Oscillation Of P(X)-Laplacian Elliptic Inequalities With Mixed Variable Exponents

Abstract

Oscillation criteria are established for p(x)-Laplacian elliptic inequalities with mixed variable nonlinearities of the form u[del.(A(x)vertical bar del u vertical bar(p(x)-2) del u) + < b(x), vertical bar del u vertical bar(p(x)-2)del u > - h(x, u) + g(x, u)] <= 0, x is an element of Omega, where beta(x) > p(x) > gamma(x) > 1, Omega is an exterior domain in R-N , and h(x, u) = ln vertical bar u vertical bar vertical bar del u vertical bar(p(x)-2) (A(x) del u). del p(x), g(x, u) = c(x) vertical bar u vertical bar(p(x)-2) u + c(1)(x) vertical bar u vertical bar(beta(x)-2) u + c(2)(x) vertical bar u vertical bar(gamma(x)-2)u + f(x). The function h(x, u) recently introduced in [N. Yoshida, Nonlinear Anal. 74 (2011) 2563-2575] allows employing the Riccati transformation technique commonly used in the oscillation theory of ordinary differential equations. It should be noted that the results obtained are new for one dimensional case as well. Examples are given to illustrate the results.