We introduce new measures of non-compactness for the embedding operator Ep,q(Ω):Lp1(Ω)→Lq(Ω) and describe their relations with the essential norm of Ep, q(Ω), 'local' isoperimetric and isocapacitary constants. An explicit formula for the essential norm of Ep, q(Ω) is obtained for domains with a power cusp on the boundary and bounded C1 domains. The Neumann problem for a particular Schrödinger operator is discussed on domains with a power cusp.