EXISTENCE AND NONEXISTENCE OF SIGN-CHANGING SOLUTIONS FOR SINGULAR SUPERLINEAR EQUATIONS ON EXTERIOR DOMAINS

In this paper we study radial solutions of Delta u + K(vertical bar x vertical bar)f(u) = 0 in the exterior of the ball of radius R > 0 in R-N where f grows superlinearly at infinity and is singular at 0 with f(u) similar to - 1/vertical bar u vertical bar(q-1)u and 0 < q < 1 for small u. We assume K(vertical bar x vertical bar) similar to vertical bar x vertical bar(-alpha) for large vertical bar x vertical bar and establish existence of an infinite number of sign-changing solutions when N + q(N - 2) < alpha < 2(N - 1). We also prove nonexistence for 0 < alpha < N + q(N - 2).