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

Abstract

In this work we consider the following class of fractional $p\&q$ Laplacian problems \begin{equation*} (-Δ)_{p}^{s}u+ (-Δ)_{q}^{s}u + V(\varepsilon x) (|u|^{p-2}u + |u|^{q-2}u)= f(u) \mbox{ in } \mathbb{R}^{N}, \end{equation*} where $\epsilon >0$ is a parameter, $s\in (0,1)$, $1<p<q<\frac{N}{s}$, $(-\Delta {)}_t^s$, with $t\in \{p,q\}$, is the fractional $t$-Laplacian operator, $V:{\mathbb{R}}^N\to \mathbb{R}$ is a continuous potential and $f:\mathbb{R}\to \mathbb{R}$ 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 $\epsilon$ is sufficiently small.

arXiv admin note: text overlap with arXiv:1709.03737