Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 38, pp. 1-29.

Title: Principal eigenvalues for the fractional  p-Laplacian with unbounded sign-changing  weights
       
Authors: Oumarou Asso (Univ. d'Abomey-Calavi, Benin)
         Mabel Cuesta (Univ. du Littoral, Calais, France)
         Jonas Tele Doumate (Univ. d'Abomey-Calavi, Benin)
         Liamidi Leadi (Univ. d'Abomey-Calavi, Benin)

Abstract:
Let $\Omega$  be a bounded  regular domain of $ \mathbb{R}^N$, $N\geqslant 1$,
$p\in (1,+\infty)$, and $ s\in (0,1) $. We consider the eigenvalue problem
$$\displaylines{
(-\Delta_p)^s u + V|u|^{p-2}u= \lambda m(x)|u|^{p-2}u \quad\hbox{in } \Omega \cr
u=0 \quad \hbox{in } \mathbb{R}^N \setminus \Omega,
}$$
where the potential V and the weight m are possibly unbounded and are
sign-changing. After establishing the  boundedness and regularity of
weak solutions, we prove that this problem admits principal eigenvalues under
certain conditions. We also show that when such eigenvalues  exist,
they are simple and isolated in the spectrum of the operator.

Submitted July 20, 2022. Published June 19, 2023.
Math Subject Classifications: 35J70, 35P30.
Key Words: Fractional p-Laplacian; fractional Sobolev space; indefinite weight;
\hfill\break\indent principal eigenvalues.