## Existence and uniqueness of multi-bump bound states of nonlinear Schrödinger equations with electromagnetic fields.(English)Zbl 1357.35258

Summary: We are concerned with the existence and uniqueness of multi-bump bound states of the nonlinear Schrödinger equations with electromagnetic potential $i\hbar\frac{\partial\psi}{\partial t}=\left (\frac {\hbar}{i}\nabla-A(x)\right )^2\psi +V(x)\psi - |\psi |^{p-2}\psi ,\quad x\in\mathbb R^N,$ for sufficiently small $$\hbar>0$$, where $$i$$ is the imaginary unit, $$2<p<\frac{2N}{N-2}$$ for $$N\geq 3$$ and $$2<p<+\infty$$ for $$N=1,2$$. $$V(x)$$ is a bounded real function on $$\mathbb{R}^N$$, and $$A(x)=(A_1(x),A_2(x),\dots,A_N(x))$$ is such that $$A_j(x)$$ is a bounded real function on $$\mathbb{R}^N$$ for $$j=1,2,\dots N$$. For any finite collection of $$\{a_1,a_2,\dots,a_k\}$$ of non-degenerate critical points of $$V(x)$$, we show that there exists a solution $$\psi(x,t)=\exp(-iEt/\hbar)u(x)$$ which is a small perturbation of a sum of one-bump solutions concentrated at $$a_1,a_2,\dots,a_k$$ and $$u(x)$$ is unique up to rotations for small $$\hbar>0$$.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 81V10 Electromagnetic interaction; quantum electrodynamics 35J60 Nonlinear elliptic equations 35B33 Critical exponents in context of PDEs
Full Text:

### References:

 [1] Ambrosetti, A.; Badiale, M.; Cingolani, S., Semiclassical states of nonlinear Schrödinger equations, Arch. rational mech. anal., 140, 285-300, (1997) · Zbl 0896.35042 [2] Ambrosetti, A.; Malchiodi, A.; Secchi, S., Multiplicity results for some nonlinear Schrödinger equations with potentials, Arch. rational mech. anal., 159, 253-271, (2001) · Zbl 1040.35107 [3] Arioli, G.; Szulkin, A., A semilinear Schrödinger equation in the presence of a magnetic field, Arch. rational mech. anal., 170, 277-295, (2003) · Zbl 1051.35082 [4] Bahri, A.; Coron, J.M., On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. pure appl. math., 41, 255-294, (1998) [5] Bahri, A.; Li, Y.Y.; Rey, O., On a variational problem with lack of compactness: the topological effect of the critical points at infinity, Cal. var. PDE, 3, 67-93, (1995) · Zbl 0814.35032 [6] Cao, D.; Dancer, E.N.; Noussair, E.S.; Yan, S., On the existence and profile of multi-peaked solutions to singularly perturbed semilinear Dirichlet problems, Discrete continuous dynamic systems, 2, 221-236, (1996) · Zbl 0947.35073 [7] Cao, D.; Heinz, H.P., Uniqueness of positive multi-bump bound states of nonlinear Schrödinger equations, Math. Z., 243, 599-642, (2003) · Zbl 1142.35601 [8] Cao, D.; Noussair, E.S.; Yan, S., Solutions with multiple “peaks” for nonlinear elliptic equations, Proc. royal soc. Edinburgh sect. A, 129, 235-264, (1999) · Zbl 0928.35048 [9] Cingolani, S., Semiclassical stationary states of nonlinear Schrödinger equation with external magnetic field, J. differential equations, 188, 52-79, (2003) · Zbl 1062.81056 [10] Cingolani, S.; Lazzo, M., Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions, J. differential equations, 160, 118-138, (2000) · Zbl 0952.35043 [11] Cingolani, S.; Nolasco, M., Multi-peaks periodic semiclassical states for a class of nonlinear Schrödinger equations, Proc. roy. soc. Edinburgh, 128, 1249-1260, (1998) · Zbl 0922.35158 [12] Cingolani, S.; Secchi, S., Semiclassical limit for nonlinear Schrödinger equation with electromagnetic fields, J. math. anal. appl., 275, 108-130, (2002) · Zbl 1014.35087 [13] Del Pino, M.; Felmer, P., Multi-peak bound states for nonlinear Schrödinger equations, J. funct. anal., 149, 245-265, (1997) · Zbl 0887.35058 [14] Del Pino, M.; Felmer, P., Semi-classical states for nonlinear Schrödinger equations, Ann. inst. H. Poincaré, anal. non linéaire, 15, 127-149, (1998) · Zbl 0901.35023 [15] Esteban, M.; Lions, P.L., Stationary solutions of nonlinear Schrödinger equations with an external magnetic field, (), 369-408 [16] Floer, A.; Weinstein, A., Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. funct. anal., 69, 397-408, (1986) · Zbl 0613.35076 [17] Gidas, B.; Ni, W.-M.; Nirenberg, L., Symmetry and related properties via the maximum principle, Comm. math. phys., 68, 209-243, (1979) · Zbl 0425.35020 [18] Glangetas, L., Uniqueness of positive solutions of a nonlinear elliptic equation involving the critical exponent, Nonlinear anal. TMA, 20, 571-603, (1993) · Zbl 0797.35048 [19] Gui, C., Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method, Comm. partial differential equations, 21, 787-820, (1996) · Zbl 0857.35116 [20] Kurata, K., Existence and semi-classical limit of the least energy solution to a nonlinear Schrödinger equation with electromagnetic fields, Nonlin. anal. TMA, 41, 763-778, (2000) · Zbl 0993.35081 [21] Kwong, M.K., Uniqueness of positive solutions of $$\operatorname{\Delta} u - u + u^p = 0$$ in $$\mathbb{R}^n$$, Arch. rational mech. anal., 105, 243-266, (1989) · Zbl 0676.35032 [22] Li, Y.Y., On a singularly perturbed elliptic equation, Adv. differential equations, 2, 955-980, (1997) · Zbl 1023.35500 [23] Li, Y.Y.; Nirenberg, L., The Dirichlet problem for singularly perturbed elliptic equations, Comm. pure appl. math., 51, 1445-1490, (1998) · Zbl 0933.35083 [24] Lions, P.L., On positive solutions of semilinear elliptic equations in unbounded domains, () · Zbl 0685.35039 [25] Lu, G.; Wei, J., On nonlinear Schrödinger equations with totally degenerate potentials, C. R. acad. sci. Paris, ser. I, 326, 1-7, (1998) [26] Oh, Y.-G., Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of class $$(V)_a$$, Comm. partial differential equations, 13, 1499-1519, (1988) · Zbl 0702.35228 [27] Oh, Y.-G., On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Comm. math. phys., 131, 223-253, (1990) · Zbl 0753.35097 [28] Rabinowitz, P.H., On a class of nonlinear Schrödinger equations, Z. angew. math. phys., 43, 270-291, (1992) · Zbl 0763.35087 [29] Rey, O., The role of the Green’s function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. funct. anal., 89, 1-52, (1990) · Zbl 0786.35059 [30] Wang, X., On the concentration of positive bound states of nonlinear Schrödinger equations, Comm. math. phys., 153, 229-244, (1993) · Zbl 0795.35118 [31] Wang, X.; Zeng, B., On the concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions, SIAM J. math. anal., 28, 633-655, (1997) · Zbl 0879.35053
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.