Existence and Regularity for Minimizers of Some Noncoercive Integral Functionals

Abstract

This paper deals with minimizers for noncoercive integral functional of the type

$\mathcal{J}(v) = \int_{\Omega} a(x)|\nabla v(x)|^p dx - \int_{\Omega} f(x)v(x)dx,\quad v\in W_0^{1,p}(\Omega), $

with $1 < p < n, 0 < a(x) \leq \beta, a.e. \Omega$ and $\frac{1}{a(x)}$ and $f(x)$ belong to some Lebesgue or Marcinkiewicz spaces. It is shown by Weierstrass Theorem that such a functional has a minimizer in a larger space $W_0^{1,q}(\Omega)$ for an appropriate exponent $1 < q < p$. Some regularity properties are given by using Stampacchia Lemma. This paper also considers regularizing effect of an interplay between the coefficient of zero order term and the datum in noncoercive integral functional of the type

$ \mathcal{I}(v)=\int_{\Omega} a(x)|\nabla v(x)|^p dx + \int_{\Omega} b(x)|v(x)|^p dx - \int_{\Omega} f(x)v(x)dx,\quad v\in W_0^{1,p}(\Omega). $

It is shown that, even if $0 < b(x)$ and $f(x)$ belong only to $L^1(\Omega)$, the interplay 

$ |f(x)| \leq 2Q b(x) $

implies the existence of a minimizer $u\in W_0^{1,q}(\Omega)$ satisfying $|u| \leq Q$.