Dirichlet problem for a class of nonlinear degenerate elliptic operators with critical growth and logarithmic perturbation

n this paper, we investigate the existence of weak solutions for a class of degenerate ellipticDirichlet problems with critical nonlinearity and a logarithmic perturbation, i.e.{-(Delta(x)u+(alpha+1)2|x|(2 alpha)Delta(y)u)=uQ+2/Q-2+lambda u logu(2),u=0 on partial derivative Omega, (0.2) where (x, y) is an element of Omega subset of R-N=R(m)xR(n )with m >= 1, n >= 0, Omega boolean AND {x=0} not equal & empty;is a bounded domain, the parameter alpha >= 0 and Q = m+n(alpha+1) denotes the "homogeneous dimension" of R-N. When lambda=0, we know that from [23] the problem (0.2) has a Poho & zcaron;aev-type non-existence result. Then for lambda is an element of R\{0}, we establish the existences of non-negative ground state weak solutions and non-trivial weak solutions subject to certain conditions