In this article, we show the existence of a nonnegative solution to the singular problem (๎ผ ๐ ) posed in a bounded domain ฮฉ in R 2 (see below). We achieve this by approximating the singular function u โ๐ฝ log(u) by a function l ๐ (u), which pointwisely converges to โu ๐ฝ log(u) as ๐ โ 0. Using variational techniques, the perturbed equation) is shown to have a solution u ๐ โ H 1 0 (ฮฉ) when the parameter ๐ > 0 is small enough. Letting ๐ โ 0 and proving a pointwise gradient estimate, we show that the solution u ๐ converges to a nontrivial nonnegative solution of the original problem (๎ผ ๐ ).
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