Nonlinear elliptic equations and intrinsic potentials of Wolff type

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• 作者：Dat Cao ; Igor Verbitsky ; ^{verbitskyi@missouri.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 35J92 ; 42B37 ; secondary ; 35B09
• 刊名：Journal of Functional Analysis
• 出版年：2017
• 出版时间：1 January 2017
• 年：2017
• 卷：272
• 期：1
• 页码：112-165
• 全文大小：724 K

文摘

We give necessary and sufficient conditions for the existence of weak solutions to the model equation in the case 353d8aa65959917" title="Click to view the MathML source">0<q<p−1$0<q<p-1$, where σ≥0$\sigma \ge 0$ is an arbitrary locally integrable function, or measure, and Δ_pu=div(∇u|∇u|^p−2)${\mathrm{\Delta }}_{p}u=\mathrm{div}(\mathrm{\nabla }u|\mathrm{\nabla }u{|}^{p-2})$ is the p-Laplacian. Sharp global pointwise estimates and regularity properties of solutions are obtained as well. As a consequence, we characterize the solvability of the equation where b>0$b>0$. These results are new even in the classical case p=2$p=2$.

Our approach is based on the use of special nonlinear potentials of Wolff type adapted for “sublinear” problems, and related integral inequalities. It allows us to treat simultaneously several problems of this type, such as equations with general quasilinear operators $\text{div}\mathcal{A}(x,\mathrm{\nabla }u)$, fractional Laplacians 359b" title="Click to view the MathML source">(−Δ)^α${(-\mathrm{\Delta })}^{\alpha }$, or fully nonlinear k-Hessian operators.