# American Institute of Mathematical Sciences

• Previous Article
Initial value problem for the fourth order nonlinear Schrödinger type equation on torus and orbital stability of standing waves
• CPAA Home
• This Issue
• Next Article
Uniform stability of the Boltzmann equation with an external force near vacuum
May  2015, 14(3): 825-842. doi: 10.3934/cpaa.2015.14.825

## Standing wave concentrating on compact manifolds for nonlinear Schrödinger equations

 1 Department of Mathematical Sciences, KAIST, 335 Gwahangno, Yuseong-gu Daejeon, 305-701, South Korea 2 Center for Partial Differential Equations, East China Normal University, 500 Dongchuan Road, Shanghai, China 3 Department of Mathematics, Okayama University, 3-1-1 Tsushima-naka, Okayama 700-8530, Japan

Received  April 2014 Revised  December 2014 Published  March 2015

For $k =1,\cdots,K,$ let $M_k$ be a $q_k$-dimensional smooth compact framed manifold in $R^N$ with $q_k \in \{1,\cdots,N-1\}$. We consider the equation $-\varepsilon^2\Delta u + V(x)u - u^p = 0$ in $R^N$ where for each $k \in \{1,\cdots,K\}$ and some $m_k > 0,$ $V(x)=|\textrm{dist}(x,M_k)|^{m_k}+O(|\textrm{dist}(x,M_k)|^{m_k+1})$ as $\textrm{dist}(x,M_k) \to 0$. For a sequence of $\varepsilon$ converging to zero, we will find a positive solution $u_{\varepsilon}$ of the equation which concentrates on $M_1\cup \dots \cup M_K$.
Citation: Jaeyoung Byeon, Ohsang Kwon, Yoshihito Oshita. Standing wave concentrating on compact manifolds for nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2015, 14 (3) : 825-842. doi: 10.3934/cpaa.2015.14.825
##### References:
 [1] A. Ambrosetti, A. Malchiodi and W.-M Ni, Singularly perturves elliptic equation with symmetry: existence of solutions concentrating on spheres. I,, \emph{Comm. Math. Phys.}, 235 (2003), 427. doi: 10.1007/s00220-003-0811-y. Google Scholar [2] A. Ambrosetti, A. Malchiodi and D. Ruiz, Bound states of nonlinear Schrödinger equations with potentials vanishing at infinity,, \emph{J. d'Anayse Math.}, 98 (2006), 317. doi: 10.1007/BF02790279. Google Scholar [3] A. Ambrosetti, A. Malchiodi and A. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials,, \emph{Arch. Rational Mech. Anal.}, 159 (2001), 253. doi: 10.1007/s002050100152. Google Scholar [4] A. Ambrosetti and D. Ruiz, Radial solutions concentrating on sphere of nonlinear Schrödinger equations with vanishing potentials,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 136 (2006), 889. doi: 10.1017/S0308210500004789. Google Scholar [5] M. Badiale. and T. D'Aprile, Concentration around a sphere for a singulary perturbed Schrödinger equation,, \emph{Nonlinear Anal.}, 49 (2002), 947. doi: 10.1016/S0362-546X(01)00717-9. Google Scholar [6] V. Benci. and T. D'Aprile, The semiclassical limit of the nonlinear Schrödinger equation in a radial potential,, \emph{J. Diff. Equat.}, 184 (2002), 109. doi: 10.1006/jdeq.2001.4138. Google Scholar [7] J. Byeon, Standing Waves for nonlinear Schrödinger equations with a radial potential,, \emph{Nonlinear Anal.}, 50 (2002), 1135. doi: 10.1016/S0362-546X(01)00805-7. Google Scholar [8] J. Byeon and Y. Oshita, Existence of multi-bump standing waves with a critical frequency for nonlinear Schrödinger equations,, \emph{Comm. Partial Differential Equations}, 29 (2004), 1877. doi: 10.1081/PDE-200040205. Google Scholar [9] J. Byeon and Y. Oshita, Uniqueness of a standing wave for nonlinear Schrödinger equations,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 138A (2008), 975. doi: 10.1017/S0308210507000236. Google Scholar [10] J. Byeon and Z.-Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations,, \emph{Arch. Rational Mech. Anal.