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 ) = λ m ( x ) φ ( | u | ) u . We work in the Orlicz–Sobolev space W 1 , G ( Ω ) , where G ( t ) is the convex function ∫ 0 t φ ( | s | ) s d s . We also derive a nodal set estimate.