Multiple solutions to a nonlinear elliptic equation involving Paneitz type operators

Abstract

Introduction.In this paper, we study a nonlinear elliptic equation involving Paneitz type operators on compact Riemannian manifolds. The nonlinearity considered here is concave-convex. The simultaneous effect of the concave and convex terms has been initially investigated by Ambrosetti et al. [1] in the Euclidian case for the Laplace operator. Since then, elliptic problems with this kind of nonlinearities were extensively studied by several authors with different classes of domains and with more general differential operators like the p-Laplacian. We can refer the reader to the valuable survey article by Ambrosetti et al. [2] and the references therein.The aim of this paper is to establish nonlocal and multiple existence results (with respect to a real parameter) to an elliptic equation involving the Paneitz-Branson operator with concave-convex nonlinear terms. Also, characteristic values of the real parameter are introduced (under variational form) and some of their specific properties are carried out. Now, let (M, g) be a smooth 4-dimensional Riemannian manifold and let S g , Rc g be the scalar curvature and the Ricci curvature of g, respectively. The Paneitz operator, introduced by Paneitz [23], defined on (M, g) is the fourth-order operator