}, 165 (2002), 295. doi: 10.1007/s00205-002-0225-6. Google Scholar [11] J. Byeon and Z.-Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations. II,, \emph{Calc. Var. Partial Differential Equations}, 18 (2003), 207. doi: 10.1007/s00526-002-0191-8. Google Scholar [12] D. Cao, E. S. Noussair and S. Yan, Multiscale-bump standing waves with a critical frequency for nonlinear Schrödinger equations,, to appear in \emph{Trans. of AMS}., (). doi: 10.1090/S0002-9947-08-04348-1. Google Scholar [13] D. Cao and S. Peng, Multi-bump bound states of Schrödinger equations with a critical frequency,, \emph{Math. Ann}, 336 (2006), 925. doi: 10.1007/s00208-006-0021-y. Google Scholar [14] E. N. Dancer, The effect of domain shape on the number of positive solutions of certain nonlinear equations,, \emph{J. Differential Equations}, 74 (1998), 120. doi: 10.1016/0022-0396(88)90021-6. Google Scholar [15] E. N. Dancer and S. Yan, On the existence of multipeak solutions for nonlinear field equations on $R^N$,, \emph{Discrete Contin. Dynam. Systems, 6 (2000), 39. Google Scholar [16] 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,, \emph{Abstr. Apply. Anal.}, 3 (1998), 293. doi: 10.1155/S1085337598000578. Google Scholar [17] M. del Pino and P. L. Felmer, Local mountaion passes for semilinear elliptic problems in unbounded domains,, \emph{Calc. Var. Partial Differential Equations}, 4 (1996), 121. doi: 10.1007/BF01189950. Google Scholar [18] M. del Pino and P. L. Felmer, Semi-classical states for nonlinear Schrödinger equations,, \emph{J. Funct. Anal.}, 149 (1997), 245. doi: 10.1006/jfan.1996.3085. Google Scholar [19] M. del Pino and P. L. Felmer, Multi-peak bound states for nonlinear Schrödinger equations,, \emph{Ann. Inst. Henri Poincar\'e}, 15 (1998), 127. doi: 10.1016/S0294-1449(97)89296-7. Google Scholar [20] M. del Pino and P. L. Felmer, Semi-classical states of nonlinear Schrödinger equations: a variational reduction method,, \emph{Math. Ann.}, 324 (2002), 1. doi: 10.1007/s002080200327. Google Scholar [21] M. del Pino, P. L. Felmer and O. H. Miyagaki, Existence of positive bound states of nonlinear Schrödinger equations with saddle-like potential,, \emph{Nonlinear Anal. TMA, 34 (1998), 979. doi: 10.1016/S0362-546X(97)00593-2. Google Scholar [22] M. del Pino, M. Kowalczyk and J.-C. Wei, Concentration on curves for nonlinear Schrödinger equations,, \emph{Comm. Pur. App. Math.}, LX (2007), 113. doi: 10.1002/cpa.20135. Google Scholar [23] A. Floer and A. Weinstein, Non spreading wave packets for the cubic Schrödinger equations with a bounded potential,, \emph{J. Funct. Anal.}, 69 (1986), 397. doi: 10.1016/0022-1236(86)90096-0. Google Scholar [24] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, \emph{Comm. Math. Phys., 68 (1979), 209. Google Scholar [25] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, 2nd ed., (1983). doi: 10.1007/978-3-642-61798-0. Google Scholar [26] C. Gui, Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method,, \emph{Comm. Partial Differential Equations}, 21 (1996), 787. doi: 10.1080/03605309608821208. Google Scholar [27] Y. Kabeya and K. Tanaka, Uniqueness of positive solutions of semilinear elliptic equations in $R^N$ and Séré's non-degeneracy condition,, \emph{Comm. Partial Differential Equations, 24 (1999), 563. doi: 10.1080/03605309908821434. Google Scholar [28] X. Kang and J. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations,, \emph{Adv. Differential Equations}, 5 (2000), 899. Google Scholar [29] T. Kato, Perturbation Theory for Linear Operators,, second ed., (1976). Google Scholar [30] O. Kwon, Existence of multi-bump standing waves with a critical frequency for nonlinear Schrödinger equations with potentials vanishing at infinity,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 139A (2009), 833. doi: 10.1017/S0308210508000309. Google Scholar [31] C. Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains,, \emph{Comm. Partial Differential Equations, 16 (1996), 585. doi: 10.1080/03605309108820770. Google Scholar [32] Y. Y. Li, On a singular perturbed elliptic equation,, \emph{Adv. Differential Equations}, 2 (1997), 955. Google Scholar [33] F. Mahmoudi and A. Malchiodi, Concentration on minimal submainifolds for a singularly perturbed Neumann problem,, \emph{Adv. Math.}, 209 (2007), 460. doi: 10.1016/j.aim.2006.05.014. Google Scholar [34] F. Mahmoudi, R. Mazzeo and F. Pacard, Constant mean curvature hypersurfaces condensing on a submanifold,, \emph{Geom. Funct. Anal.}, 16 (2006), 924. doi: 10.1007/s00039-006-0566-7. Google Scholar [35] F. Mahmoudi, F. S. Sánchez and W. Yao, On the Ambrosetti-Malchiodi-Ni conjecture for general submanifolds,, \emph{J. Differential Equations}, 258 (2015), 243. doi: 10.1016/j.jde.2014.09.010. Google Scholar [36] A. Malchiodi, Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains,, \emph{G.A.F.A.}, 15-16 (2005), 15. doi: 10.1007/s00039-005-0542-7. Google Scholar [37] A. Malchiodi and M. Montenegro, Boundary concentration phenomena for a singularly perturbed elliptic problem,, \emph{Comm. Pure Appl. Math.}, 15 (2002), 1507. doi: 10.1002/cpa.10049. Google Scholar [38] A. Malchiodi and M. Montenegro, Multidimensional boundary-layers for a singularly perturbed Neumann problem,, \emph{Duke Math. J.}, 124 (2004), 105. doi: 10.1215/S0012-7094-04-12414-5. Google Scholar [39] S. Minakshisundaram and and A. Pleijel, Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds,, \emph{Canad. J. Math.}, 1 (1949), 242. Google Scholar [40] Y. G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_a$,, \emph{Comm. Partial Differential Equations}, 13 (1988), 1499. doi: 10.1080/03605308808820585. Google Scholar [41] Y. G. Oh, Correction to: Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_a$,, \emph{Comm. Partial Differential Equations}, 14 (1989), 833. doi: 10.1080/03605308908820631. Google Scholar [42] Y. G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multile well potential,, \emph{Comm. Math. Phys.}, 131 (1990), 223. Google Scholar [43] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations,, \emph{Z. Angew. Math. Phys.}, 43 (1992), 270. doi: 10.1007/BF00946631. Google Scholar [44] Y. Sato, Multi-peak positive solutions for nonlinear Schrödinger equations with critical frequency,, \emph{Calc. Var. Partial Differential Equations}, 29 (2007), 365. doi: 10.1007/s00526-006-0070-9. Google Scholar [45] B. Sirakov, Standing wave solutions of the nonlinear Schrödinger equation in $R^N$,, \emph{Ann. Mat. Pura Appl.}, 184 (2002), 73. doi: 10.1007/s102310200029. Google Scholar [46] W. Strauss, Existence of solitary waves in higher demensions,, \emph{Comm. Math. Phys.}, 55 (1977), 149. Google Scholar [47] J. Su, Z.-Q. Wang and M. Willem, Weighted Sobolev embedding with unbounded and decaying radial potentials,, \emph{J. Differential Equations}, 238 (2007), 201. doi: 10.1016/j.jde.2007.03.018. Google Scholar [48] L. P. Wang, J. Wei and J. Yang, On Ambrosetti-Malchiodi-Ni conjecture for general hypersurfaces,, \emph{Comm. Partial Differential Equations}, 36 (2011), 2117. doi: 10.1080/03605302.2011.580033. Google Scholar [49] X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations,, \emph{Comm. Math. Phys.}, 153 (1993), 229. Google Scholar [50] Z.-Q. Wang, Existence and symmetry of multi-bump solutions for nonlinear Schrödinger equations,, \emph{J. Differential Equations}, 159 (1999), 102. doi: 10.1006/jdeq.1999.3650. Google Scholar

