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# Zeitschrift für Analysis und ihre Anwendungen

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Volume 34, Issue 3, 2015, pp. 309–319
DOI: 10.4171/ZAA/1541

Published online: 2015-07-08

Multiplicity of Nontrivial Solutions of a Class of Fractional $p$-Laplacian Problem

Ghanmi Abdeljabbar

(1) Université de Tunis, Tunisia

In this paper, we deal with existence of nontrivial solutions to the fractional $p$-Laplacian problem of the type $$\left\{ \begin{array}{rll} (-\triangle)^{\alpha}_{p} u \! \!&= \ \frac{1}{r}\frac{\partial F(x,u)}{\partial u} + \lambda a(x)|u|^{q-2}u &\mbox{ in} \ \Omega, \\[1mm] u \! \!&= \ 0 &\mbox{ in}\ \mathbb{R}^{n}\setminus\Omega, \end{array} \right.$$ where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with smooth boundary $\partial \Omega$, $a\in C(\Omega)$, $p\geq 2$, $\alpha \in (0,1)$ such that $p \alpha < n$, $1 < q < p < r < \frac{np}{n-\alpha p}$, and $F \in C^{1}(\overline{\Omega} \times \mathbb{R}, \mathbb{R})$. Using the decomposition of the Nehari manifold, we prove that the non-local elliptic problem has at least two nontrivial solutions.

Keywords: Nontrivial solutions, sign-changing weight function, Nehari manifold

Abdeljabbar Ghanmi: Multiplicity of Nontrivial Solutions of a Class of Fractional $p$-Laplacian Problem. Z. Anal. Anwend. 34 (2015), 309-319. doi: 10.4171/ZAA/1541