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Register a new userLocal estimates for conformal $Q$-curvature equations
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We derive local estimates of positive solutions to the conformal $Q$-curvature equation $$ (-Delta)^m u = K(x) u^{frac{n+2m}{n-2m}} ~~~~~~ in ~ Omega backslash Lambda $$ near their singular set $Lambda$, where $Omega subset mathbb{R}^n$ is an open set, $K(x)$ is a positive continuous function on $Omega$, $Lambda$ is a closed subset of $mathbb{R}^n$, $2 leq m < n/2$ and $m$ is an integer. Under certain flatness conditions at critical points of $K$ on $Lambda$, we prove that $u(x) leq C [{dist}(x, Lambda)]^{-(n-2m)/2}$ when the upper Minkowski dimension of $Lambda$ is less than $(n-2m)/2$.
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This paper is focused on the local interior $W^{1,infty}$-regularity for weak solutions of degenerate elliptic equations of the form $text{div}[mathbf{a}(x,u, abla u)] +b(x, u, abla u) =0$, which include those of $p$-Laplacian type. We derive an explicit estimate of the local $L^infty$-norm for the solutions gradient in terms of its local $L^p$-norm. Specifically, we prove begin{equation*} | abla u|_{L^infty(B_{frac{R}{2}}(x_0))}^p leq frac{C}{|B_R(x_0)|}int_{B_R(x_0)}| abla u(x)|^p dx. end{equation*} This estimate paves the way for our forthcoming work in establishing $W^{1,q}$-estimates (for $q>p$) for weak solutions to a much larger class of quasilinear elliptic equations.
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