Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power. (English) Zbl 0808.35141

The generalized nonlinear Schrödinger (NLS) equation of the following form is considered: \[ i {{\partial u} \over {\partial t}}=- \Delta u- | u|^{4/N} u,\tag{1} \] where \(u\) is a complex variable, and \(\Delta\) is the Laplacian in the \(N\)-dimensional space. It is well known that the particular nonlinear term in equation (1) corresponds to the weak collapse in the generalized NLS equation; while in the case of the nonlinear term with a smaller power a generic solution of the initial- value problem remains bounded up to \(t=\infty\), in the case of a larger power the solution demonstrates the strong collapse, i.e., it gives rise to a singularity at a finite value of \(t\). Notice that equation (1) has the integral of motion (“mass”) \[ M=\int | u({\mathbf x},t)|^ 2 d{\mathbf x}. \tag{2} \] In the case of the strong collapse, a large part of the mass of a generic initial state is involved into the collapse. In the boundary case, corresponding exactly to equation (1), a collapse also takes place at a finite time, but only a small part of the mass of a generic initial condition is involved into it, that is why it is called the weak collapse. In this work, a solution with a minimum mass (2) leading to the weak collapse is constructed. This solution is expressed in terms of the so-called ground-state solution \(Q\) of the elliptic problem corresponding to equation (1). \(Q\) is a real positive solution to the equation \[ \Delta u+| u|^{4/N}u =u \tag{3} \] such that it depends only upon the radial variable \(r= (x_ 1^ 2+\dots+ x_ N^ 2)^{1/2}\), and any other nonzero solution to equation (3) has an \(L^ 2\)-norm larger than that of \(Q\). A whole family of more general ground- state solutions can be obtained by application of the conformal transformations, which constitute a natural group of symmetry of equation (1), to the solution \(Q\). The main result obtained in this work is that, given the solution \(Q\), a fundamental minimum-mass collapsing solution to equation (1) is \[ | t|^{-N/2} e^{(| x|^ 2/ 4t)- i/t} Q\bigl( {\textstyle {{\mathbf x} \over t}} -{\mathbf x}_ 1 \bigr), \tag{4} \] where \({\mathbf x}_ 1\) is an arbitrary constant, and \(t=0\) is the collapse moment. More general collapsing minimum-mass solutions can be obtained form (4) by means of the conformal transformations. The proof is based on variational estimates for solutions, obtained in terms of the \(L^ 2\) norm.


35Q55 NLS equations (nonlinear Schrödinger equations)
35B40 Asymptotic behavior of solutions to PDEs
Full Text: DOI


[1] H. Berestycki, P. L. Lions, and I. A. Peletier, An ODE approach to the existence of positive solutions for semilinear problems in \(\mathbbR^N\) , Indiana Univ. Math. J. 30 (1981), no. 1, 141-157. · Zbl 0522.35036
[2] 1 H. Berestycki and P. L. Lions, Existence d’ondes solitaires dans des problèmes non-linéaires du type Klein-Gordon , C. R. Acad. Sci. Paris Sér. A-B 287 (1978), no. 7, A503-A506. · Zbl 0391.35055
[3] 2 H. Berestycki and P. L. Lions, Existence d’ondes solitaires dans des problèmes nonlinéaires du type Klein-Gordon , C. R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 7, A395-A398. · Zbl 0397.35024
[4] T. Cazenave and F. Weissler, Some remarks on the nonlinear Schrödinger equation in the critical case , Nonlinear semigroups, partial differential equations and attractors (Washington, DC, 1987), Lecture Notes in Math., vol. 1394, Springer, Berlin, 1989, pp. 18-29. · Zbl 0694.35170
[5] W. Craig, T. Kappeler, and W. Strauss,
[6] J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations, I: The Cauchy problem, general case , J. Funct. Anal. 32 (1979), no. 1, 1-32. · Zbl 0396.35028
[7] J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Schrödinger equation revisited , Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), no. 4, 309-327. · Zbl 0586.35042
[8] R. T. Glassey, On the blowing-up of solutions to the Cauchy problem for nonlinear Schrödinger equations , J. Math. Phys. 18 (1977), no. 9, 1794-1797. · Zbl 0372.35009
[9] T. Kato, On nonlinear Schrödinger equations , Ann. Inst. H. Poincaré Phys. Théor. 46 (1987), no. 1, 113-129. · Zbl 0632.35038
[10] M. K. Kwong, Uniqueness of positive solutions of \(\Delta u-u+u^ p=0\) in \(\mathbbR^N\) , Arch. Rational Mech. Ann. 105 (1989), no. 3, 243-266. · Zbl 0676.35032
[11] M. Landman, G. C. Papanicolaou, C. Sulem, and P. L. Sulem, Rate of blow-up for solutions of the nonlinear Schrödinger equation at critical dimension , Phys. Rev. A (3) 38 (1988), no. 8, 3837-3843.
[12] D. W. McLaughlin, G. Papanicolaou, C. Sulem, and P. L. Sulem, Focusing singularity of the cubic Schrödinger equation , Phys. Rev. A 34 (1986), 1200-1210.
[13] F. Merle, Construction of solutions with exactly \(k\) blow-up points for the Schrödinger equation with critical nonlinearity , Comm. Math. Phys. 129 (1990), no. 2, 223-240. · Zbl 0707.35021
[14] F. Merle, On uniqueness and continuation properties after blow-up time of self-similar solutions of nonlinear Schrödinger equation with critical exponent and critical mass , Comm. Pure Appl. Math. 45 (1992), no. 2, 203-254. · Zbl 0767.35084
[15] F. Merle and Y. Tsutsumi, \(L^ 2\) concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power nonlinearity , J. Differential Equations 84 (1990), no. 2, 205-214. · Zbl 0722.35047
[16] L. Nirenberg, On elliptic partial differential equations , Ann. Sc. Norm. Sup. Pisa (3) 13 (1959), 115-162. · Zbl 0088.07601
[17] T. Ogawa and Y. Tsutsumi, Blow-up of \(H^1\) solution for the nonlinear Schrödinger equation , · Zbl 0739.35093
[18] W. A. Strauss, Existence of solitary waves in higher dimensions , Comm. Math. Phys. 55 (1977), no. 2, 149-162. · Zbl 0356.35028
[19] M. I. Weinstein, The nonlinear Schrödinger equation-singularity formation, stability and dispersion , The connection between infinite-dimensional and finite-dimensional dynamical systems (Boulder, CO, 1987), Contemp. Math., vol. 99, Amer. Math. Soc., Providence, RI, 1989, pp. 213-232. · Zbl 0703.35159
[20] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates , Comm. Math. Phys. 87 (1983), no. 4, 567-576. · Zbl 0527.35023
[21] M. I. Weinstein, On the structure and formation of singularities in solutions to nonlinear dispersive evolution equations , Comm. Partial Differential Equations 11 (1986), no. 5, 545-565. · Zbl 0596.35022
[22] V. E. Zakharov, V. V. Sobolev, and V. S. Synach, Character of the singularity and stochastic phenomena in self-focusing , Zh. Èksper. Teoret. Fiz. 14 (1971), 390-393.
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.