# Sign changing solutions of p-fractional equations with concave-convex nonlinearities

## TL;DR

This paper proves the existence of sign-changing solutions for a class of p-fractional equations with concave-critical nonlinearities, expanding understanding of solutions in fractional PDEs with critical exponents.

## Contribution

It establishes the existence of sign-changing solutions for fractional p-Laplacian equations involving concave and critical nonlinearities, a novel result in fractional PDE analysis.

## Key findings

- Existence of sign-changing solutions proven.
- Solutions exist under specific parameter conditions.
- Advances understanding of fractional PDEs with critical nonlinearities.

## Abstract

In this article we study the existence of sign changing solution of the following p-fractional problem with concave-critical nonlinearities: \begin{eqnarray*} (-\Delta)^s_pu &=& \mu |u|^{q-1}u + |u|^{p^*_s-2}u \quad\mbox{in}\quad \Omega, u&=&0\quad\mbox{in}\quad\mathbb{R}^N\setminus\Omega, \end{eqnarray*} where $s\in(0,1)$ and $p\geq 2$ are fixed parameters, $0<q<p-1$, $\mu\in\mathbb{R}^+$ and $p_s^*=\frac{Np}{N-ps}$. $\Omega$ is an open, bounded domain in $\mathbb{R}^N$ with smooth boundary with $N>ps$ .