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

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.


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
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