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Given two continuous functions $V\left(r \right)\geq 0$ and $K\left(r\right)>0$ ($r>0$), which may be singular or vanishing at zero as well as at infinity,we study the quasilinear elliptic equation \[ -\Delta w+ V\left( \left|x\right| \right) w - w \left( \Delta w^2 \right)= K(|x|) g(w) \quad \text{in}\mathbb{R}^{N}, \] where $N\geq3$. To study this problem we apply a change ofvariables $w=f(u)$, already used by several authors, and find existence resultsfor nonnegative solutions by the application of variational methods. The mainfeatures of our results are that they do not require any compatibility betweenhow the potentials $V$ and $K$ behave at the origin and at infinity, and thatthey essentially rely on power type estimates of the relative growth of $V$ and$K$, not of the potentials separately. Our solutions satisfy a weakformulations of the above equation, but we are able to prove that they are infact classical solutions in $\mathbb{R}^{N} \backslash \{ 0\}$. To applyvariational methods, we have to study the compactness of the embedding of asuitable function space into the sum of Lebesgue spaces$L_{K}^{q_{1}}+L_{K}^{q_{2}}$, and thus into $L_{K}^{q}$($=L_{K}^{q}+L_{K}^{q}$) as a particular case.  The nonlinearity $g$ has a double-power behavior, whose standard example is$g(t) = \min \{ t^{q_1 -1}, t^{q_2 -1} \}$, recovering the usual case of asingle-power behavior when $q_1 = q_2$.

Existence results for a class of quasilinear Schr\"odinger equations with singular or vanishing potentials