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

We consider the p-Laplace eigenvalue problem -Delta(p)u =lambda w(x) u lu in 2. u=0 on partial derivative Omega. having a nonlocal term [u(q)Omega integral(q,w)a w/u/9dx > 0. The first eigenvalue is well known as the best constant of the Sobolev-Poicar & eacute; inequality. In this paper, we give the existence of the least eigenvalue mu" such that the equation has a nodal solution, We show that u coincides with the second eigenvalue in p-sublinear case (1 <q<p) provided w >= 0 a.e. in 2. In p-superlinear case (p <q<p(2)), we characterize " by the minimax value using the Rayleigh quotient.