Nonlinear Elliptic Equations with Perturbed Symmetry

. In this paper, we are interested in studying multiplicity of solutions for nonlinear elliptic equations with perturbed symmetry. Existence of infinitely many solutions of superlinear problems with perturbed symmetry was considered by several mathematicians in the 1980s under some restrictive growth conditions on both the unperturbed nonlinear term and the perturbing term which is fixed. While we have a small parameter epsilon to drive the perturbing term, we only need very weak growth conditions on the unperturbed nonlinear term and the perturbing term. We prove that the equations have as many solutions as prescribed when epsilon is suitably small. We give a further extension of the classical result of Berestycki and Lions [ARMA, 82 (1983), 347-375], and we also improve the famous symmetric mountain pass theorem due to Ambrosetti and Rabinowitz [JFA, 14 (1973), 349-381]. A key ingredient of our proof is to find an infinite number of nondegenerate critical values of the unperturbed energy functionals.