show all references

##### References:
 [1] A. Ambrosetti, A. Malchiodi and W.-M Ni, Singularly perturves elliptic equation with symmetry: existence of solutions concentrating on spheres. I,, \emph{Comm. Math. Phys.}, 235 (2003), 427. doi: 10.1007/s00220-003-0811-y. Google Scholar [2] A. Ambrosetti, A. Malchiodi and D. Ruiz, Bound states of nonlinear Schrödinger equations with potentials vanishing at infinity,, \emph{J. d'Anayse Math.}, 98 (2006), 317. doi: 10.1007/BF02790279. Google Scholar [3] A. Ambrosetti, A. Malchiodi and A. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials,, \emph{Arch. Rational Mech. Anal.}, 159 (2001), 253. doi: 10.1007/s002050100152. Google Scholar [4] A. Ambrosetti and D. Ruiz, Radial solutions concentrating on sphere of nonlinear Schrödinger equations with vanishing potentials,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 136 (2006), 889. doi: 10.1017/S0308210500004789. Google Scholar [5] M. Badiale. and T. D'Aprile, Concentration around a sphere for a singulary perturbed Schrödinger equation,, \emph{Nonlinear Anal.}, 49 (2002), 947. doi: 10.1016/S0362-546X(01)00717-9. Google Scholar [6] V. Benci. and T. D'Aprile, The semiclassical limit of the nonlinear Schrödinger equation in a radial potential,, \emph{J. Diff. Equat.}, 184 (2002), 109. doi: 10.1006/jdeq.2001.4138. Google Scholar [7] J. Byeon, Standing Waves for nonlinear Schrödinger equations with a radial potential,, \emph{Nonlinear Anal.}, 50 (2002), 1135. doi: 10.1016/S0362-546X(01)00805-7. Google Scholar [8] J. Byeon and Y. Oshita, Existence of multi-bump standing waves with a critical frequency for nonlinear Schrödinger equations,, \emph{Comm. Partial Differential Equations}, 29 (2004), 1877. doi: 10.1081/PDE-200040205. Google Scholar [9] J. Byeon and Y. Oshita, Uniqueness of a standing wave for nonlinear Schrödinger equations,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 138A (2008), 975. doi: 10.1017/S0308210507000236. Google Scholar [10] J. Byeon and Z.-Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations,, \emph{Arch. Rational Mech. Anal.}, 165 (2002), 295. doi: 10.1007/s00205-002-0225-6. Google Scholar [11] J. Byeon and Z.-Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations. II,, \emph{Calc. Var. Partial Differential Equations}, 18 (2003), 207. doi: 10.1007/s00526-002-0191-8. Google Scholar [12] D. Cao, E. S. Noussair and S. Yan, Multiscale-bump standing waves with a critical frequency for nonlinear Schrödinger equations,, to appear in \emph{Trans. of AMS}., (). doi: 10.1090/S0002-9947-08-04348-1. Google Scholar [13] D. Cao and S. Peng, Multi-bump bound states of Schrödinger equations with a critical frequency,, \emph{Math. Ann}, 336 (2006), 925. doi: 10.1007/s00208-006-0021-y. Google Scholar [14] E. N. Dancer, The effect of domain shape on the number of positive solutions of certain nonlinear equations,, \emph{J. Differential Equations}, 74 (1998), 120. doi: 10.1016/0022-0396(88)90021-6. Google Scholar [15] E. N. Dancer and S. Yan, On the existence of multipeak solutions for nonlinear field equations on $R^N$,, \emph{Discrete Contin. Dynam. Systems, 6 (2000), 39. Google Scholar [16] 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,, \emph{Abstr. Apply. Anal.}, 3 (1998), 293. doi: 10.1155/S1085337598000578. Google Scholar [17] M. del Pino and P. L. Felmer, Local mountaion passes for semilinear elliptic problems in unbounded domains,, \emph{Calc. Var. Partial Differential Equations}, 4 (1996), 121. doi: 10.1007/BF01189950. Google Scholar [18] M. del Pino and P. L. Felmer, Semi-classical states for nonlinear Schrödinger equations,, \emph{J. Funct. Anal.}, 149 (1997), 245. doi: 10.1006/jfan.1996.3085. Google Scholar [19] M. del Pino and P. L. Felmer, Multi-peak bound states for nonlinear Schrödinger equations,, \emph{Ann. Inst. Henri Poincar\'e}, 15 (1998), 127. doi: 10.1016/S0294-1449(97)89296-7. Google Scholar [20] M. del Pino and P. L. Felmer, Semi-classical states of nonlinear Schrödinger equations: a variational reduction method,, \emph{Math. Ann.}, 324 (2002), 1. doi: 10.1007/s002080200327. Google Scholar [21] M. del Pino, P. L. Felmer and O. H. Miyagaki, Existence of positive bound states of nonlinear Schrödinger equations with saddle-like potential,, \emph{Nonlinear Anal. TMA, 34 (1998), 979. doi: 10.1016/S0362-546X(97)00593-2. Google Scholar [22] M. del Pino, M. Kowalczyk and J.