# Existence, multiplicity and concentration for a class of fractional $p\&q$ Laplacian problems in $\mathbb{R}^{N}$

## TL;DR

This paper studies fractional p&q Laplacian problems in rica, proving existence, multiplicity, and concentration of solutions using variational methods, especially for small psilons, with applications to nonlinear PDEs.

## Contribution

It introduces new results on the existence and multiplicity of solutions for a class of fractional p&q Laplacian equations in rica, employing minimax and topological methods.

## Key findings

- Existence of solutions for small psilons.
- Multiple solutions under certain conditions.
- Solutions concentrate as psilons tend to zero.

## Abstract

In this work we consider the following class of fractional $p\&q$ Laplacian problems \begin{equation*} (-\Delta)_{p}^{s}u+ (-\Delta)_{q}^{s}u + V(\varepsilon x) (|u|^{p-2}u + |u|^{q-2}u)= f(u) \mbox{ in } \mathbb{R}^{N}, \end{equation*} where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $1< p<q<\frac{N}{s}$, $(-\Delta)^{s}_{t}$, with $t\in \{p,q\}$, is the fractional $t$-Laplacian operator, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a continuous potential and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a $\mathcal{C}^{1}$-function with subcritical growth. Applying minimax theorems and the Ljusternik-Schnirelmann theory, we investigate the existence, multiplicity and concentration of nontrivial solutions provided that $\varepsilon$ is sufficiently small.