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

摘要

We derive the inequality ∫_R|f'(x)|~ph(f(x))dx≤((P-1)(1/2))~p∫_R((|f''(x)Th(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 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)(1/2))p ∫_R((|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.