Annulus Criteria For Mixed Nonlinear Elliptic Differential Equations

Abstract

New oscillation criteria are obtained for forced second order elliptic partial differential equations with damping and mixed nonlinearities of the form del . (A(x)vertical bar del y vertical bar(alpha-1)del y) + < b(x), vertical bar del y vertical bar(alpha-1)del y > + f(x, y) = e(x), x is an element of Omega where Omega is an exterior domain in R-N, f(x, y) = c(x)vertical bar y vertical bar(alpha-1)y + c(1)(x)vertical bar y vertical bar(beta-1)y + c(2)(x)vertical bar y vertical bar(gamma-1)y and beta > alpha > gamma > 0. It is assumed that A = (aij)(NxN) is a real symmetric positive definite matrix function, b = (b(i))(Nx1) is a real vector function, a(ij) is an element of C-loc(1+mu) (Omega, R), and b(i), c, c(1), c(2), e is an element of C-loc(mu)(Omega, R) for all i, j, for some mu is an element of (0, 1). Examples are given to illustrate the results. (C) 2011 Elsevier Ltd. All rights reserved.