On some quasilinear equations involving the p-Laplacian with Robin boundary conditions

Let O???R N be a smooth-bounded domain, let f and g be continuous functions on , g???0. Let a?>?0 be some real number and let p???(1, N?) and q???(p, p*] be two real numbers, where p*?=?np/(n?-?p). We are interested in proving the results of existence and non-existence for the non-negative solution of the equationWe consider the case ??<??1 (coercive case) and the case ?????1, where ?1 is the principal eigenvalue for the operator ? p ?+?g with Robin boundary conditions from which we recall the definition and some properties.