Proc. Amer. Math. Soc., 131 (2003), 2399. doi: 10.1090/S0002-9939-02-06821-1. Google Scholar [24] L. Jeanjean and K. Tanaka, Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities,, Calculus of Variations and PDE, 21 (2004), 287. doi: 10.1007/s00526-003-0261-6. Google Scholar [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. doi: 10.1017/S0308210508000309. Google Scholar [26] V. Kondratiev, V. Liskevich and Z. Sobol, Second-order semilinear elliptic inequalities in exterior domains,, J. Differential Equations, 187 (2003), 429. doi: 10.1016/S0022-0396(02)00036-0. Google Scholar [27] X. Kang and J. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations,, Adv. Differential Equations, 5 (2000), 899. Google Scholar [28] Y. Y. Li, On a singularly perturbed elliptic equation,, Adv. Differential Equations, 2 (1997), 955. Google Scholar [29] P. L. Lions, The concentration -compactness principle in the calculus of variations. The locally compact case, part II ,, Ann. Inst. Henri Poincar\'e, 1 (1984), 223. Google Scholar [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. Google Scholar [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. doi: 10.1007/s00526-009-0249-y. Google Scholar [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, (1985), 171. Google Scholar [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. doi: 10.1080/03605308808820585. Google Scholar [34] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations,, Z. Angew. Math. Phys., 43 (1992), 270. doi: 10.1007/BF00946631. Google Scholar [35] X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations,, Comm. Math. Phys., 153 (1993), 229. doi: 10.1007/BF02096642. Google Scholar [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. doi: 10.1016/j.jde.2009.03.002. Google Scholar

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. doi: 10.1007/s002050050067. Google Scholar [2] A. Ambrosetti, A. Malchiodi and S. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials,, Arch. Ration. Mech. Anal., 159 (2001), 253. doi: 10.1007/s002050100152. Google Scholar [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. doi: 10.4171/JEMS/24. Google Scholar [4] A. Ambrosetti and A. Malchiodi, "Perturbation Methods and Semilinear Elliptic Problems on $R^N$,", Progress in Mathematics 240, (2006). doi: 10.1007/3-7643-7396-2. Google Scholar [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. doi: 10.1007/BF02790279. Google Scholar [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. doi: 10.1017/S0308210500004789. Google Scholar [7] H. Berestycki and P.L. Lions, Nonlinear scalar field equations I existence of a ground state,, Arch. Ration. Mech. Anal., 82 (1983), 313. doi: 10.1007/BF00250555. Google Scholar [8] M. Bidaut-Veron, Local and global behavior of solutions of quasilinear equations of Emden-Fowler type,, Arch. Rational Mech. Anal., 107 (1989), 293. doi: 10.1007/BF00251552. Google Scholar [9] J. Byeon, and L. Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity,, Arch. Ration. Mech. Anal., 185 (2007), 185. doi: 10.1007/s00205-006-0019-3. Google Scholar [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. doi: 10.1080/03605300701518174. Google Scholar [11] J. Byeon and Z.-Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations,, Arch. Ration. Mech. Anal., 165 (2002), 295. doi: 10.1007/s00205-002-0225-6. Google Scholar [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. doi: 10.1007/s00526-002-0191-8. Google Scholar [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. doi: 10.1155/S1085337501000276. Google Scholar [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. doi: 10.3934/dcds.2000.6.39. Google Scholar [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. doi: 10.1007/BF01189950. Google Scholar [16] M. Del Pino and P. L. Felmer, Semi-classical states for nonlinear Schrödinger equations,, J. Functional Analysis, 149 (1997), 245. doi: 10.1006/jfan.1996.3085. Google Scholar [17] M. Del Pino and P. L. Felmer, Multi-peak bound states for nonlinear Schrödinger equations,, Ann. Inst. Henri Poincar\'e, 15 (1998), 127. doi: 10.1016/S0294-1449(97)89296-7. Google Scholar [18] M. Del Pino and P. L. Felmer, Semi-classical states for nonlinear Schrödinger equations: a variational reduction method,, Math. Ann., 324 (2002), 1. doi: 10.1007/s002080200327. Google Scholar [19] A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equations with a bounded potential,, J. Functional Analysis, 69 (1986), 397. doi: 10.1016/0022-1236(86)90096-0. Google Scholar [20] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", 2$^{nd}$ edition, (1983). Google Scholar [21] M. Guedda and L. Veron, Local and global properties of solutions of quasilinear elliptic equations,, J. Differential Equations, 76 (1988), 159. doi: 10.1016/0022-0396(88)90068-X. Google Scholar [22] C. Gui, Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method,, Comm. Partial Differential Equations, 21 (1996), 787. doi: 10.1080/03605309608821208. Google Scholar [23] L. Jeanjean and K. Tanaka, A remark on least energy solutions in $R^N$,, Proc. Amer. Math. Soc., 131 (2003), 2399. doi: 10.1090/S0002-9939-02-06821-1. Google Scholar [24] L. Jeanjean and K. Tanaka, Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities,, Calculus of Variations and PDE, 21 (2004), 287. doi: 10.1007/s00526-003-0261-6. Google Scholar [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. doi: 10.1017/S0308210508000309. Google Scholar [26] V. Kondratiev, V. Liskevich and Z. Sobol, Second-order semilinear elliptic inequalities in exterior domains,, J. Differential Equations, 187 (2003), 429. doi: 10.1016/S0022-0396(02)00036-0. Google Scholar [27] X. Kang and J. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations,, Adv. Differential Equations, 5 (2000), 899. Google Scholar [28] Y. Y. Li, On a singularly perturbed elliptic equation,, Adv. Differential Equations, 2 (1997), 955. Google Scholar [29] P. L. Lions, The concentration -compactness principle in the calculus of variations. The locally compact case, part II ,, Ann. Inst. Henri Poincar\'e, 1 (1984), 223. Google Scholar [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. Google Scholar [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. doi: 10.1007/s00526-009-0249-y. Google Scholar [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, (1985), 171. Google Scholar [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. doi: 10.1080/03605308808820585. Google Scholar [34] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations,, Z. Angew. Math. Phys., 43 (1992), 270. doi: 10.1007/BF00946631. Google Scholar [35] X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations,, Comm. Math. Phys., 153 (1993), 229. doi: 10.1007/BF02096642. Google Scholar [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. doi: 10.1016/j.jde.2009.03.002. Google Scholar
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