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 the nonlinear elliptic problem {-div(a(x, u, del u)) + A(t) (x, u, del u) = g(x, u) + h(x) in Omega, u = 0 on partial derivative Omega, where Omega subset of R-N is an open bounded domain, N >= 3, and A(x, t, xi), g(x, t), h(x) are given functions, with A(t) = partial derivative A/partial derivative t , a = del(xi)A, such that A(x, ., .) is even and g(x, .) is odd. 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 A(x, t, xi) grows fast enough with respect to t, then the nonlinear term related to g(x, t) may have also a supercritical growth.