-C. Wei, Concentration on curves for nonlinear Schrödinger equations,, \emph{Comm. Pur. App. Math.}, LX (2007), 113. doi: 10.1002/cpa.20135. Google Scholar [23] A. Floer and A. Weinstein, Non spreading wave packets for the cubic Schrödinger equations with a bounded potential,, \emph{J. Funct. Anal.}, 69 (1986), 397. doi: 10.1016/0022-1236(86)90096-0. Google Scholar [24] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, \emph{Comm. Math. Phys., 68 (1979), 209. Google Scholar [25] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, 2nd ed., (1983). doi: 10.1007/978-3-642-61798-0. Google Scholar [26] C. Gui, Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method,, \emph{Comm. Partial Differential Equations}, 21 (1996), 787. doi: 10.1080/03605309608821208. Google Scholar [27] Y. Kabeya and K. Tanaka, Uniqueness of positive solutions of semilinear elliptic equations in $R^N$ and Séré's non-degeneracy condition,, \emph{Comm. Partial Differential Equations, 24 (1999), 563. doi: 10.1080/03605309908821434. Google Scholar [28] X. Kang and J. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations,, \emph{Adv. Differential Equations}, 5 (2000), 899. Google Scholar [29] T. Kato, Perturbation Theory for Linear Operators,, second ed., (1976). Google Scholar [30] O. Kwon, Existence of multi-bump standing waves with a critical frequency for nonlinear Schrödinger equations with potentials vanishing at infinity,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 139A (2009), 833. doi: 10.1017/S0308210508000309. Google Scholar [31] C. Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains,, \emph{Comm. Partial Differential Equations, 16 (1996), 585. doi: 10.1080/03605309108820770. Google Scholar [32] Y. Y. Li, On a singular perturbed elliptic equation,, \emph{Adv. Differential Equations}, 2 (1997), 955. Google Scholar [33] F. Mahmoudi and A. Malchiodi, Concentration on minimal submainifolds for a singularly perturbed Neumann problem,, \emph{Adv. Math.}, 209 (2007), 460. doi: 10.1016/j.aim.2006.05.014. Google Scholar [34] F. Mahmoudi, R. Mazzeo and F. Pacard, Constant mean curvature hypersurfaces condensing on a submanifold,, \emph{Geom. Funct. Anal.}, 16 (2006), 924. doi: 10.1007/s00039-006-0566-7. Google Scholar [35] F. Mahmoudi, F. S. Sánchez and W. Yao, On the Ambrosetti-Malchiodi-Ni conjecture for general submanifolds,, \emph{J. Differential Equations}, 258 (2015), 243. doi: 10.1016/j.jde.2014.09.010. Google Scholar [36] A. Malchiodi, Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains,, \emph{G.A.F.A.}, 15-16 (2005), 15. doi: 10.1007/s00039-005-0542-7. Google Scholar [37] A. Malchiodi and M. Montenegro, Boundary concentration phenomena for a singularly perturbed elliptic problem,, \emph{Comm. Pure Appl. Math.}, 15 (2002), 1507. doi: 10.1002/cpa.10049. Google Scholar [38] A. Malchiodi and M. Montenegro, Multidimensional boundary-layers for a singularly perturbed Neumann problem,, \emph{Duke Math. J.}, 124 (2004), 105. doi: 10.1215/S0012-7094-04-12414-5. Google Scholar [39] S. Minakshisundaram and and A. Pleijel, Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds,, \emph{Canad. J. Math.}, 1 (1949), 242. Google Scholar [40] Y. G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_a$,, \emph{Comm. Partial Differential Equations}, 13 (1988), 1499. doi: 10.1080/03605308808820585. Google Scholar [41] Y. G. Oh, Correction to: Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_a$,, \emph{Comm. Partial Differential Equations}, 14 (1989), 833. doi: 10.1080/03605308908820631. Google Scholar [42] Y. G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multile well potential,, \emph{Comm. Math. Phys.}, 131 (1990), 223. Google Scholar [43] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations,, \emph{Z. Angew. Math. Phys.}, 43 (1992), 270. doi: 10.1007/BF00946631. Google Scholar [44] Y. Sato, Multi-peak positive solutions for nonlinear Schrödinger equations with critical frequency,, \emph{Calc. Var. Partial Differential Equations}, 29 (2007), 365. doi: 10.1007/s00526-006-0070-9. Google Scholar [45] B. Sirakov, Standing wave solutions of the nonlinear Schrödinger equation in $R^N$,, \emph{Ann. Mat. Pura Appl.}, 184 (2002), 73. doi: 10.1007/s102310200029. Google Scholar [46] W. Strauss, Existence of solitary waves in higher demensions,, \emph{Comm. Math. Phys.}, 55 (1977), 149. Google Scholar [47] J. Su, Z.