ftp ejde.math.txstate.edu (login: ftp) VARIATIONAL METHODS FOR A RESONANT PROBLEM WITH THE p-LAPLACIAN IN RN

Abstract. The solvability of the resonant Cauchy problem −∆pu = λ1m(|x|)|u|p−2u+ f(x) in RN; u ∈ D1,p(RN), in the entire Euclidean space RN (N ≥ 1) is investigated as a part of the Fredholm alternative at the first (smallest) eigenvalue λ1 of the positive p-Laplacian −∆p on D1,p(RN) relative to the weight m(|x|). Here, ∆p stands for the p-Laplacian, m: R+ → R+ is a weight function assumed to be radially symmetric, m 6 ≡ 0 in R+, and f: RN → R is a given function satisfying a suit-able integrability condition. The weightm(r) is assumed to be bounded and to decay fast enough as r → +∞. Let ϕ1 denote the (positive) eigenfunction as-sociated with the (simple) eigenvalue λ1 of −∆p. If RN fϕ1 dx = 0, we show that problem has at least one solution u in the completionD1,p(RN) of C1c (RN) endowed with the norm ( RN |∇u|p dx)1/p. To establish this existence result, we employ a saddle point method if 1 < p < 2, and an improved Poincaré inequality if 2 ≤ p < N. We use weighted Lebesgue and Sobolev spaces with weights depending on ϕ1. The asymptotic behavior of ϕ1(x) = ϕ1(|x|) as |x | → ∞ plays a crucial role. 1