MULTIPLE SOLUTIONS FOR NONHOMOGENEOUS ELLIPTIC-EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENT

We consider the following problem: [GRAPHICS] where p* = (Np)/(N-p), D-1,D-p(R(N)) = (u:u is an element of L(p)* (R(N)), del u is an element of (L(p)(R(N)))(N)}, h(x) is an element of (D-1,D-p(R(N)))* is continuous on R(N) and h(x) not equal 0. By using Ekeland's variational principle and the Mountain Pass Theorem without (PS) conditions, through a careful inspection of the energy balance for the approximated solutions, we show that the probelm (*) has at least two solutions for some lambda* > 0 and lambda is an element of (0, lambda*). In particular, if p = 2, in a different way we prove that problem (*) with I = 1 and h(x) greater than or equal to 0 has at least two positive solutions as \\h\\* < CNSN/4, where C-N = 4/N-2(N-2/N+2)((N+2)/4).