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The existence of positive solutions to Schrödinger-Poisson type systems in R 3 with critically growing nonlocal term is proved by using variational method which does not require usual compactness conditions. A key ingredient of the proof is a new Brézis-Lieb type convergence result.

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In this paper, we discuss the existence of positive periodic solutions to the nonlinear differential equationwhere a : R → [0, +∞) is an ω-periodic continuous function with a(t) ≡ 0, f : R × [0, +∞) → [0, +∞) is continuous and f (·, u) : R → [0, +∞) is also an ω-periodic function for each u ∈ [0, +∞). Using the fixed point index theory in a cone, we get an essential existence result because of its involving the first positive eigenvalue of the linear equation with regard to the above equation.