An existence and stability result for standing waves of nonlinear Schrödinger equations (Q930317)

Considered is the nonlinear Schrödinger equation \(i u_t + \Delta u = V(x) g(u)\), where \(V(x)\) is a real-valued potential and \(g\) satisfies \(g(e^{i\theta}s) = e^{i\theta} g(s)\) for real \(s,\theta\). Standing waves are solutions of the form \(u(t,x) = e^{i\lambda t} \phi(x)\), where \(\phi\) is a real solution of the elliptic equation \(-\Delta \phi + \lambda \phi = V(x) g(\phi)\). The authors use mountain pass type arguments to prove the existence of non-negative solutions \(\phi_\lambda\) for small positive values of \(\lambda\) under sufficiently general conditions on the behaviour of \(V(x)\) at infinity and \(g(s)\) near zero. Next, stability of the corresponding solution \(e^{i\lambda t} \phi_\lambda\) of the nonlinear Schrödinger equation is considered under additional conditions on the behaviour of the derivative \(g'\) near zero and at infinity.