## Topological Methods in Nonlinear Analysis

- Topol. Methods Nonlinear Anal.
- Volume 49, Number 2 (2017), 445-461.

### Multiplicity results for nonlocal fractional $p$-Kirchhoff equations via Morse theory

Binlin Zhang, Giovanni Molica Bisci, and Mingqi Xiang

#### Abstract

In this paper, we apply Morse theory and local linking to study the existence of nontrivial solutions for Kirchhoff type equations involving the nonlocal fractional $p$-Laplacian with homogeneous Dirichlet boundary conditions: \[ \begin{cases} \bigg[M\bigg(\displaystyle\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\,dxdy\bigg)\bigg]^{p-1} (-\Delta)_p^su(x)=f(x,u)&\mbox{in }\Omega,\\ u=0&\mbox{in } \mathbb{R}^{N}\setminus\Omega, \end{cases} \] where $\Omega$ is a smooth bounded domain of $\mathbb{R}^N$, $(-\Delta)_p^s$ is the fractional $p$-Laplace operator with $0<s<1<p<\infty$ with $sp<N$, $M \colon \mathbb{R}^{+}_{0}\rightarrow \mathbb{R}^{+}$ is a continuous and positive function not necessarily satisfying the increasing condition and $f$ is a Carathéodory function satisfying some extra assumptions.

#### Article information

**Source**

Topol. Methods Nonlinear Anal., Volume 49, Number 2 (2017), 445-461.

**Dates**

First available in Project Euclid: 14 March 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.tmna/1489457023

**Digital Object Identifier**

doi:10.12775/TMNA.2016.081

**Mathematical Reviews number (MathSciNet)**

MR3670468

**Zentralblatt MATH identifier**

1370.35270

#### Citation

Zhang, Binlin; Molica Bisci, Giovanni; Xiang, Mingqi. Multiplicity results for nonlocal fractional $p$-Kirchhoff equations via Morse theory. Topol. Methods Nonlinear Anal. 49 (2017), no. 2, 445--461. doi:10.12775/TMNA.2016.081. https://projecteuclid.org/euclid.tmna/1489457023