Perturbations near resonance for the \(p\)-Laplacian in \({\mathbb R}^{N}\) (Q1608808)

Summary: We study a multiplicity result for the perturbed \(p\)-Laplacian equation \(-\Delta_p u-\lambda g(x) | u|^{p-2} u = f(x,u)+h(x)\) in \({\mathbb R}^{N}\), where \(1<p<N\) and \(\lambda\) is near \(\lambda{1}\), the principal eigenvalue of the weighted eigenvalue problem \[ -\Delta_p u = \lambda g(x) | u|^{p-2}u \] in \({\mathbb R}^{N}\). Depending on which side \(\lambda\) is from \(\lambda{1}\), 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.