Quasilinear and Hessian equations of Lane–Emden type

Nguyen Cong Phuc¹ ; Igor E. Verbitsky²

¹ Department of Mathematics<br/>Purdue University<br/>West Lafayette, IN 47907<br/>United States
² Department of Mathematics<br/>University of Missouri<br/>Columbia, MO 65211<br/>United States

Annals of mathematics, Tome 168 (2008) no. 3, pp. 859-914.

Résumé

The existence problem is solved, and global pointwise estimates of solutions are obtained for quasilinear and Hessian equations of Lane-Emden type, including the following two model problems: \[ -\Delta_p u = u^q + \mu, \qquad F_k[-u] = u^q + \mu, \qquad u \ge 0, \] on $\mathbb{R}^n$, or on a bounded domain $\Omega \subset \mathbb{R}^n$. Here $\Delta_p$ is the $p$-Laplacian defined by $\Delta_p u = {\rm div} \, ( \nabla u |\nabla u|^{p-2})$, and $F_k[u]$ is the $k$-Hessian defined as the sum of $k\times k$ principal minors of the Hessian matrix $D^2 u$ ($k=1,2, \dots, n$); $\mu$ is a nonnegative measurable function (or measure) on $\Omega$.

Détail

DOI : 10.4007/annals.2008.168.859