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Existence and multiplicity of solutions for a fractional $p$-Laplacian problem of Kirchhoff type via Krasnoselskii's genus

Ghania Benhamida, Toufik Moussaoui

Received January 28, 2017.   First published August 14, 2017.

Abstract: We use the genus theory to prove the existence and multiplicity of solutions for the fractional $p$-Kirchhoff problem $\begin{cases} \displaystyle-\biggl[M \biggl(\int_Q\frac{\vert u(x)-u(y)\vert^p}{\vert x-y \vert^{N+ps}} {\rm d}x {\rm d}y\biggr)\biggr]^{p-1} (-\Delta)_p^su=\lambda h(x,u) \quad\text{in}\ \Omega, \\ u=0 \quad\text{on}\ \mathbb{R}^N \setminus\Omega, \end{cases} $ where $\Omega$ is an open bounded smooth domain of $\mathbb{R}^N$, $p>1$, $N>ps$ with $s\in(0,1)$ fixed, $Q = \mathbb{R}^{2N}\setminus(C\Omega\times C\Omega)$, $\lambda> 0$ is a numerical parameter, $M$ and $h$ are continuous functions.

Keywords: existence results; genus theory; fractional $p$-Kirchhoff problem

Classification (MSC 2010): 35A15, 34A08, 35B38

DOI: 10.21136/MB.2017.0010-17

Full text available as PDF.

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