Gagliardo-Nirenberg inequalities in regular Orlicz spaces involving nonlinear expressions

We consider a triple of N-functions (M,H,J) that satisfy the Δ′-condition, μ=|x|αdx and suppose that an additive variant of interpolation inequality holds∫RnM(|∇u|)μ(dx)⩽C(∫RnH(|u|)μ(dx)+∫RnJ(|∇(2)u|)μ(dx)), where u∈R⊆Wloc2,1(Rn), R is an arbitrary set invariant with respect to external and internal dilations. We show that the above inequality implies its certain nonlinear variant involving the expressions ∫RnH(|u|)μ(dx) and ∫RnJ(|∇(2)u|)μ(dx). Various generalizations of this inequality to the more general class of N-functions, measures and to higher order derivatives are also discussed and the examples are presented.