Strongly singular problems with unbalanced growth and without summability of the weight function

Abstract In this paper we study existence, uniqueness, nonexistence and asymptotic behavior of solutions to the following class of elliptic problems with strongly singular nonlinearities $$\begin{aligned} \left\{ \begin{aligned} -&\Delta u -\frac{1}{2}\left( x\cdot \nabla u\right) =\lambda u + h(x)u^{-\alpha } \quad \text{ in }\quad \mathbb {R}^{N},\\ u&>0 \quad \text{ in }\quad \mathbb {R}^{N}\text{, } \ \ u(x)\rightarrow 0 \quad \text{ as }\quad |x|\rightarrow \infty , \end{aligned} \right. \end{aligned}$$ - Δ u - 1 2 x · ∇ u = λ u + h ( x ) u - α in R N , u > 0 in R N , u ( x ) → 0 as | x | → ∞ , where $$N\ge 3,$$ N ≥ 3 , $$\lambda \in \mathbb {R}$$ λ ∈ R and $$\alpha >1$$ α > 1 is a real parameter and $$h: \mathbb {R}^N\rightarrow \mathbb {R}$$ h