Global bifurcation of the \(p\)-Laplacian and related operators (Q2484521)

The author studies the problem \[ \Delta_p u(x) + \mu_0 | u(x)| ^{p-2}u(x) = q(\lambda,x,u(x),\nabla u(x)) \] with zero Dirichlet boundary condition in a bounded smooth domain in \(\mathbb R^N\), where \(\Delta_p\) is the \(p\)-Laplacian, \(q\) is a function satisfying some growth and continuity assumptions. The existence of an unbounded connected set of weak solutions \((u,\lambda)\) containing a couple of the type \((u,0)\) is proved under the assumption that \(\mu_0\) is not an eigenvalue of \(\Delta_p\). In the case \(q(0,\cdot,\cdot)=0\), the branch mentioned emanates from \((0,0)\). An abstract result guaranteeing the existence of global branches of solutions of equations with very general operators in normal spaces is proved, the result for the boundary value problem metioned above being its simple consequence. In fact, this very general theorem can be considered as the main result of the paper.