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Search: MSC category 35D30 ( Weak solutions )

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1. CJM Online first

Du, Zhuoran; Fang, Yanqin; Gui, Changfeng
A class of degenerate elliptic equations with nonlinear boundary conditions
We consider positive solutions of the problem \begin{equation} (*)\qquad \left\{ \begin{array}{l}-\mbox{div}(x_{n}^{a}\nabla u)=bx_{n}^{a}u^{p}\;\;\;\;\;\mbox{in}\;\;\mathbb{R}_{+}^{n}, \\ \frac{\partial u}{\partial \nu^a}=u^{q} \;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mbox{on}\;\;\partial \mathbb{R}_{+}^{n}, \\ \end{array} \right. \end{equation} where $a\in (-1,0)\cup(0,1)$, $b\geq 0$, $p, q\gt 1$ and $\frac{\partial u}{\partial \nu^a}:=-\lim_{x_{n}\rightarrow 0^+}x_{n}^{a}\frac{\partial u}{\partial x_{n}}$. In special case $b=0$, it is associated to fractional Laplacian equation $(-\Delta)^{s}u=u^{q} $ in entire space $\mathbb{R}^{n-1}$. We obtain the existence of positive axially symmetric solutions to ($*$) for the case $a\in (-1,0)$ in $n\geq3$ for supercritical exponents $p\geq\frac{n+a+2}{n+a-2}, \;\;q\geq\frac{n-a}{n+a-2}$. The nonexistence is obtained for the case $a\in (-1,0)$, $b\geq 0$ and any $p,~q\gt 1$ in $n=2$ as well.

Keywords:existence, non-existence, positive solutions, degenerate elliptic equation, nonlinear boundary conditions, symmetry, monotonicity
Categories:35D30, 35J70, 35J25

2. CJM 2012 (vol 64 pp. 1395)

Rodney, Scott
Existence of Weak Solutions of Linear Subelliptic Dirichlet Problems With Rough Coefficients
This article gives an existence theory for weak solutions of second order non-elliptic linear Dirichlet problems of the form \begin{align*} \nabla'P(x)\nabla u +{\bf HR}u+{\bf S'G}u +Fu &= f+{\bf T'g} \text{ in }\Theta \\ u&=\varphi\text{ on }\partial \Theta. \end{align*} The principal part $\xi'P(x)\xi$ of the above equation is assumed to be comparable to a quadratic form ${\mathcal Q}(x,\xi) = \xi'Q(x)\xi$ that may vanish for non-zero $\xi\in\mathbb{R}^n$. This is achieved using techniques of functional analysis applied to the degenerate Sobolev spaces $QH^1(\Theta)=W^{1,2}(\Theta,Q)$ and $QH^1_0(\Theta)=W^{1,2}_0(\Theta,Q)$ as defined in previous works. Sawyer and Wheeden give a regularity theory for a subset of the class of equations dealt with here.

Keywords:degenerate quadratic forms, linear equations, rough coefficients, subelliptic, weak solutions
Categories:35A01, 35A02, 35D30, 35J70, 35H20

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