The boundary value problems for quasilinear higher order elliptic equations with a small parameter (Q1057419)

This paper considers the singular perturbation of the general boundary value problem for quasilinear higher order elliptic equations of the form: \[ N_{\epsilon}u_{\epsilon}\equiv \epsilon^{2\ell_ 1}N_ 1u_{\epsilon}+N_ 0u_{\epsilon}=f(x)\quad (0<\epsilon \ll 1,\quad x\in \Omega \subset R^ n), \] \[ B_ ju_{\epsilon}|_{\partial \Omega}=h_ j(x)|_{\partial \Omega}\quad (j=0,1,...,\ell_ 1+\ell_ 0-1), \] where \(N_ 1\) and \(N_ 0\) are strongly elliptic differential operators of order \(2(\ell_ 1+\ell_ 0)\) and \(2\ell_ 0\) on \(\Omega\) respectively, \[ N_ 1u_{\epsilon}\equiv \sum_{| \beta | \leq 2(\ell_ 1+\ell_ 0)}A_{\beta}(x,u_{\epsilon})D^{\beta}u_{\epsilon}, \] \[ N_ 0u_{\epsilon}\equiv \sum_{| \beta | \leq 2\ell_ 0}a_{\beta}(x,u_{\epsilon})D^{\beta}u_{\epsilon} \] where \[ (- 1)^{\ell_ 1+\ell_ 2}\sum_{| \beta | =2(\ell_ 1+\ell_ 0)}A_{\beta}(x,w_ 0)\xi^{\beta}\leq \delta_ 1| \xi |^{2(\ell_ 1+\ell_ 0)}, \] \[ (-1)^{\ell_ 0}\sum_{| \beta | =2\ell_ 0}a_{\beta}(x,w_ 0)\xi^{\beta}\geq \delta_ 0| \xi |^{2\ell_ 0}. \] By using the method of modified multiple scales proposed by the author and \textit{R. Gao} [see Kexue Tongbao 24, No.23, 1057-1061 (1979; Zbl 0423.35016) and the author, Appl. Math. Mech., Engl. Ed. 2, No.5, 505-518 (1981)], the asymptotic expansion of the solution has been constructed. The remainder term has been estimated by using the contraction mapping principle.