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

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We derive inequality [∫ |f'(x)|ᵖh(f(x))dx ≤ (√p-1)ᵖ∫(√|f"(x) Tₕ(f(x))|)ᵖh(f(x))dx,] where f belongs locally to Sobolev space W²,¹ and f' has bounded support. Here h(...) is a given function and Tₕ(...) is its given transform, it is independent of p. In case when h≡ 1 we retrieve the well known inequality: (∫ |f'(x)|ᵖdx ≤ (√p-1)ᵖ ∫(√|f"(x)f(x)|)ᵖdx.) Our inequalities have form similar to the classical second order Oppial inequalites. 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 apriori estimates for nonlinear eigenvalue problems.