\(L^p\)-weak regularity and asymptotic behavior of solutions for critical equations with singular potentials on Carnot groups
Let \(\mathbb{G}= (\mathbb{R}^N, 0)\) be a homogeneous Carnot group with homogeneous dimension \(Q\), and let \(\Delta_{\mathbb{G}}\) be a fixed sub-Laplacian on \(\mathbb{G}\). The author considers the following semilinear subelliptic problem \[ -\Delta_{\mathbb{G}}u= \sum_{1\leq i\leq n} a_i(\xi)|u|^{p_i-2} u,\quad u\in S^1_a(\Omega),\tag{\(*\)} \] where \(\Omega\subset\mathbb{R}^N\) is an open unbounded domain, \(n\) is a positive integer, \(2< p_i\leq 2^*\) for \(1\leq i\leq n\), where \(2^*= 2Q/(Q- 2)\), and \(S^1_0(\Omega)\) is the appropriate Sobolev space, defined as the completion of \(C^\infty_0(\Omega)\) with respect to the norm \(\| u\|=\|\nabla_{\mathbb{G}}u\|_2\). The coefficients \(a_i\) satisfy the following summability assumptions: \(a_i\in M^{q_i}(\Omega)\), \(1/q_i+ p_i/2= 1\), \(i= 1,\dots, n\), where \(M^{q_i}\) denotes the classical Marcinkiewicz space. The first result is a sharp regularity result: a weak solution \(u\in S^1_0(\Omega)\) to problem \((*)\) belongs not only to \(L^p(\Omega)\) for any \(2^*/2< p\leq\infty\), but also to the weak space \(M^{2^*/2}(\Omega)= L^{2^*/2,\infty}(\Omega)\). This result is a crucial ingredient in order to prove the main theorem of the paper, concerning the behavior of solutions at infinity: if \(u\in S^1_0(\Omega)\) is a weak solution of the Dirichlet problem \((*)\), then there exist \(C,R> 0\) such that \(|u(\xi)|\leq Cd(\xi)^{2-Q}\), \(d(\xi)\geq R\), where \(d\) is any homogeneous norm on \(\mathbb{G}\). These results apply to the study of qualitative properties of the extremal functions for some relevant Hardy-Sobolev type inequalities on stratified groups.
scientific article; zbMATH DE number 5808444
scientific article