Existence of at least four solutions for Schrodinger equations with magnetic potential involving and sign-changing weight function

Abstract

We consider the elliptic problem $$ - \Delta_A u + u = a_{\lambda}(x) |u|^{q-2}u+b_{\mu}(x) |u|^{p-2}u , $$ for \(x \in \mathbb{R}^N\), \( 1 < q < 2 < p < 2^*= 2N/(N-2)\), \(a_{\lambda}(x)\) is a sign-changing weight function, \(b_{\mu}(x)\) satisfies some additional conditions, \(u \in H^1_A(\mathbb{R}^N)\) and \(A:\mathbb{R}^N \to \mathbb{R}^N\) is a magnetic potential. Exploring the Bahri-Li argument and some preliminary results we will discuss the existence of a four nontrivial solutions to the problem in question.