Multiplicity and Concentration of Positive Solutions for (p, q)-Kirchhoff Type Problems

Abstract

This paper is devoted to the study of the (p, q)-Kirchhoff type problem $-\left( 1+a\int _{{\mathbb {R}}^{N}}|\nabla u|^{p}dx\right) \Delta _{p} u -\left( 1+b\int _{{\mathbb {R}}^{N}}|\nabla u|^{q}dx\right) \Delta _{q} u +V_{\varepsilon }(x) \left( |u|^{p-2}u + |u|^{q-2}u\right) = f(u)\quad \hbox {in}\quad {\mathbb {R}}^{N},$ where $\varepsilon >0$ is a small parameter, $a,\,b>0$, $1< p<q<N<2q$, $\Delta _{s}u={{\,\textrm{div}\,}}(|\nabla u|^{s-2}\nabla u)$, with $s\in \{p, q\}$, is the s-Laplacian, $V:{\mathbb {R}}^{N}\rightarrow {\mathbb {R}}$ is a continuous potential, $V_{\varepsilon }(x)=V(\varepsilon x)$ and $f:{\mathbb {R}}\rightarrow {\mathbb {R}}$ is a continuous non-linearity without satisfying usual Ambrosetti–Rabinowitz condition. By using Nehari manifold techniques and perturbation methods, we establish the existence of positive ground state solution, and by those methods combined with Ljusternik–Schnirelmann theory, we investigate the multiplicity and concentration of solutions. In particular, our results extend and improve the works of He and Zou (J Differ Equ 252:1813–1834, 2012).