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We consider a nonlinear Dirichlet problem driven by the (p, q)-Laplacian and with a reaction which is dependent on the gradient. We look for positive solutions and we do not assume that the reaction is nonnegative. Using a mixture of variational and topological methods (the "frozen variable" technique), we prove the existence of a positive smooth 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.

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 bySλthe set of positive solutions of the problem corresponding to the parameterλ, we establish the following essential properties ofSλ:there exists a smallest element$\begin{array}{} u_\lambda^* \end{array}$inSλ, and the mappingλ↦$\begin{array}{} u_\lambda^* \end{array}$is (strictly) increasing and left continuous;the set-valued mappingλ↦Sλis sequentially continuous.