×

zbMATH — the first resource for mathematics

On the symmetry of the ground states of nonlinear Schrödinger equation with potential. (English) Zbl 1223.35133
Summary: We investigate the minimizers of the energy functional \[ {\mathcal E}(u)= \frac12 \int_{\mathbb R^N}|\nabla u|^2\,dx+ \frac12\int_{\mathbb R^N}V|u|^2\,dx- \frac{1}{p+1}\int_{\mathbb R^N} b|u|^{p+1}\,dx \]
under the constraint of the \(L^2\)-norm. We show that, for the case when the \(L^2\)-norm is small, the minimizer is unique and, for the case when the \(L^2\)-norm is large, the minimizer concentrates at the maximum point of \(b\) and decays exponentially. By this result, we can show that, if \(V\) and \(b\) are radially symmetric but \(b\) does not attain its maximum at the origin, then the symmetry breaking occurs as the \(L^2\)-norm increases. Further, we show that, for the case when \(b\) has several maximum points, the minimizer concentrates at a point which minimizes a function which is denned by \(b\), \(V\) and the unique positive radial solution of \(-\Delta\varphi+\varphi-\varphi^p=0\). For the case when \(V\) and \(b\) are radially symmetric, we show that, if the minimizer concentrates at the origin, then the minimizer is radially symmetric. Further, we construct an energy functional such that the minimizer breaks its symmetry once, but after that it recovers to be symmetric as the \(L^2\)-norm increases.

MSC:
35J20 Variational methods for second-order elliptic equations
35J61 Semilinear elliptic equations
35Q40 PDEs in connection with quantum mechanics
81R40 Symmetry breaking in quantum theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Pino, del Multi - peak bound states for nonlinear Schro dinger equations Non Line aire no, Inst Anal 15 pp 127– (1998)
[2] Floer, Nonspreading wave packets for the cubic Schro dinger equation with a bounded potential no, Funct Anal pp 69– (1986)
[3] Wang, On concentration of positive bound states of nonlinear Schro dinger equations with competing potential functions SIAM no, Math Anal 28 pp 633– (1997)
[4] Gidas, Symmetry of positive solutions of nonlinear elliptic equations in Rn Mathematical analysis and applications Part A in vol Academic Press New York, Math Suppl Stud 7 pp 369– (1981)
[5] Pino, del Semi - classical states of nonlinear Schro dinger equations : a variational reduction method Math no, Ann pp 324– (2002)
[6] Cazenave, Orbital stability of standing waves for some nonlinear Schro dinger equations Comm no, Math Phys pp 85– (1982)
[7] Cid, Orbital stability and standing waves for the nonlinear Schro dinger equation with potential no, Rev Math Phys 12 pp 13– (2001)
[8] Pino, del Semi - classical states for nonlinear Schro dinger equa - tions no, Funct Anal pp 149– (1997)
[9] Wang, On concentration of positive bound states of nonlinear Schro dinger equa - tions Comm no, Math Phys pp 153– (1993)
[10] Kwong, Uniqueness of positive solutions of u u up in Rn Arch Rational Mech no, Anal pp 105– (1989)
[11] Ambrosetti, Singularly perturbed elliptic equations with symmetry : existence of solutions concentrating on spheres Comm no, Math Phys pp 235– (2003) · Zbl 1072.35019
[12] Kirr, Symmetry - breaking bifurcation in nonlinear Schro dinger Gross - Pitaevskii equations SIAM no, Math Anal pp 40– (2008)
[13] Aschbacher, Fro hlich Symmetry breaking regime in the nonlinear Hartree equation no, Math Phys pp 43– (2002)
[14] Sirakov, Standing wave solutions of the nonlinear Schro dinger equation in RN no, Ann Mat Appl pp 181– (2002)
[15] Rabinowitz, On a class of nonlinear Schro dinger equations no, Angew Math Phys pp 43– (1992)
[16] Ambrosetti, Semiclassical states of nonlinear Schro dinger equations Arch Rational no, Mech Anal pp 140– (1997)
[17] Grossi, On the number of single - peak solutions of the nonlinear Schro dinger equa - tion Non Line aire no On nonlinear Schro dinger equation with potential Locating the peak of ground states of nonlinear Schro dinger equations no, Inst Anal J Math 19 pp 261– (2002)
[18] Okazawa, An Lp theory for Schro dinger operators with nonnegative potentials Japan no, Math Soc 36 pp 675– (1984) · Zbl 0556.35032 · doi:10.2969/jmsj/03640675
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.