Liu, Shibo; Li, Shujie Existence of solutions for asymptotically ‘linear’ \(p\)-Laplacian equations. (English) Zbl 1088.35025 Bull. Lond. Math. Soc. 36, No. 1, 81-87 (2004). The authors consider the Dirichlet problem for the \(p\)-Laplacian equation \(-\Delta_ p u = f(x,u)\) in \(\Omega\), \(u=0\) on \(\partial \Omega\) where \(\Omega\) is a bounded domain on \(\mathbb R^ N\) with smooth boundary \(\partial \Omega\), \(-\Delta u = - \text{div} (| \nabla u|^{p-2} \nabla u )\), \(1<p<N\). Denote by \(\sigma (\Delta_ p)\) the spectrum of \(-\Delta_ p\) on \(W^{1,p}_ 0 (\Omega)\) and assume the following conditions: (\(f_ 1\)) \(f\colon \Omega \times \mathbb R \to \mathbb R\) is continuous, and \(\lim _{| u|\to \infty}f(x,u)/(| u|^{p-2}u )=\lambda\). (\(f_ 2\)) There exists \(r>0\) and \(\nu \in (1,p)\), such that for \(x\in \Omega\) and \(0<| u |\leq r\), the following statements hold: \[ f(x,u)u >0, \quad\text{and}\quad f(x,u)u/\nu \leq F(x,u) := \int_ 0 ^ u f(x,s) ds. \] (\(f_ 3\)) There exist \(a>0\) and \(\tau \in (1,p)\) such that \[ \lim_{| u|\to 0} (f(x,u) -a | u |^{\tau-2}u)/ (| u|^{p-2}u)=0 \]The author prove, using Morse theory: (1) If condition (\(f_ 1\)) holds and \(\lambda \not\in \sigma (-\Delta_ p)\), then the Dirichlet problem has a solution. (2) If condition (\(f_ 1\)) and (\(f_ 2\)) hold and \(\lambda\not\in \sigma (-\Delta_ p)\), then the Dirichlet problem has a nonzero solution. (3) If condition (\(f_ 1\)) and (\(f_ 3\)) hold and \(\lambda\not\in \sigma (-\Delta_ p)\), then the Dirichlet problem has a nonzero solution. Reviewer: Youssef Jabri (Oujda) Cited in 17 Documents MSC: 35J60 Nonlinear elliptic equations 35J20 Variational methods for second-order elliptic equations 47J30 Variational methods involving nonlinear operators 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces Keywords:Dirichlet problem fot the p-Laplacian; Morse theory; Tang index; existence PDFBibTeX XMLCite \textit{S. Liu} and \textit{S. Li}, Bull. Lond. Math. Soc. 36, No. 1, 81--87 (2004; Zbl 1088.35025) Full Text: DOI