\lambda_1$, even in space dimension one (N=1). We will restrict ourselves to the case $\lambda = \lambda_1$, the Fredholm alternative for the p-Laplacian at the first eigenvalue. Even if the functional $\mathcal{J}_{\lambda_1}$ is no longer coercive on $W_0^{1,p}(\Omega)$, for $p>2$ we will show that it is bounded from below and does possess a global minimizer. For $1

0$ small enough). A crucial ingredient in our proofs are rather precise asymptotic estimates for possible "large" solutions to problem (P) obtained from the linearization of problem (P) about the eigenfunction $\varphi_1$. These will be briefly discussed. Naturally, the (linear selfadjoint) Fredholm alternative for the linearization of problem (P) about $\varphi_1$ (with $\lambda = \lambda_1$) appears in the proofs. Published July 10, 2010. Math Subject Classifications: 35J20, 49J35, 35P30, 49R50. Key Words: Nonlinear eigenvalue problem; Fredholm alternative; degenerate or singular quasilinear Dirichlet problem; p-Laplacian; global minimizer; minimax principle.