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We deal with existence and multiplicity for the following class of nonhomogeneous Schrödinger–Poisson systems { −∆u + V(x)u + K(x)φ(x)u = f (x, u) + g(x) in R3, −∆φ = K(x)u2 in R3, where V, K : R3 → R+ are suitable potentials and f : R3 ×R → R satisfies sublinear growth assumptions involving a finite number of positive weights Wi, i = 1, . . . , r with r ≥ 1. By exploiting compact embeddings of the functional space on which we work in every weighted space Li Wi (R 3), wi ∈ (1, 2), we establish existence by means of a generalized Weierstrass theorem. Moreover, we prove multiplicity of solutions if f is odd in u and g(x) ≡ 0 thanks to a variant of the symmetric mountain pass theorem stated by R. Kajikiya for subquadratic functionals.

On existence and multiplicity for Schrödinger–Poisson systems involving weighted sublinear nonlinearities