Principal eigenvalues for the fractional p-Laplacian with unbounded sign-changing weights

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.