# zbMATH — the first resource for mathematics

Normalized solutions to the mixed dispersion nonlinear Schrödinger equation in the mass critical and supercritical regime. (English) Zbl 1420.35343
Summary: In this paper, we study the existence of solutions to the mixed dispersion nonlinear Schrödinger equation $\gamma \Delta ^2 u -\Delta u + \alpha u=\vert u\vert^{2 \sigma} u, \quad u \in H^2(\mathbb{R}^N),$ under the constraint $\int _{\mathbb{R}^N}\vert u\vert^2 \, dx =c>0.$ We assume that $$\gamma >0, N \geq 1, 4 \leq \sigma N < \frac {4N}{(N-4)^+}$$, whereas the parameter $$\alpha \in \mathbb{R}$$ will appear as a Lagrange multiplier. Given $$c \in \mathbb{R}^+$$, we consider several questions including the existence of ground states and of positive solutions and the multiplicity of radial solutions. We also discuss the stability of the standing waves of the associated dispersive equation.

##### MSC:
 35Q55 NLS equations (nonlinear Schrödinger equations) 35J30 Higher-order elliptic equations 35J50 Variational methods for elliptic systems 35B35 Stability in context of PDEs 35Q60 PDEs in connection with optics and electromagnetic theory 35Q40 PDEs in connection with quantum mechanics 35B09 Positive solutions to PDEs
Full Text:
##### References:
 [1] AcWeN. Ackermann and T. Weth, \em Unstable normalized standing waves for the space periodic NLS, arXiv:1706.06950 (2017). [2] Alsholm, P.; Schmidt, G., Spectral and scattering theory for Schr\"odinger operators, Arch. Rational Mech. Anal., 40, 281-311, (1970/1971) · Zbl 0226.35076 [3] Ambrosetti, Antonio; Malchiodi, Andrea, Nonlinear analysis and semilinear elliptic problems, Cambridge Studies in Advanced Mathematics 104, xii+316 pp., (2007), Cambridge University Press, Cambridge, England · Zbl 1125.47052 [4] Bartsch, Thomas; de Valeriola, S\'ebastien, Normalized solutions of nonlinear Schr\"odinger equations, Arch. Math. (Basel), 100, 1, 75-83, (2013) · Zbl 1260.35098 [5] Bartsch, Thomas; Jeanjean, Louis, Normalized solutions for nonlinear Schr\"odinger systems, Proc. Roy. Soc. Edinburgh Sect. A, 148, 2, 225-242, (2018) · Zbl 1393.35035 [6] Bartsch, Thomas; Jeanjean, Louis; Soave, Nicola, Normalized solutions for a system of coupled cubic Schr\"odinger equations on $$\mathbb{R}^3$$, J. Math. Pures Appl. (9), 106, 4, 583-614, (2016) · Zbl 1347.35107 [7] Bartsch, Thomas; Soave, Nicola, A natural constraint approach to normalized solutions of nonlinear Schr\"odinger equations and systems, J. Funct. Anal., 272, 12, 4998-5037, (2017) · Zbl 06714264 [8] Bartsch, Thomas; Soave, Nicola, Correction to: “A natural constraint approach to normalized solutions of nonlinear Schr\"odinger equations and systems” $$[$$J. Funct. Anal. 272 $$($$12$$)$$ $$($$2017$$)$$ 4998–5037$$]$$ $$[$$MR3639521$$]$$, J. Funct. Anal., 275, 2, 516-521, (2018) · Zbl 1434.35011 [9] BaSo2 T. Bartsch and N. Soave, \em Multiple normalized solutions for a competing system of Schr\"odinger equations, arXiv:1703.02832 (2017). [10] Baruch, G.; Fibich, G., Singular solutions of the $$L^2$$-supercritical biharmonic nonlinear Schr\"odinger equation, Nonlinearity, 24, 6, 1843-1859, (2011) · Zbl 1230.35125 [11] Baruch, G.; Fibich, G.; Mandelbaum, E., Ring-type singular solutions of the biharmonic nonlinear Schr\"odinger equation, Nonlinearity, 23, 11, 2867-2887, (2010) · Zbl 1202.35294 [12] Baruch, G.; Fibich, G.; Mandelbaum, E., Singular solutions of the biharmonic nonlinear Schr\"odinger equation, SIAM J. Appl. Math., 70, 8, 3319-3341, (2010) · Zbl 1210.35224 [13] Bellazzini, Jacopo; Frank, Rupert L.; Visciglia, Nicola, Maximizers for Gagliardo-Nirenberg