On Some Nonhomogeneous Elliptic Problems in Unbounded Domains

In this paper we consider the problem {-Delta u + u = vertical bar u vertical bar(p-2) u + f(x) in Omega u = u(0) on partial derivative Omega, where Omega = R(N) \ (omega) over bar, omega being a nonempty open bounded domain of R(N) having smooth boundary partial derivative omega = partial derivative Omega, N >= 3, 2 < p < 2N/N-2, u(0) is an element of H(1/2)(partial derivative Omega), f is an element of L(2)(Omega). We prove that, if u(0) and f are nonnegative and satisfy suitable conditions, there exist at least two positive solutions. Moreover, the existence of a solution is shown to hold even when no condition on the sign of u(0) and f is assumed.