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)−1v=μin Ω,v=0on ∂Ω, where Ω   is a smooth bounded domain in RN(N⩾2), μ is a bounded Radon measure. Our results are twofold: when μ   is absolutely continuous with respect to the p(⋅)p(⋅)-capacity, the problem admits a unique solution; when μ   is concentrated on a set of zero r(⋅)r(⋅)-capacity (r(⋅)>p(⋅)r(⋅)>p(⋅) in Ω¯), then the problem does not admit a solution if q(⋅)q(⋅) is large enough.