Quasilinear and Hessian equations of Lane-Emden type

TL;DR

The paper studies solvability and pointwise behavior of Lane-Emden type equations in both quasilinear and Hessian forms on $\mathbb{R}^n$ or bounded domains, focusing on two models: $-\Delta_p u = u^q + \mu$ and $F_k[-u] = u^q + \mu$ with nonnegative data. It develops a unified potential-theoretic framework based on Wolff potentials, dyadic models, and capacity techniques to derive sharp existence criteria and global pointwise and integral estimates; it identifies the critical exponents $s = n(q-p+1)/pq$ and $s = n(q-k)/2kq$ governing solvability for measure data. It provides two-sided pointwise bounds, capacity characterizations, and complete removable singularity results, with solvability in renormalized (entropy) or viscosity senses. The approach extends to discrete models and fully nonlinear Hessian theory, enabling treatment of singular and nonlocal solutions, and establishes a coherent theory for Lane-Emden type problems with measure data.

Abstract

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$ and $F_k[-u] = u^q + \mu$, $u > 0$, on $R^n$, or on a bounded domain $\Omega$. Here $\Delta_p$ is the p-Laplacian, and $F_k[u]$ is the $k$-Hessian defined as the sum of $k\times k$ principal minors of the Hessian matrix $D^2 u$; $\mu$ is a nonnegative measurable function (or measure) on $\Omega$. The solvability of these classes of equations in the renormalized (entropy) or viscosity sense has been an open problem even for good data $\mu \in L^s (\Omega)$, $s>1$. Such results are deduced from our existence criteria with the sharp exponents $s = n(q-p+1)/pq$ for the first equation, and $s = n(q-k)/2kq$ for the second one. Furthermore, a complete characterization of removable singularities is given. Our methods are based on systematic use of Wolff's potentials, dyadic models, and nonlinear trace inequalities. We make use of recent advances in potential theory and PDE due to Kilpelainen and Maly, Trudinger and Wang, and Labutin. This enables us to treat singular solutions, nonlocal operators, and distributed singularities, and develop the theory simultaneously for quasilinear equations and equations of Monge-Ampere type.

Theorems & Definitions (81)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Corollary 2.4
• Remark 2.5
• Theorem 2.6
• Corollary 2.7
• Corollary 2.8
• Remark 2.9
• Theorem 2.10