Existence and non-existence results for nonlinear elliptic equations with nonstandard growth

We consider the existence and non-existence of solutions for nonlinear elliptic equations whose model is { − div ( | ∇ v | p ( x ) − 2 ∇ v ) + | v | q ( x ) − 1 v = μ in Ω , v = 0 on ∂ Ω , where Ω is a smooth bounded domain in R N ( N ⩾ 2 ) , μ is a bounded Radon measure. Our results are twofold: when μ is absolutely continuous with respect to the p ( ⋅ ) -capacity, the problem admits a unique solution; when μ is concentrated on a set of zero r ( ⋅ ) -capacity ( r ( ⋅ ) > p ( ⋅ ) in Ω ¯ ), then the problem does not admit a solution if q ( ⋅ ) is large enough.