Given ≥ 3, 1 < p < N , two measurable functions V (r) ≥ 0, K (r) > 0, and a continuous function A(r) > 0 (r > 0), we study the quasilinear elliptic equationWe find existence of nonegative solutions by the application of variational methods, for which we have to study the compactness of the embedding of a suitable function space X into the sum of Lebesgue spaces L q 1 K + L q 2 K , and thus into L q K (= L q K + L q K ) as a particular case. Our results do not require any compatibility between how the potentials A, V and K behave at the origin and at infinity, and essentially rely on power type estimates of the relative growth of V and K, not of the potentials separately. The nonlinearity f has a double-power behavior, whose standard example is f (t) = min{t q 1 −1 , t q 2 −1 }, recovering the usual case of a single-power behavior when q1 = q2.

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