×

Nonlinear Schrödinger equations with steep potential well. (English) Zbl 1076.35037

Summary: We investigate nonlinear Schrödinger equations like the model equation \[ -\Delta u+V_\lambda(x)u= |u|^{p-2}u, \quad x\in \mathbb R^N,\;2< p< 2^*, \] where the potential \(V_\lambda\) has a potential well with bottom independent of the parameter \(\lambda > 0\). If \(\lambda\to\infty\) the infimum of the essential spectrum of \(-\Delta + V_\lambda\) in \(L^2(\mathbb R^N)\) converges towards \(\infty\) and more and more eigenvalues appear below the essential spectrum. We show that as \(\lambda\to\infty\) more and more solutions of the nonlinear Schrödinger equation exist. The solutions lie in \(H^1(\mathbb R^N)\) and are localized near the bottom of the potential well, but not near local minima of the potential. We also investigate the decay rate of the solutions as \(|x|\to\infty\) as well as their behaviour as \(\lambda\to\infty\).

MSC:

35J60 Nonlinear elliptic equations
35P05 General topics in linear spectral theory for PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1016/0022-1236(72)90025-0 · Zbl 0266.30024 · doi:10.1016/0022-1236(72)90025-0
[2] Bartsch T., Meth. and Appl. 20 pp 1205– (1993)
[3] DOI: 10.1080/03605309508821149 · Zbl 0837.35043 · doi:10.1080/03605309508821149
[4] Bartsch T., Top. Meth. Nonlin. Anal. 13 pp 191– (1999) · Zbl 0961.35150 · doi:10.12775/TMNA.1999.010
[5] DOI: 10.1016/0022-1236(86)90096-0 · Zbl 0613.35076 · doi:10.1016/0022-1236(86)90096-0
[6] DOI: 10.1215/S0012-7094-96-08423-9 · Zbl 0866.35039 · doi:10.1215/S0012-7094-96-08423-9
[7] DOI: 10.1007/BF01181619 · Zbl 0236.31010 · doi:10.1007/BF01181619
[8] Kondrat’ev V., Advances and Applications 110 pp 185– (1999)
[9] Li Y. Y., Adv. in Differential Equations 2 pp 955– (1997)
[10] Lions P., Anal. Non Linéaire 1 pp 109– (1984) · Zbl 0541.49009 · doi:10.1016/S0294-1449(16)30428-0
[11] Molchanov A. M., Trudy Mosk. Matem. Obshchestva 2 pp 169– (1953)
[12] Persson A., Math. Scand. 8 pp 143– (1960) · Zbl 0145.14901 · doi:10.7146/math.scand.a-10602
[13] M, Anal. Non Linéaire 15 pp 127– (1998)
[14] DOI: 10.1512/iumj.1972.21.21082 · Zbl 0242.47029 · doi:10.1512/iumj.1972.21.21082
[15] DOI: 10.1007/BF00946631 · Zbl 0763.35087 · doi:10.1007/BF00946631
[16] Simon B., S.) 7 pp 447– (1982)
[17] DOI: 10.1007/s005260000010 · Zbl 0977.35049 · doi:10.1007/s005260000010
[18] DOI: 10.1007/BF02096642 · Zbl 0795.35118 · doi:10.1007/BF02096642
[19] DOI: 10.1006/jdeq.1999.3650 · Zbl 1005.35083 · doi:10.1006/jdeq.1999.3650
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.