-Q. Wang and M. Willem, Weighted Sobolev embedding with unbounded and decaying radial potentials,, \emph{J. Differential Equations}, 238 (2007), 201. doi: 10.1016/j.jde.2007.03.018. Google Scholar [48] L. P. Wang, J. Wei and J. Yang, On Ambrosetti-Malchiodi-Ni conjecture for general hypersurfaces,, \emph{Comm. Partial Differential Equations}, 36 (2011), 2117. doi: 10.1080/03605302.2011.580033. Google Scholar [49] X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations,, \emph{Comm. Math. Phys.}, 153 (1993), 229. Google Scholar [50] Z.-Q. Wang, Existence and symmetry of multi-bump solutions for nonlinear Schrödinger equations,, \emph{J. Differential Equations}, 159 (1999), 102. doi: 10.1006/jdeq.1999.3650. Google Scholar
 [1] Vincenzo Ambrosio. Concentration phenomena for critical fractional Schrödinger systems. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2085-2123. doi: 10.3934/cpaa.2018099 [2] Xudong Shang, Jihui Zhang. Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2239-2259. doi: 10.3934/cpaa.2018107 [3] Liren Lin, Tai-Peng Tsai. Mixed dimensional infinite soliton trains for nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 295-336. doi: 10.3934/dcds.2017013 [4] Guoyuan Chen, Youquan Zheng. Concentration phenomenon for fractional nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2359-2376. doi: 10.3934/cpaa.2014.13.2359 [5] Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential. Mathematical Control & Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119 [6] Nakao Hayashi, Pavel Naumkin. On the reduction of the modified Benjamin-Ono equation to the cubic derivative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 237-255. doi: 10.3934/dcds.2002.8.237 [7] Hongzi Cong, Lufang Mi, Yunfeng Shi, Yuan Wu. On the existence of full dimensional KAM torus for nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6599-6630. doi: 10.3934/dcds.2019287 [8] Teresa D'Aprile. Some existence and concentration results for nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2002, 1 (4) : 457-474. doi: 10.3934/cpaa.2002.1.457 [9] D.G. deFigueiredo, Yanheng Ding. Solutions of a nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 563-584. doi: 10.3934/dcds.2002.8.563 [10] Qing Xu. Backward stochastic Schrödinger and infinite-dimensional Hamiltonian equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5379-5412. doi: 10.3934/dcds.2015.35.5379 [11] Chang-Lin Xiang. Remarks on nondegeneracy of ground states for quasilinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5789-5800. doi: 10.3934/dcds.2016054 [12] Shuya Kanagawa, Ben T. Nohara. The nonlinear Schrödinger equation created by the vibrations of an elastic plate and its dimensional expansion. Conference Publications, 2013, 2013 (special) : 415-426. doi: 10.3934/proc.2013.2013.415 [13] Myeongju Chae, Sunggeum Hong, Sanghyuk Lee. Mass concentration for the $L^2$-critical nonlinear Schrödinger equations of higher orders. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 909-928. doi: 10.3934/dcds.2011.29.909 [14] Vincenzo Ambrosio, Teresa Isernia. Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional p-Laplacian. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5835-5881. doi: 10.3934/dcds.2018254 [15] César E. Torres Ledesma. Existence and concentration of solutions for a non-linear fractional Schrödinger equation with steep potential well. Communications on Pure & Applied Analysis, 2016, 15 (2) : 535-547. doi: 10.3934/cpaa.2016.15.535 [16] Pavel I. Naumkin, Isahi Sánchez-Suárez. On the critical nongauge invariant nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 807-834. doi: 10.3934/dcds.2011.30.807 [17] Younghun Hong. Scattering for a nonlinear Schrödinger equation with a potential. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1571-1601. doi: 10.3934/cpaa.2016003 [18] Alexander Komech, Elena Kopylova, David Stuart. On asymptotic stability of solitons in a nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1063-1079. doi: 10.3934/cpaa.2012.11.1063 [19] Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109 [20] Walter Dambrosio, Duccio Papini. Multiple homoclinic solutions for a one-dimensional Schrödinger equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1025-1038. doi: 10.3934/dcdss.2016040

2018 Impact Factor: 0.925

## Metrics

• HTML views (0)
• Cited by (1)

• on AIMS