EXISTENCE OF SOLUTIONS TO SUPERLINEAR P-LAPLACE EQUATIONS WITHOUT AMBROSETTI-RABINOWIZT CONDITION

We study the existence of non-trivial weak solutions in W-0(1,p)(Omega) of the super-linear Dirichlet problem -div(vertical bar del u vertical bar(p-2)del u) = f(x, u) in Omega, u = 0 on partial derivative Omega, where f satisfies the condition vertical bar f(x, t)vertical bar <= vertical bar omega(x)t vertical bar(r-1) + b(x) for all(x, t) is an element of Omega x R, where r is an element of(p,N-p/N-p), b is an element of Lr/r-1(Omega) and vertical bar omega vertical bar(r-1) mmay be non-integrable on Omega.