The eigenvalue problem for quasilinear elliptic operators with general growth

This note is concerned with the existence and isolation of the first eigenvalue of the weighted equation −div(φ(|∇u|)∇u)=λm(x)φ(|u|)u. We work in the Orlicz–Sobolev space W1,G(Ω)W1,G(Ω), where G(t)G(t) is the convex function ∫0tφ(|s|)sds. We also derive a nodal set estimate.