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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
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