Pointwise estimates of Brezis-Kamin type for solutions of sublinear elliptic equations

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• 作者：Dat T. Cao^a ; ^{dcao4@utk.edu" class="auth_mail" title="E-mail the corresponding author} ; Igor E. Verbitsky^b ; ^{verbitskyi@missouri.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35B05 ; 35J92
• 刊名：Nonlinear Analysis
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：146
• 期：Complete
• 页码：1-19
• 全文大小：732 K

文摘

We study quasilinear elliptic equations of the type where Δ_pu=∇⋅(∇u|∇u|^p−2)${\Delta }_{p}u=\nabla \cdot \left(\nabla u{\left|\nabla u\right|}^{p-2}\right)$ is the p$p$-Laplacian (or a more general A$\mathcal{A}$-Laplace operator $\text{div}\mathcal{A}(x,\nabla u)$), 0<q<p−1$0<q<p-1$, and σ≥0$\sigma \ge 0$ is an arbitrary locally integrable function or measure on Rⁿ${\mathbb{R}}^n$.

We obtain necessary and sufficient conditions for the existence of positive solutions (not necessarily bounded) which satisfy global pointwise estimates of Brezis–Kamin type given in terms of Wolff potentials. Similar problems with the fractional Laplacian (−Δ)^α${(-\Delta )}^{\alpha }$ for $0<\alpha <\frac{n}{2}$ are treated as well, including explicit estimates for radially symmetric σ$\sigma$. Our results are new even in the classical case p=2$p=2$ and α=1$\alpha =1$.