inequalities and related non-local problems, Math. Ann., 360, 3-4, 653-673, (2014) · Zbl 1320.46026 [14] Bellazzini, Jacopo; Jeanjean, Louis, On dipolar quantum gases in the unstable regime, SIAM J. Math. Anal., 48, 3, 2028-2058, (2016) · Zbl 1352.35157 [15] Bellazzini, Jacopo; Jeanjean, Louis; Luo, Tingjian, Existence and instability of standing waves with prescribed norm for a class of Schr\"odinger-Poisson equations, Proc. Lond. Math. Soc. (3), 107, 2, 303-339, (2013) · Zbl 1284.35391 [16] Ben-Artzi, Matania; Koch, Herbert; Saut, Jean-Claude, Dispersion estimates for fourth order Schr\"odinger equations, C. R. Acad. Sci. Paris S\'er. I Math., 330, 2, 87-92, (2000) · Zbl 0942.35160 [17] Berestycki, H.; Lions, P.-L., Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82, 4, 313-345, (1983) · Zbl 0533.35029 [18] Berestycki, H.; Lions, P.-L., Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Rational Mech. Anal., 82, 4, 347-375, (1983) · Zbl 0556.35046 [19] BoCaGoJeD. Bonheure, J.-B. Casteras, T. Gou, and L. Jeanjean, \em Strong instability of ground states to a fourth order Schr\" odinger equation, arXiv:1703.07977 (2017). Int. Math. Res. Not. IMRN, DOI 10.1093/imrn/rnx273 (to appear). [20] BCMD. Bonheure, J.-B. Casteras, and R. Mandel, \em On a fourth order nonlinear Helmholtz equation, J. Lond. Math. Soc. (to appear). [21] Bonheure, Denis; Casteras, Jean-Baptiste; dos Santos, Ederson Moreira; Nascimento, Robson, Orbitally stable standing waves of a mixed dispersion nonlinear Schr\"odinger equation, SIAM J. Math. Anal., 50, 5, 5027-5071, (2018) · Zbl 1404.35403 [22] Bonheure, Denis; Moreira dos Santos, Ederson; Ramos, Miguel, Ground state and non-ground state solutions of some strongly coupled elliptic systems, Trans. Amer. Math. Soc., 364, 1, 447-491, (2012) · Zbl 1250.35093 [23] Bonheure, Denis; Nascimento, Robson, Waveguide solutions for a nonlinear Schr\`‘odinger equation with mixed dispersion. Contributions to nonlinear elliptic equations and systems, Progr. Nonlinear Differential Equations Appl. 86, 31-53, (2015), Birkh\'’auser/Springer, Cham · Zbl 1334.35310 [24] Boulenger, Thomas; Lenzmann, Enno, Blowup for biharmonic NLS, Ann. Sci. \'Ec. Norm. Sup\'er. (4), 50, 3, 503-544, (2017) · Zbl 1375.35476 [25] Busca, J\'er\^ome; Sirakov, Boyan, Symmetry results for semilinear elliptic systems in the whole space, J. Differential Equations, 163, 1, 41-56, (2000) · Zbl 0952.35033 [26] Catto, I.; Dolbeault, J.; S\'anchez, O.; Soler, J., Existence of steady states for the Maxwell-Schr\"odinger-Poisson system: Exploring the applicability of the concentration-compactness principle, Math. Models Methods Appl. Sci., 23, 10, 1915-1938, (2013) · Zbl 1283.35117 [27] Cazenave, Thierry, Semilinear Schr\"odinger equations, Courant Lecture Notes in Mathematics 10, xiv+323 pp., (2003), New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI · Zbl 1055.35003 [28] Colin, Mathieu; Jeanjean, Louis; Squassina, Marco, Stability and instability results for standing waves of quasi-linear Schr\"odinger equations, Nonlinearity, 23, 6, 1353-1385, (2010) · Zbl 1192.35163 [29] Evequoz, Gilles; Weth, Tobias, Dual variational methods and nonvanishing for the nonlinear Helmholtz equation, Adv. Math., 280, 690-728, (2015) · Zbl 1317.35030 [30] Fibich, Gadi; Ilan, Boaz; Papanicolaou, George, Self-focusing with fourth-order dispersion, SIAM J. Appl. Math., 62, 4, 1437-1462, (2002) · Zbl 1003.35112 [31] Ghoussoub, Nassif, Duality and perturbation methods in critical point theory, Cambridge Tracts in Mathematics 107, xviii+258 pp., (1993), Cambridge University Press, Cambridge, England · Zbl 0790.58002 [32] Gui, Changfeng; Luo, Xue; Zhou, Feng, Asymptotic behavior of oscillating radial solutions to certain nonlinear equations, Part II, Methods Appl. Anal., 16, 4, 459-468, (2009) · Zbl 1214.34030 [33] Jeanjean, Louis, Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal., 28, 10, 1633-1659, (1997) · Zbl 0877.35091 [34] Jeanjean, Louis; Luo, Tingjian, Sharp nonexistence results of prescribed $$L^2$$-norm solutions for some class of Schr\"odinger-Poisson and quasi-linear equations, Z. Angew. Math. Phys., 64, 4, 937-954, (2013) · Zbl 1294.35140 [35] Karpman, V. I., Stabilization of soliton instabilities by higher order dispersion: KdV-type equations, Phys. Lett. A, 210, 1-2, 77-84, (1996) · Zbl 1072.35577 [36] Karpman, V. I.; Shagalov, A. G., Stability of solitons described by nonlinear Schr\"odinger-type equations with higher-order dispersion, Phys. D, 144, 1-2, 194-210, (2000) · Zbl 0962.35165 [37] Lions, P.-L., The concentration-compactness principle in the calculus of variations. The locally compact case. II, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 1, 4, 223-283, (1984) · Zbl 0704.49004 [38] Mandel, Rainer; Montefusco, Eugenio; Pellacci, Benedetta, Oscillating solutions for nonlinear Helmholtz equations, Z. Angew. Math. Phys., 68, 6, Art. 121, 19 pp., (2017) · Zbl 1383.35060 [39] Moroz, Vitaly; Van Schaftingen, Jean, Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains, J. Differential Equations, 254, 8, 3089-3145, (2013) · Zbl 1266.35083 [40] Natali, F\'abio; Pastor, Ademir, The fourth-order dispersive nonlinear Schr\"odinger equation: Orbital stability of a standing wave, SIAM J. Appl. Dyn. Syst., 14, 3, 1326-1347, (2015) · Zbl 1331.35325 [41] Nirenberg, L., On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa (3), 13, 115-162, (1959) · Zbl 0088.07601 [42] Noris, Benedetta; Tavares, Hugo; Verzini, Gianmaria, Existence and orbital stability of the ground states with prescribed mass for the $$L^2$$-critical and supercritical NLS on bounded domains, Anal. PDE, 7, 8, 1807-1838, (2014) · Zbl 1314.35168 [43] Noris, Benedetta; Tavares, Hugo; Verzini, Gianmaria, Stable solitary waves with prescribed $$L^2$$-mass for the cubic Schr\"odinger system with trapping potentials, Discrete Contin. Dyn. Syst., 35, 12, 6085-6112, (2015) · Zbl 1336.35321 [44] Pausader, Benoit, Global well-posedness for energy critical fourth-order Schr\"odinger equations in the radial case, Dyn. Partial Differ. Equ., 4, 3, 197-225, (2007) · Zbl 1155.35096 [45] Pausader, Benoit; Shao, Shuanglin, The mass-critical fourth-order Schr\"odinger equation in high dimensions, J. Hyperbolic Differ. Equ., 7, 4, 651-705, (2010) · Zbl 1232.35156 [46] Pausader, Benoit; Xia, Suxia, Scattering theory for the fourth-order Schr\"odinger equation in low dimensions, Nonlinearity, 26, 8, 2175-2191, (2013) · Zbl 1319.35240 [47] Pierotti, Dario; Verzini, Gianmaria, Normalized bound states for the nonlinear Schr\"odinger equation in bounded domains, Calc. Var. Partial Differential Equations, 56, 5, Art. 133, 27 pp., (2017) · Zbl 1420.35374 [48] Sirakov, Boyan, Some estimates and maximum principles for weakly coupled systems of elliptic PDE, Nonlinear Anal., 70, 8, 3039-3046, (2009) · Zbl 1173.35391 [49] Stein, Elias M., Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, xiv+290 pp., (1970), Princeton University Press, Princeton, NJ · Zbl 0207.13501 [50] Weinstein, Michael I., Nonlinear Schr\"odinger equations and sharp interpolation estimates, Comm. Math. Phys., 87, 4, 567-576, (1982/83) · Zbl 0527.35023
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.