## Abstract

We study the concentration behavior of positive bound states of the nonlinear Schrödinger equation

Under certain condition of*V*, we show that positive ground state solutions must concentrate at global minimum points of*V* as*h*→0^{+}; moreover, a point at which a sequence of positive bound states concentrates must be a critical point of*V*. In cases that*V* is radial, we prove that the positive radial solutions with least energy among all nontrivial radial solutions must concentrate at the origin as*h*→0^{+}.

### Similar content being viewed by others

## References

[CGS] Cafferelli, L., Gidas, B., Spruck, J.: Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Comm. Pure Appl. Math.

**42**, 271–297 (1989)[CL] Chen, W., Li, C.: Classification of solutions of some semilinear elliptic equations. Duke Math. J.

**63**, No. 3, 615–622 (1991)[CR] Coti Zelati, V., Rabinowitz, P.H.: Homoclinic type solutions for a semilinear elliptic PDE on ℝ

^{n}. Preprint[DN] Ding, W.-Y., Ni, W.-M.: On the existence of positive entire solutions of a semilinear elliptic equation. Arch. Rational Mech. Anal.

**91**, 283–308 (1986)[FW] Floer, A., Weinstein, A.: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal.

**69**, 397–408 (1986)[G] Changfeng Gui: A remark on nonlinear Schrödinger equations. Preprint

[GNN] Gidas, B., Ni, W.-M., Nirenberg, L.: Symmetry of positive solutions of nonlinear elliptic equations in ℝ

^{n}. Adv. Math. Suppl. Stud.**7A**, Math. Anal. Appl. Part A, 369–402 (1981)[GT] Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order. 2nd ed., New York, Berlin: Springer 1983

[H] Han, Z.-C.: Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent. Analyse Nonlin.

**8**, 159–174 (1991)[Ka] Kato, T.: Growth properties of solutions of the reduced wave equation with a variable coefficient. Comm. Pure Appl. Math.

**12**, 403–425 (1959)[K] Kwong, M.: Uniqueness of positive solutions of Δ

*u*−*u*+*u*^{p}=0 in ℝ^{n}. Arch. Rational Mech. Anal.**105**, 243–266 (1989)[L

_{1}] Lions, P.L.: On positive solutions of semilinear elliptic equations in unbounded domains. Nonlinear Diffusion Equations and their Equilibrium States, Vol.**II**(W.-M. Ni, L.A. Peletier, J. Serrin, eds.), MSRI Publications**13**, 85–122 (1988)[L

_{2}] Lions, P.L.: The concentration compactness principles in the calculus of variations. The locally compact case, Part 2. Analyse Nonlin,**1**, 223–283 (1984)[MW] Mawhim, J., Willem, M.: Critical Point Theory and Hamiltonian Systems. Berlin, Heidelberg, New York: Springer 1989

[N] Ni, W.-M.: Some aspects of semilinear elliptic equations in ℝ

^{n}. Nonlinear Diffusion Equations and their Equilibrium States, Vol. II (W.-M. Ni, L.A. Peletier, J. Serrin, eds.), MSRI Publications**13**, 171–205 (1988)[NT] Ni, W.-M., Takagi, I.: On the shape of least-energy solutions to a semilinear Neumann problem. Comm. Pure Appl. Math.

**XLIV**, 819–851 (1991)[O

_{1}] Oh, Y.-G.: Existence of semi-classical bound states of nonlinear Schrödinger equations with potential of the class (*V*)_{(a }. Comm. Partial Diff. Eq.**13**, 1499–1519 (1988)[O

_{2}] Oh, Y.-G.: Correction to Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class (*V*)_{ a }. Comm. Partial Diff. Eq.**14**, 833–834 (1989)[O

_{3}] Oh, Y.-G.: On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential. Commun. Math. Phys.**131**, 223–253 (1990)[R] Rabinowitz, P.H.: On a class of nonlinear Schrödinger equations. to appear in ZAMP

[S] Strauss, W.A.: Existence of solitary waves in higher dimensions. Commun. Math. Phys.

**55**, 149–162 (1977)[T] Trudinger, N.S.: On Harnack type inequalities and their applications to quasilinear elliptic equations. Comm. Pure Appl. Math.

**20**, 721–747 (1967)

## Author information

### Authors and Affiliations

## Additional information

Communicated by A. Jaffe

Research supported in part by NSF Grant DMS-9105172.

## Rights and permissions

## About this article

### Cite this article

Wang, X. On concentration of positive bound states of nonlinear Schrödinger equations.
*Commun.Math. Phys.* **153**, 229–244 (1993). https://doi.org/10.1007/BF02096642

Received:

Revised:

Issue Date:

DOI: https://doi.org/10.1007/BF02096642