The space of Hardy-weights for quasilinear equations: Maz'ya-type characterization and sufficient conditions for existence of minimizers (Q6622470)

Let \(p\in (1,\infty)\), let \(\Omega\) be a domain in \(\mathbb{R}^N\) (\(N\geq 2\)), and let \(A\in L_{\text{loc}}^\infty(\Omega,\mathbb{R}^N\times \mathbb{R}^N)\) be a symmetric locally uniformly positive defined matrix.\N\NIn this paper, the authors consider a potential \(V:\Omega \rightarrow\mathbb{R}\) belonging to a certain local Morrey space satisfying\N\[\NQ_{p,A,V}(\phi):= \int_\Omega \big[|\nabla \phi|_A^p+V|\phi|^p\big]dx\geq 0 \text{ for all } \phi\in C_c(\Omega)\cap W^{1,p}(\Omega),\N\]\Nwhere \(|\xi|_A^2=\langle A(x)\xi,\xi\rangle\), and introduce the vector space \(\mathcal{H}_p(\Omega,V)\) of all functions \(g\in L^1_{\text{loc}}(\Omega)\) such that, for some \(C>0\), the following Hardy-type inequality holds \N\[\N \int_\Omega |g||\phi|^pdx\leq C\cdot Q_{p,A,V}(\phi) \text{ for all } \phi\in C_c(\Omega)\cap W^{1,p}(\Omega).\tag{1}\N\]\NThen, after introducing the following definition of \(Q_{p,A,V}\)-capacity \N\[\N\text{Cap}_{\mathbf{u}}(F,\Omega):=\inf\Big\{Q_{p,A,V}(\phi)\;\Big\vert\; \phi\in C_c(\Omega)\cap W^{1,p}(\Omega) \text{ and } \phi \geq {\mathbf{u}} \text{ on } F\Big\}, \N\]\Nwhere \({\mathbf{u}}\in W^{1,p}_{\text{loc}}(\Omega)\cap C(\Omega)\) is a positive solution of \(Q'_{p,A,V}(u)=0\), and the functional \(\|\cdot\|_{\mathcal{H}_p(\Omega,V)}: \mathcal{H}_p(\Omega,V)\rightarrow \mathbb{R}\) (which is known to be a norm on \(\mathcal{H}_p(\Omega,V)\)) defined by \N\[\N\|g\|_{\mathcal{H}_p(\Omega,V)}:=\sup\bigg\{\frac{\int_F|g||{\mathbf{u}}|^pdx}{\text{Cap}_{\mathbf{u}}(F,\Omega)}\;\bigg\vert\;F\Subset \Omega \text{ is compact and } \text{Cap}_{\mathbf{u}}(F,\Omega)\neq 0 \bigg\}, \N\]\Nthe authors give a characterization of the space \(\mathcal{H}_p(\Omega,V)\) by showing that a function \(g\in L_{\text{loc}}^1(\Omega)\) belongs to \(\mathcal{H}_p(\Omega,V)\) if and only if \(\|g\|_{\mathcal{H}_p(\Omega,V)}<\infty\). They also prove that, if \(\mathcal{B}_g(\Omega,V)\) denotes the best constant for \((1)\), then \(\mathcal{B}_g(\Omega,V)\) is a norm on \(\mathcal{H}_p(\Omega,V)\) equivalent to \(\|\cdot \|_{\mathcal{H}_p(\Omega,V)}\) and, in particular, there exists \(C_p>0\) depending only on \(p\), such that \(\|g\|_{\mathcal{H}_p(\Omega,V)}\leq \mathcal{B}_g(\Omega,V)\leq C_p\|g\|_{\mathcal{H}_p(\Omega,V)}\).\N\NSufficient conditions on \(V\) and \(g\) are also provided in order the best constant for \((1)\) is attained in an appropriate Beppo Levi space.