Existence and multiplicity of solutions for a supercritical elliptic problem in unbounded cylinders

u = 0 on partial derivative Omega, in an unbounded cylindrical domain Omega := {(y,z) epsilon Rm+1 x R-N,R-m-1; o < A < |y| < B < infinity}, where 1 <= m < N - p, q = q(a, b) := Np/N-p(a+1-b), p>1 and A,B epsilon R+. Let p*(N,m) :=p(N-m)/N-m-p. We show that p*(N,m) is the true critical exponent for this problem. The starting point for a variational approach to this problem is the known Mazja's inequality (Sobolev Spaces, 1980) which guarantees, for the q previously defined, that the energy functional associated with this problem is well defined. This inequality generalizes the inequalities of Sobolev (p = 2, a = 0 and b = 0) and Hardy ( p = 2, a = 0 and b = 1). Under certain conditions on the parameters a and b, using the principle of symmetric criticality and variational methods, we prove that the problem has at least one solution in the case f = 0 and at least two solutions in the case f not equivalent to 0, if p < q < p*(N,m).