Remarks on the non-local eigenvalue problems for the p-Laplacian

We consider the p-Laplace eigenvalue problem $$ -\Delta _{p} u=\lambda |u|_{q, w}^{p-q} w(x)|u|^{q-2} u \quad {\text{in}} \ \Omega , \qquad u=0 \quad {\text{on}}\ \partial \Omega , $$ having a nonlocal term $|u|_{q, w}^{q}=\int _{\Omega} w|u|^{q} d x>0$ . The first eigenvalue is well known as the best constant of the Sobolev–Poicaré inequality. In this paper, we give the existence of the least eigenvalue $\mu _{q}^{w}$ such that the equation has a nodal solution. We show that $\mu _{q}^{w}$ coincides with the second eigenvalue in p-sublinear case ( $1< q< p$ ) provided $w \geq 0$ a.e. in Ω. In p-superlinear case ( $p< q< p^{*}$ ), we characterize $\mu _{q}^{w}$ by the minimax value using the Rayleigh quotient.