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We characterize the principal eigenvalues and eigenfunctions in R N , and present comparison results, for higher dimensional p-Laplacian. Our main tool is Picone's identity. In this way we extend several recent results on spectral and comparison properties for differential equations. Academic Press

Abstract. In this paper we consider the bifurcation problemin R N with p > 1. We show that a continuum of positive solutions bifurcates out from the principal eigenvalue λ 1 of the problemHere both functions f and g may change sign.

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We are concerned with the existence of solutions ofwhere A p is the p-Laplacian, p € (1, oo), and Ci is a bounded smooth domain in K". For h(x) = 0 and f(x, u) satisfying proper asymptotic spectral conditions, existence of a unique positive solution is obtained by invoking the sub-supersolution technique and the spectral method. For h(x) ^ 0, with assumptions on asymptotic behavior of f{x, u) as u -»• ±oo, an existence result is also proved.