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We consider nonlinear Dirichlet problems driven by the p-Laplacian, which are resonant at +∞ with respect to the principal eigenvalue. Using a variational approach based on the critical point theory, we show that the problem has three nontrivial smooth solutions, two of which have constant sign (one positive, the other negative). In the semilinear case, assuming stronger regularity on the nonlinear perturbation f (z, · ) and using Morse theory, we show that the problem has at least fourdoi:10.2140/pjm.2009.241.309 fatcat:kn4hvjxj3jcrbndct6lv7kp4bi