Preservation of nodal structure on global bifurcation solution branches of elliptic equations with symmetry (Q1313490)

The paper concerns parameter depending \((\lambda \in \mathbb{R})\) quasilinear elliptic equations of the type \[ G(\lambda,u):=\sum^ n_{i,j=1} a_{ij} (\nabla u,u) \partial_ i \partial_ j u+g(\lambda,\nabla u,u)=0 \text{ in } \mathbb{R}^ n, \tag{1} \] where \(u:\mathbb{R}^ n \to \mathbb{R}\) enjoys the multiple periodicity of some lattice \({\mathcal L} \subset \mathbb{R}^ n\). Assume that \({\mathcal L}\) possesses a nontrivial holohedral point group \(H\) and that \(G(\lambda,\gamma u)=\gamma G(\lambda,u)\) and \(G(\lambda,-u)=- G(\lambda,u)\) for all \(\lambda,u\) and \(\gamma \in H\) (where \((\gamma u) (x):=u(\gamma^ Tx))\). Moreover, suppose that for some isolated parameter value \(\lambda=\lambda_ 0\) the linearized problem (about the trivial solution \(u=0)\) admits an eigenfunction \(h\) whose nodal set \(N:=\{x \in \mathbb{R}^ n:u(x)=0\}\) is complementary to a so-called monohedral tiling of \(\mathbb{R}^ n\) and that \(N\) is the minimal nodal set for all functions belonging to the fixed-point subspace of the isotropy subgroup of \(h\). Then, if the restriction of (1) to this fixed-point subspace satisfies a crossing condition along the trivial solution at \(\lambda=\lambda_ 0\), there exists a global bifurcation from the solution \(\lambda=\lambda_ 0\), \(u=0\) of (1), and the nodal sets of all members of the global solution branch equal to \(N\). Finally, the authors provide several explicit examples, mostly associated with lattices in \(\mathbb{R}^ 2\).