On resonant Neumann problems: multiplicity of solutions
type: Item
instance of: scholarly article
title: On resonant Neumann problems: multiplicity of solutions
zbMATH Open document ID: 1234.35081
author: Nikolaos S. Papageorgiou
author: Ana Isabel Santos Coelho Rodrigues
author: Vasile Staicu
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2011-10-17T00:00:00Z
review text: The authors consider the following semilinear Neumann problem \[ \begin{cases} -\Delta u = f(z,u(z)) & \text{in } \Omega ,\\ \frac{\partial u }{\partial n} =0 & \text{in }\Omega, \end{cases}\tag{1} \] where \(\Omega \subset {\mathbb R}^N\) is a bounded domain with \(C^2\) boundary \(\partial \Omega \). Let \(\{ \lambda _m\}_{m=1}^{\infty }\) be the distinct eigenvalues of \((-\Delta ,H^1(\Omega ))\). Assumptions on the function \(f(z,x)\) \((z \in \Omega ,x \in {\mathbb R}^N)\) are as follows: {\parindent=8mm \begin{itemize}\item[(i)] for all \(x \in {\mathbb R}\), \(z \to f(z,x)\) is measurable; \item[(ii)] for almost all \(z \in \Omega \), \(x \to f(z,x)\) is \(C^1\) and \(f(z,0)=0\); \item[(iii)] for almost all \(z \in \Omega \), and all \(x \in {\mathbb R}\), \(| f_x'(z,x)| \leq a(z)+ c| x | ^{r-2}\) with \(a \in L^{\infty }(\Omega )\), \(c>0\) and \(2<r<2^*\) where \(2^*= 2N/(N-2)\) if \(N\geq 3\) and \(+ \infty \) if \(N\in \{1,2\}\); \item[(iv)] there exists an integer \(m\geq 1\) such that \[ \lim _{| x | \to \infty } \frac{f(z,x)}{x} = \lambda _m \quad \text{uniformly for a.e. } z\in \Omega ; \] \item[(v)] there exist functions \(\xi ,\widehat{\xi }\in L^{\infty }(\Omega )_+\) such that \(\xi (z)\leq \lambda _m\) a.e. on \(\Omega \), the inequality is strict on a set of positive measure and \[ -\widehat{\xi }(z)(x-x')^2 \leq (f(z,x)- f(z,x'))(x-x')\leq \xi (z) (x-x')^2 \] for a.e. \( z\in \Omega \), and \( x,x' \in {\mathbb R}\); \item[(vi)] if \(F(z,x)= \int _0^x f(z,s)ds\), then \(\lim _{| x | \to \infty }(f(z,x)x-2F(z,x)) = -\infty \) for a.e. \(z \in \Omega \); \item[(vii)] there exists a function \(\eta \in L^{\infty }(\Omega )\) such that \(\eta (z) \leq 0\) a.e. on \(\Omega \), the inequality is strict on a set of positive measure and \[ \lim _{x \to 0} \frac{2F(z,x)}{x^2} \leq \eta (z) \] uniformly for a.e. \(z \in \Omega \). \end{itemize}} Here (iv) and (vi) imply that (1) is resonant at infinity with respect to \(\lambda _m \, (m\geq 1)\) from the…
reviewed by: Junichi Aramaki
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35J61
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 5959706
zbMATH Keywords: asymptotically linear problem
zbMATH Keywords: resonant problems
zbMATH Keywords: reduction technique
zbMATH Keywords: critical groups
zbMATH Keywords: mountain pass theorem
zbMATH Keywords: Morse theory
MaRDI profile type: Publication
full work available at URL: j.na.2011.06.031
OpenAlex ID: W2024989170
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DOI: 10.1016/J.NA.2011.06.031
description: scientific article; zbMATH DE number 5959706