Annals of the University of Bucharest (mathematical series) 3 (LXI) (2012), 227–232 Multiple solutions for Lane–Emden equations with mixed nonlinearities Vicent ¸iu D. R˘ adulescu This paper is dedicated with esteem to Professor Jean Mawhin on his 70th birthday anniversary Abstract - We are concerned with a linear perturbation of the Lane-Emden equation with different growths near the origin and at infinity. By means of a version of the Pucci–Serrin three critical points theorem, we establish the existence of at least two nontrivial solutions in the case of large values of the parameter. Key words and phrases : Lane–Emden equation, eigenvalue problem, multiple solutions. Mathematics Subject Classification (2010) : 35J60, 58J05. 1. Introduction The Lane–Emden equation describes naturally many physical phenomena. For example, super-diffusivity equations of this type have been proposed by de Gennes (see [8]) as a model for long-range Van der Waals interac- tions in thin films spreading on solid surfaces. This equation also appears in the study of cellular automata and interacting particle systems with self- organized criticality (see [7]), as well as to describe the flow over an imper- meable plate (see [6]). Our main purpose in the present paper is to connect a general class of Lane–Emden equations with the eigenvalue problem for the Laplace operator in order to establish a striking multiplicity result for large values of a certain real parameter. The proof of this existence prop- erty for the perturbed equation relies on simple variational tools, namely on a version of the celebrated Pucci–Serrin three critical points theorem (see [14]). 227