Infinitely many solutions for some nonlinear supercritical problems with break of symmetry

In this paper, we prove the existence of infinitely many weak bounded solutions of a nonlinear elliptic problem with Dirichlet boundary conditions on an open bounded domain when there is a break of symmetry. To this aim, we use variational arguments and the Rabinowitz's perturbation method which is adapted to our setting and exploits a weak version of the Cerami-Palais-Smale condition. Furthermore, if the principal part grows fast enough, then the nonlinear term may have also a supercritical growth.