Hidden symmetry of fully nonlinear boundary conditions in elliptic equations: Global bifurcation and nodal structure
The nonlinear elliptic boundary value problem under consideration has the form \[ -\nabla\cdot{\mathbf g}(\nabla u,u)=f(\nabla u,u,\lambda)\quad\text{in }\Omega,\quad{\mathbf g}(\nabla u,u)\cdot{\mathbf n}|_{\partial\Omega}=0 \] in a rectangle \(\Omega\), where \(f,{\mathbf g}\) are prescribed functions, \({\mathbf n}\) denotes the unit external normal to \(\partial\Omega\), and \(\lambda\) corresponds to a control parameter in a steady-state diffusion process. The ``hidden symmetry'' of the boundary condition permits application of the technique used previously for analogous problems with linear boundary conditions, leading to generalizations of results of \textit{M. G. Crandall} and \textit{P. H. Rabinowitz} [J. Math. Mech. 19, 1083-1102 (1970; Zbl 0206.097)]. In particular, an existence proof is given for bifurcating solution branches along which nodal configurations of linearized eigenfunctions are preserved.
scientific article; zbMATH DE number 62156