Quasilinear Schrödinger equations involving concave and convex nonlinearities (Q1006197)

The authors of this interesting paper study quasilinear Schrödinger equations of the form \[ i \partial \psi / \partial t = -\Delta \psi +W(x)\psi - \eta (|\psi |^2)\psi - k \{\Delta \rho (|\psi |^2)\}\rho '(|\psi |^2)\psi , \] where \(\psi :\mathbb R\times\mathbb R ^N\to \mathbb C\), \(k\) is a positive constant, \(W:\mathbb R ^N\to \mathbb R \) is a given potential and \(\rho ,\eta :\mathbb R^+\to \mathbb R \) are suitable functions. Such kind Schrödinger equations arise in many areas of the quantum physics, superfluid theory, plasma physics, etc. The case of interest here is when \(\rho (s)=s\), \(k>0\). The authors study the positive solutions to the Schrödinger equation stated above that does not depend on \(t\), \[ -\Delta u+V(x)u-[\Delta (u^2)]u=\lambda h (x,u) +g(x,u), \] where \(\lambda \geq 0\), \(N\geq 3\), the potential \(V\) is continuous, uniformly positive and satisfies an appropriate integrability condition. The nonlinear term \(\lambda h (x,u) +g(x,u)\) involves a combination of concave and convex terms in \(u\). The mountain-pass theorem and the Ekeland variational principle in a suitable Orlicz space are employed to establish the existence of positive standing wave solutions in the class \(H^1(\mathbb R ^N)\).