Advanced Nonlinear Studies 8 (2008), 179–190 A Note on The Schr¨ odinger-Poisson-Slater Equation on Bounded Domains David Ruiz * SISSA, Via Beirut 2-4, Trieste 34014, Italy Permanent address: Dpto. An´ alisis Matem´atico, Univ. Granada 18071 Granada, Spain e-mail: daruiz@ugr.es Gaetano Siciliano † Dpto di Matematica, Univ. degli Studi di Bari Via E. Orabona 4, Bari 70125, Italy e-mail: siciliano@dm.uniba.it Received 6 June 2007 Communicated by Antonio Ambrosetti Abstract In this note we consider the problem  -Δu + λφu = |u| p-1 u, -Δφ = u 2 , where u, φ ∈ H 1 0 (Ω) and Ω ⊂ R 3 is a smooth bounded domain. We are interested in existence and multiplicity of solutions depending on the parameters p and λ. Our results extend previous work made in R 3 . 2000 Mathematics Subject Classification. 35J10, 35J20, 35J60,35Q55.. Key words. Schr¨ odinger-Poisson equation, variational methods, monotonicity trick. * David Ruiz has been supported by M.U.R.S.T within the PRIN 2004 “Variational methods and nonlinear differential equations”, by the Spanish Ministry of Science and Technology under Grant MTM2005-01331 and by J. Andaluc´ ıa (FQM 116). † Gaetano Siciliano has been supported by M.I.U.R. Cofin. ”Metodi variazionali e topologici ed equazioni differenziali nonlineari”. 179