On the existence and uniqueness of minimizers for a class of integral functionals

摘要

We study the solvability of the minimization problem $$mathop {min }limits_{eta in mathcal{K}_alpha } int_0^T {alpha (t)left[ {fleft( {|eta '(t)|} right) + gleft( {eta (t)} right)} right]} ,dt,$$ where $mathcal{K}_alpha $ is a subset of AC loc[0, T] depending on the weight function α. Neither the convexity nor the superlinearity of f are required. The main application concerns the existence and uniqueness of minimizers to integral functionals on convex domains $Omega subset mathbb{R}^N ,$ defined in the class of functions in $W_0 ^{1,1} left( Omega right)$ depending only on the distance from the boundary of Ω. As a corollary, when Ω is a ball we obtain the existence of radially symmetric solutions to nonconvex and noncoercive functionals.