Concentration for a biharmonic Schrödinger equation

Dong Wang

Vol. 289 (2017), No. 2, 469–487

DOI: 10.2140/pjm.2017.289.469

Abstract

We consider the fourth-order problem

$$\left\{\begin{array}{c}\hfill \\ \hfill \\ {ϵ}^4{\Delta }^{2}u+V\left(x\right)u=P\left(x\right)f\left(\left|u\right|\right)u,\hfill & x\in {ℝ}^N,\hfill \\ \hfill \\ \hfill \\ u\left(x\right)\to 0\hfill & \text{ as }\left|x\right|\to \infty ,\hfill \\ \hfill \\ \hfill \\ \hfill \end{array}\right.$$

where $V$ and $P$ are spatial distributions of external potentials. We study the concentration phenomena of the solutions as $ϵ\to 0$ using variational methods.

Keywords

nonlinear biharmonic Schrödinger equations, standing waves, critical point theory

Mathematical Subject Classification 2010

Primary: 35J10, 35Q40, 47J30

Milestones

Received: 31 July 2015

Revised: 20 December 2016

Accepted: 20 December 2016

Published: 19 June 2017