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

We derive inequality ∫ R |f ′(x)|ph(f(x))dx ≤ (√ p− 1 )p ∫ R (√ |f ′′(x)Th(f(x))| )p h(f(x))dx, where f belongs locally to Sobolev space W 2,1 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: ∫ R |f ′(x)|pdx ≤ (√ p− 1 )p ∫ R (√ |f (x)f(x)| )p dx. 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. The work is supported by the Polish Ministry of Science grant no. N N201 397837 (years 2009-2012). The work is supported by EU FP6 Marie Curie RTN programme CODY