Electron. J. Differential Equations, Vol. 2023 (2023), No. 38, pp. 1-29.
Principal eigenvalues for the fractional p-Laplacian with unbounded sign-changing weights
Oumarou Asso, Mabel Cuesta, Jonas Tele Doumate, Liamidi Leadi:
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;
principal eigenvalues.
DOI: https://doi.org/10.58997/ejde.2023.38
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Oumarou Asso
Institut de Mathématiques et de Sciences Physiques
Université d'Abomey-Calavi, 613 Porto-Novo, Bénin
email: oumarou.asso@imsp-uac.org
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Mabel Cuesta
Université du Littoral Côte d'Opale (ULCO), LMPA
50 rue F. Buisson 62220 Calais, France
email: mabel.cuesta@univ-littoral.fr
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Jonas Têlé Doumatè
Département de Mathématiques
Faculté des Sciences et Techniques
Institut de Mathématiques et de Sciences Physiques
Université d'Abomey-Calavi, Benin
email: jonas.doumate@fast.uac.bj
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Liamidi Leadi
Département de Mathématiques
Faculté des Sciences et Techniques
Institut de Mathématiques et de Sciences Physiques
Université d'Abomey-Calavi, Benin
email: leadiare@imsp-uac.org
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