Existence of ground states for a modified nonlinear Schrödinger equation. (English) Zbl 1189.35316

Ruiz, David; Siciliano, Gaetano

Nonlinearity 23, No. 5, 1221-1233 (2010).

Summary: We prove the existence of ground state solutions of the modified nonlinear Schrödinger equation: \[ -\Delta u +V(x)u-\frac 1 2 u\Delta u^2 = |u|^{p-1}u,\quad x \in \mathbb R^N, \quad N\geqslant 3, \] under some hypotheses on \(V(x)\). This model has been proposed in the theory of superfluid films in plasma physics. As a main novelty with respect to some previous results, we are able to deal with exponents \(p \in (1, 3)\). The proof is accomplished by minimization under a convenient constraint.

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
76A20	Thin fluid films
82D10	Statistical mechanics of plasmas
82D50	Statistical mechanics of superfluids
35A15	Variational methods applied to PDEs

Keywords:

thin fluid films; superfluids; nonlinear Schrödinger equations; plasmas