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

We consider the following elliptic problem: ? div( |?u| p?2?u |y| ap ) = |u| q?2u |y| bq + f(x) in , u = 0 on ?, in an unbounded cylindrical domain := (y,z) ? Rm+1 ? RN?m?1;0< A < |y| < B < ? , where 1 ? m < N ? p, q = q(a, b) := Np N?p(a+1?b) , p > 1 and A, B ? 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 Maz?ja?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 ? 0, if p < q < p? N,m