Perturbations near resonance for thep-Laplacian inℝN

Abstract

We study a multiplicity result for the perturbedp-Laplacian equation−Δpu−λg(x)|u|p−2u=f(x,u)+h(x) in ℝN, where1<p<Nandλis nearλ 1, the principal eigenvalue of the weighted eigenvalue problem−Δpu=λg(x)|u|p−2uinℝN. Depending on which sideλis fromλ 1, we prove the existence of one or three solutions. This kind of result was firstly obtained by Mawhin and Schmitt (1990) for a semilinear two-point boundary value problem.