Nehari manifold for a Schrödinger equation with magnetic potential involving sign-changing weight function

Abstract

We consider the following class of elliptic problems $-{\mathrm{\Delta }}_{A}u+u=a_{\lambda }(x)|u{|}^{q-2}u+b_{\mu }(x)|u{|}^{p-2}u,x\in {\mathbb{R}}^N,$ where $1<q<2<p<2^{\ast }=\frac{2N}{N-2}$ $N\ge 3$, $a_{\lambda }(x)$ is a sign-changing weight function, $b_{\mu }(x)$ is continuous, $\lambda >0$ and $\mu >0$ are real parameters, $u\in H_A^1({\mathbb{R}}^N)$ and $A:{\mathbb{R}}^N\to {\mathbb{R}}^N$ is a magnetic potential. Exploring the relationship between the Nehari manifold and fibering maps, we will discuss the existence, multiplicity and regularity of solutions.

Keywords:

• Sign-changing weight function
• magnetic potential
• Nehari manifold
• fibering map

2010 Mathematics Subject Classifications:

• 35Q60
• 35Q55
• 35B38
• 35B33

Disclosure statement

The authors have no conflicts of interest to declare. Also, all co-authors have seen and agree with the contents of the manuscript and there is no financial interest to report. We certify that the submission is original work and is not under review at any other publication.

Additional information

Funding

Francisco Odair de Paiva received research grants from FAPESP 17/16108-6. Sandra Machado de Souza Lima was supported by CAPES/Brazil and the paper was completed while she was visiting the Dept of Math of UFJF, whose hospitality she gratefully acknowledges. Olímpio Hiroshi Miyagaki was supported by FAPESP/Brazil grant 2019/24901-3 and CNPq/Brazil grant 307061/2018-3.