No abstract

This paper is concerned with the existence and qualitative property of standing wave solutions ψ(t, x) = e −iEt/h v(x) for the nonlinear Schrödinger equationWe show that there exists a standing wave which is trapped in a neighbourhood of isolated minimum points of V and whose amplitude goes to 0 as h → 0. Moreover, depending upon the local behaviour of the potential function V (x) near the minimum points, the limiting profile of the standing-wave solutions will be shown to exhibit quite different characteristic features. This is in striking contrast with the non-critical frequency case (inf R N V (x) > E) which has been extensively studied in recent years.

Variational techniques are applied to prove the existence of standing wave solutions for quasilinear Schrödinger equations containing strongly singular nonlinearities which include derivatives of the second order. Such equations have been derived as models of several physical phenomena. The nonlinearity here corresponds to the superfluid film equation in plasma physics. Direct methods of the calculus of variations and minimax methods like the Mountain Pass Theorem are used. The difficulties introduced by the nonconvex functional Φ(u) = |∇u| 2 u 2 are substantially different from the semilinear case. Mathematics Subject Classification (1991):35J20, (35Q55, 35J60)

The paper is concerned with the local and global bifurcation structure of positiveThe system arises in nonlinear optics and in the Hartree-Fock theory for a double condensate. Local and global bifurcations in terms of the nonlinear coupling parameter β of the system are investigated by using spectral analysis and by establishing a new Liouville type theorem for nonlinear elliptic systems which provides a-priori bounds of solution branches. If the domain is radial, possibly unbounded, then we also control the nodal structure of a certain weighted difference of the components of the solutions along the bifurcating branches. Mathematics Subject Classification (2000)35B05 · 35B32 · 35J50 · 35J55 · 58C40 · 58E07 Dedicated to Paul Rabinowitz on the occasion of his 70th birthday.

We investigate nonlinear Schrödinger equations like the model equationwhere the potential V λ has a potential well with bottom independent of the parameter λ > 0. If λ → ∞ the infimum of the essential spectrum of −∆ + V λ in L 2 (R N ) converges towards ∞ and more and more eigenvalues appear below the essential spectrum. We show that as λ → ∞ more and more solutions of the nonlinear Schrödinger equation exist. The solutions lie in H 1 (R N ) and are localized near the bottom of the potential well, but not near local minima of the potential. We also investigate the decay rate of the solutions as |x| → ∞ as well as their behaviour as λ → ∞.