Existence and non-existence results for a quasilinear problem with nonlinear boundary condition

We study the problem - div(a(x)\del u \(p-2) del u) = lambda(1 + \ x \)(alpha 1)\ u \(q-2)u - h( x)\ u \(r-2)u in Omega subset of R-N, a(x)\del u \(p-2) del u.n + b(x).\ u \(p-2)u = theta g(x,u) on Gamma, u greater than or equal to 0 in Omega, where Omega is an unbounded domain with smooth boundary Gamma, n denotes th e unit outward normal vector on Gamma, and lambda > 0, theta a are real par ameters. We assume throughout that p < q < r < p* = pN/N-p, 1 < p < N, -N < alpha(1) < q . N-p/p -N, while a, b, and h are positive functions. We show that there exist an open interval I and lambda* > 0 such that the problem has no solution if theta epsilon I and lambda epsilon (0, lambda*). Further more, there exist an open interval J subset of I and lambda(0) > 0 such tha t, for any theta epsilon J, the above problem has at least a solution if la mbda greater than or equal to lambda(0), but it has no solution provided th at lambda epsilon (0, lambda(0)). Our paper extends previous results obtain ed by J. Chabrowski and K. Pfluger. (C) 2000 Academic Press.