×

Existence of solutions for asymptotically ‘linear’ \(p\)-Laplacian equations. (English) Zbl 1088.35025

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.

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
PDFBibTeX XMLCite
Full Text: DOI