论文标题
NLS接地状态在半线上与点相互作用
NLS ground states on the half-line with point interactions
论文作者
论文摘要
我们研究了在原点存在点相互作用的情况下，在半线上固定质量的NLS基态的存在和独特性。非线性是权力类型的，该制度是$ l^2 $ - 缩写或$ l^{2} $ - 关键，而点相互作用则具有吸引力或令人反感。在$ l^{2} $ - 亚临界案例中，我们证明，如果相互作用有吸引力，则每个质量价值都存在，而基态仅存在于相互作用的相互作用的情况下，仅存在足够大的质量。在后一种情况下，如果功率较小或等于四个，则基态与唯一的约束状态一致。相反，如果功率大于四，那么存在两个结合状态的质量值，而两个质量都不是基态，并且两个结合状态存在的质量值，其中一个是基态。在$ l^{2} $ - 关键案例中，我们证明，在有吸引力的情况下，群众严格低于临界质量价值，而基态在排斥性案例中不存在。
We investigate the existence and the uniqueness of NLS ground states of fixed mass on the half-line in the presence of a point interaction at the origin. The nonlinearity is of power type, and the regime is either $L^2$-subcritical or $L^{2}$-critical, while the point interaction is either attractive or repulsive. In the $L^{2}$-subcritical case, we prove that ground states exist for every mass value if the interaction is attractive, while ground states exist only for sufficiently large masses if the interaction is repulsive. In the latter case, if the power is less or equal to four, ground states coincide with the only bound state. If instead, the power is greater than four, then there are values of the mass for which two bound states exist, and neither of the two is a ground state, and values of the mass for which two bound states exist, and one of them is a ground state. In the $L^{2}$-critical case, we prove that ground states exist for masses strictly below a critical mass value in the attractive case, while ground states never exist in the repulsive case.
