Multiplicity and concentration results for a $(p, q)$-Laplacian problem in $\mathbb{R}^{N}$

TL;DR

This paper investigates the existence, multiplicity, and concentration of positive solutions for a $(p, q)$-Laplacian problem in $ ext{R}^N$, showing how the solutions relate to the topology of the potential's minimum set as a small parameter tends to zero.

Contribution

The paper introduces new variational methods combined with Ljusternik-Schnirelmann theory to connect solution multiplicity with the topology of the potential's minimum set.

Findings

Multiple positive solutions exist for small $ ext{\varepsilon}$.

Solutions concentrate near the minimum set of the potential.

The number of solutions relates to the topological complexity of the minimum set.

Abstract

In this paper we study the multiplicity and concentration of positive solutions for the following $(p, q)$-Laplacian problem: \begin{equation*} \left\{ \begin{array}{ll} -\Delta_{p} u -\Delta_{q} u +V(\varepsilon x) \left(|u|^{p-2}u + |u|^{q-2}u\right) = f(u) &\mbox{ in } \mathbb{R}^{N}, \\ u\in W^{1, p}(\mathbb{R}^{N})\cap W^{1, q}(\mathbb{R}^{N}), \quad u>0 \mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a small parameter, $1< p<q<N$, $\Delta_{r}u=\mbox{div}(|\nabla u|^{r-2}\nabla u)$, with $r\in \{p, q\}$, is the $r$-Laplacian operator, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a continuous function satisfying the global Rabinowitz condition, and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a continuous function with subcritical growth. Using suitable variational arguments and Ljusternik-Schnirelmann category theory, we investigate the relation between the number of positive solutions and the topology of the set where $V$ attains its minimum for small $\varepsilon$.