## 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.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35B40 Asymptotic behavior of solutions to PDEs

### Keywords:

weak collapse; conformal invariance
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### References:

  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  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  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  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  W. Craig, T. Kappeler, and W. Strauss,  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  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  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  T. Kato, On nonlinear Schrödinger equations , Ann. Inst. H. Poincaré Phys. Théor. 46 (1987), no. 1, 113-129. · Zbl 0632.35038  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  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.  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.  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  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  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  L. Nirenberg, On elliptic partial differential equations , Ann. Sc. Norm. Sup. Pisa (3) 13 (1959), 115-162. · Zbl 0088.07601  T. Ogawa and Y. Tsutsumi, Blow-up of $$H^1$$ solution for the nonlinear Schrödinger equation , · Zbl 0739.35093  W. A. Strauss, Existence of solitary waves in higher dimensions , Comm. Math. Phys. 55 (1977), no. 2, 149-162. · Zbl 0356.35028  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  M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates , Comm. Math. Phys. 87 (1983), no. 4, 567-576. · Zbl 0527.35023  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  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.
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