Branches of positive solutions of quasilinear elliptic equations on non-smooth domains

Abstract

We prove existence of an unbounded global branch (i.e. connected set) of weak solutions of a second order quasilinear equation depending on a real parameter λ on an arbitrary (possibly non-smooth) bounded domain in R N , with a Leray–Lions operator as the leading part. Here, we can allow lower order nonlinearities which depend on first derivatives, satisfying appropriate growth conditions including the critical case. Furthermore, we give sufficient conditions for the existence of a branch consisting entirely of nonnegative solutions for positive λ . Our approach also yields a new existence result in the case of critical growth in derivatives of lower order.