Existence at least four solutions for a Schrödinger equation with magnetic potential involving sign-changing weight function

Abstract

In this paper we consider the following class of elliptic problems $$-{\Delta }_{A}u+u=a_{\lambda }(x)|u{|}^{q-2}u+b_{\mu }(x)|u{|}^{p-2}u,$$ for $x\in {\mathbb{R}}^N$, $1<q<2<p<2^{\ast }-1=\frac{N+2}{N-2}$, $a_{\lambda }(x)$ is a sign-changing weight function, $b_{\mu }(x)$ has some aditional conditions, $u\in H_A^1({\mathbb{R}}^N)$ and $A:{\mathbb{R}}^N\to {\mathbb{R}}^N$ is a magnetic potential. Exploring the Bahri Li argument and some preliminar results we will discuss the existence of four solution to the problem in question.

14 pages