On some nonlinear extensions of the Gagliardo–Nirenberg inequality with applications to nonlinear eigenvalue problems

We derive the inequality

∫_R|f′(x)|^ph(f(x)) dx≤(\sqrt{p−1})^p∫_R(\sqrt{|f″(x)𝒯_h(f(x))|})^ph(f(x)) dx,

where f belongs locally to the Sobolev space W^2,1 and f′ has bounded support. Here h(·) is a given function and 𝒯_h(·) is its given transform, it is independent of p. In case when h≡1 we retrieve the well-known inequality: ∫_R|f′(x)|^p dx≤(\sqrt{p−1})^p∫_R(\sqrt{|f″(x)f(x)|})^p dx. Our inequalities have a form similar to the classical second-order Opial inequalities. They also extend certain class of inequalities due to Mazya, used to obtain second-order isoperimetric inequalities and capacitary estimates. We apply them to obtain new a priori estimates for nonlinear eigenvalue problems.