Resonance for quasilinear elliptic higher order partial differential equations at the first eigenvalue
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 \(v\in W^{m,p}(\Omega)\).
scientific article; zbMATH DE number 1309723
scientific article