Harnack inequalities and quantization properties for the \(n\)-Liouville equation
Consider the \(n\)-Laplacian equation \N\[\N-\Delta_n u=h(x)e^u,\text{ in }\Omega,\tag{1}\N\]\Nwhere \(\Omega\subseteq \mathbb{R}^n\), \(n\geq 2\), is a bounded open set and \(h:\Omega\rightarrow \mathbb{R}\) is a continuous function. Equation \((1)\) is known to satisfy the following ``concentration-compactness'' alternative: assume that \(h_k:\Omega \rightarrow \mathbb{R}\) is a sequence of continuous functions such that \(0\leq h_k\leq b\) in \(\Omega\), for each \(k\in \mathbb{N}\) and some \(b>0\), and let \(u_k\) be a solution to \N\[\N-\Delta_n u=h_k(x)e^u, \text{ in }\Omega, \N\]\Nwith \(\sup_{k\in \mathbb{N}}\int_\Omega e^{u_k}dx<+\infty\). Then, up to a subsequence, the following alternative holds:\N\begin{itemize}\N\item[i)] \(u_k\) is bounded in \(L_{\text{loc}}^\infty(\Omega)\);\N\item[ii)] \(u_k\rightarrow -\infty\) locally uniformly in \(\Omega\);\N\item[iii)] the set \(S:=\{p\in \Omega: \exists x_k\in \Omega \ \text{such that} \ \ x_k\rightarrow p, \ u_k(x_k)\rightarrow \infty\}\) is finite, \(u_k\rightarrow -\infty\) locally uniformly in \(\Omega\setminus S\), and \(h_ke^{u_k}\rightarrow \sum_{p\in S}\beta_p\delta_p\) weakly in the sense of measures, where \(\beta_p\geq n^n\omega_n\) and \(\omega_n\) is the volume of the unit ball of \(\mathbb{R}^n\).\N\end{itemize}\NIn the two-dimensional case and with the additional assumption that \N\[\N0\leq h_k\rightarrow h \text{ in } C_{\text{loc}}(\Omega),\tag{2}\N\]\Nit is known that, for any \(p\in S\), the following holds: \(h(p)>0\), the concentration mass \(\beta_p\) is a multiple of \(8\pi\), and \(8\pi\) is the mass of each solution of \(-\Delta u=h(p)e^U\) in \(\mathbb{R}^2\), \(\int_{\mathbb{R}^2}e^Udx<\infty\).\N\NOne of the authors of this paper recently extended this latter result to the case of the \(n\)-Laplacian equation \(-\Delta_n u=h(p)e^U\) in \(\mathbb{R}^n\), proving that the mass of each solution is \(c_n\omega_n\), where \(c_n=n\left(\frac{n^2}{n-1}\right)^{n-1}\).\N\NThus, for \(n\geq 3\), a natural question is to see whether, under assumption \((2)\), one has \(h(p)>0\) and \(\beta_p\in c_n\omega_n\mathbb{N}\), for any \(p\in S\).\N\NThe authors, after proving a Harnack inequality of ``sup+inf'' type for the solutions of \((1)\), use this inequality and a blow-up analysis to give a positive answer to the above question.
scientific article; zbMATH DE number 7877910
scientific article