In this paper, we study the following critical elliptic problem with a variable exponent:
$$\left\{ {\matrix{{ - \Delta u = {u^{p + \epsilon a\left( x \right)}}} \hfill & {{\rm{in}}\,\,\Omega ,} \hfill \cr {u > 0} \hfill & {{\rm{in}}\,\,\Omega ,} \hfill \cr {u = 0} \hfill & {{\rm{on}}\,\partial \Omega ,} \hfill \cr } } \right.$$
where
$$a\left( x \right) \in {C^2}\left( {\overline \Omega } \right),\,p = {{N + 2} \over {N - 2}},\,\,\epsilon > 0$$
, and Ω is a smooth bounded domain in ℝN (N ≽ 4). We show that for ∊ small eno...
