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 the unit outward normal vector on Gamma, and lambda > 0, theta a are real parameters. 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*). Furthermore, there exist an open interval J subset of I and lambda(0) > 0 such that, for any theta epsilon J, the above problem has at least a solution if lambda greater than or equal to lambda(0), but it has no solution provided that lambda epsilon (0, lambda(0)). Our paper extends previous results obtained by J. Chabrowski and K. Pfluger. (C) 2000 Academic Press.