A singular eigenvalue problem for the Dirichlet (p, q)-Laplacian
Yunru Bai, Nikolaos S. Papageorgiou, Shengda Zeng
Abstract
Abstract We consider a parametric nonlinear, nonhomogeneous Dirichlet problem driven by the ( p , q )-Laplacian with a reaction involving a singular term plus a superlinear reaction which does not satisfy the Ambrosetti–Rabinowitz condition. The main goal of the paper is to look for positive solutions and our approach is based on the use of variational tools combined with suitable truncations and comparison techniques. We prove a bifurcation-type theorem describing in a precise way the dependence of the set of positive solutions on the parameter $$\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>λ</mml:mi></mml:math> . Moreover, we produce minimal positive solutions and determine the monotonicity and continuity properties of the minimal positive solution map.