A class of integral functionals with a $W_0^{1,1}$-minimum

Abstract

In this paper, we study the minimization of the integral functionals of the type

where $0<B$, $0<a_0\le a(x)\in L_{\mathrm{loc}}^1(\Omega )$, $0\not\equiv {\rho }^{+}(x)\in L^{\frac{2}{2-q}}(\Omega )$, and $0<q<2$. The degeneracy of the principal part of the functional implies that it is not coercive in $W_0^{1,2}(\Omega )$ and pushes to set and solve the minimization problem in $W_0^{1,1}(\Omega )$ instead of the space $W_0^{1,2}(\Omega )$. In addition, when the coefficients $a(x)$ and $\rho (x)$ are merely in $L^1(\Omega )$, but satisfy for some $Q>0$ the condition $|\rho (x)|\le Qa(x)$, we show the existence of a minimum of the functional which belongs to $W_0^{1,2}(\Omega )\cap L^{\infty }(\Omega )\setminus \{0\}$.