Resonance for quasilinear elliptic higher order partial differential equations at the first eigenvalue
type: Item
instance of: scholarly article
title: Resonance for quasilinear elliptic higher order partial differential equations at the first eigenvalue
zbMATH Open document ID: 0929.35038
author: Martha Contreras
published in: Rocky Mountain Journal of Mathematics
publication date: 2000-01-25T00:00:00Z
full work available at URL: cont282.html
review text: Let \(mp>N\), \(1<p<\infty\) and \(\Omega\subset\mathbb{R}^N\) be a bounded domain with the cone property. In \(\Omega\) the \(2m\)-th-order quasilinear differential equation \[ Qu(x)= g(x, u(x))+ h(x)\tag{\(*\)} \] is considered, where \[ \begin{aligned} Qu & = \sum_{1\leq|\alpha|\leq m}(-1)^{|\alpha|} D^\alpha A_\alpha(x, \xi_m(u)),\\ \xi_m & = \{D^\alpha u\mid 0\leq|\alpha|\leq m\},\;h(x)\in L^{p'}(\Omega),\;p'= {p\over p-1}\end{aligned} \] and \(g(x,t): \Omega\times\mathbb{R}\to \mathbb{R}\) meets a Carathéodory condition, grows superlinearily, that is, for all \(\varepsilon> 0\), there exists a \(g_\varepsilon\in L^{p'}(\Omega)\) such that \[ | g(x,t)|\leq \varepsilon| t|^{p- 1}+ g_\varepsilon(x),\quad g_\varepsilon(x)\geq 0 \] and the one-sided growth condition \[ tg(x,t)\geq -c(x)| t|- d(x),\quad c(x), d(x)\geq 0. \] In case \(A_\alpha\) satisfies a Carathéodory condition, a monotonicity condition for \(|\alpha|= m\) and there exists a positive constant \(c_0>0\) such that \[ \sum_{1\leq|\alpha|\leq m} A_\alpha(x, \xi_m)\xi_\alpha\geq c_0\Biggl\{\sum_{1\leq|\alpha|\leq m}|\xi_\alpha|^2 \Biggr\}^{p/2}, \] it is shown that \((*)\) has a weak solution, i.e. there exists a \(u\in W^{m,p}(\Omega)\) such that \[ \sum_{1\leq|\alpha|\leq m} \int_\Omega A_\alpha(x, \xi_m(u)) D^\alpha v dx= \int_\Omega g(x,u) v dx+ \int_\Omega hv dx \] for all \(v\in W^{m,p}(\Omega)\).
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35D05
zbMATH DE Number: 1309723
zbMATH Keywords: superlinear growth
zbMATH Keywords: monotonicity condition
zbMATH Keywords: Carathéodory condition
MaRDI profile type: Publication
OpenAlex ID: W2100648796
cites work: Q4073917
cites work: Q5518037
cites work: Q5596338
cites work: Q3994949
cites work: Q3785223
cites work: Q5593890
cites work: Q4405392
cites work: Quasilinear ellipticity and the ist eigenvalue
cites work: Resonance and the Second BVP
DOI: 10.1216/RMJM/1181071779
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description: scientific article; zbMATH DE number 1309723