Nonlinear SchrÖdinger equations with sum of periodic and vanishing potentials and sign-changing nonlinearities

Bartosz Bieganowski; Jaros law Mederski

doi:10.3934/cpaa.2018009

Article Contents

2018, Volume 17, Issue 1: 143-161. Doi: 10.3934/cpaa.2018009

This issue Previous Article The Riemann problem and the limit solutions as magnetic field vanishes to magnetogasdynamics for generalized Chaplygin gas Next Article Stability of standing waves for a nonlinear SchrÖdinger equation under an external magnetic field

Nonlinear SchrÖdinger equations with sum of periodic and vanishing potentials and sign-changing nonlinearities

Bartosz Bieganowski^1, and
Jaros law Mederski^1,2, ,

1.
Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland
2.
Institute of Mathematics, Polish Academy of Sciences, ul. Sniadeckich 8, 00-956 Warszawa, Poland

* Corresponding author
* Corresponding author

Received: February 2017

Revised: February 2017

Published: January 2018

The second author was supported by the National Science Centre, Poland (Grant No. 2014/15/D/ST1/03638).

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Abstract

We look for ground state solutions to the following nonlinear Schrödinger equation

$-Δ u + V(x)u = f(x,u)-Γ(x)|u|^{q-2}u\hbox{ on }\mathbb{R}^N,$

where $V=V_{per}+V_{loc}∈ L^{∞}(\mathbb{R}^N)$ is the sum of a periodic potential $V_{per}$ and a localized potential $V_{loc}$, $Γ∈ L^{∞}(\mathbb{R}^N)$ is periodic and $Γ(x)≥ 0$ for a.e. $x∈\mathbb{R}^N$ and $2≤q <2^*$. We assume that $\inf σ(-Δ+V)>0$, where $σ(-Δ+V)$ stands for the spectrum of $-Δ +V$ and $f$ has the subcritical growth but higher than $Γ(x)|u|^{q-2}u$, however the nonlinearity $f(x, u)-Γ(x)|u|^{q-2}u$ may change sign. Although a Nehari-type monotonicity condition for the nonlinearity is not satisfied, we investigate the existence of ground state solutions being minimizers on the Nehari manifold.

Keywords:

Mathematics Subject Classification: Primary:35Q60;Secondary:35J20, 35Q55, 58E05, 35J47.

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