In this paper we study implicit obstacle problems driven by a nonhomogenous differential operator, called double phase operator, and a multivalued term which is described by Clarke’s generalized gradient. Based on a surjectivity theorem for multivalued mappings, Kluge’s fixed point principle and tools from nonsmooth analysis, we prove the existence of at least one solution.

In the present paper, we introduce a family of the approximating problems corresponding to an elliptic obstacle problem with a double phase phenomena and a multivalued reaction convection term. Denoting by 𝓢 the solution set of the obstacle problem and by 𝓢 _n the solution sets of approximating problems, we prove the following convergence relation ∅ ≠ w - lim sup n → ∞ S n = s - lim sup n → ∞ S n ⊂ S , $$\begin{array}{} \displaystyle \emptyset\neq w\text{-}\limsup\limits_{n\to\infty}{\mathcal S}_n=s\text{-}\limsup\limits_{n\to\infty}{\mathcal S}_n\subset \mathcal S, \end{array}$$ where w -lim sup _{n →∞} 𝓢 _n and s -lim sup _{n →∞} 𝓢 _n denote the weak and the strong Kuratowski upper limit of 𝓢 _n , respectively.

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 $$\lambda $$λ . Moreover, we produce minimal positive solutions and determine the monotonicity and continuity properties of the minimal positive solution map.

In this paper we study from a qualitative point of view the nonlinear singular Dirichlet problem depending on a parameterλ> 0 that was considered in [32]. Denoting byS_λthe set of positive solutions of the problem corresponding to the parameterλ, we establish the following essential properties ofS_λ: there exists a smallest element $\begin{array}{} u_\lambda^* \end{array}$ inS_λ, and the mappingλ↦ $\begin{array}{} u_\lambda^* \end{array}$ is (strictly) increasing and left continuous; the set-valued mappingλ↦S_λis sequentially continuous.