In this paper, we study parametric quasilinear elliptic equations driven by the double phase operator, where the right-hand side consists of a singular term and a sublinear term. By combining a new Hopf's Lemma with truncation techniques and an abstract critical point theorem, we establish the existence of three bounded positive solutions and provide an explicit upper bound for the parameter.

In this paper we consider a class of anisotropic $(\overrightarrow{p},\overrightarrow{q})$-Laplacian problems with nonlinear right-hand sides that are superlinear at $\pm \mathrm{\infty }$. We prove the existence of two nontrivial weak solutions to this kind of problem by applying an abstract critical point theorem under very general assumptions on the data without supposing the Ambrosetti-Rabinowitz condition.