A short review on (p,q)-equations with Carathéodory perturbation

We review some recent works dealing with \((p, q)\)-Laplacian equations in the setting of Sobolev spaces and Dirichlet boundary condition. We aim to underline the key role of growth conditions on the Carathéodory perturbation, in establishing both the existence and multiplicity of positive weak solutions. We focus on \((p − 1)\)-superlinear perturbations which do not satisfy the Ambrosetti-Rabinowitz condition, and a special attention is paid to those problems involving a singular term in the reaction. We refer both to variational tools and topological tools, and point out the dependence of the multiplicity result on a real parameter, when possible.