Fractional Morrey-Sobolev type embeddings and nonlocal subelliptic problems with oscillating nonlinearities on stratified Lie groups

Abstract

In this paper, we establish the fractional Morrey-Sobolev type embeddings on stratified Lie groups. This extends and complements the Sobolev type embeddings derived in \cite{GKR}. As an application of the results, we study the following nonlocal subelliptic problem, \begin{equation} \begin{cases} (-Δ_{\mathbb{G}, p})^s u= λβ(x) g(u) & \text{in} \quad Ω, \\ u=0\quad & \text{in}\quad \mathbb{G}\backslash Ω, \end{cases} \end{equation} where $0<s<1<p<\infty$ with $ps\ge Q$, $Q$ is the homogeneous dimension of the stratified Lie group $\mathbb{G}$, $(-\hat{I}{”}_{\mathbb{G},p}{)}^s$ is the fractional $p$-sub-Laplacian on $\mathbb{G},$ $\hat{I}\text{c◯}$ is an open bounded subset of $\mathbb{G}$, $\hat{I}»$ is a positive real parameter, ${\hat{I}}^2\in L^{\infty }(\hat{I}\text{c◯},{\mathbb{R}}_{>0})$ and $g\in C(\mathbb{R},\mathbb{R})$ oscillates near the origin or at infinity. By using the variational principle of Ricceri, we prove the existence and asymptotic behaviors of infinitely many solutions to the problem under consideration. We emphasize that the results obtained here are also novel for $\mathbb{G}$ being the Heisenberg group and $p=2$.