Existence and multiplicity of solutions for nonlocal Schrödinger–Kirchhoff equations of convex–concave type with the external magnetic field

Abstract

We are concerned with the following elliptic equations $ \begin{equation*} K(|z|^p_{s, {A}})(-\Delta)^s_{p, A}z+ V(x)|z|^{p-2}z = a(x)|z|^{r-2}z+\lambda f(x, |z|)z \quad {\rm{in}} \; \; \mathbb{R}^{N}, \end{equation*} $ where $ (-\Delta)^{s}_{p, A} $ is the fractional magnetic operator, $ K:\mathbb{R}_0^+ \to\mathbb{R}^+_0 $ is a Kirchhoff function, $ A : \Bbb R^N \rightarrow \Bbb R^N $ is a magnetic potential and $ V:\Bbb R^{N}\to(0, \infty) $ is continuous potential. The main purpose is to show the existence of infinitely many large- or small- energy solutions to the problem above. The strategy of the proof for these results is to approach the problem variationally by employing the variational methods, namely, the fountain and the dual fountain theorem with Cerami condition.

Keywords

• schrödinger-kirchhoff equation
• fractional magnetic operators
• variational methods