Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation

Abstract

We investigate the following quasilinear and singular problem, { − pu = λ uδ + uq in ; u|∂ = 0, u > 0 in , (P) where is an open bounded domain with smooth boundary, 1 0, and 0 N . We employ variational methods in order to show the existence of at least two distinct (positive) solutions of problem (P) in W 1,p 0 ( ). While following an approach due to Ambrosetti-Brezis-Cerami, we need to prove two new results of separate interest: a strong comparison principle and a regularity result for solutions to problem (P) in C1,β ( ) with some β ∈ (0, 1). Furthermore, we show that δ < 1 is a reasonable sufficient (and likely optimal) condition to obtain solutions of problem (P) in C1( ). Mathematics Subject Classification (2000): 35J65 (primary); 35J20, 35J70 (secondary).