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摘要
We determine and study the ground states of a focusing Schrodinger equation in dimension one with a power nonlinearity |psi|(2 mu) psi and a strong inhomogeneity represented by a singular point perturbation, the so-called (attractive) delta%26apos; interaction, located at the origin. The time-dependent problem turns out to be globally well posed in the subcritical regime, and locally well posed in the supercritical and critical regime in the appropriate energy space. The set of the (nonlinear) ground states is completely determined. For any value of the nonlinearity power, it exhibits a symmetry breaking bifurcation structure as a function of the frequency (i.e., the nonlinear eigenvalue) omega. More precisely, there exists a critical value omega* of the nonlinear eigenvalue omega, such that: if omega(0) %26lt; omega %26lt; omega*, then there is a single ground state and it is an odd function; if omega %26gt; omega* then there exist two non-symmetric ground states. We prove that before bifurcation (i.e., for omega %26lt; omega*) and for any subcritical power, every ground state is orbitally stable. After bifurcation (omega = omega(*) + 0), ground states are stable if mu does not exceed a value that lies between 2 and 2.5, and become unstable for mu %26gt; mu*. Finally, for mu %26gt; 2 and , all ground states are unstable. The branch of odd ground states for omega %26lt; omega* can be continued at any omega %26gt; omega*, obtaining a family of orbitally unstable stationary states. Existence of ground states is proved by variational techniques, and the stability properties of stationary states are investigated by means of the Grillakis-Shatah-Strauss framework, where some non-standard techniques have to be used to establish the needed properties of linearization operators.
- 出版日期2013-2
