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This paper consists of two main parts. The first deals with a perturbative method in critical point theory and can be seen as the generalisation and completion of some earlier results. The second part is concerned with applications of the abstract setup to the existence of bound states of a class of elliptic differential equations that branch off from the infimum of the essential spectrum.

Abstract.We study the sum of weighted Lebesgue spaces, by considering an abstract measure space (Ω, A, µ) and investigating the main properties of both the Banach spaceand the Nemytskiȋ 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 of the formwhere V is a nonnegative measurable potential, possibly singular and vanishing at infinity, and f is a Carathéodory function satisfying a double-power growth condition in u. Mathematics Subject Classification (2010). Primary 35J62; Secondary 46E30.