NONLINEAR ELLIPTIC-EQUATIONS WITH SUPERCR ITICAL SOBOLEV GROWTH

Let S-n be the unit sphere of R(n+1), Omega a smooth bounded domain of R(n) and a (x), f (x) two functions of C-infinity (Omega) or C-infini ty (S-n). We study the existence and the multiplicity of positive solu tions and of nodal solutions for the equations Delta u + a (x)u = f(x) \u\(p-2)u on S-n or on Omega with u = 0 on partial derivative Omega. S ymmetry assumptions allow us to take supercritical values p greater th an or equal to 2n/(n - 2) of the exponent.