On the Bahri–Lions conjecture for elliptic equations with non-symmetric nonlinearities

We prove the existence of infinitely many solutions for a class of elliptic Dirichlet problems with non-symmetric nonlinearities. In particular, in suitable domains of R-n with n = 3, this result gives a positive answer to a well known conjecture formulated by A. Bahri and P.L. Lions. The proof is based on a minimization method which does not require the use of techniques of deformation from the symmetry. This method allows us to piece together solutions of Dirichlet problems in suitable subdomains, so we obtain infinitely many nodal solutions with a prescribed nodal structure.