The alternative theorems of nonlinear weakly continuous mappings and their application. (Chinese) Zbl 0682.35045

Consider the nonlinear weakly continuous operator equation \[ (1)\quad D_ i[a_{ij}(x,u)D_ ju+b_ i(x,u)]+f(x,u)=\lambda h(x,u);\quad x\in \Omega, \] and the boundary condition \(u|_{\partial \Omega}=0.\)
Alternative theorems for (1) are established. These theorems are applied to discuss a characteristic problem of a Dirichlet problem of the quasilinear elliptic equation of nonforced type \[ D_ i[a_{ij}(x,u)D_ ju+b_ i(x,u)]+D_ ic_ i(x,u)+f(x,u)=\lambda h(x,u),\quad x\in \Omega;\quad u|_{\partial \Omega}=0. \] It is proved that either the existence of a nontrivial solution of (1) when \(\lambda =0\) or the local solvability of (1) when \(\lambda\neq 0\) must be valid.
Reviewer: J.Tian


35J65 Nonlinear boundary value problems for linear elliptic equations