Abstract
Abstract We give conditions on the variable exponent q that ensure the existence and nonexistence of a positive solution to the elliptic equation Δu = uq(x) on ℝN (N ≥ 3) which satisfies lim|x|→∞ u(x) = ∞. The nonnegative function q is required to be locally Hölder continuous on ℝN. We prove existence for q > 1 provided q(x) decays to unity rapidly as |x| → ∞. 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.
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