On a semilinear equation in ℝ 2 involving bounded measures

Source abstract

Synopsis We study the semilinear equation –Δ u + β( u ) = f in ℝ 2 , where β is a continuous increasing real function with β(0) = 0 and f is a bounded Radon measure. We show the existence of a solution, which is unique in the appropriate class, provided that each of the point masses contained in f does not exceed some critical value denned in terms of the growth of (β at ∞ This condition is shown to be necessary for the existence of solutions, even locally. The one-dimensional situation is also discussed.