Nonlinear perturbations of a periodic Schrödinger equation with supercritical growth
The main focus in the paper under review is on the study of the existence of positive solutions for the quasilinear elliptic problem \[ -\Delta u - \Delta(u^2)u +V(x) u=p(u),\quad x\in\mathbb{R}^N, \quad N\geq 3, \] where \(V:\mathbb{R}^N\to \mathbb{R}\) is a continuos function positive function and \(p:\mathbb{R}\to \mathbb{R}\) has the form \(p=f_0 + \varepsilon g\). Moreover, both functions \(V\) and \(p\) satisfy certain additional assumptions. As their main result, the authors establish the existence of a positive solution for all sufficiently small values of a positive parameter \(\varepsilon\) in two cases: (i) \(V\) is a periodic function, and (ii) \(V\) is a small perturbation of a periodic potential.
scientific article; zbMATH DE number 6513335
scientific article