## On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential.(English)Zbl 0753.35097

The author considers nonlinear Schrödinger equations with an additional linear potential $$V$$ of class $$(V)_ a$$ in the sense of Kato and an attractive power law nonlinearity. He proves the existence of special solutions of type $$\exp(-iEt/h)\cdot v(x)$$ (called multi-lump bound states) for each finite collection of nondegenerate critical points of $$V$$; here $$v(x)$$ is a real small perturbation of a sum of one-lump solutions concentrated near one critical point resp. of the potential $$V$$.
This generalizes results of A. Floer and A. Weinstein on one- lump solutions for bounded potentials [J. Funct. Anal. 69, 397-408 (1986; Zbl 0613.35076)]. The crucial technical point is a new estimate of the norm of a certain Fredholm inverse which is not needed in the one-lump case.
Reviewer: H.Lange (Köln)

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics

Zbl 0613.35076
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### References:

 [1] [FW.a] Floer, A., Weinstein, A.: Nonspreading wave packets for the cubic Schrödinger equations with a bounded potential. J. Funct. Anal.69, 397–408 (1986) · Zbl 0613.35076 [2] [GlJa] Gilmm, J., Jaffe, A.: Quantum physics. Berlin, Heidelberg, New York: Springer 1981 [3] [Gr] Grillakis, M.: Linearized instability for nonlinear Schrödinger and Klein-Gordon equations. Commun. Pure Appl. Math.41, 747–774 (1988) · Zbl 0632.70015 [4] [GrSS] Grillakis, M., Shatah, J., Strauss, W.: Stability theory of solitary waves in the presence of symmetry, I. J. Funct. Anal.74, 160–197 (1987) · Zbl 0656.35122 [5] [JaT] Jaffe, A., Taubes, C. H.: Vortices and Monopoles, Boston: Birkhäuser 1980 [6] [Jo] Jones, C.: Instability of standing waves for nonlinear Schrödinger type equations. Ergodic Theory and Dynamical Systems8, 119–138 (1988) · Zbl 0636.35017 [7] [K] Kato, K.: Remarks on holomorphic families of Schrödinger and Dirac operators. In: Differential Equations. Knowles, I., Lewis, R. (eds.) pp. 341–352. Amsterdam: North Holland 1984 · Zbl 0565.47011 [8] [Kw] Kwong, M. K.: Uniqueness of positive solutions of {$$\Delta$$}u+u p=0 in $$\mathbb{R}$$n. Arch. Rational Mech. Anal.105, 243–266 (1989) · Zbl 0676.35032 [9] [O1] Oh, Y.-G.: Existence of semi-classical bound states of nonlinear Schrödinger equations with potentials of the class(V) a. Commun. Partial Diff. Eq.13, 1499–1519 (1988) · Zbl 0702.35228 [10] [O2] –: Correction to ”Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class(V) a”. Commun. Partial Diff. Eq.14, 833–834 (1989) · Zbl 0714.35078 [11] [O3] –: Stability of semi-classical bound states of nonlinear Schrödinger equations with potentials. Commun. Math. Phys.121, 11–33 (1989) · Zbl 0693.35132 [12] [ReS] Reed, M., Simon, B.: Methods of modern mathematical physics II, IV. New York: Academic Press 1978 · Zbl 0401.47001 [13] [RW.m] Rose, H., Weinstein, M.: On the bound states of the nonlinear Schrödinger equation with a linear potential. Physica D.30, 207–218 (1988) · Zbl 0694.35202 [14] [T] Taubes, C. H.: The existence of multi-monopole solutions to the non-abelian Yang-Mills-Higgs equations for arbitrary simple gauge groups. Commun. Math. Phys.80, 343–367 (1981) · Zbl 0486.35072 [15] [W.a] Weinstein, A.: Nonlinear stabilization of quasimodes. Proc. A.M.S. Symposium on Geometry of the Laplacian, Hawaii, 1979, AMS Colloq. Publ.36, 301–318 (1980) [16] [W.m] Weinstein, M.: Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal.16, 567–576 (1985) · Zbl 0583.35028
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