Eigenvalue problems with weights in Lorentz spaces

Abstract

Given V ,w locally integrable functions on a general domainwith V ≥ 0b ut wallowedtochangesign,westudytheexistenceofgroundstatesforthenonlineareigenvalue problem: −� u + Vu = λw|u| p−2 u, u|∂� = 0, with p subcritical. Theseareminimizers ofthe associatedRayleigh quotientwhoseexistence is ensured under suitable assumptions on the weight w. In the present paper we show that an admissible space of weight functions is provided by the closure of smooth functions with compact support in the Lorentz space L( ˜ p, ∞) with 1p + p 2� = 1. This generalizes previous results and gives new sufficient conditions ensuring existence of extremals for generalized Hardy-Sobolev inequalities. The existence in such a generality of a principal eigenfunction in the linear case p = 2 is applied to study the bifurcation for semilinear problems of the type