Positive solutions for some generalized $ p $–Laplacian type problems

In this paper, we prove the existence of nontrivial weak bounded solutions of the nonlinear elliptic problem{-div(a(x, u, del u)) + A(t)(x, u, del u) = f(x, u) in Omega, u >= 0 in Omega, u = 0 on partial derivative Omega,where Omega subset of R-N is an open bounded domain, N >= 3, and A(x, t, xi), f(x, t) are given functions, with A(t) = partial derivative A/partial derivative t, a = del xi A,To this aim, we use variational arguments which are adapted to our setting and exploit a weak version of the Cerami-Palais-Smale condition.Furthermore, if A(x, t, xi) grows fast enough with respect to t, then the nonlinear term related to f(x, t) may have also a supercritical growth.