Entire Large Solutions to Elliptic Equations of Power Non-linearities with Variable Exponents

We give conditions on the variable exponent q that ensure the existence and nonexistence of a positive solution to the elliptic equation Delta u = u(q(x)) on R-N (N >= 3) which satisfies lim(vertical bar x vertical bar ->) (infinity) u(x) = infinity. The nonnegative function q is required to be locally Holder continuous on R-N. We prove existence for q > 1 provided q(x) decays to unity rapidly as vertical bar x vertical bar -> infinity. We treat the case q <= 1 as a special case of q - 1 changing signs and show that a solution exists provided q is asymptotically radial. In addition, we give an example to show that our results are nearly optimal.