Sum of wheighted Lebesgue spaces and nonlinear elliptic equations

We study the sum of weighted Lebesgue spaces, by considering an abstract measure space and the Nemytskii operator defined on it. Then we apply our general results to prove existence and multiplicity of solutions to a class of nonlinear p-Laplacian equations in R^n; with a nonnegative measurable potential, possibly singular and vanishing at infinity, and Carathéodory functions satisfying a double-power growth condition in u