Existence and multiplicity results for a class of non-linear Schrödinger equations with magnetic potential involving sign-changing non linearity

Abstract

In this work we consider the following class of elliptic problems $$- Δ_A u + u = a(x) |u|^{q-2}u+b(x) |u|^{p-2}u , \mbox{ in } \mathbb{R}^N, $$ $u\in H_A^1({\mathbb{R}}^N)$, with $2<q<p<2^{\ast }=\frac{2N}{N-2}$, $a(x)$ and $b(x)$ are functions that can change signal and satisfy some additional conditions; $u\in H_A^1({\mathbb{R}}^N)$ and $A:{\mathbb{R}}^N\to {\mathbb{R}}^N$ is a magnetic potential. Also using the Nehari method in combination with other complementary arguments, we discuss the existence of infinite solutions to the problem in question, varying the assumptions about the weight functions.

19 pages, 4 figures